tag:blogger.com,1999:blog-57614102539720374352016-06-05T17:43:17.883-07:00MatematicaNassif Nabeal (Nassif Medici)http://www.blogger.com/profile/13708191074451322062noreply@blogger.comBlogger8125tag:blogger.com,1999:blog-5761410253972037435.post-50531177136208513542015-07-21T18:08:00.001-07:002015-07-21T18:08:30.550-07:00Benoît Mandelbrot's contribution to Finance<div class="MsoNormal"><span style="font-family: inherit;"><span style="color: #3d85c6;"> </span><span style="font-size: 12pt; line-height: 18.3999996185303px;">Other mathematicians of probability like Kolmogorov may be more academic or progressive but Mandelbrot was unique he proved that mathematicians actually understand randomness, he is called by Nassim Nicholas Taleb as <span style="color: #3d85c6;">'poet of randomness'</span>. Before Nassim Taleb ,Black Swans were dealt by him in a philosophical and aesthetic way.</span><span class="Apple-style-span" style="font-size: 16px; line-height: 18px;">Mandelbrot was initially a probability guy but later went into other fields of maths and made his name in other fields.</span><span style="font-size: 12pt; line-height: 18px;">In 1960s Mandelbrot presented his ideas on prices of commodity and stock prices and made a contribution on mathematics of randomness in economic theory.<span style="color: #3d85c6;">Mandelbrot also knew the pitfalls in Louis Bachelier's model.</span></span><span class="Apple-style-span" style="font-size: 16px; line-height: 18px;">Mandelbrot linked randomness to geometry and made randomness a more natural science.</span>If stock markets were Gaussian then stock market crashed would have happen once in a Billion years. Mandelbrot's randomness methods make the statistics methods look useless. </span></div><span style="font-family: inherit;"></span><br /><br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td><a href="http://4.bp.blogspot.com/-TTgB5NNfrWM/UYYhLQMxKBI/AAAAAAAAVv0/5rSLyEg8KGw/s1600/na24oc-frontiers.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="266" src="http://4.bp.blogspot.com/-TTgB5NNfrWM/UYYhLQMxKBI/AAAAAAAAVv0/5rSLyEg8KGw/s400/na24oc-frontiers.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="font-size: 12.8000001907349px;"><span style="font-size: small;">Benoît Mandelbrot , the late Sterling Professor of Mathematical Sciences at Yale University</span></td></tr></tbody></table><br />The first formal model for security price changes was put forward by Louis Bachelier (1900). His price difference process in essence sets out the mathematics of Brownian Motion before Einstein and Wiener rediscovered his results in 1905 and 1923 in the context of physical particles, and in particular generates a Normal (i.e. Gaussian) distribution where variance increases proportionally with time. A crucial assumption of Bachelier’s approach is that successive price changes are independent. His dissertation, which was awarded only a “mention honorable” rather than the “mention très honorable” that was essential for recognition in the academic world, remained unknown to the financial world until M.F. M. Osborne , who made no reference to Bachelier’s work, rediscovered Brownian Motion as a plausible model for security price changes.<br /><br /><span style="color: #cc0000;">But in 1963 the famous mathematician Mandelbrot produced a paper pointing out that the tails of security price distributions are far fatter than those of normal distributions (what he called the “Noah effect” in reference to the deluge in biblical times) and recommending instead a class of independent and identically distributed “alpha-stable” Paretian distributions with infinite variance. </span>Towards the end of the paper Mandelbrot observes that the independence assumption in his suggested model does not fully reflect reality in that “on closer inspection … large changes tend to be followed by large changes – of either sign – and small changes tend to be followed by small changes.” Mandelbrot later called this the “Joseph effect” in reference to the biblical account of seven years of plentiful harvests in Egypt followed by seven years of famine. Such a sequence of events would have had an exceptionally low probability of taking place if harvest yields in successive years were independent. While considering how best to model this dependence effect, Mandelbrot came across the work of Hurst (1951, 1955) which dealt with a very strong dependence in natural events such as river flows (particularly in the case of the Nile) from one year to another and developed the Hurst exponent H as a robust statistical measure of dependence. Mandelbrot’s new model of Fractional Brownian Motion, which is described in detail in Mandelbrot & van Ness (1968), is defined by an equation which incorporates the Hurst exponent H. Many financial economists, particularly Cootner (1964), were highly critical of Mandelbrot’s work, mainly because – if he was correct about normal distributions being seriously inconsistent with reality – most of their earlier statistical work, particularly in tests of the Capital Asset Pricing Model and the Efficient Market Hypothesis, would be invalid. Indeed, in his seminal review work on stockmarket efficiency, Fama (1970) describes how non-normal stable distributions of precisely the type advocated by Mandelbrot are more realistic than standard distributions .<br /><br />Partly because of estimation problems with alpha-stable Paretian distributions and the mathematical complexity of Fractional Brownian Motion, and partly because of the conclusion in Andrew Lo's work (1991) that standard distributions might give an adequate representation of reality, Mandelbrot’s two suggested new models failed to make a major impact on finance theory, and he essentially left the financial scene to pursue other interests such as fractal geometry. However, in his “Fractal Geometry of Nature”, Mandelbrot (1982) commented on what he regarded as the “suicidal” statistical methodologies that were standard in finance theory: “Faced with a statistical test that rejects the Brownian hypothesis that price changes are Gaussian, the economist can try one modification after another until the test is fooled. A popular fix is censorship, hypocritically called ‘rejection of outliers’. One distinguishes the ordinary ‘small’ price changes from the large changes that defeat Alexander’s filters. The former are viewed as random and Gaussian, and treasures of ingenuity are devoted to them .The latter are handled separately, as ‘nonstochastic’.”<br /><br />Shortly after the “Noah effect” manifested itself with extreme severity in the collapse of Long-Term Capital Management, Mandelbrot (1999) produced a brief article, the cover story of the February 1999 issue of “Scientific American”, in which he used nautical analogies to highlight the foolhardy nature of standard risk models that assumed independent normal distributions. He also pointed out that a more realistic depiction of market fluctuations, namely Fractional Brownian Motion in multifractal trading time, already existed.<br /><br /><div class="MsoNormal"><span style="font-size: 12pt; line-height: 18.3999996185303px;"></span></div><div class="MsoNormal"><span style="font-size: 12pt; line-height: 18.3999996185303px;"><span style="color: #cc0000;">Fractals are linked with power laws, Mandelbrot worked on it and applied it to randomness. Mandelbrot designed the mathematical object called "Mandelbrot set" and later worked on shapes and fractals of maths and also worked on Chaos Theory.</span> These objects play an important role on aesthetics , music , architecture , poetry , gestures and tones are derived from fractals . Mandelbrot's book "Fractal Geometry of Nature" it made a fame in arts , visual arts and every artistic circle. Many artists used to call Mandelbrot "The Rock Star of Mathematics". Mandelbrot became famous because of the number of applications of mathematics in our society.</span><br /><br /></div><span style="font-size: 12pt; line-height: 18.3999996185303px;"></span><br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td><a href="http://1.bp.blogspot.com/-GFiDy1wUo5Q/T5kdKd_VlBI/AAAAAAAALyI/gM54FGVUGZg/s1600/0465043550.jpg" style="margin-left: auto; margin-right: auto;"><span class="Apple-style-span" style="font-size: xx-small;"><img border="0" height="320" src="http://1.bp.blogspot.com/-GFiDy1wUo5Q/T5kdKd_VlBI/AAAAAAAALyI/gM54FGVUGZg/s320/0465043550.jpg" width="211" /></span></a></td></tr><tr style="color: #3d85c6;"><td class="tr-caption" style="font-size: 12.8000001907349px;"><span style="font-size: small;"><span class="Apple-style-span" style="font-family: Arial,Helvetica,sans-serif; line-height: 15px;">Mandelbrot used his fractal theory to explain the presence of extreme events in Wall Street.</span></span></td></tr></tbody></table><span class="Apple-style-span" style="font-family: Georgia, 'Times New Roman', serif;"><br /></span><span class="Apple-style-span" style="font-family: Georgia, 'Times New Roman', serif;"><b><span class="Apple-style-span" style="color: #0b5394;"><br /></span></b></span><span class="Apple-style-span" style="color: #cc0000; font-size: 16px; line-height: 18px;">In fact he was one of the pioneers in studying the variation of financial prices even before Bchelier's Brownian model became widely accepted in academia and </span><span class="Apple-style-span" style="font-size: 16px; line-height: 18px;"><span class="Apple-style-span" style="color: #cc0000;">Mandelbrot also knew the pitfalls in Bachelier's model.</span></span><span class="Apple-style-span" style="color: #cc0000; font-size: 16px; line-height: 18px;">For this reason many call him as the "father of Quantitative Finance".</span>Mandelbrot has been best known since the early 1960s as one of the pioneers in studying the variation of financial prices.He pointed out that two features of Bachelier's model are unacceptable (in 1960s when Bachelier's model got accepted by academia and financial world).These flaws were based on power-law distributions and so Mandelbrot scaled these both by fractal theory and thus correcting the errors and flaws.Since then scaling by use of fractal theory has become important in finance and as well as in Physics.In fact Nassim Nicholas Taleb's "Black Swan Theory" is inspired by work of Mandelbrot as Mandelbrot was much concerned about high-risk rare events (Black Swans).Nassim and Mandelbrot collaborated in research projects related to risk and randomness.<br /><br /><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><br /></span></span><div style="color: #3d85c6;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: large;"><b>Mandelbrot's contribution in finance fall into three main stages:</b></span></span></div><br />He was the first to stress the essential importance, even in a first approximation, of large variations that may occur as sudden price discontinuities. The Brownian model is unjustified in neglecting them. They are not “outliers” one can safely disregard or study separately. To the contrary, their distribution is much more important than that of the "background noise" constituted by the small changes of Brownian motion. He followed this critique in by showing in 1963 that the big discontinuities and the small "noise" fall on a single power-law distribution and represented them by a scenario based on Levy stable distributions. He and Taylor introduced in 1967 the new notion of intrinsic "trading time." In recent years, fractal trading time and his 1963 model have gained wide acceptance.<br /><br />Secondly, Mandelbrot tackled the fact that the “background noise” of small price changes is of variable “volatility.” This feature was ordinarily viewed as a symptom of non-stationality that must be studied separately. To the contrary, Mandelbrot interpreted this variability as indicating that price changes are far from being statistically independent. In fact, for all practical purposes, their interdependence should be viewed as continuing to an infinitely long term. In particular, it is not limited to the short term that is studied by Markov processes and more recently ARCH and its variants. In fact, it too follows a power-law side of dependence. He followed this critique and illustrated long-dependence by introducing in 1965 a process called fractional Brownian motion which has become very widely used.<br /><br />Thirdly, he introduced the new notion of multifractality that combines long power-law tails and long power-law dependence. Early on, his work was motivated by the context of turbulence, but he immediately observed and pointed out that in 1972 the same ideas also apply to finance. After a long hiatus while he was developing other aspects of fractal geometry, he returned to finance in the mid-1990s and developed the multifractal scenario theory in detail in his 1997 book "Fractals and Scaling in Finance". The concept of scaling invariance used by Mandelbrot started by being perceived as suspect, because at that time other fields did not use it. However the period after 1972 also saw the growth of a new subfield of statistical physics concerned with “criticality.” The concepts used in that field are similar to those Mandelbrot had been using in finance.In "The Misbehavior of Markets", another popular book by Mandelbrot,he argues that the Gaussian models for financial risk used by economists like William Sharpe and Harry Markowitz should be discarded, since these models do not reflect reality. Mandelbrot argues that fractal techniques may provide a more powerful way to analyze risk.Nassif Nabeal (Nassif Medici)http://www.blogger.com/profile/13708191074451322062noreply@blogger.com0tag:blogger.com,1999:blog-5761410253972037435.post-19136511609098261062014-01-04T19:27:00.000-08:002015-06-24T22:12:07.653-07:00John von Neumann<span style="color: red;"><b>Man and Machine </b></span><br /><br />John von Neumann, one of 20th century’s preeminent scientists, along with being a great mathematician and physicist, was an early pioneer in fields such as game theory, nuclear deterrence, and modern computing<span style="color: #674ea7;">. <span style="color: #3d85c6;">He was a polymath who possessed fearsome technical prowess and is considered "the last of the great mathematicians"</span></span><span style="color: #3d85c6;">.A mathematician obsessed with making contribution to every branch of mathematics.</span> His was a mind comfortable in the realms of both man and machine. His kinship with the logical machine was displayed at an early age by his ability to compute the product of two eight-digit numbers in his head. His strong and lasting influence on the human world is apparent through his many friends and admirers who so often had comments as to von Neumann’s greatness as a man and a scientist.He made major contributions to the field of set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics, linear programming, game theory, computer science, numerical analysis, hydrodynamics, nuclear physics and statistics.<br /><br />Although he is often well known for his dominance of logic and rigorous mathematical science, von Neumann’s genius can be said to have grown from a comfortable and relaxed upbringing.<br /><span style="color: red;"><br /></span><span style="color: red;"><b> </b></span><br /><span style="color: red;"><b>Early Life and Education in Budapest</b></span><br /><br />He was born Neumann Janos on December 28, 1903, in Budapest, the capital of Hungary. He was the first born son of Neumann Miksa and Kann Margit. In Hungarian, the family name appears before the given name. So, in English, the parent’s names would be Max Neumann and Margaret Kann. Max Neumann purchased a title early in his son’s life, and so became von Neumann.Max Neumann, born 1870, arrived in Budapest in the late 1880s. He was a non-practicing Hungarian Jew with a good education. He became a doctor of laws and then worked as a lawyer for a bank. He had a good marriage to Margaret, who came from a prosperous family.In 1903, Budapest was growing rapidly, a booming, intellectual capital. It is said that the Budapest that von Neumann was born into "was about to produce one of the most glittering single generations of scientists, writers,artists, musicians, and useful expatriate millionaires to come from one small community since the city-states of the Italian Renaissance. Indeed, John von Neumann was one of those who,through his natural genius and prosperous family, was able to excel in the elitist educational system of the time.<br /><br />At a very young age, von Neumann was interested in math, the nature of numbers and the logic of the world around him. Even at age six, when his mother once stared aimlessly in front of her, he asked, "What are you calculating?" thus displaying his natural affinity for numbers. The young von Neumann was not only interested in math, though. Just as in his adult life he would claim fame in a wide range of disciplines (and be declared a genius in each one), he also had varying interests as a child. At age eight he became fascinated by history and read all forty-four volumes of the universal history, which resided in the family’s library. Even this early, von Neumann showed that he was comfortable applying his mind to both the logical and social world.<br /><br />His parents encouraged him in every interest, but were careful not to push their young son, as many parents are apt to do when they find they have a genius for a child. This allowed von Neumann to develop not only a powerful intellect but what many people considered a likable personality as well.It was never in question that von Neumann would attend university, and in 1914, at the age of 10, the educational road to the university started at ,the Lutheran Gymnasium. This was one of the three best institutions of its kind in Budapest at the time and gave von Neumann the opportunity to develop his great intellect. Before he would graduate from this high school he would be considered a colleague by most of the university mathematicians. His first paper was published in 1922, when he was 17, in the <i> </i><i>Journal of the German<i> </i>Mathematical Society,</i> dealing with the zeros of certain minimal polynomials.<br /><span style="color: red;"><br /></span><span style="color: red;"><b> </b></span><br /><span style="color: red;"><b>University — Berlin, Zurich and Budapest</b></span><br /><br />In 1921 von Neumann was sent to become a chemical engineer at the University of Berlin and then to Zurich two years later. Though John von Neumann had little interest in either chemistry or engineering, his father was a practical man and encouraged this path. At that time chemical engineering was a popular career that almost guaranteed a good living, in part due to the success of German chemists from 1914 to 1918. So, von Neumann set on the road planned in part by his father Max. He would spend two years in Berlin in a non-degree chemistry program. After this he would take the entrance exam for second year standing in the chemical engineering program at the prestigious Eidgennossische Technische Hochschule (ETH) in Zurich, where Einstein had failed the entrance exam in 1895 and then gained acceptance a year later.<br /><br />During this time of practical undergraduate study, von Neumann was executing another plan that was more in tune with his interests. In the summer after his studies at Berlin and before he went to Zurich he enrolled at the Budapest University as a candidate for an advanced doctorate in mathematics. His Ph.D. thesis was to attempt the axiomatization of set theory, developed by George Cantor. At the time, this was one of the hot topics in mathematics and had already been studied by great professors, causing a great deal of trouble to most of them. None the less, the young von Neumann, devising and executing this plan at the age of 17, was not one to shy away from great intellectual challenges.<br /><br />Von Neumann breezed through his two years at Berlin and then set himself to the work on chemical engineering at the ETH and his mathematical studies in Budapest. He received excellent grades at the ETH, even for classes he almost never attended. He received a perfect mark of 6 in each of his courses during his first semester in the winter of 1923-24; courses including organic chemistry, inorganic chemistry, analytical chemistry, experimental physics, higher mathematics and French language.From time to time he would visit Budapest University when his studies there required his presence and to visit his family. He worked on his set theory thesis in Zurich while completing classes for the ETH. After finishing his thesis he took the final exams in Budapest to receive his Ph.D with highest honors. This was just after his graduation from the ETH, so in 1926 he had two degrees, one an undergraduate degree in chemical engineering and the other a Ph.D. in mathematics, all by the time he was twenty-two.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><span style="font-size: small;"><a href="http://1.bp.blogspot.com/-yZugev7bil8/T4stqywHdUI/AAAAAAAALjY/U46loNkBGyQ/s1600/12_von_neumann_stamp.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="302" src="http://1.bp.blogspot.com/-yZugev7bil8/T4stqywHdUI/AAAAAAAALjY/U46loNkBGyQ/s400/12_von_neumann_stamp.jpg" width="400" /></a></span></td></tr><tr><td class="tr-caption" style="text-align: center;"><span class="phototitles3" style="font-size: small;">This von Neumann stamp, issued in Hungary in 1992, honors his contributions to mathematics and computing.</span></td></tr></tbody></table><br /><span style="color: red;"><b> Game Theory </b></span><br /><br />Von Neumann is commonly described as a practical joker and always the life of the party. John and his wife held a party every week or so, creating a kind of salon at their house. Von Neumann used his phenomenal memory to compile an immense library of jokes which he used to liven up a conversation. <span style="color: blue;">Von Neumann loved games and toys, which probably contributed in great part to his work in Game Theory. </span><br /><br />An occasional heavy drinker, Von Neumann was an aggressive and reckless driver, supposedly totaling a car every year or so. According to William Poundstone's <i>Prisoner's Dilemma</i>, "an intersection in Princeton was nicknamed "Von Neumann Corner" for all the auto accidents he had there." <br /><br />His colleagues found it "disconcerting" that upon entering an office where a pretty secretary worked, von Neumann habitually would "bend way way over, more or less trying to look up her dress." (Steve J. Heims, John Von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life and Death, 1980, quoted in <i>Prisoner's Dilemma</i>, p.26) Some secretaries were so bothered by Von Neumann that they put cardboard partitions at the front of their desks to block his view.Despite his personality quirks, no one could dispute that Von Neumann was brilliant. Beginning in 1927, Von Neumann applied new mathematical methods to quantum theory. His work was instrumental in subsequent "philosophical" interpretations of the theory. <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-Pr_VWLhxcCU/T5kmET5MmOI/AAAAAAAAL0g/aJfoq_Kuqkg/s1600/k7802.gif" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://4.bp.blogspot.com/-Pr_VWLhxcCU/T5kmET5MmOI/AAAAAAAAL0g/aJfoq_Kuqkg/s320/k7802.gif" width="211" /></a></div><br /><span style="color: blue;"><span style="color: #cc0000;">For Von Neumann, the inspiration for game theory was poker, a game he played occasionally and not terribly well.</span> </span>Von Neumann realized that poker was not guided by probability theory alone, as an unfortunate player who would use only probability theory would find out. Von Neumann wanted to formalize the idea of "bluffing," a strategy that is meant to deceive the other players and hide information from them.In his 1928 article, "Theory of Parlor Games," Von Neumann first approached the discussion of game theory, and proved the famous Minimax theorem. From the outset, Von Neumann knew that game theory would prove invaluable to economists. He teamed up with Oskar Morgenstern, an Austrian economist at Princeton, to develop his theory.<br /><br />Their book, <span style="color: red;"><i>Theory of Games and Economic Behavior</i></span>, revolutionized the field of economics. Although the work itself was intended solely for economists, its applications to psychology, sociology, politics, warfare, recreational games, and many other fields soon became apparent.Although Von Neumann appreciated Game Theory's applications to economics, he was most interested in applying his methods to politics and warfare, perhaps stemming from his favorite childhood game, Kriegspiel, a chess-like military simulation. He used his methods to model the Cold War interaction between the U.S. and the USSR, viewing them as two players in a zero-sum game. <br />From the very beginning of World War II, Von Neumann was confident of the Allies' victory. He sketched out a mathematical model of the conflict from which he deduced that the Allies would win, applying some of the methods of game theory to his predictions. <br /><br />In 1943, Von Neumann was invited to work on the Manhattan Project. Von Neumann did crucial calculations on the implosion design of the atomic bomb, allowing for a more efficient, and more deadly, weapon. Von Neumann's mathematical models were also used to plan out the path the bombers carrying the bombs would take to minimize their chances of being shot down. The mathematician helped select the location in Japan to bomb. Among the potential targets he examined was Kyoto, Yokohama, and Kokura."Of all of Von Neumann's postwar work, his development of the digital computer looms the largest today." (Poundstone 76) After examining the Army's ENIAC during the war, Von Neumann came up with ideas for a better computer, using his mathematical abilities to improve the computer's logic design. Once the war had ended, the U.S. Navy and other sources provided funds for Von Neumann's machine, which he claimed would be able to accurately predict weather patterns.Capable of 2,000 operations a second, the computer did not predict weather very well, but became quite useful doing a set of calculations necessary for the design of the hydrogen bomb. Von Neumann is also credited with coming up with the idea of basing computer calculations on binary numbers, having programs stored in computer's memory in coded form as opposed to punchcards, and several other crucial developments. Von Neumann's wife, Klara, became one of the first computer programmers.<br /><br />Von Neumann later helped design the SAGE computer system designed to detect a Soviet nuclear attack in 1948, Von Neumann became a consultant for the RAND Corporation. RAND (Research And Development) was founded by defense contractors and the Air Force as a "think tank" to "think about the unthinkable." Their main focus was exploring the possibilities of nuclear war and the possible strategies for such a possibility.<br /><br />Von Neumann was, at the time, a strong supporter of "preventive war." Confident even during World War II that the Russian spy network had obtained many of the details of the atom bomb design, Von Neumann knew that it was only a matter of time before the Soviet Union became a nuclear power. He predicted that were Russia allowed to build a nuclear arsenal, a war against the U.S. would be inevitable. He therefore recommended that the U.S. launch a nuclear strike at Moscow, destroying its enemy and becoming a dominant world power, so as to avoid a more destructive nuclear war later on. "With the Russians it is not a question of whether but of when," he would say. An oft-quoted remark of his is, "If you say why not bomb them tomorrow, I say why not today? If you say today at 5 o'clock, I say why not one o'clock?"Just a few years after "preventive war" was first advocated, it became an impossibility. By 1953, the Soviets had 300-400 warheads, meaning that any nuclear strike would be effectively retaliated.In 1954, Von Neumann was appointed to the Atomic Energy Commission. A year later, he was diagnosed with bone cancer. William Poundstone's <i>Prisoner's Dilemma</i> suggests that the disease resulted from the radiation Von Neumann received as a witness to the atomic tests on Bikini atoll. "A number of physicists associated with the bomb succumbed to cancer at relatively early ages.<br /><br /><br /><span style="color: red;"><b>Quantum Mechanics</b></span><br /><br />Von Neumann was a creative and original thinker, but he also had the ability to take other people’s suggestions and concepts and in short order turn them into something much more complete and logical. This is in a way what he did with quantum mechanics after he went to the university in Göttingen, Germany after receiving his degrees in 1926.Quantum mechanics deals with the nature of atomic particles and the laws that govern their actions. Theories of quantum mechanics began to appear to confront the discrepancies that occurred when one used purely Newtonian physics to describe the observations of atomic particles.<br /><br />One of these observations has to do with the wavelengths of light that atoms can absorb and emit. For example, hydrogen atoms absorb energy at 656.3 nm, 486.1 nm, 434.0 nm or 410.2 nm, but not the wavelengths in between.This was contrary to the principles of physics as they were at the end of the nineteenth century, which would predict that an electron orbiting the nucleus in an atom should radiate all wavelengths of light, therefore losing energy and quickly falling into the nucleus. This is obviously not what is observed, so a new theory of quanta was introduced by Berliner Max Plank in 1900 that said energy could only be emitted in certain definable packets.This lead to two competing theories describing the nature of the atom, which could only absorb and emit energy in specific quanta. One of these, developed by Erwin Schrödinger, suggested that the electron in hydrogen is analogous to a string in a musical instrument. Like a string, which emits aspecific tone along with overtones, the electron would have a certain "tone" at which it would emit energy. Using this theory, Schrödinger developed a wave equation for the electron that correctly predicted the wavelengths of light that hydrogen would emit.<br /><br />Another theory, developed by physicists at Göttingen including Werner Heisenberg, Max Born, and Pascual Jordan, focused on the position and momentum of an electron in an atom. They contested that these values were not directly observable (only the light emitted by the atom could be observed) and so could behave much differently from the motion of a particle in Newtonianphysics. They theorized that the values of position and momentum should be described by mathematical constructs other than ordinary numbers. The calculations they used to describe the motion of the electron made use of matrices and matrix algebra.These two systems, although apparently very different, were quickly determined to be mathematically equivalent, two forms of the same principle. The proponents of the two systems, none the less, denounced the others theories and claimed their own to be superior. It is in this environment, in 1926, that von Neumann appears on the scene and quickly went to work reconciling and advancing the theories of quantum mechanics.<br /><br />Von Neumann wanted to find what the two systems, wave mechanics and matrix mechanics, had in common. Through a more rigorous mathematical approach he wanted to find a new theory, more fundamental and powerful than the other two. He abstracted the two systems using an axiomatic approach, in which each logical state is the definite consequence of the previous state. Von Neumann constructed the rules of "abstract Hilbert space" to aid in his development of a mathematical structure for quantum theory. His formalism of the subject allowed considerable advances to be made by others and even predicted strange new consequences, one that consciousness and observations alone can affect electrons in a Labaratory.<br /><br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-PXDHe8ZC3yk/T5kmPNIXeDI/AAAAAAAAL0o/HQN92UClLPc/s1600/693795.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="http://3.bp.blogspot.com/-PXDHe8ZC3yk/T5kmPNIXeDI/AAAAAAAAL0o/HQN92UClLPc/s400/693795.jpg" width="260" /></a></div>.<br /><br /><span style="color: red;"><b>Marriages and America</b></span><br /><br />From 1927-29, after his formalization of quantum mechanics, von Neumann traveled extensively to various academic conferences and colloquia and turned out mathematical papers at the rate of one a month at times. By the end of 1929 he had 32 papers to his name, all of them in German, and each written in a highly logical and orderly manner so that other mathematicians could easily incorporate von Neumann’s ideas into their own work.<br /><br />Von Neumann was now a rising star in the academic world, lecturing on new ideas, assisting other great minds of the time with their own works, and creating an image for himself as a likable and witty young genius in his early twenties. He would often avoid arguments with the more confrontational of his colleagues by telling one of his many jokes or stories, some of which he couldnot tell in the presence of ladies (though there were few women at these mathematical seminars). Other times he would bring up some interesting fact from ancient history, changing the subject and making von Neumann seem surprisingly learned for his age and professional interests.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-ruq5p8GTQAk/T4Gavx2C1LI/AAAAAAAALGk/E1qslYpcJGU/s1600/jon+von+neumann.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="240" src="http://1.bp.blogspot.com/-ruq5p8GTQAk/T4Gavx2C1LI/AAAAAAAALGk/E1qslYpcJGU/s320/jon+von+neumann.jpg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;">Neumann at Princeton</span></td></tr></tbody></table><br />Near the end of 1929 he was offered a lectureship at Princeton in an America that was trying to stimulate its mathematical sciences by seeking out the best of Europe. At this same time, von Neumann decided to marry Mariette Kovesi, whom he had known since his early childhood. Their honeymoon was a cruise across the Atlantic to New York, although most of their trip was subdued by Mariette’s unexpected seasickness.They had a daughter, Marina, in 1935. Von Neumann was affectionate with his new daughter, but did not contribute to the care of her or to the housework, which he considered to be the job of the wife. The gap between the lively 26-year-old Mariette and the respectable 31-year-old John von Neumann began to increase and in 1936 they broke up, Mariette going home to Budapest and von Neumann, after drifting around Europe to various engagements, went to the United States. Soon after, on a trip to Budapest, he met Klari Dan and they were married in 1938.<br /><br />Although this marriage lasted longer than his first, von Neumann was often distant from his personal life, obsessed and engrossed in his thoughts and work. In this personal tradeoff of von Neumann’s the world of science profited tremendously, and much of his work changed all of our lives. Two of the most influential and well known of von Neumann’s interests during his time in America, from 1933 (when he was appointed as one of the few original members of the Institute for Advanced Studies at Princeton) to 1957 (when he died of cancer), were the development of nuclear weapons and the invention of the modern digital computer.<br /><br /><br /><span style="color: red;"><b>Von Neumann’s Role in Nuclear Development</b></span><br /><br />In the biography of a genius such as von Neumann it would be easy to overestimate his role in the development of nuclear weapons in Los Alamos in 1943. It is important to remember that there was a collection of great minds there, recruited by the American government to produce what many saw as a necessary evil. The fear that Germany would produce an atomic bomb before the US drove the effort at Los Alamos.<br /><br />Von Neumann’s two main contributions to the Los Alamos project were the mathematicization of development and his contributions to the implosion bomb.The scientists at Los Alamos were used to doing scientific experiments but it’s difficult to do many experiments when developing weapons of mass destruction. They needed some way to predict what was going to happen in these complex reactions without actually performing them. Von Neumann therefore was a member of the team that invented modern mathematical modeling. He applied his math skills at every level, from helping upper officials to make logical decisions to knocking down tough calculations for those at the bottom of the ladder.<br /><br />The atomic bombs that were eventually dropped were of two kinds, one using uranium-235 as its fissionable material, the other using plutonium. An atomic chain reaction occurs when the fissionable material present in the bomb reaches a critical mass, or density. In the uranium-235 bomb, this was done using the gun method. A large mass of uranium-235, still under the critical mass, would have another mass of uranium-235 shot into a cavity. The combined masses would then reach critical mass, where an uncontrolled nuclear fission reaction would occur. This process was known to work and was a relatively simple procedure. The difficult part was obtaining the uranium-235, which has to be separated from other isotopes of uranium, which are chemically identical.<br /><br />Plutonium, on the other hand, can be separated using chemical means, and so production of plutonium based bombs could progress more quickly. The problem here was that plutonium bombs could not use the gun method. The plutonium would need to reach critical mass through another technique, implosion. Here, a mass of plutonium is completely surrounded by high explosives that are ignited simultaneously to cause the plutonium mass to compress to supercritical levels and explode.<br /><br />Although von Neumann did not arrive first at the implosion technique for plutonium, he was the one who made it work, developing the "implosion lens" of high explosives that would correctly compress the plutonium.This is just one more example of von Neumann’s ability to pick up an idea and advance it where others had gotten stuck.<br /><br /><br /><span style="color: red;"><b>Development of Modern Computing</b></span><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/-lgNa_Sqxkdg/T4GaVjfRXFI/AAAAAAAALGc/g1_ovdGCe5Y/s1600/Von_Neumann_5.jpeg" style="margin-left: auto; margin-right: auto;"><img border="0" height="216" src="http://3.bp.blogspot.com/-lgNa_Sqxkdg/T4GaVjfRXFI/AAAAAAAALGc/g1_ovdGCe5Y/s320/Von_Neumann_5.jpeg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;">Von Neumann with first super Computer</span></td></tr></tbody></table><br /><br />Just like the project at Los Alamos, the development of the modern computer was a collaborative effort including the ideas and effort of many great scientists. Also like the development of nuclear weaponry, there have been many volumes written about the development of modern computer. With so much involved in the process and von Neumann himself being involved in so much of it, only a few contributions can be covered here.<br /><br />A<span style="color: red;"> <b>von Neumann language</b></span> is any of those programming languages that are high-level abstract isomorphic copies of von Neumann architectures. As of 2009, most current programming languages fit into this description, likely as a consequence of the extensive domination of the von Neumann computer architecture during the past 50 years.<br /><br />Von Neumann’s experience with mathematical modeling at Los Alamos, and the computational tools he used there, gave him the experience he needed to push the development of the computer. Also, because of his far reaching and influential connections, through the IAS, Los Alamos, a number of Universities and his reputation as a mathematical genius, he was in a position to secure funding and resources to help develop the modern computer. In 1947 this was a very difficult task because computing was not yet a respected science. Most people saw computing only as making a bigger and faster calculator. Von Neumann on the other hand saw bigger possibilities.<br /><br />Von Neumann wanted computers to be put to use in all fields of science, bringing a more logical and precise nature to those fields as he had tried to do. With his contributions to the architecture of the computer, which describe how logical operations are represented by numbers that can then be read and processed, many von Neumann’s dreams have come true. Today we have extremely powerful computing machines used in scores of scientific fields, as well many more non-scientific fields.<br /><br />In von Neumann’s later years, however, he worked and dreamed of applications for computers that have not yet been realized. He drew from his many other interests and imagined powerful combinations of the computer’s ability to perform logically and quickly with our brain’s unique ability to solve ill defined problems with little data, or life’s ability to self-reproduce and evolve.In this vein, von Neumann developed a theory of artificial automata. Von Neumann believed that life was ultimately based on logic, and so any construct that supports logic should be able to support life. Artificial automata, like their natural counter parts, process information and proceed in their actions based on data received from their environment in light of rules and instructions they hold internally. Cellular automata are a class of automata that exist in an infinite plane that is covered by square cells, much like a sheet of graph paper. Each of these cells can rest in a number of states. The whole plane of cells will go through time steps, where the new state of each cell is determined by its own state and the state of the cells neighboring it. In these simple actions there lies a great complexity and the basis for life like actions.<br /><span style="color: red;"><br /></span><span style="color: red;"><b>Untimely End</b></span><br /><br />Perhaps all deaths can be considered to come too early; John von Neumann’s own death came far too early. He died on February 8, 1957, 18 months after he was diagnosed with cancer.He never finished his work on automata theory, although he worked as long as he possibly could. He attended ceremonies held in his honor using a wheelchair, and tried to keep up appearances with his family and friends. Though he had accomplished so much in his years he could not accept death, could not consider a world that existed without his mind constantly thinking and solving. But today, his ideas live on and affect our lives in more ways than the few examples given here can demonstrate.<br /><br />Nassif Nabeal (Nassif Medici)http://www.blogger.com/profile/13708191074451322062noreply@blogger.com0tag:blogger.com,1999:blog-5761410253972037435.post-74203244073864543182013-10-26T05:48:00.003-07:002015-06-25T22:10:08.594-07:00Logic:Ludwig Wittgenstein & Bertrand Russell<br /><span style="color: red;"><span style="font-size: small;"><b>Ludwig Wittgenstein</b> </span></span>was born in Vienna in 1889 and died in Cambridge in 1951<span style="font-family: inherit;">.</span><span style="font-family: inherit; font-size: 11pt; line-height: 115%;">Wittgenstein found the final solution to philosophy, a man who put end to philosophy by changing it into logic. </span>His father, Karl, a friend of Johannes Brahms, was the most acute industrialist in the Austrian steel industry; he made the family the Austrian equivalent of the Carnegies or Rothschilds. He had five sons and three daughters by a Roman Catholic wife, and baptized all of them into the Catholic faith. He set out to educate the sons in a very severe regime which would turn them into captains of industry. He did not succeed. Three of the sons committed suicide; the fourth, Paul, became (despite the loss of an arm in World War I) a concert pianist; the fifth, the youngest child, born in Vienna in 1889, was the philosopher. Wittgenstein studied engineering, first in Berlin and then in Manchester, and he soon began to ask himself philosophical questions about the foundations of mathematics. What are numbers? What sort of truth does a mathematical equation possess? What is the force of proof in pure mathematics? In order to find the answers to such questions, he went to Cambridge in 1911 to work with Russell, who had just produced in collaboration with Whitehead (1861-1947) <i>Principia Mathematica</i> (1910-1913), a monumental treatise which bases mathematics on logic<span style="font-family: inherit;">. </span><span style="font-family: inherit; font-size: 11pt; line-height: 115%;">While in Cambridge he worked his way up to publish on logic and philosophy and became a top philosopher (he never read a philosophy book before) .Russell later was convinced by him and later worked with him. Wittgenstein in 2 years was able to read all Russell has to teach him ,so Wittgenstein choose his father (Russell)and killed him in philosophy.</span>But on what is logic based? Wittgenstein's attempt to answer this question convinced Russell that he was a genius. During the 1914-8 war he served in the Austrian army and in spare moments continued the work on the foundations of logic which he had begun in 1912.<span style="font-family: inherit;"> </span><span style="font-family: inherit; font-size: 11pt; line-height: 115%;">Later he went on to know the meaning of life, which is knowing the world and knowing God. Later he wrote <span style="color: red;">“Tractus Logico Philosophicus”</span> the most prolific work of philosophy ever written and even its first few pages tell you how different and powerful this work is. He said that some things cannot be proved to be true but can be showed to be true. We cannot say anything about God, but we believe and we cannot see it and though nature suggests that something is governing everything.</span><br /><br /><span style="color: #0b5394;"><b>"He was perhaps the most perfect example I have known of genius as traditionally conceived, passionate, profound, intense, and dominating".</b></span><br /><span style="color: #0b5394;"><b>Bertrand Russell on Wittgenstein</b></span><br /><br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-pIjTv5Zm2Sc/T4LyFX1aLZI/AAAAAAAALP8/QBgauVEu5KY/s1600/kefpqcer.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="300" src="http://1.bp.blogspot.com/-pIjTv5Zm2Sc/T4LyFX1aLZI/AAAAAAAALP8/QBgauVEu5KY/s400/kefpqcer.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b><span style="font-family: Georgia,'Times New Roman',serif; font-size: small;">Ludwig Wittgenstein, <b><span style="line-height: 115%;">Wittgenstein had the pride of Lucifer and arrogance which he inherited from his father but there was a saint in him,he was perfectionist who killed philosophy from its root.</span></b></span></b><br /></td></tr></tbody></table>He then abandoned philosophy for a life of a very different kind. He taught in a village school in Austria and after that worked in the garden of a monastery. He returned to philosophy in the late 1920s, drawn back into it by discussions with some of the members of the Vienna Circle and with the Cambridge philosopher Frank Ramsey (1903-30)<span style="font-family: inherit;">.</span><span style="font-family: Calibri,sans-serif; font-size: 11pt; line-height: 115%;"><span style="font-family: inherit;"><b><span style="font-family: inherit;"> <span style="color: #0b5394;">He was considered “God” by top intellectuals like Maynard Keynes, he was becoming eccentric though as he was conquering philosophy and logic. Probably only Russell understood his later </span></span><span style="color: #0b5394;">work on logic.</span></b><span style="color: blue;"> </span></span></span>They wanted him to explain the <i><span class="Apple-style-span" style="color: red;">Tractatus</span></i>, but their request for elucidations soon produced second thoughts. When he returned to Cambridge, he developed a different philosophy which made its first public appearance in 1953 in his posthumous book, <span class="Apple-style-span" style="color: red;"><i>Philosophical Investigations</i>. </span>Before its publication, direct acquaintance with his new ideas had been confined to those who attended his lectures and seminars in Cambridge. <br />In the second period of his philosophy, as in the first, his notes are the key to his published work. Written continuously from 1929 until his death, they would occupy many metres of shelves if they were all edited and published as books. Philosophers usually intend what they write to be read by others, but these notes are a kind of thinking on paper. It is true that some of the sets are evidently being steered towards publication, but in most of them Wittgenstein is facing problems alone. Since his death many books have been extracted from this material.<br />In 1939 he was elected as Professor of Philosophy in Cambridge, and after years of teaching he went to live in a cottage and preferred to live like a saint. He was later diagnosed with Cancer and died in Cambridge.<br /><span style="color: #3d85c6;"><span style="font-family: Georgia,'Times New Roman',serif;"><br /></span></span><br /><div><span style="color: #3d85c6;"><span style="font-family: Georgia,'Times New Roman',serif;"><span style="font-size: small;"><b><span style="line-height: 115%;">Wittgenstein outlined his legacy, he said that his philosophy is written only for those who value his depth and aspiration. Wittgenstein had the pride of Lucifer and arrogance which he inherited from his father but there was a saint in him,he was perfectionist who killed philosophy from its root.He wanted to know life,God and he lived life of a sait ,went into isolation as well and also got eccentric.</span></b></span></span></span></div><div class="MsoNormal"><br /></div><span style="color: red;"><b>2. The General Character of Wittgenstein's Philosophy</b></span> <br />Wittgenstein's philosophy is difficult to place in the history of ideas largely because it is anti-theoretical. It is true that in his early work he did produce a theory of logic and language, but it was a theory which demonstrated its own meaninglessness. That was a paradox which he presented, appropriately enough, in a metaphor borrowed from the Greek sceptic, Sextus Empiricus (c.150-c.225): 'Anyone who understands me eventually recognises [my propositions] as nonsensical, when he has used them as steps - to climb up beyond them. (He must, so to speak, throw away the ladder after he has climbed up it)' (Wittgenstein, 1922, 6.54). After 1929 he completely avoided theorising. The task of philosophy, as he now saw it, was never to explain but only to describe. Since western philosophy had mainly been conceived as a search for explanations at a very high level of generality, his work stood to one side of the tradition. <br /><br />Wittgenstein was not a sceptic. The reason why he rejected philosophical theorising was not that he thought it too risky and liable to error, but because he believed that it was the wrong way for philosophers to work. Philosophy could not, and should not try, to emulate science. That is a point of affinity with Kant, but while Kant's critique resulted in a system in which each of the many forms of human experience found a place, Wittgenstein attempted no such thing. His method was to lead any philosophical theory back to the point where it originated, which might be some very simple routine, observable even in the life of animals but rendered unintelligible by the demand for an intellectual justification. Or it might start from the 'crossing of two pictures' - for example, we construe sensations (pp. 180-2) as objects that are not essentially dependent on their links with the physical world, and so we attribute to them a basic independence modelled on the basic independence of physical objects. His aim was to cure this kind of illusion by a therapy that would gradually lead the sufferer to recognise, and almost to recreate its origin, and so to escape from its domination. <br />Philosophers are expected to be able to abstract the general from the particular, but Wittgenstein's gift was the opposite - a rare ability to see the particular in the general. He could demolish a theory with a few appropriate counter-examples. His method was to describe an everyday situation which brings a philosophical speculation down to earth. When he used imagery, it was carefully chosen to reveal the structure of the problem under examination. All this helps to explain why his later philosophical writings have been read and appreciated by people with very little philosophical training. However, the explanation of the wide appeal of his later work is not just stylistic. He is evidently taking apart a philosophical tradition that goes back to antiquity. That is a way of treating the past which can be found in many other disciplines today, and even when the scene that he is dismantling has not been precisely identified he can still be read with sympathy and with intuitive understanding.<br /><span style="color: red;"><br /></span><span style="color: red;"><b>3. Wittgenstein's Early Philosophy</b></span> <br />In spite of the wider appeal of his later work, Wittgenstein was without a doubt a philosopher's philosopher. In the <i>Tractatus</i> he developed a theory of language that was designed to explain something that Russell had left unexplained in <i>Principia Mathematica</i>, the nature of logical necessity. The marginal status of theories in his early philosophy did not deffect this theory from its main goal, which was to show that logically necessary propositions are a kind of by-product of the ordinary use of propositions to state facts. A factual proposition, according to Wittgenstein, is true or false with no third alternative. For he agreed with Russell's theory of definite descriptions: failure of a complex reference simply makes a proposition false. So if two propositions are combined to form a third, compound, proposition, its truth of falsity will simply depend on the truth or falsity of its two components. Now suppose that we want to find out if it really is a contingent, factual proposition, like its two components. What we have to do is to take the two components and run through all their combinations of truth and falsity, and we will find that there are three possible outcomes to this test. The compound proposition may be true for some combinations of the truth-values of its components but false for others, in which case it is a contingent factual proposition. Or it may come out false for all combinations, in which case it is a contradiction. Or, finally, it may be a tautology, true for all combinations. Contradictions and tautologies say nothing. However, they achieve this distinction in two opposite ways, the former by excluding, and the latter by allowing, every state of affairs. So propositions whose truth or falsity are guaranteed by logical necessity are limiting cases, extreme developments of the essential nature of factual propositions. <br /><br />This explanation of logical necessity answered a question that the axiomatization of logic by Frege and Russell had left unanswered: why should we accept the axioms and rules of inference with which such calculi start? In fact, if every proposition can be tested independently for logical necessity, there does not seem to be any need to promote some of them as axioms and to deduce the others from them. Schopenhauer (1788-1860) had taken the same view of the axiomatization of geometry, which seemed to him to be rendered superfluous by spatial intuition. <br />Wittgenstein's early account of the foundations of logic relies on semantic insight. If a formula is logically necessary, we can see that it is. There is no need to prove its status from axioms, because a truth-functional analysis will reveal it. If one factual proposition follows from another, we can see that it does, and there is no need to demonstrate that it does by a proof that starts from a logically necessary formula. The two propositions will simply show that their senses are connected in a way that validates our inference. This treatment of logic leaves proof without any obvious utility - a problem that Wittgenstein took up later in his philosophy of mathematics. <br />The pictorial character of propositions is a theme with many developments in the <span style="color: red;"><i>Tractatus</i></span>. Not only is it used to explain the foundations of logic and the internal structure of factual language; it also has implications for scientific theories, it yields a treatment of the self which carries the ideas of Hume, Kant and Schopenhauer one stage further, and it allows Wittgenstein to demonstrate that factual language has definite limits. <br /><br />Wittgenstein believed that Ethics, Aesthetic and religious discourse lie beyond those limits. This has started a long-running controversy about the implications of his placing of the 'softer' kinds of discourse. Was he a positivist, as the philosophers of the Vienna Circle later assume? Or could the opposite conclusion be drawn from what he said about the <i>Tractatus</i> in a letter to L. Ficker: 'The book's point is an ethical one . . . My work consists of two parts: the one presented here plus all that I have not written. And it is precisely this second part that is the important one. My book draws limits to the sphere of the ethical from the inside, as it were' - that is, from inside factual language. If this epigram is taken literally, it appears to put a low value on his solutions to the problems of language and logic. However, nearly all his later work is concerned with the same problems, and there is really no need to subject the <i>Tractatus</i> to such a simple dilemma. Its point is Kantian: ethics (and the whole softer side of discourse, including philosophy itself) must not be assimilated to science.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-XEajT0b5Ifc/T5-Q_QrentI/AAAAAAAAL6E/n9gFmld-mqk/s1600/103068415.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="400" src="http://1.bp.blogspot.com/-XEajT0b5Ifc/T5-Q_QrentI/AAAAAAAAL6E/n9gFmld-mqk/s400/103068415.jpg" width="261" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><i><b>"The world is the totality of facts,not of things" </b></i><i><b>Tractatus </b><b><span class="st">Logico Philosophicus</span></b></i></td><td class="tr-caption" style="text-align: center;"><br /></td><td class="tr-caption" style="text-align: center;"><i><b><br /></b></i></td><td class="tr-caption" style="text-align: center;"><br /></td></tr></tbody></table><span style="color: #3d85c6;">“Death is not an event in life: we do not live to experience death. If we take eternity to mean not infinite temporal duration but timelessness, then eternal life belongs to those who live in the present. Our life has no end in the way in which our visual field has no limits.” </span><br /><span style="color: #3d85c6;">Ludwig Wittgenstein, <span class="st"> <b><i>Tractatus .</i></b></span></span><br /><span class="st"><b><i> </i></b></span><i> </i><br /><span style="color: red;"><b>4. Propositions as Pictures</b></span> <br />Wittgenstein's early theory of language was also developed in another direction. If it threw light on the foundations of logic, it ought also to throw light on the structure of ordinary factual discourse. In order to understand this development, we have to go back one step and ask why he thought that factual propositions must be true or false with no third alternative. His reason was that he took them to be a kind of picture. If the points on the canvas of a landscape-painter were not correlated with points in space, no picture that he painted would succeed in saying anything. Similarly, if the words in a factual proposition were not correlated with things, no sentence constructed out of them would say anything. In both cases alike the constructions would lack sense. But, given the necessary correlations, the painting and the proposition have sense and what they say can only be true or false. <br />Now this runs up against an obvious objection. Many words designate complex things, which do not have to exist in order that the propositions containing them should make sense, and this casts doubt on the analogy between points on a canvas and words. The obvious response would be to claim that such words are complex and that the simpler words out of which they are compounded do have to designate things. Wittgenstein went further and argued that it must be possible to continue this kind of analysis to a point at which no more subdivision would be possible. His argument for this extreme version of logical atomism had nothing to do with empiricism. What he argued was that, if analysis stopped short of that terminus, the sense of a proposition containing a word which designated something complex would depend on the truth of a further proposition. This further proposition would say that things had been combined to form the complex but it would not be part of the sense of the original proposition. That, he argued, was an unacceptable result both for pictures and propositions. Their senses must be complete, self-contained and independent of one another.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://4.bp.blogspot.com/-7397ciXMRmc/T4LyLNJuAmI/AAAAAAAALQE/vVh8beBDn2E/s1600/wittgenstein_brothers.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="http://4.bp.blogspot.com/-7397ciXMRmc/T4LyLNJuAmI/AAAAAAAALQE/vVh8beBDn2E/s320/wittgenstein_brothers.jpg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;"><span style="color: #3d85c6;">What can be said at all can be said clearly, and what we cannot talk about we must pass over in silence.</span></span><br /><span style="font-size: small;"><span style="color: #3d85c6;">—Ludwig Wittgenstein in<i> <b>Tractatus</b></i><b>.</b></span></span></td></tr></tbody></table><br /><span style="color: red;"><b>5. Transition</b></span> <br />The first modification of the system of the <i>Tractatus</i> appeared in 1929 in Wittgenstein's article, 'Some Remarks on Logical Form' (the only other piece of work that he published - everything else is posthumous). He no longer believed in the extreme version of logical atomism for which he had argued in 1922. The requirement, that elementary propositions be logically independent of one another, now struck him as excessive. The reason for his change of mind was simple: singular factual propositions always contain predicates belonging to ranges of contraries. So colour-predicates are incompatible with one another and there is no hope of analysing them into simpler predicates that would not be incompatible with one another. Position, length, velocity, and, in general, all measurable properties, show the same recalcitrance to the analysis required by the <i>Tractatus</i>. He therefore dropped the requirement. <br /><br />There are two things that make this change of mind important. Firstly, though the <i>Tractatus</i> contains an atomistic theory of language, there are passages that reveal an underlying holism. For example, he says: 'A proposition can determine only one place in logical space: nevertheless the whole of logical space must already be given with it' (1922, 3.42). But the new view of elementary propositions is an open move towards holism. What is now said to be 'laid against reality like a ruler' (1922, 2.1512) is not a single, independent, elementary proposition, but, rather, a set of logically incompatible elementary propositions. For when one predicate in a group of contraries is ascribed to a thing, the others are necessarily withheld. It is plausible to regard this holism as the natural tendency of Wittgenstein's mind, and the atomism as something that he took over from Russell and eventually repudiated.The change of mind also has a more general importance. <span style="color: #3d85c6;">The atomism of the <i>Tractatus</i> was offered not as a theory that was supposed to fit the observable surface of factual language, but as a theoretical deduction about its deep structure. </span>Indeed, Wittgenstein was so confident of the validity of the deduction that he was not worried by his inability to produce a single example of a logically independent elementary proposition. This dogmatism evaporated when it occurred to him that the logical structure of language might be visible on its surface and might actually be gathered from the ordinary uses that we make of words in ordinary situations. This was the point of departure of his later philosophy. <br /><br /><span style="color: red;"><b>6. His Later Philosophy: The Blue Book</b></span> <br />The most accessible exposition of the leading ideas of Wittgenstein's later philosophy is to be found in the <span style="color: red;"><i>Blue</i> <i>Book</i> (1958)</span>, a set of lecture-notes that he dictated to his Cambridge pupils in 1933-4. What he then did was so far out of line with the tradition that we may at first feel inclined to question whether it really is philosophy. His answer was that it might be called 'one of the heirs of the subject which used to be called "philosophy".' The conspicuous novelty is the absolute refusal to force all the multifarious variety of thought and language into the mould of a single theory. He criticised the 'contempt for the particular case' that any such attempt would involve, and he systematically repressed the craving for generality that has characterised western philosophy since Socrates first instigated the search for the essences of things. We may, of course, ask such questions as 'What is knowledge?', but we must not expect to find the answer wrapped up in the neat package of a definition. There will be many different cases, and though they will show a family resemblance to one another, they will not be linked by the possession of a single set of common properties. Socrates asked for a conjunction of properties, but we must be content with a disjunction. <br />That is an accurate placing of Wittgenstein's new philosophy in the history of ideas, but it leaves an important question unanswered. Why should a catalogue of examples be regarded as a solution to a philosophical problem? Is it not just a collection of the kind of material that poses the problem? The point of these questions is that 'the heir to philosophy' needs to be something more than well-documented negative advice not to theorise: it ought to teach us to see philosophical problems from the inside and to find a more positive way of laying them to rest. <br />There are, in fact, two discussions in the <i>Blue Book</i> that demonstrate that Wittgenstein's later work was a positive continuation of the philosophy of the past. One is the long investigation of meaning and the other is the treatment of the self. Both are very illuminating. <br />The discussion of meaning is a development of a point made in the <i>Tractatus</i>: 'In order to recognise a symbol by its sign, we must observe how it is used with a sense' (1922, 3.326). This remark consorts uneasily with the picture theory of propositions, which derives meaning from the original act of correlating name with object. The theory implies that meaning is rigid, because it is based on a single, self-contained connection which, once made, remains authoritative, without any need for interpretation or any possibility of revision. The remark points the way to a more flexible account of meaning which will accommodate all the different uses that we make of words and leaves room for plasticity. This is the difference between treating language as a fossil and treating it as a living organism. <br /><br />The discussion of meaning in the Blue Book develops the isolated remark in the <i>Tractatus</i> and criticise the rigidity of the theory offered elsewhere in the book. Ostensive definition, which was supposed to attach a word to its object, is shown up as a very inscrutable performance, compatible with many different interpretations of a word's meaning: the underlying assimilation of all descriptive words to names designating objects is rejected; and so too is the assumption that the meaning of a word is something that belongs to it intrinsically, and, therefore, independently of its use. This last point proved to be important. For if meaning never belongs to a word intrinsically, it will never be possible to explain the regularity of a person's use of a word by citing the rule that he or she is following. For the meanings of the words in which the rule is expressed will themselves need to be interpreted. This line of thought is developed in <span style="color: red;"><i>Philosophical Investigations</i> (1953)</span>. <br />The treatment of the self in the Blue Book is very clear and strongly argued. As in the Notebooks and the <i>Tractatus</i>, it is presented as part of an examination of solipsism, but it is much easier to discern the structure of the later version of the argument. The central point is that the solipsist's claim 'Only what I see exists', is not what it seems to be. The solipsist seems to be referring to himself as a person, but really he or she is using the pronoun 'I' to refer to something entirely abstract which is introduced merely as 'the subject which is living this mental life' or 'the subject which is having these visual impressions'. But if the subject is not given any independent criterion of identity, there is no point from which the reference to 'these impressions' can be made. The solipsist constructs something which looks like a clock, except that he pins the hand to the dial, so that they both go round together. Saying what exists, like telling the time, must be a discriminating performance. The idea that the subject is a vanishing point, which was developed by Hume, Kant and Schopenhauer, is here put to a new use. <br /><br /><span style="color: red;"><b>7. <i>Philosophical Investigations</i>: the Private Language Argument</b></span> <br />The so-called 'private language argument' of <i>Philosophical Investigations</i> is closely related to the rejection of a solipsism which is based on an ego without a criterion of identity. What the two critiques have in common is a requirement which was later expressed very concisely by W. V. O. Quine, 'No entity without identity'. The solipsist's ego lacks any criterion of personal identity, and similarly, if the quality of a sensory experience were completely disconnected from everything in the physical world - not only from any stimulus but also from any response - it would lack any criterion of type identity. The parallelism of the two critiques is very close in the lecture-notes in which Wittgenstein first developed the so-called 'private language argument'. Against the solipsist who says, 'But I am in a favoured position. I am the centre of the world', he objects, 'suppose I saw myself in a mirror saying this and pointing to myself, would it still be all right?' (Notes for Lectures on 'Private Experience' and 'Sense-data', 1968, p. 299). Against the phenomenalist who argues for detached sensation-types and says, 'But it seems as if you are neglecting something', he objects, 'What more can I do than <i>distinguish</i> the case of saying "I have tooth-ache" when I really have toothache, and the case of saying the words without having tooth-ache? I am also (further) ready to talk of any x behind my words so long as it keeps its identity' (1968, p. 297).The interpretation of this important line of thought in his later work is difficult. The parallelism between the two critiques is always a helpful clue, a thread which we must never relinquish in the labyrinth of confusing indications. <br /><br />One source of confusion is hardly Wittgenstein's fault. He himself never used the phrase 'private language argument'. It is his commentators who use it and, by doing so, they have created the illusion that a single, formal argument ought to be extractable from the text of <i>Philosophical Investigations</i> is not the structure of his critique. He argued dialectically, and when his adversary tries to introduce the 'neglected x' behind the words reporting a sensation, he always tries to show his adversary that, if this x is not covered by ordinary criteria of identity based on the physical world, it will not have any criterion of identity at all.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-MYs1nBdsH18/T4LydrIkhYI/AAAAAAAALQU/c4ezm1C1W6c/s1600/180px-Wittgenstein-investigations.JPG" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://3.bp.blogspot.com/-MYs1nBdsH18/T4LydrIkhYI/AAAAAAAALQU/c4ezm1C1W6c/s320/180px-Wittgenstein-investigations.JPG" width="214" /></a></div><br />At this point it is a good idea to ask who his adversary is. Evidently, ,his adversary is a philosopher who supports some kind of sense datum theory. B t what kind? One suggestion that has been made is that it is the <i>sense datum</i> theory which he himself adopted in the <i>Tractatus</i> (not very openly). Since he extended his critique to include other mental entities besides sensations, this suggestion has been generalised, and he has been taken to be criticising the 'mentalistic theory of meaning of the <i>Tractatus</i>' (Malcolm, 1986, ch. 4). But a brief review of the development of his philosophy of mind will show that these suggestions are mistaken. <br /><br />Anyone who compares what Wittgenstein said about simple objects in the <i>Notebooks</i> and in the <i>Tractatus</i> will see immediately that he was uncertain of their category in the former and in the latter was convinced that his uncertainty did not endanger his logical atomism. Maybe they were material particles or perhaps they were Russellian sense data. He did not care, because his argument for their existence did not depend on their category and he did not think that it needed verification by the actual discovery of examples. However, the possibility that they were sense data was worth exploring, especially after his abandonment in 1929 of the extreme version of logical atomism. So when he returned to these problems after the long interval that followed the publication of the <i>Tractatus</i>, he worked out the consequences of identifying them with sense data. This fitted in very well with the programme of the Vienna Circle philosophers. They were interested in the philosophy of science and predisposed to accept a simple stratification of language, with the phenomenal vocabulary on the basic level and the physical vocabulary on the upper level and complete inter-translatability between the two levels. <br /><br />It is notorious that this kind of phenomenalism looks impregnable until we examine the route that led us into it. That is what Wittgenstein did, and he found an obstacle which seemed to him to make the route impassable. The original position from which it started was supposed to be one in which people spoke a phenomenal language with a vocabulary completely disconnected from the physical world. It is obvious that such a language would be necessarily unteachable, and since Wittgenstein used the word 'private' to mean 'necessarily unteachable', it would be a private language. What is not so obvious is how he thought that he could show that such a language would be impossible. <br />The primary target of his critique of private language is the sense-datum language that phenomalists claimed that each of us could set up independently; of anything in the physical world and, therefore, in isolation from one another. But the scope of his attack is much wider, because it would show that no mental entity of any kind could ever be reported in such a language. However, neither in its narrow nor in its wide scope is it directed against anything in the <i>Tractatus</i>. For just as there was no commitment to phenomenalism in the early work, so too there was no commitment to the thesis that the meaning of a sentence is derived from the meaning of the thought behind it. If the critique of private language is related to anything in the <i>Tractatus</i>, it is to the critique of ego-based solipsism, but positively, as a further application of the same general demand for a criterion of identity. <br />It is necessary to distinguish two moves that Wittgenstein made in his dialectical critique of a necessarily unteachable sensation-language. The phenomenalist believes that we can set up this language and use it to report our sense data in complete independence from anything in the physical world. Against this, Wittgenstein's first move was the one that has already been described: he asked for the criterion of identity of the supposedly independent sensation-types. That is a purely destructive demand. His second move was to point out that a report of a sensation will usually contain an <i>expression</i> of the sensory type and seldom a <i>description</i> of it. This move was the beginning of a reconstruction of the situation, designed to lead to a better account of sensation-language. <br /><br /> The destructive move is made most perspicuously in <i>Philosophical Investigations</i>. Suppose that a word for a sensation-type had no links with anything in the physical world and, therefore, no criteria that would allow me ~ to teach anyone else its meaning. Even so, I might think that, when 'I applied it to one of my own sensations, I would know that I was using it correctly But, according to Wittgenstein, that would be an illusion, because in such an isolated situation I would have no way of distinguishing between knowing that my use of the word was correct and merely thinking that I knew that it was correct., Notice that he did not say that my claim would be wrong: his point is more radical - there would be no right or wrong in this case. (Wittgenstein, 1953, § 258). <br />The common objection to this criticism is that it simply fails to allow for the ability to recognise recurring types of things. This, it is said, is a purely intellectual ability on which we all rely in the physical world. So what is there to stop a single person relying on it in the inner world of his mind? Perhaps Carnap was right when he chose 'remembered similarity' as the foundation of his <i>Logical Structure of the World</i> (1967).Here Wittgenstein's second move is needed. If the ability to recognise types really were purely intellectual, it might be used in the way in which Carnap and others have used it, and it might be possible to dismiss Wittgenstein's objection by saying, 'We have to stop somewhere and we have to treat something as fundamental - so why not our ability to recognise sensation types?' But against this Wittgenstein argues that what looks like a purely intellectual ability is really based on natural sequences of predicament, behaviour and achievement in the physical world. Pain may seem to be ~a"clea example of a sensation-type which is independently recognisable, but the word is really only a substitute for the cry which is a natural expression of the sensation (1953, ~~ 244-6). Or, to take another example, our ability to recognise locations in our visual fields is connected with the success of our movements in physical space. Our discriminations in the inner world of the mind are, and must be, answerable to the exigencies of the physical world. <br />At this point we might begin to regret Wittgenstein's refusal to theorise. If he had offered a more systematic account of the dependence of our sensory language on the physical world, the so-called 'private language argument' might have carried more conviction. In fact, many philosophers have been convinced by it, but there is a large opposition, containing few doubters and consisting almost entirely of philosophers who feel sure that the argument is invalid. The dialectical character of Wittgenstein's argument has contributed to this result. <br /><br /><span style="color: red;"><b>8. <i>Philosophical Investigations</i>: Meaning and Rules</b></span> <br />Another, similar-sounding, but in fact very different, question is discussed in <i>Philosophical Investigations</i>. Could a person speak a language that was never used for communication with anyone else? Such a language would be private in the ordinary sense of that word, because it would be unshared; but it would not be necessarily unteachable, because it would be a language for describing the physical world, and so it would not be private in Wittgenstein's sense. The question is important, but we have to go back to the theory of meaning of the <i>Tractatus</i> in order to see why it is important. <br />A rough, but useful, distinction can be drawn between two kinds of theory of meaning, the rigid and the plastic. The theory offered in the <i>Tractatus</i> is rigid. Once names have been attached to objects everything proceeds on fixed lines. The application of the names is settled once and for all, and propositions and truth-functional combinations of propositions, including the two limiting cases, namely tautology and contradiction, all unfold without any more help from us. The theory does not actually treat the meaning of a name as something intrinsic to it, because we do have to correlate the name with an object. If we want an example of a theory that does take the further step and treats meaning as an intrinsic feature of a symbol, there is the theory that a mental image automatically stands for things that it resembles. That illustrates the extreme degree of rigidity: we would have no options. <br /><br /> A plastic theory of meaning would reject the analogy between a descriptive word and a name, and it would deny that the meaning of a descriptive word can be fixed once and for all by ostensive definition. Both these moves are made in <i>Philosophical Investigations</i>. It is, of course, not denied that our use of a descriptive word will exhibit a regularity: what is denied is that it is a regularity that is answerable to an independent authority. We do have options. We may say, if we like, that we are following a rule, but that will not be an explanation of the regularity of our practice, because it is our practice that shows how we are interpreting the words in which the rule is expressed. Are there, then, no constraints? Is the use of descriptive language pure improvisation? Evidently, there must be some limit to plasticity, and one obvious possibility is that it is imposed by the need to keep in step with other people. That is how the question, 'Does language require exchanges between members of a community of speakers?' comes to be important in <i>Philosophical Investigations</i>. When we trace the line of development from Wittgenstein's early to his late theory of meaning, we need to know how far he moved in the direction of plasticity. <br /> <br /> His first two steps were taken soon after his return to philosophy in 1929. The meaning of a symbol can never be one of its intrinsic features, and even an ostensive definition cannot saddle it with a single, definite meaning, because an ostensive definition is always compatible with many different sequels. Meaning, then, must depend on what we do next with a word - on our use of it. What limit to plasticity did Wittgenstein recognise? Did he treat agreement with the responses of other people as an absolute constraint? And did he recognise any other absolute constraints? It is difficult to extract definite answers to these questions from his later writings. Both in <i>Philosophical Investigations</i> and in Remarks on the Foundations of Mathematics (1978) he says that agreement in judgements is required if people are going to communicate with one another. But that does not rule out the possibility that a wolf-child might develop a language solely for his own use. It would, of course, be a language of written signs with which he would communicate with himself across intervals of time. But it would not require the co-operation of other people. So it looks as if Wittgenstein went no further than maintaining that, if there are other people around and if the language-user is going to communicate with them, the plasticity of his language will be limited by the exigencies of agreement with them. This squares with the fact that in several of his texts dating from the 1930s he allows that a person might set up a system of signs solely for his own use. However, though Wittgenstein does not deny this in <i>Philosophical Investigations</i> or in Remarks on the Foundations of Mathematics, he does not re-assert it either. There seems to be some ambivalence. <br />In any case, there is another, more important, constraint on plasticity which really is absolute. If we are going to discover regularities in nature, our language must exercise a certain self-discipline: it must follow routines which allow us to collect evidence, to make predictions and, later, to understand them. Sheer improvisation will not put us in a position to do these things. This obvious constraint is overlooked by those who attribute to Wittgenstein a 'community theory of language'. It is hardly likely that he overlooked it.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/-yWTXFpLNSUg/T5UtANJtfWI/AAAAAAAALuE/4dgjyxYxzSI/s1600/wittgenstein1.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="400" src="http://2.bp.blogspot.com/-yWTXFpLNSUg/T5UtANJtfWI/AAAAAAAALuE/4dgjyxYxzSI/s400/wittgenstein1.jpg" width="258" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Ludwig Wittgenstein, Professor of Philosophy, Cambridge</td></tr></tbody></table><span style="color: red;"><br /></span><span style="color: red;"><b>9. The Authority of Rules</b></span> <br />Wittgenstein's next step in the development of his theory of meaning was to argue by reduction and absurdum. Given that the meaning of a word is never contained in the word itself, either intrinsically or after an ostensive definition, it must be a mistake to hold that someone who follows a linguistic rule is obeying an independent, external authority. For any basis that we might propose for the so-called 'authority' will always leave it open what the speaker should do next, and, what is more, open between many different alternatives. If we try to remedy this situation by offering a more explicit statement of the rule that he is supposed to be following, he will still be able to interpret that statement in many different ways. So when we try to fix the right use of a word purely by precept and past applications, we fail, because we end by abolishing the distinction between right and wrong. We feel that the distinction requires a rigid external authority and so we eliminate all plasticity, but, when we do that, we find we have lost the distinction. <br />It would be absurd to suggest that the meaning of a particular instruction is determined by what a person does when he tries to obey it. If the instruction did not already stand there complete with its meaning, there would be no question of obedience. But when the same suggestion is made about a general instruction, or rule, it is not absurd but only paradoxical. In order to understand Wittgenstein's argument, we have to elucidate the paradox.<br /><br />It shocks us, because in daily life there is no doubt about what counts as obeying the instruction, 'Always take the next left turn'. However, there is an important grain of truth in the paradox. For the reason why there is never any doubt about what counts as obeying this instruction is that in the ordinary course of our lives nobody ever does take it to mean anything bizarre, like, 'Always take the opposite turn to the one you took last'. But if someone did understand it in this eccentric way, a verbal explanation of what it really meant might well fail to put him right. For he might give our verbal explanation an equally eccentric interpretation. Now because this sort of thing never happens in real life, we find Wittgenstein's argument paradoxical. It simply does not fit our picture of the independent authority of a rule. However, his point is that it could happen, and that indicates something important. It indicates that our use of language to give general instructions and state rules depends on our shared tendency to find the same responses natural. We have to agree in our practice before rules can have any independent authority. The independent authority is limited by the requirement that makes it possible. Wittgenstein's argument is not concerned with the real possibility of linguistic crankiness but with the logical structure of the situation.<br /><br />It appears, then, that the intellectual performance of following a linguistic rule is based on something outside the realm of the intellect. Its basis is the fact that we, like other animals, find if natural to divide and classify things in the same way as other members of our species. This line of thought runs parallel to the line that Wittgenstein took about sensation-language. For there too the intellectual achievement of reporting sensations was based on pre-established natural responses and behaviour of a more primitive kind. <br /><br /><span style="color: red;"><b>10. Wittgenstein's Philosophy of Mathematics</b></span> <br />Wittgenstein's Philosophy of Mathematics is another development of the same general idea. For here too our most elaborate intellectual constructions are said to be founded on basic routines that cannot be justified intellectually. When we count, we feel that we are using footholds already carved for us in a rock-like reality. 'But counting . . . is a technique that is employed daily in the most various operations of our lives. And that is why we learn to count as we do with endless practice, and merciless exactitude. . . . "But is this counting only a use, then? Isn't there also some truth corresponding to this sequence?" . . . it can't be said of the series of natural numbers - that it is true, but that it is useful, and, above all, it is used' (Wittgenstein, 1978, I, 4). <br />The application of this idea to the philosophy of mathematics has proved less fruitful than its application to the philosophy of language. The reason for this may only be the greater distance between the superstructure and the proposed basis in this case. Or some would argue that mathematics is really not amenable to this treatment. It is, for example, questionable whether it can yield a convincing account of proof in mathematics. <br /><br /><span style="color: red;"><b>Conclusion </b></span><br />Inevitably people ask what message can be extracted from Wittgenstein's philosophy. If a message is a theory, then, as we have seen, the message is that there is no message. Like any other philosopher, he pushed the quest for understanding beyond the point at which the ordinary criteria for understanding are satisfied. However, unlike others, he believed that philosophical understanding is more like the experience of a journey than the attainment of a destination. He regarded philosophy as an activity that is like Freudian therapy. You relive all the temptations to misunderstand and your cure recapitulates the stages by which it was achieved.<br /><br />So it is not the restoration of a state of unreflected health. In fact, rather than aiming to re-establish any kind of state, philosophy is concerned with the process.If there is a single structure discernible in his philosophy, it is his rejection of all illusory, independent support for our modes of thought. Rigid theories of meaning treat linguistic rules as independent authorities to which we who follow them are supposed to be wholly subservient. But he argues that this is an illusion, because the system of instruction and obedience involves a contribution from each individual and presupposes a basic like-mindedness. Similarly, the necessity of mathematics is something which we project from our practice and then mistakenly hail as the foundation of our practice. Evidently, he was rejecting realism, but his treatment of rules shows that he was not recommending conventionalism in its place. Certainly, his investigations have a structure, but it is not the structure of traditional philosophy.<br /><div style="font-family: Georgia, "Times New Roman", serif;"><span style="color: #0b5394;"><span style="font-size: small;"><br /></span></span></div><div style="font-family: Georgia, "Times New Roman", serif;"><span style="color: red;"><span style="font-size: large;"><b><span style="color: #0b5394;">Bertrand Russell</span> </b></span></span></div><div style="font-family: Georgia, "Times New Roman", serif;"><br /></div><div class="mbl notesBlogText clearfix"><div>Counted among the towering intellectual and philosophical giants of the 20th century, Bertrand Russell was an eminent mathematician, essayist, social critic, educator and political activist, as well as one of the most widely read philosophers of his time.<span style="color: #3d85c6;"> One of the fathers of analytic philosophy, he is considered so influential in his field that biographer A.C. Grayling says “he is practically its wallpaper.”</span><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-Jhd5MxT6XrU/T4MYAKJl4II/AAAAAAAALRU/w2nuQM045Io/s1600/51784.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="http://1.bp.blogspot.com/-Jhd5MxT6XrU/T4MYAKJl4II/AAAAAAAALRU/w2nuQM045Io/s320/51784.jpg" width="213" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b>"Three passions, simple but overwhelmingly strong, have governed my life: the longing for love, the search for knowledge, and unbearable pity for the suffering of mankind. These passions, like great winds, have blown me hither and thither, in a wayward course, over a deep ocean of anguish, reaching to the very verge of despair....<br />With equal passion I have sought knowledge. I have wished to understand the hearts of men. I have wished to know why the stars shine. And I have tried to apprehend the Pythagorean power by which number holds sway above the flux. A little of this, but not much, I have achieved" Russell in his Autobiography.</b></td><td class="tr-caption" style="text-align: center;"><br /></td><td class="tr-caption" style="text-align: center;"><br /></td></tr></tbody></table><br /><div><span style="color: #3d85c6;">Russell’s early life was marred by tragedy and bereavement.</span> By the time he was age six, his sister, Rachel, his parents, and his grandfather had all died, and he and Frank were left in the care of their grandmother, Countess Russell. Though Frank was sent to Winchester School, Bertrand was educated privately at home, and his childhood, to his later great regret, was spent largely in isolation from other children. Intellectually precocious, he became absorbed in mathematics from an early age and found the experience of learning Euclidean geometry at the age of 11 “as dazzling as first love,” because it introduced him to the intoxicating possibility of certain, demonstrable knowledge. This led him to imagine that all knowledge might be provided with such secure foundations, a hope that lay at the very heart of his motivations as a philosopher. His earliest philosophical work was written during his adolescence and records the skeptical doubts that led him to abandon the Christian faith in which he had been brought up by his grandmother<span style="color: #3d85c6;"><span style="color: black;">.</span>While in his late teenage he tried to commit suicide several times but that only the wish to learn more mathematics kept him from suicide.</span><br /><br />In 1890 Russell’s isolation came to an end when he entered Trinity College, University of Cambridge, to study mathematics. There he made lifelong friends through his membership in the famously secretive student society the <a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823460" name="ref823460"></a>Apostles, whose members included some of the most influential philosophers of the day. Inspired by his discussions with this group, Russell abandoned mathematics for philosophy and won a fellowship at Trinity on the strength of a thesis entitled <i>An Essay on the Foundations of Geometry,</i> a revised version of which was published as his first philosophical book in 1897. Following Kant’s <i><a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823486" name="ref823486"></a>Critique of Pure Reason </i>(1781, 1787), this work presented a sophisticated idealist theory that viewed geometry as a description of the structure of spatial intuition. <br /><br />In 1896 Russell published his first political work, <i><a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823489" name="ref823489"></a><span style="color: blue;">German Social Democracy.</span></i><span style="color: blue;"> </span>Though sympathetic to the reformist aims of the German socialist movement, it included some trenchant and farsighted criticisms of Marxist dogmas. The book was written partly as the outcome of a visit to Berlin in 1895 with his first wife, Alys Pearsall Smith, whom he had married the previous year. In Berlin, Russell formulated an ambitious scheme of writing two series of books, one on the philosophy of the sciences, the other on social and political questions. “At last,” as he later put it, “I would achieve a Hegelian synthesis in an encyclopaedic work dealing equally with theory and practice.” He did, in fact, come to write on all the subjects he intended, but not in the form that he envisaged. Shortly after finishing his book on geometry, he abandoned the metaphysical idealism that was to have provided the framework for this grand synthesis. <br /><br />Russell’s abandonment of idealism is customarily attributed to the influence of his friend and fellow Apostle G.E. Moore. A much greater influence on his thought at this time, however, was a group of German mathematicians that included Karl Weierstrass, Georg Cantor, and Richard Dedekind, whose work was aimed at providing mathematics with a set of logically rigorous foundations. For Russell, their success in this endeavour was of enormous philosophical as well as mathematical significance; indeed, he described it as “the greatest triumph of which our age has to boast.” After becoming acquainted with this body of work, Russell abandoned all vestiges of his earlier idealism and adopted the view, which he was to hold for the rest of his life, that analysis rather than synthesis was the surest method of philosophy and that therefore all the grand system building of previous philosophers was misconceived. In arguing for this view with passion and acuity, Russell exerted a profound influence on the entire tradition of English-speaking analytic philosophy, bequeathing to it its characteristic style, method, and tone. <br /><br />Inspired by the work of the mathematicians whom he so greatly admired, Russell conceived the idea of demonstrating that mathematics not only had logically rigorous foundations but also that it was in its entirety nothing but logic. The philosophical case for this point of view—subsequently known as <a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823491" name="ref823491"></a>logicism—was stated at length in<i><span style="color: blue;"> </span></i> <i><a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823467" name="ref823467"></a><span style="color: blue;">The Principles of Mathematics</span></i><span style="color: blue;">(1903).</span> There Russell argued that the whole of mathematics could be derived from a few simple axioms that made no use of specifically mathematical notions, such as number and square root, but were rather confined to purely logical notions, such as proposition and class. In this way not only could the truths of mathematics be shown to be immune from doubt, they could also be freed from any taint of subjectivity, such as the subjectivity involved in Russell’s earlier Kantian view that geometry describes the structure of spatial intuition. Near the end of his work on <i>The Principles of Mathematics,</i> Russell discovered that he had been anticipated in his logicist philosophy of mathematics by the German mathematician <a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823492" name="ref823492"></a>Gottlob Frege, whose book <i><a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823493" name="ref823493"></a>The Foundations of Arithmetic </i>(1884) contained, as Russell put it, “many things…which I believed I had invented.” Russell quickly added an appendix to his book that discussed Frege’s work, acknowledged Frege’s earlier discoveries, and explained the differences in their respective understandings of the nature of logic. <br /><br />The tragedy of Russell’s intellectual life is that the deeper he thought about logic, the more his exalted conception of its significance came under threat. He himself described his philosophical development after <i>The Principles of Mathematics</i> as a “retreat from Pythagoras.” The first step in this retreat was his discovery of a contradiction—now known as <a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823495" name="ref823495"></a><a href="http://www.britannica.com/EBchecked/topic/513243/Russells-paradox" title="Russell’s Paradox">Russell’s Paradox</a>—at the very heart of the system of logic upon which he had hoped to build the whole of mathematics. The contradiction arises from the following considerations: Some classes are members of themselves (e.g., the class of all classes), and some are not (e.g., the class of all men), so we ought to be able to construct the class of all classes that are not members of themselves. But now, if we ask of this class “Is it a member of itself?” we become enmeshed in a contradiction. If it is, then it is not, and if it is not, then it is. This is rather like defining the village barber as “the man who shaves all those who do not shave themselves” and then asking whether the barber shaves himself or not. <br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://4.bp.blogspot.com/-pgdd1uDldko/T_67FPR-mAI/AAAAAAAAOu4/JqedJAroFBI/s1600/The-Principles-of-Mathematics-Bertrand-Russell-9781603861199.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="http://4.bp.blogspot.com/-pgdd1uDldko/T_67FPR-mAI/AAAAAAAAOu4/JqedJAroFBI/s320/The-Principles-of-Mathematics-Bertrand-Russell-9781603861199.jpg" width="259" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;">The Pioneering book on Logic</span></td></tr></tbody></table><br />At first this paradox seemed trivial, but the more Russell reflected upon it, the deeper the problem seemed, and eventually he was persuaded that there was something fundamentally wrong with the notion of class as he had understood it in <i>The Principles of Mathematics.</i> Frege saw the depth of the problem immediately. When Russell wrote to him to tell him of the paradox, Frege replied, “arithmetic totters.” The foundation upon which Frege and Russell had hoped to build mathematics had, it seemed, collapsed. Whereas Frege sank into a deep depression, Russell set about repairing the damage by attempting to construct a theory of logic immune to the paradox. Like a malignant cancerous growth, however, the contradiction reappeared in different guises whenever Russell thought that he had eliminated it. Eventually, Russell’s attempts to overcome the paradox resulted in a complete transformation of his scheme of logic, as he added one refinement after another to the basic theory. In the process, important elements of his “Pythagorean” view of logic were abandoned. In particular, Russell came to the conclusion that there were no such things as classes and propositions and that therefore, whatever logic was, it was not the study of them. In their place he substituted a bewilderingly complex theory known as the ramified <a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823497" name="ref823497"></a>theory of types, which, though it successfully avoided contradictions such as Russell’s Paradox, was (and remains) extraordinarily difficult to understand. By the time he and his collaborator,<a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823473" name="ref823473"></a> Alfred North Whitehead, had finished the three volumes of <i><a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823472" name="ref823472"></a>Principia Mathematica </i>(1910–13), the theory of types and other innovations to the basic logical system had made it unmanageably complicated. Very few people, whether philosophers or mathematicians, have made the gargantuan effort required to master the details of this monumental work. It is nevertheless rightly regarded as one of the great intellectual achievements of the 20th century.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://4.bp.blogspot.com/-b8G4thAG3j0/T5Ut0PYLFdI/AAAAAAAALuM/RoPBR7UEn48/s1600/2_br_3.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="http://4.bp.blogspot.com/-b8G4thAG3j0/T5Ut0PYLFdI/AAAAAAAALuM/RoPBR7UEn48/s320/2_br_3.jpg" width="231" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Russell while a student at Cambridge</td></tr></tbody></table><br /><i>Principia Mathematica</i> is a herculean attempt to demonstrate mathematically what <i>The Principles of Mathematics</i> had argued for philosophically, namely that mathematics is a branch of logic. The validity of the individual formal proofs that make up the bulk of its three volumes has gone largely unchallenged, but the philosophical significance of the work as a whole is still a matter of debate. Does it demonstrate that mathematics is logic? Only if one regards the theory of types as a logical truth, and about that there is much more room for doubt than there was about the trivial truisms upon which Russell had originally intended to build mathematics. Moreover, Kurt Gödel’s <a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823498" name="ref823498"></a>first incompleteness theorem (1931) proves that there cannot be a single logical theory from which the whole of mathematics is derivable: all consistent theories of arithmetic are necessarily incomplete. <i>Principia Mathematica</i> cannot, however, be dismissed as nothing more than a heroic failure. Its influence on the development of mathematical logic and the philosophy of mathematics has been immense. <br /><br />Despite their differences, Russell and Frege were alike in taking an essentially Platonic view of logic. Indeed, the passion with which Russell pursued the project of deriving mathematics from logic owed a great deal to what he would later somewhat scornfully describe as a “kind of mathematical mysticism.” As he put it in his more disillusioned old age, “I disliked the real world and sought refuge in a timeless world, without change or decay or the will-o’-the-wisp of progress.” Russell, like Pythagoras and Plato before him, believed that there existed a realm of truth that, unlike the messy contingencies of the everyday world of sense-experience, was immutable and eternal. This realm was accessible only to reason, and knowledge of it, once attained, was not tentative or corrigible but certain and irrefutable. Logic, for Russell, was the means by which one gained access to this realm, and thus the pursuit of logic was, for him, the highest and noblest enterprise life had to offer. <br />In philosophy the greatest impact of <i>Principia Mathematica</i> has been through its so-called <a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823504" name="ref823504"></a>theory of descriptions. This method of analysis, first introduced by Russell in his article <span style="color: blue;">“On Denoting” (1905), </span>translates propositions containing definite descriptions (e.g., “the present king of France”) into expressions that do not—the purpose being to remove the logical awkwardness of appearing to refer to things (such as the present king of France) that do not exist. Originally developed by Russell as part of his efforts to overcome the contradictions in his theory of logic, this method of analysis has since become widely influential even among philosophers with no specific interest in mathematics. Thegeneral idea at the root of Russell’s theory of descriptions—that the grammatical structures of ordinary language are distinct from, and often conceal, the true “logical forms” of expressions—has become his most enduring contribution to philosophy.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-No2qe_qWesA/T4iE_IkEmXI/AAAAAAAALZY/Vmr6g_upE78/s1600/bertrand-russell_02_766.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="225" src="http://1.bp.blogspot.com/-No2qe_qWesA/T4iE_IkEmXI/AAAAAAAALZY/Vmr6g_upE78/s400/bertrand-russell_02_766.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span class="hasCaption"><b>Bertrand Russell</b>, </span><b>3rd Earl Russell</b></td></tr></tbody></table><br />Russell later said that his mind never fully recovered from the strain of writing <i>Principia Mathematica,</i> and he never again worked on logic with quite the same intensity. In 1918 he wrote<span style="color: blue;"> <i>An Introduction to Mathematical Philosophy,</i></span> which was intended as a popularization of <i>Principia,</i> but, apart from this, his philosophical work tended to be on epistemology rather than logic. In 1914, in <i><span style="color: blue;">Our Knowledge of the External World</span>,</i> Russell argued that the world is “constructed” out of sense-data, an idea that he refined in <i><span style="color: blue;">The Philosophy of <a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823468" name="ref823468"></a>Logical Atomism</span> </i>(1918–19). In <span style="color: blue;"><i><a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823470" name="ref823470"></a>The Analysis of Mind</i> </span>(1921) and<i><a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823469" name="ref823469"></a><span style="color: blue;"> The Analysis of Matter</span></i><span style="color: blue;"> </span>(1927), he abandoned this notion in favour of what he called<a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823506" name="ref823506"></a> neutral monism, the view that the “ultimate stuff” of the world is neither mental nor physical but something “neutral” between the two. Although treated with respect, these works had markedly less impact upon subsequent philosophers than his early works in logic and the philosophy of mathematics, and they are generally regarded as inferior by comparison.<br /><br />Russell’s intellectual fame was firmly established when he coauthored one of the most influential and widely discussed books on the subject of mathematics, <span style="color: black;"><i>Principia Mathematica</i></span>, which theorized that mathematics and logic are identical and that mathematical principles can be deduced by a handful of logical ideas.<span style="color: blue;"> <span style="color: black;"><i>Principia Mathematica</i></span></span> did not generate financial rewards but did earn Russell an Olympian scholarly stature, evidenced by his election as a fellow of the Royal Society at the remarkably young age of 35.Growing bored with mathematics, Russell became more and more involved with philosophy and social activism. In 1912, following an unsuccessful run for parliament as a candidate for the Women’s Suffragette Society, he completed <span style="color: blue;"><i>The Problems of Philosophy</i></span>, the first of numerous works on philosophy. He later combined his idealist notions with his logical methodology in an attempt to bring philosophy closer to science. The result was his most popular work, <span style="color: blue;"><i>The History of Western Philosophy</i> (1945).</span><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-74F9DryZj3Q/T_68MqtB_oI/AAAAAAAAOvI/xMWcunMEdnw/s1600/20008d78v04Mobi300x400.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="400" src="http://1.bp.blogspot.com/-74F9DryZj3Q/T_68MqtB_oI/AAAAAAAAOvI/xMWcunMEdnw/s400/20008d78v04Mobi300x400.jpg" width="300" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">One of the best book ever written</td></tr></tbody></table><br />Russell felt that all organized religions were relics from humanity’s barbaric history. Despite his ultimate rejection of God, however, he admitted in <span style="color: blue;"><i>The Elements of Ethics</i></span> (1910) that obedience to rules like the Ten Commandments “will in almost all cases have better consequences than disobedience.” In his 1927 work,<span style="color: blue;"> <i>Why I Am Not a Christian</i>,</span> Russell confessed his agnostic and anti-Christian views.<br />It comes as no surprise, therefore, that in 1929, in <span style="color: blue;"><i>Marriage and Morality</i></span>, he argued for sexual liberation and the rejection of monogamy and morality. Indeed, his own life reflected his beliefs, as he engaged in numerous adulteries during the course of his four marriages, three of which ended in divorce. His assertion that unfaithfulness should not “be treated as something terrible” was profoundly shocking to a still largely conservative and traditional public.As a result of <i>Marriage and Morality</i>, Russell was terminated from a teaching position in the United States. On the other hand, the views he expressed were very popular with the vanguard of 1920s and ’30s left-wing intellectuals, who championed him as an emancipating hero. And it was this same work that a decade later won him a Nobel Prize for literature.<br /><br />Connected with the change in his intellectual direction after the completion of <span style="color: blue;"><i>Principia</i></span> was a profound change in his personal life. Throughout the years that he worked single-mindedly on logic, Russell’s private life was bleak and joyless. He had fallen out of love with his first wife, Alys, though he continued to live with her. In 1911, however, he fell passionately in love with <a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823474" name="ref823474"></a>Lady Ottoline Morrell. Doomed from the start (because Morrell had no intention of leaving her husband), this love nevertheless transformed Russell’s entire life. He left Alys and began to hope that he might, after all, find fulfillment in romance. Partly under Morrell’s influence, he also largely lost interest in technical philosophy and began to write in a different, more accessible style. Through writing a best-selling introductory survey called <i><a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823507" name="ref823507"></a><span style="color: blue;">The Problems of Philosophy</span></i> <span style="color: blue;">(1911)</span>, Russell discovered that he had a gift for writing on difficult subjects for lay readers, and he began increasingly to address his work to them rather than to the tiny handful of people capable of understanding <i>Principia Mathematica.</i> <br />In the same year that he began his affair with Morrell, Russell met <a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823476" name="ref823476"></a>Ludwig Wittgenstein, a brilliant young Austrian who arrived at Cambridge to study logic with Russell. Fired with intense enthusiasm for the subject, Wittgenstein made great progress, and within a year Russell began to look to him to provide the next big step in philosophy and to defer to him on questions of logic. <span style="color: red;">However, Wittgenstein’s own work, eventually published in 1921 as <i>Logisch-philosophische Abhandlung</i> ( <i><a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823477" name="ref823477"></a>Tractatus Logico-Philosophicus , </i>1922), undermined the entire approach to logic that had inspired Russell’s great contributions to the philosophy of mathematics. </span>It persuaded Russell that there were no “truths” of logic at all, that logic consisted entirely of tautologies, the truth of which was not guaranteed by eternal facts in the Platonic realm of ideas but lay, rather, simply in the nature of language. This was to be the final step in the retreat from Pythagoras and a further incentive for Russell to abandon technical philosophy in favour of other pursuits.<br /><br /><span id="freeText16715551502337455020">The key to human nature that Marx found in wealth and Freud in sex, Bertrand Russell finds in power. Power, he argues, is man's ultimate goal, and is, in its many guises, the single most important element in the development of any society. Writting in the late 1930s when Europe was being torn apart by extremist ideologies and the world was on the brink of war, Russell set out to found a 'new science' to make sense of the traumatic events of the day and explain those that would follow. <br />The result was <span style="color: blue;"><b>Power</b></span>, a remarkable book that Russell regarded as one of the most important of his long career. Countering the totalitarian desire to dominate, Russell shows how political enlightenment and human understanding can lead to peace - his book is a passionate call for independence of mind and a celebration of the instinctive joy of human life.</span><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/-6JCVQo7ivDE/T_2mkBseFGI/AAAAAAAAOuY/jLnpq-v_p_8/s1600/cover.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="400" src="http://3.bp.blogspot.com/-6JCVQo7ivDE/T_2mkBseFGI/AAAAAAAAOuY/jLnpq-v_p_8/s400/cover.jpg" width="252" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="color: #3d85c6;"><span style="font-size: small;"><span id="freeText3700016194583259162">The key to human nature that Marx found in wealth and Freud in sex, Bertrand Russell finds in power. Power, he argues, is man's ultimate goal, and is, in its many guises, the single most important element in the development of any society. Writting in the late 1930s when Europe was being torn apart by extremist ideologies and the world was on the brink of war, Russell set out to found a 'new science' to make sense of the traumatic events of the day and explain those that would follow. The result was <b>Power</b>, a remarkable book that Russell regarded as one of the most important of his long career. Countering the totalitarian desire to dominate, Russell shows how political enlightenment and human understanding can lead to peace - his book is a passionate call for independence of mind and a celebration of the instinctive joy of human life</span></span></span></td></tr></tbody></table><br />During World War I Russell was for a while a full-time political agitator, campaigning for peace and against conscription. His activities attracted the attention of the British authorities, who regarded him as subversive. He was twice taken to court, the second time to receive a sentence of six months in prison, which he served at the end of the war. In 1916, as a result of his antiwar campaigning, Russell was dismissed from his lectureship at Trinity College. Although Trinity offered to rehire him after the war, he ultimately turned down the offer, preferring instead to pursue a career as a journalist and freelance writer. The war had had a profound effect on Russell’s political views, causing him to abandon his inherited liberalism and to adopt a thorough-going socialism, which he espoused in a series of books including <span style="color: blue;"><i>Principles of Social Reconstruction</i> (1916),<i>Roads to Freedom</i> (1918), and <i>The Prospects of Industrial Civilization</i> (1923)</span>. He was initially sympathetic to the Russian Revolution of 1917 , but a visit to the Soviet Union in 1920 left him with a deep and abiding loathing for Soviet communism, which he expressed in <i><a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823475" name="ref823475"></a><span style="color: blue;">The Practice and Theory of Bolshevism</span></i><span style="color: blue;"> (1920)</span>. <br />In 1921 Russell married his second wife, Dora Black, a young graduate of Girton College, Cambridge, with whom he had two children, John and Kate. In the interwar years Russell and Dora acquired a reputation as leaders of a progressive socialist movement that was stridently anticlerical, openly defiant of conventional sexual morality, and dedicated to educational reform. Russell’s published work during this period consists mainly of journalism and popular books written in support of these causes. Many of these books—such as <span style="color: blue;"><i>On Education</i> (1926), <i>Marriage and Morals</i> (1929), and <i>The Conquest of Happiness</i> (1930)</span>—enjoyed large sales and helped establish Russell in the eyes of the general public as a philosopher with important things to say about the moral, political, and social issues of the day. His public lecture “Why I Am Not a Christian,” delivered in 1927 and printed many times, became a popular locus classicus of atheistic rationalism. In 1927 Russell and Dora set up their own school, Beacon Hill, as a pioneering experiment in primary education. To pay for it, Russell undertook a few lucrative but exhausting lecture tours of the United States. <br />During these years Russell’s second marriage came under increasing strain, partly because of overwork but chiefly because Dora chose to have two children with another man and insisted on raising them alongside John and Kate. In 1932 Russell left Dora for Patricia (“Peter”) Spence, a young University of Oxford undergraduate, and for the next three years his life was dominated by an extraordinarily acrimonious and complicated divorce from Dora, which was finally granted in 1935. In the following year he married Spence, and in 1937 they had a son, Conrad. Worn out by years of frenetic public activity and desiring, at this comparatively late stage in his life (he was then age 66), to return to academic philosophy, Russell gained a teaching post at the University of Chicago. From 1938 to 1944 Russell lived in the United States, where he taught at Chicago and the University of California at Los Angeles, but he was prevented from taking a post at the City College of New York because of objections to his views on sex and marriage. On the brink of financial ruin, he secured a job teaching the history of philosophy at the Barnes Foundation in Philadelphia. Although he soon fell out with its founder, <a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823510" name="ref823510"></a>Albert C. Barnes, and lost his job, Russell was able to turn the lectures he delivered at the foundation into a book, <span style="color: red;"><i><a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823478" name="ref823478"></a>A History of Western Philosophy</i> (1945), which proved to be a best-seller and was for many years his main source of income</span>.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-FJChi7TCHf8/T5kbntlDq6I/AAAAAAAALyA/dTjQdg_hDog/s1600/845041.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://4.bp.blogspot.com/-FJChi7TCHf8/T5kbntlDq6I/AAAAAAAALyA/dTjQdg_hDog/s320/845041.jpg" width="211" /></a></div><br /><br />In 1944 Russell returned to Trinity College, where he lectured on the ideas that formed his last major contribution to philosophy, <i><span style="color: blue;"><a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823465" name="ref823465"></a>Human Knowledge: Its Scope and Limits</span></i> (1948). During this period Russell, for once in his life, found favour with the authorities, and he received many official tributes, including the Order of Merit in 1949 and the Nobel Prize for Literature in 1950. His private life, however, remained as turbulent as ever, and he left his third wife in 1949. For a while he shared a house in Richmond upon Thames, London, with the family of his son John and, forsaking both philosophy and politics, dedicated himself to writing short stories. Despite his famously immaculate prose style, Russell did not have a talent for writing great fiction, and his short stories were generally greeted with an embarrassed and puzzled silence, even by his admirers. <br /><br />In 1952 Russell married his fourth wife, Edith Finch, and finally, at the age of 80, found lasting marital harmony. Russell devoted his last years to campaigning against nuclear weapons and the <a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" id="ref823511" name="ref823511"></a>Vietnam War, assuming once again the role of gadfly of the establishment. The sight of Russell in extreme old age taking his place in mass demonstrations and inciting young people to civil disobedience through his passionate rhetoric inspired a new generation of admirers. Their admiration only increased when in 1961 the British judiciary system took the extraordinary step of sentencing the 89-year-old Russell to a second period of imprisonment. <br /><br />When he died in 1970 Russell was far better known as an antiwar campaigner than as a philosopher of mathematics. In retrospect, however, it is possible to see that it is for his great contributions to philosophy that he will be remembered and honoured by future generations.A prodigious talent, Russell wrote constantly. His views were published in more than 70 books and booklets and over 3,000 articles. His endeavors in the social sciences in particular earned him both the wide acclaim of a Nobel Prize and the reprimand of two prison sentences. </div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td class="tr-caption" style="text-align: center;"><br /></td></tr></tbody></table>A militant pacifist appalled by the carnage of World War I, Russell became a very vocal critic of the war, even to the point of a six-month imprisonment for his participation in antiwar protests.<br />He then traveled to Russia and China to search for societies that he felt would transcend the warlike Western European standard. The pursuit proved disillusioning, but as a result, Russell came to believe that only through education could the world be transformed. Accordingly he authored several books on the subject. He returned to England and attempted to revolutionize education by starting his own progressive school, Beacon Hill. But because the school could never seem to find and apply what he considered to be the right balance between discipline and freedom, he eventually deemed the experiment a failure.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/-JgFkliU0KfI/T4GcJOhy55I/AAAAAAAALG0/yeLSPihSZOs/s1600/00001036.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="307" src="http://3.bp.blogspot.com/-JgFkliU0KfI/T4GcJOhy55I/AAAAAAAALG0/yeLSPihSZOs/s320/00001036.jpg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Russell in a rally</td></tr></tbody></table><span style="color: #3d85c6;"><br /></span><span style="color: #3d85c6;">Russell devoted the final two decades of his life primarily to combating nuclear proliferation.</span> Feeling strongly that unilateral disarmament was the key to world peace, in 1954 he broadcast on the BBC his famous “Man’s Peril” speech—dire warnings about the Bikini Atoll hydrogen bomb tests. Later he joined Albert Einstein in calling for the curtailment of nuclear weapons in the <span style="color: blue;"><span style="color: #0b5394;">Russell-Einstein Manifesto</span>. </span>Russell was also president of the Campaign for Nuclear Disarmament, and in 1957 he organized the first Pugwash Conference, which gathered together an extensive collection of leading scientists anxious about the nuclear issue. He advocated world government as the only cause worth fighting for and the only alternative to nuclear war.<br /><br />Russell’s controversial rebellion against what he saw as Victorian repression turned him into an apostle of personal liberation from any moral code. This approach, added to his ardent social activism, helps account for his great popularity in the liberal period of the 1960s. His social perspectives were an early expression of present attitudes. He played an important part in shaping a revolution in social mores with his arguments in support of sexual freedom, radical feminism, euthanasia and abolition of the death penalty.One of Russell’s motives in life was to determine whether anything could be known with absolute certainty.<br /><br /><span style="color: #cc0000;"> <b>At a dinner party at the age of 90, he is said to have been asked how he would respond if, after he died, he found himself facing the Almighty. According to the story, Russell cavalierly replied, “I would ask him why He hadn’t given me more evidence.”</b></span><br /><br />With his mind closed to belief and faith, however, it is questionable whether any kind of evidence would have been enough to convince him of the existence of a Creator God. Russell, though raised to believe in God, became profoundly skeptical. Like many, he felt that such belief was irrational and based on fear.<span style="color: #0b5394;"> Russell claimed that Socrates was more worthy of reverence than Jesus Christ, so found his only religious satisfaction in an evolving idealistic philosophy.</span><br /><br /></div></div>Nassif Nabeal (Nassif Medici)http://www.blogger.com/profile/13708191074451322062noreply@blogger.com0tag:blogger.com,1999:blog-5761410253972037435.post-36363808416279922762013-10-21T06:59:00.003-07:002015-07-10T20:47:22.139-07:00Nassim Nicholas Taleb against Gaussian CurveNote:This article is made from theories and books of <b><span style="color: #0b5394;">Nassim Nicholas Taleb.</span></b><br /><br /><span style="color: red;">Gaussian or Bell curve</span> is named because Friedrich Gauss proposed this curve . Bell curve, or Gaussian model, has since pervaded our business and scientific culture terms like sigma, variance, standard deviation, correlation, R-square and Sharpe.<br /><br /><span style="color: #3d85c6;"><span class="Apple-style-span">Bell curve is meaningless in money and stocks but you may see it on German notes , financial markets run on the theories which are based on Gaussian Distribution .</span></span> Bell curve is used in risk management though in many banks by Black Suit wearing officer. Assume that average height of men has a mean of 1.5 m and standard deviation 0.22 m and they follow a Normal /Bell distribution. Note that 220 cm standard deviation(.220 m) is randomness here and a very high one for a computer programmer.Gaussian yields its properties rather rapidly (a way to get a solution rather accuracy ), standard deviation in Bell curve faces a head wing where probabilities move rapidly as you move far away from mean. But my way of calculation does not make probability change ,it stays same over a range(unlike Bell curve). If I tell you that combined height of 2 men is 14 feet then you will think of 7 feet for each not 11 and 3 feet for them. People like to think in an easiest way and avoid randomness as 7 feet as for frequent and mind see it easier to conceive. <span style="color: red;">Bell curves used in extreme events may cause a lot of disaster. Measures of uncertainty that are based on Bell curve disregard the impact the sharp jumps and inequalities and using them is like getting grass (grass disaster) and missing out the trees (Big Black Swans).</span> This is why economics is based on Equilibrium , it allows you to treat economics as Gaussian . Assume you have a sample of 1000 people(giants and dwarfs) , your average will not be changed if you add another giant as your average will not be altered but if you add a mega giant it may be. So a single event will not change anything. <br /><br />Randomness if Gaussian is tameable and is not altered by a single addition or removal. Casino people make such calculation and sleep well in night, no single gambler with a big hit will not change it and you will never see one gambler getting 1 Billion. The Gaussian family also contains Poisson and the distribution are the ones where mean and standard deviation describe everything and you don't need any other thing.In fact, while the occasional and unpredictable large deviations are rare, they cannot be dismissed as “outliers” because, cumulatively,their impact in the long term is so dramatic.<span style="color: red;">Mediocre events get fine or acceptable with Gaussian Distribution because big trees are not present in such events. <span style="color: red;">I say that one should not use Gaussian in extreme events. But once you get Bell curve in head it's hard to avoid.</span></span><br /><b> </b> <br />A group of thinkers consisting Karl Marx and others , they all worked on Socialism and were looking for "Golden mean" of everything ,like height ,wealth and economy etc. A golden saying my Dad said once :Virtue lies in moderation ,all should embrace mediocrity. People have a golden mean and so people deviate from these like a normal mind and steady health is best but many deviate .Some men are sick and some are intellectual and some are bulky and intelligent. Being an average man means that one should be mediocre in thinking but God has made every man equal and have given gifts to few and given flaws so to tell people as in Equilibrium. Though divergent society does change people like wars disable people and prosper some.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-Tx5BfTZBWcI/T4iN7RXxP_I/AAAAAAAALZo/xzXmmg1GO90/s1600/nassim_taleb_01.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="262" src="http://1.bp.blogspot.com/-Tx5BfTZBWcI/T4iN7RXxP_I/AAAAAAAALZo/xzXmmg1GO90/s400/nassim_taleb_01.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;"><b>Nassim Nicholas Taleb, known for his aggressive attitude towards Financial Industry.</b><b><span class="st">Taleb is a modern day <i>Nietzsche.</i></span></b></span> <b><span style="font-size: small;">This is a man who suffers fools impatiently, and his intellect makes his hauteur largely justified.</span></b></td></tr></tbody></table><span style="color: #0b5394;"><i><b>Nassim Taleb is a guy who made a very groundbreaking theory of "Black Swans" , an original idea which few economists and financial advisers understand (as all theories are based on Gaussian Distribution) and his idea is based on Probability of Wild Events(which even few mathematicians are aware of).</b></i></span><br /><br /><br />Henri Poincare was suspicious of Gaussian curve,Gaussian was initially established for cosmology and atomic uncertainty. But apart from physicists mathematicians began to use it because maths people trusted physics people. Doing science for sake of knowledge does not mean you will be successful. <span style="color: red;">Gentlemen scientists like Lord Cavendish ,Lord Kelvin ,Ludwig Wittgenstein and </span><br /><span style="color: red;">Uber philosopher Bertrand Russell are those who will think twice in using Gaussian curve. Bell curves are used in medicine in yes -no events because they are mediocre events.Even Sir Karl Popper also considered how new observations affected knowledge - such as spotting a black swan when it was thought all swans were white.</span><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://4.bp.blogspot.com/-vWE7B28tUKs/T4dnbMLwxLI/AAAAAAAALVg/Bdp7vPiE4x0/s1600/bell-curve-joke.png" style="margin-left: auto; margin-right: auto;"><img border="0" height="300" src="http://4.bp.blogspot.com/-vWE7B28tUKs/T4dnbMLwxLI/AAAAAAAALVg/Bdp7vPiE4x0/s400/bell-curve-joke.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Gaussian fallacies are everywhere</td></tr></tbody></table><br /><span style="font-size: 12pt; line-height: 115%;"><span class="Apple-style-span" style="font-family: inherit;">Gaussian yields its properties rather rapidly (a way to get a solution rather accuracy ), standard deviation in Bell curve face a head wing where probabilities move rapidly as you move far away from mean. But my way of calculation thus not make probability change ,it stays remain same over a range(unlike Bell curve). If I tell you that combined height of 2 men is 14 feet then you will think of 7 feet for each not 11 and 3 feet for them. People like to think in an easiest way and avoid randomness as 7 feet as for frequent and mind see it easier to conceive.<span style="color: red;"> Bell curves used in extreme events may cause a lot of disaster</span>. Measures of uncertainty that are based on Bell curve disregard the impact the sharp jumps and inequalities and using them is like getting grass (grass disaster) and missing out the trees<span style="color: #3d85c6;"> (<b>Black Swans,black swans are rare events that carry a massive impact</b>).</span></span></span><br /><span style="font-size: 12pt; line-height: 115%;"><span style="color: blue; font-family: Verdana,sans-serif;"><br /></span></span><span style="color: #0b5394;"><span style="font-size: 12pt; line-height: 115%;"><span class="Apple-style-span" style="font-family: inherit;">Nassim Taleb is trying since 2005-2006 that financial markets do not follow Gaussian curves and that academics around the world , all the financial theories and all the monetary policy makers do not understand this concept and so cannot access its validity and that only guy to promote such thinking was Beniot Mandelbrot.</span></span></span><br /><br /><span style="font-size: 12pt; line-height: 115%;"><span class="Apple-style-span" style="color: red; font-family: inherit;">So, while weight, height and calorie consumption are Gaussian, wealth is not. Nor are income, market returns, size of hedge funds, returns in the financial markets, number of deaths in wars or casualties in terrorist attacks. Almost all man-made variables are wild or carry massive randomness(Black Swans).</span></span><span class="Apple-style-span" style="font-size: 16px; line-height: 18px;"><span class="Apple-style-span" style="color: red;">The unknown process and factors influence the financial markets and that according to the Central Limit Theorem ,these unknown influences become or accumulate to normal distribution. The reason that systematic risk is based on Normal distribution, so systematic risk is what rules financial risk and Bell curve does not follow it as it is evident from the 2007-2008 Financial crises.</span></span><br /><span style="font-size: 12pt; line-height: 115%;"><span class="Apple-style-span" style="font-family: inherit;"><br /></span></span><span style="font-size: 12pt; line-height: 115%;"><span class="Apple-style-span" style="font-family: inherit;">The problem is that measures of uncertainty using the bell curve simply disregard the possibility of sharp jumps or discontinuities and, therefore, have no meaning or consequence.Using them is like focusing on the grass and missing out on the (gigantic) trees .This is why economics is based on Equilibrium , it allows you to treat economics as Gaussian . Assume you have a sample of 1000 people(giants and dwarfs) , your average will not be changed if you add another giant as your average will not be altered but if you add a mega giant it may be. So a single event will not change anything.</span></span><span class="Apple-style-span" style="font-family: inherit;"><span style="font-size: 12pt; line-height: 115%;">Mediocre events get fine or acceptable with Gaussian Distribution because big trees are not present in such events. <b style="mso-bidi-font-weight: normal;"> </b></span></span><br /><br /><span style="color: #3d85c6;"><b>Financial Crisis of 2008</b></span>:A key factor that led to the collapse of the banking industry in 2008 was the increasing use of mathematical models, spurred by the desire to exert total control over risk. These models, in all its elegance and beauty, badly underestimated the occurrence of extreme events.<br /><br /><b><br /></b><b>Of particular note is a modeling technique called the <span style="color: #3d85c6;">Gaussian copula</span>, which puts a price on the risk of multiple assets (or in this case, mortgages) defaulting at the same time. Upon its introduction by </b><b>a quant named David Li, the popularity of this model sky-rocketed and the banking industry</b><b>embraced it gleefully as the final piece to the risk management jigsaw that the industry had been </b><b>piecing together. Ratings agencies such as Moody’s and the S&P readily adopted it in </b><b>formulating company credit ratings, and the model finally found its way into the Basel II </b><b>regulatory framework, as the guideline to calculate capital requirements for banks based on </b><b>structured credit that they hold.</b><br /><b><br /></b><br />If you use Bell curve in top stocks and genetic measures, extreme events if calculated Bell curve may cause disaster. Head -tail on a coin is a random walk (left or right or win or loose) is a mediocre event so we use Gaussian curve. Tree diagrams are based on multiple tosses or multiple balls chosen etc or two or more dices tossed etc. In tree diagram if net is one Win it can have many cases (2^3=8) .Use of this tree diagram makes us closer to Normal/Bell curve as we know that condition of Gaussian is that N (no of objects) should be greater than 10 etc and also Poisson is followed by these tosses of coins but it will get to Normal after some time.<br /><br /><br />We have moved from observation to mathematics ,something abstract is like thermometer where 25 degree Celsius is pleasant and 40 degrees is hot and you do not need to know what temperature is. Also remember that standard deviation is not average standard deviation of a curve. Standard deviation is between +1 to -1 ,it is a scale ,Standard deviation and Variance(standard deviation ^2) or sigma variates dramatically when you get away from average .Scaling to a sigma is used as well.In real life people don't take account of past probability ,though past winning has effect on future probability but Bell curve doesn't take into account of it. Models are made to scale standard deviation.<br /><br />A major theme of Nassim Taleb is that models of uncertainty are too precise, and this thread has a long history. Taleb's sometime co-author Benoit Mandelbrot has been trying to sell the world on the big idea of fractals in finance for several decades. James Gleick’s <i><a href="http://www.amazon.com/Chaos-Making-Science-James-Gleick/dp/0140092501">Chaos</a></i> outlined the essence of Benoit Mandelbrot’s fractals, which takes a simple few lines of inputs to create graphics of insane complexity yet also beautiful recursive symmetry, in many cases eerily similar to nature (eg, ferns, snowflakes). In dynamic systems, you have chaotic systems that are purely deterministic though sufficiently complex that they appear random. These systems have large jumps, or phase shifts, reminiscent of market crashes or sudden bankruptcies; they have butterfly effects where small changes produce big differences in outcomes. Mandelbrot and others have been trying to apply these ideas to financial markets for many decades now (since 1962!), and the effort has not gained any traction, in spite of many papers applying this concept (search skew or kurtosis in any financial journal and you will see many papers). Mandelbrot’s big idea in finance is that finance relies on a profoundly flawed assumption, mainly that market prices are normally distributed.<br /><br /><br /><b>The markets are non-linear, dynamic systems, subject to the rules of Chaos Theory. Market prices are highly random, with a short to intermediate term trend component. They are highly dependent on initial conditions. Markets also show qualities of fractals -- self-similar in the sense that the individual parts are related to the whole.Due to the non-Gaussian behavior of the markets the methods from Chas Theory, Fractals and Quantum Physics(probability calculations from quantum mechanics) are being used in Finance.</b> <br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-nFpCG6b1D-Y/UAWLXMSkKoI/AAAAAAAAO5U/vSXSkABI_Mo/s1600/professions_iq_graphic.jpeg" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://1.bp.blogspot.com/-nFpCG6b1D-Y/UAWLXMSkKoI/AAAAAAAAO5U/vSXSkABI_Mo/s1600/professions_iq_graphic.jpeg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;"><b>Nassim Taleb and <span style="font-family: inherit;">Beniot Mandelbrot suggest that Gaussian curve is not so useful for calculating randomness of man-made variables and especially the financial market.</span></b></span></td></tr></tbody></table><br /><span style="color: red;"><i><b>There were only 2 Mathematicians who were actually able to understand randomness in a practical way and understand the flaws in them<i><b>:</b></i> 1.Beniot Mandelbrot 2.Henri Poincare </b></i></span><br /><br /><br /><span style="color: #3d85c6;"><b>The poet of randomness</b> </span>: <span style="color: red;"><b><i>Beniot Mandelbrot</i></b> </span>: French philosopher, other mathematicians of probability like Kolmogorov may be more academic or progressive but Mandelbrot was unique he proved that mathematicians actually understand randomness.In "The Misbehavior of Markets", another popular book by Mandelbrot,he argues that the Gaussian models for financial risk used by economists like William Sharpe and Harry Markowitz should be discarded, since these models do not reflect reality. Mandelbrot argues that fractal techniques may provide a more powerful way to analyze risk. Black Swans were dealt by him in a philosophical and aesthetic way. Dr Mandelbrot claimed that financial-market movements, too, have fractal forms, rather than the familiar bell shapes of “normal” distribution that Gauss described.<span class="Apple-style-span" style="font-size: 16px; line-height: 18px;"><span class="Apple-style-span" style="font-family: inherit;">Fractals are linked with power laws, Mandelbrot worked on it and applied it to randomness. Mandelbrot designed the mathematical object called "Mandelbrot set" and later worked on shapes and fractals of maths and also worked on Chaos Theory.</span></span><br /><br /><span class="Apple-style-span" style="font-size: 16px; line-height: 18px;"><span class="Apple-style-span" style="font-family: inherit;"> Alternative to Gaussian/Bell curve would be using power laws and fractals instead of the Gaussian distribution. The idea of power laws and fractals in the financial markets is first pioneered by Benoit Mandelbrot, and subsequently popularized by Nassim Nicholas Taleb. This theory states<br />that the markets are not just random—they are turbulent.Randomness associated with Gaussian distributions is too polite, too courteous, and is too unrealistic. Turbulent markets, on the other hand, incorporate a “wild” kind of randomness into consideration, which is characterized by<br />sudden large jumps in volatility.EMH(Efficient Market Hypothesis ),the core concept of Finance also assumes Gaussian curve for its validity,another flaw in Finance.</span></span><br /><span class="Apple-style-span" style="font-size: 16px; line-height: 18px;"><span class="Apple-style-span" style="font-family: inherit;"><br /></span></span><br /><div class="MsoNormal"><span style="font-size: 12pt; line-height: 115%;"><span class="Apple-style-span" style="font-family: inherit;">If stock markets were Gaussian then stock market crashes would have happened once in a Billion years. Mandelbrot's randomness methods make the statistics methods look useless. After the stock market crash William Sharpe and Markowitz model was given a Nobel Prize and this portfolio model was based on Gaussian Distribution. If in this world such method can get Noble then anything in this world is possible , anyone can become President etc. </span></span></div><br /><br /><div class="MsoNormal"><span style="font-size: 12pt; line-height: 115%;">Fractals distributions do better than Bell curve in avoiding the Big Black Swans .Sometimes a Fractal can make you believe it is Gaussian. Normally extreme events fit into Fractal category , fractals thus have very high standard deviation . Statistical Physics is what is good for use in Fractals Methods and Econometric and Gaussian methods are not to be used in Fractal Distribution. Businessmen have big egos, the Fractal Ego. So Fractal is any event described mathematically and is an extreme event and has high standard deviation, just like Black Swanevent<span style="color: red;">.(<i>Nassim Taleb also worked with Mandelbrot on randomness of Black Swan events ).</i></span></span></div><i><br /></i><span style="color: red;"><b><i>The Gaussian bell curve variations face a headwind that makes probabilities drop at a faster and faster rate, as you move away from the mean, while “scalables” or Mandelbrotian variations<br />do not have such restriction. </i></b></span><br /><br />True total intellectual people are what I look for, erudition is what I look for in people.Mandelbrot linked randomness to geometry and made randomness a more natural science.Fractals are linked with power laws, Mandelbrot worked on it and applied it to randomness.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://4.bp.blogspot.com/-IowGJqiHqnI/T3wstTlm3aI/AAAAAAAAK6I/Q-6hcBmDLDk/s1600/benoit-e1287435335385-580x435.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="300" src="http://4.bp.blogspot.com/-IowGJqiHqnI/T3wstTlm3aI/AAAAAAAAK6I/Q-6hcBmDLDk/s400/benoit-e1287435335385-580x435.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><b><i>Beniot Mandelbrot</i></b></td></tr></tbody></table><br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-Oqk7VZEH_aA/T3ws0zbL27I/AAAAAAAAK6Y/dJxlPKmKP4s/s1600/Poincare_2.jpeg" style="margin-left: auto; margin-right: auto;"><img border="0" height="400" src="http://1.bp.blogspot.com/-Oqk7VZEH_aA/T3ws0zbL27I/AAAAAAAAK6Y/dJxlPKmKP4s/s400/Poincare_2.jpeg" width="328" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Henri Poincare</td></tr></tbody></table><br /><br /><br /><span style="color: red;"><b>Henri Poincare</b></span><span style="color: #0b5394;"><b> is said to be underrated, he was the best mathematical thinker of all time,a true polymath and the man who published in every branch of math and science.</b></span> Every time I see picture of Einstein I think of Poincare because I think Poincare was better than Einstein(another form of narrative fallacy).It took almost a century to understand his theories ,Poincare was the first thinker to go against Gaussian or Normal Bell curve<span class="Apple-style-span" style="font-family: inherit;">.<span class="Apple-style-span" style="font-size: 16px; line-height: 18px;">Poincare was suspicious of of Gaussian </span></span><span style="font-family: inherit;">as he knew that extreme events don't follow Bell curve.Poincare was the master of theory of relativity and atomic structure and even Einstein had to read him before he published as he was foremost authority on relativity.</span>Many claim that Poincare was the first one to give idea of relativity but he never made it big to get prominence.Poincare also started the study of fractals and their use in physics,math and randomness etc,Poincare worked on Theory of Probability and also on Geometry,Chaos Theory and Astronomy/Mathematical Physics.Mandlbrolt later continued his work 100 years after his death.<span style="color: #cc0000;"><b>Poincre was first big gun to understand mathematical techniques and limits involved in randomness and hence forecasting limits.</b></span> Poincare's research on solar system got a Prize which was the highest academic prize at that time. <span style="color: #cc0000;"><b>Poincare was the advisor of Louis Bachelier, the pioneer of Financial Maths.</b> </span>Poincare suggested that as you project in the future you may need an increasing amount of precision ,near precision is not possible . Think of forecasting as in terms of tree branches, this grows in multiple ways and doubling every time so such increasing amount requires a lot of precision.<br /><br /><span style="font-size: 12pt; line-height: 115%;">Merton made his famous formulae based on Gaussian and so a flattering thing. Steve Ross an economist ,famed to be more intellectual than Merton gave Nassim applause on his Black Swan theory work in a seminar in U.S. <span style="color: blue;"> <span style="color: #3d85c6;"><b>Portfolio theory users</b> can't tell me how can they accept the use of Gaussian curve with large deviations(high standard deviations) in stocks.Gaussian and high sigma cannot go together but all economists have been using it since a long time.</span></span></span><br /><br /><span class="Apple-style-span" style="font-family: inherit;"><span class="Apple-style-span" style="font-size: 16px; line-height: 18px;">Robert Merton and Scholes made their company</span><span class="Apple-style-span" style="font-size: 16px; line-height: 18px;"> </span><span class="Apple-style-span" style="font-size: 16px; line-height: 18px;">LTCM (Long Term Capital Management), they employed top quants and used complex methods based on portfolio theory. Later in Russia when there was market crash and it made big impact on U.S market and thus making extreme event in U.S market and everything got busted along with LTCM. Someone using Gaussian in our U.S market or Wall Street (market which can experience extreme events)</span><span class="Apple-style-span" style="font-size: 16px; line-height: 18px;"> is a</span><span class="Apple-style-span" style="font-size: 16px; line-height: 18px;"> madman in my world</span></span><span class="Apple-style-span" style="font-family: Calibri,sans-serif; font-size: 16px; line-height: 18px;">.</span>Nassif Nabeal (Nassif Medici)http://www.blogger.com/profile/13708191074451322062noreply@blogger.com0tag:blogger.com,1999:blog-5761410253972037435.post-40024507766713416632013-10-21T06:58:00.000-07:002015-07-05T20:30:50.364-07:00Henri Poincare<span style="color: red;"><b>Jules Henri Poincaré (1854-1912)</b></span><br /><br /><span style="color: blue;"><span style="color: #3d85c6;">Poincaré was an influential French philosopher of science and mathematics, a distinguished scientist</span></span><span style="color: blue;"><span style="color: #3d85c6;"><span style="color: blue;"><span style="color: #3d85c6;">,theoretical physicist</span></span> and </span></span><span style="color: blue;"><span style="color: #3d85c6;"><span style="color: blue;"><span style="color: #3d85c6;"> one of the greatest </span></span>mathematician.Henri Poincaré was called the "last universalist" because he was a great contributor to all fields of mathematics that existed at that time.His work was also broad in physics, philosophy, and psychology.</span> </span> In the foundations of mathematics he argued for conventionalism, against formalism, against logicism, and against Cantor’s treating his new infinite sets as being independent of human thinking. Poincaré stressed the essential role of intuition in a proper constructive foundation for mathematics. He believed that logic was a system of analytic truths, whereas arithmetic was synthetic and a priori, in Kant‘s sense of these terms. Mathematicians can use the methods of logic to check a proof, but they must use intuition to create a proof, he believed.He maintained that non-Euclidean geometries are just as legitimate as Euclidean geometry, because all geometries are conventions or “disguised” definitions. Although all geometries are about physical space, a choice of one geometry over others is a matter of economy and simplicity, not a matter of finding the true one among the false ones.<br /><br />For Poincaré, the aim of science is prediction rather than, say, explanation. Although every scientific theory has its own language or syntax, which is chosen by convention, it is not a matter of convention whether scientific predictions agree with the facts. For example, it is a matter of convention whether to define gravitation as following Newton’s theory of gravitation, but it is not a matter of convention as to whether gravitation is a force that acts on celestial bodies, or is the only force that does so. So, Poincaré believed that scientific laws are conventions but not arbitrary conventions.Poincaré had an especially interesting view of scientific induction. Laws, he said, are not direct generalizations of experience; they aren’t mere summaries of the points on the graph. Rather, the scientist declares the law to be some interpolated curve that is more or less smooth and so will miss some of those points<span style="font-family: Arial,Helvetica,sans-serif;"><span style="color: #3d85c6;">.</span></span>Thus a scientific theory is not directly falsifiable by the data of experience; instead, the falsification process is more indirect. <br /><br /><br /><span style="color: #3d85c6;"><span style="font-family: Georgia,"Times New Roman",serif;"><span style="font-size: small;"><b>Poincare also proposed the theory of relativity before Einstein but never published it vastly, infact Einstein had to read Poincare about relativity(as Poincare at that time was the only pioneer of the theory of relativity).Einstein's work are not completely original, they are based on previous works of Hendrik Lorentz and Henri Poincaré. </b></span></span></span><br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://4.bp.blogspot.com/-67CSJ3J71Eg/T4GfM77aHRI/AAAAAAAALG8/2bSVosVBewM/s1600/JH_Poincare.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="http://4.bp.blogspot.com/-67CSJ3J71Eg/T4GfM77aHRI/AAAAAAAALG8/2bSVosVBewM/s320/JH_Poincare.jpg" width="236" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span class="Apple-style-span" style="font-family: Verdana, sans-serif; font-size: small;">Henri Poincare</span></td></tr></tbody></table><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: #3d85c6;"><br /></span></span><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: #3d85c6;"><b>With 49 proposals between 1901 and 1912, Poincare still remains the most nominated scientist of Noble Prize in Physics. Untimely death of Poincare prevented him to add the Noble prize to his amazing list of awards.</b></span></span><br /><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: #3d85c6;"><br /></span></span><br /><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: #3d85c6;"><b><span class="Apple-style-span">"Every time I see picture of Einstein I think of Poincare because I think Poincare was supreme and better than Einstein.It took almost a century to understand his theories".</span></b></span></span></div><div style="margin-bottom: 0px; margin-left: 0px; margin-right: 0px; margin-top: 0px;"><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: #3d85c6;"><b><span class="Apple-style-span">Nassim Nicholas Taleb</span></b></span></span></div><div><span style="color: #cc0000;"><b><br /></b></span></div><b><span style="color: red;"><br /></span></b><b><span style="color: red;">1. Life</span></b><br />Poincaré was born on April 29,1854 in Nancy and died on July 17, 1912 in Paris. Poincaré’s family was influential. His cousin Raymond was the President and the Prime Minister of France, and his father Leon was a professor of medicine at the University of Nancy. His sister Aline married the spiritualist philosopher Emile Boutroux.<br /><br />Poincaré studied mining engineering, mathematics and physics in Paris.Beginning in 1881, he taught at the University of Paris.There he held the chairs of Physical and Experimental Mechanics, Mathematical Physics and Theory of Probability, and Celestial Mechanics and Astronomy.At the beginning of his scientific career, in his doctoral dissertation of 1879, Poincaré devised a new way of studying the properties of functions defined by differential equations. He not only faced the question of determining the integral of such equations, but also was the first person to study the general geometric properties of these functions. He clearly saw that this method was useful in the solution of problems such as the stability of the solar system, in which the question is about the qualitative properties of planetary orbits (for example, are orbits regular or chaotic?) and not about the numerical solution of gravitational equations.During his studies on differential equations, Poincaré made use of Lobachevsky’s non-Euclidean geometry. Later, Poincaré applied to celestial mechanics the methods he had introduced in his doctoral dissertation. His research on the stability of the solar system opened the door to the study of chaotic deterministic systems; and the methods he used gave rise to algebraic topology.<br /><br />Poincaré sketched a preliminary version of the special theory of relativity and stated that the velocity of light is a limit velocity and that mass depends on speed. He formulated the principle of relativity, according to which no mechanical or electromagnetic experiment can discriminate between a state of uniform motion and a state of rest, and he derived the Lorentz transformation. His fundamental theorem that every isolated mechanical system returns after a finite time [the Poincaré Recurrence Time] to its initial state is the source of many philosophical and scientific analyses on entropy. Finally, he clearly understood how radical is quantum theory’s departure from classical physics.<br /><br />Poincaré was deeply interested in the philosophy of science and the foundations of mathematics. He argued for conventionalism and against both formalism and logicism.Cantor’s set theory was also an object of his criticism. He wrote several articles on the philosophical interpretation of mathematical logic. During his life, he published three books on the philosophy of science and mathematics. A fourth book was published posthumously in 1913.<br /><br /><br /><span style="color: red;"><b>2. Chaos and the Solar System</b></span><br />In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system. Given the law of gravity and the initial positions and velocities of the only three bodies in all of space, the subsequent positions and velocities are fixed–so the three-body system is deterministic.However, Poincaré found that the evolution of such a system is often chaotic in the sense that a small perturbation in the initial state such as a slight change in one body’s initial position might lead to a radically different later state than would be produced by the unperturbed system. If the slight change isn’t detectable by our measuring instruments, then we won’t be able to predict which final state will occur. So, Poincaré’s research proved that the problem of determinism and the problem of predictability are distinct problems.From a philosophical point of view, Poincaré’s results did not receive the attention that they deserved.Also the scientific line of research that Poincaré opened was neglected until meteorologist Edward Lorenz, in 1963, rediscovered a chaotic deterministic system while he was studying the evolution of a simple model of the atmosphere. Earlier, Poincaré had suggested that the difficulties of reliable weather predicting are due to the intrinsic chaotic behavior of the atmosphere. Another interesting aspect of Poincaré’s study is the real nature of the distribution in phase space of stable and unstable points, which are so mixed that he did not try to make a picture of their arrangement. Now we know that the shape of such distribution is fractal-like. However, the scientific study of fractals did not begin until Benoit Mandelbrot’s work in 1975, a century after Poincaré’s first insight.<br /><br />Why was Poincaré’s research neglected and underestimated? The problem is interesting because Poincaré was awarded an important scientific prize for his research; and his research in celestial mechanics was recognized to be of fundamental importance.Probably there were two causes.Scientists and philosophers were primarily interested in the revolutionary new physics of relativity and quantum mechanics, but Poincaré worked with classical mechanics.Also, the behavior of a chaotic deterministic system can be described only by means of a numerical solution whose complexity is staggering. Without the help of a computer the task is almost hopeless.<br /><br /><br /><span style="color: red;"><b>3. Arithmetic, Intuition and Logic</b></span><br />Logicists such as Bertrand Russell and Gottlob Frege believed that mathematics is basically a branch of symbolic logic, because they supposed that mathematical terminology can be defined using only the terminology of logic and because, after this translation of terms, any mathematical theorem can be shown to be a restatement of a theorem of logic.Poincaré objected to this logicist program.He was an intuitionist who stressed the essential role of human intuition in the foundations of mathematics.According to Poincaré, a definition of a mathematical entity is not the exposition of the essential properties of the entity, but it is the construction of the entity itself; in other words, a legitimate mathematical definition creates and justifies its object. For Poincaré, arithmetic is a synthetic science whose objects are not independent from human thought.Poincaré made this point in his investigation of Peano’s axiomatization of arithmetic.Italian mathematician Giuseppe Peano (1858-1932) axiomatized the mathematical theory of natural numbers. This is the arithmetic of the nonnegative integers. Apart from some purely logical principles, Peano employed five mathematical axioms. Informally, these axioms are:<br /><ol><li>Zero is a natural number.</li><li>Zero is not the successor of any natural number.</li><li>Every natural number has a successor, which is a natural number.</li><li>If the successor of natural number <i>a</i> is equal to the successor of natural number <i>b</i>, then <i>a</i> and <i>b</i> are equal.</li><li>Suppose:<br />(i) zero has a property P;<br />(ii) if every natural number less than <i>a</i> has the property P then <i>a</i> also has the property P.<br />Then every natural number has the property P. (This is the principle of complete induction.)</li></ol>Bertrand Russell said Peano’s axioms constitute an implicit definition of natural numbers, but Poincaré said they do only if they can be demonstrated to be consistent.They can be shown consistent only by showing there is some object satisfying these axioms. From a general point of view, an axiom system can be conceived of as an implicit definition only if it is possible to prove the existence of at least one object that satisfies all the axiom Proving this is not an easy task, for the number of consequences of Peano axioms is infinite and so a direct inspection of each consequence is not possible. Only one way seems adequate: we must verify that if the premises of an inference in the system are consistent with the axioms of logic, then so is the conclusion. Therefore, if after <i>n</i> inferences no contradiction is produced, then after <i>n</i>+1 inferences no contradiction will be either.Poincaré argues that this reasoning is a vicious circle, for it relies upon the principle of complete induction, whose consistency we have to prove. (In 1936, Gerhard Gentzen proved the consistency of Peano axioms, but his proof required the use of a limited form of transfinite induction whose own consistency is in doubt). As a consequence, Poincaré asserts that if we can’t noncircularly establish the consistency of Peano’s axioms, then the principle of complete induction is surely not provable by means of general logical laws; thus it is not analytic, but it is a synthetic judgment, and logicism is refuted.It is evident that Poincaré supports Kant’s epistemological viewpoint on arithmetic.For Poincaré, the principle of complete induction, which is not provable via analytical inferences, is a genuine synthetic a priori judgment. Hence arithmetic cannot be reduced to logic; the latter is analytic, while arithmetic is synthetic.<br /><br />The synthetic character of arithmetic is also evident if we consider the nature of mathematical reasoning. Poincaré suggests a distinction between two different kinds of mathematical inference: <i>verification</i> and <i>proof</i>. Verification or proof-check is a sort of mechanical reasoning, while proof-creation is a fecund inference. For example, the statement “2+2 = 4″ is verifiable because it is possible to demonstrate its truth with the help of logical laws and the definition of sum; it is an analytical statement that admits a straightforward verification. On the contrary, the general statement (the commutative law of addition)<br /><blockquote>For any natural numbers x and y, <i>x</i> + <i>y</i> = <i>y</i> + <i>x</i></blockquote>is not directly verifiable. We can choose an arbitrary pair of natural numbers <i>a</i> and <i>b</i>, and we can verify that <i>a</i>+<i>b</i> = <i>b</i>+<i>a</i>; but there is an infinite number of admissible choices of pairs, so the verification is always incomplete. In other words, the verification of the commutative law is an analytical method by means of which we can verify every particular instance of a general theorem, but the proof of the theorem itself is synthetic reasoning which really extends our knowledge, Poincaré believed.Another aspect of mathematical thinking that Poincaré analyzes is the different roles played by intuition and logic. Methods of formal logic are elementary and certain, and we can surely rely on them.However, logic does not teach us how to build a proof. It is intuition that helps mathematicians find the correct way of to assemble basic inferences into a useful proof. Poincaré offers the following example. An unskilled chess player who watches a game can verify whether a move is legal, but he does not understand why players move certain pieces, for he does not see the plan which guides players’ choices. In a similar way, a mathematician who uses only logical methods can verify every inference in a given proof, but he cannot find an original proof.In other words, every elementary inference in a proof is easily verifiable through formal logic, but the invention of a proof requires the understanding -grasped by intuition of the general scheme, which directs mathematician’s efforts towards the final goal.<br /><br />Logic is — according to Poincaré the study of properties which are common to all classifications.There are two different kinds of classifications: <i>predicative</i> classifications, which are not modified by the introduction of new elements; and <i>impredicative</i> classifications, which are modified by new elements.Definitions as well as classifications are divided into predicative and impredicative. A set is defined by a law according to which every element is generated. In the case of an infinite set, the process of generating elements is unfinished; thus there are always new elements. If their introduction changes the classification of already generated objects, then the definition is impredicative. For example, look at phrases containing a finite number of words and defining a point of space. These phrases are arranged in alphabetical order and each of them is associated with a natural number: the first is associated with number 1, the second with 2, etc. Hence every point defined by such phrases is associated with a natural number. Now suppose that a new point is defined by a new phrase.To determine the corresponding number it is necessary to insert this phrase in alphabetical order; but such an operation modifies the number associated with the already classified points whose defining phrase follows, in alphabetical order, the new phrase. Thus this new definition is impredicative.<br /><br />For Poincaré, impredicative definitions are the source of antinomies in set theory, and the prohibition of impredicative definitions will remove such antinomies.To this end, Poincaré enunciates the vicious circle principle: a thing cannot be defined with respect to a collection that presupposes the thing itself. In other words, in a definition of an object, one cannot use a set to which the object belongs, because doing so produces an impredicative definition. Poincaré attributes the vicious circle principle to a French mathematician J. Richard. In 1905, Richard discovered a new paradox in set theory, and he offered a tentative solution based on the vicious circle principle.<br /><br />Poincaré’s prohibition of impredicative definitions is also connected with his point of view on infinity. According to Poincaré, there are two different schools of thought about infinite sets; he called these schools <i>Cantorian</i> and <i>Pragmatist</i>.Cantorians are realists with respect to mathematical entities; these entities have a reality that is independent of human conceptions.The mathematician discovers them but does not create them. Pragmatists believe that a thing exists only when it is the object of an act of thinking, and infinity is nothing but the possibility of the mind’s generating an endless series of finite objects. Practicing mathematicians tend to be realists, not pragmatists or intuitionists.This dispute is not about the role of impredicative definitions in producing antinomies, but about the independence of mathematical entities from human thinking.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-XTh0gR65KV4/T4Gf8u-VWMI/AAAAAAAALHE/DSumRVpklvM/s1600/38487-004-4DDEF34E.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="http://1.bp.blogspot.com/-XTh0gR65KV4/T4Gf8u-VWMI/AAAAAAAALHE/DSumRVpklvM/s320/38487-004-4DDEF34E.jpg" width="307" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;">Henri Poincare</span></td></tr></tbody></table><br /><span style="color: red;"><b>4. Conventionalism and the Philosophy of Geometry</b></span><br />The discovery of non-Euclidean geometries upset the commonly accepted Kantian viewpoint that the true structure of space can be known apriori.To understand Poincaré’s point of view on the foundation of geometry, it helps to remember that, during his research on functions defined by differential equations, he actually used non-Euclidean geometry. He found that several geometric properties are easily provable by means of Lobachevsky geometry, while their proof is not straightforward in Euclidean geometry. Also, Poincaré knew Beltrami’s research on Lobachevsky’s geometry. Beltrami (Italian mathematician, 1835-1899) proved the consistency of Lobachevsky geometry with respect to Euclidean geometry, by means of a translation of every term of Lobachevsky geometry into a term of Euclidean geometry. The translation is carefully chosen so that every axiom of non-Euclidean geometry is translated into a theorem of Euclidean geometry. Beltrami’s translation and Poincaré’s study of functions led Poincaré to assert that:<br /><ul><li>Non-Euclidean geometries have the same logical and mathematical legitimacy as Euclidean geometry.</li><li>All geometric systems are equivalent and thus no system of axioms may claim that it is the true geometry.</li><li>Axioms of geometry are neither synthetic a priori judgments nor analytic ones; they are conventions or ‘disguised’ definitions.</li></ul>According to Poincaré, all geometric systems deal with the same properties of space, although each of them employs its own language, whose syntax is defined by the set of axioms. In other words, geometries differ in their language, but they are concerned with the same reality, for a geometry can be translated into another geometry. There is only one criterion according to which we can select a geometry, namely a criterion of economy and simplicity. This is the very reason why we commonly use Euclidean geometry: it is the simplest. However, with respect to a specific problem, non-Euclidean geometry may give us the result with less effort. In 1915, Albert Einstein found it more convenient, the conventionalist would say, to develop his theory of general relativity using non-Euclidean rather than Euclidean geometry. Poincaré’s realist opponent would disagree and say that Einstein discovered space to be non-Euclidean.<br /><br />Poincaré’s treatment of geometry is applicable also to the general analysis of scientific theories. Every scientific theory has its own language, which is chosen <i>by convention</i>. However, in spite of this freedom, the agreement or disagreement between predictions and facts is not conventional but is substantial and objective. Science has an objective validity. It is not due to chance or to freedom of choice that scientific predictions are often accurate.<br />These considerations clarify Poincaré’s conventionalism.There is an objective criterion, independent of the scientist’s will, according to which it is possible to judge the soundness of the scientific theory, namely the accuracy of its predictions. Thus the principles of science are not set by an arbitrary convention. In so far as scientific predictions are true, science gives us objective, although incomplete, knowledge. The freedom of a scientist takes place in the choice of language, axioms, and the facts that deserve attention.<br />However, according to Poincaré, every scientific law can be analyzed into two parts, namely a <i>principle</i>, that is a conventional truth, and an <i>empirical law</i>. The following example is due to Poincaré. The law:<br /><blockquote>Celestial bodies obey Newton’s law of gravitation</blockquote>The law consists of two elements:<br /><ol><li>Gravitation follows Newton law.</li><li>Gravitation is the only force that acts on celestial bodies.</li></ol>We can regard the first statement as a principle, as a convention; thus it becomes the definition of gravitation. But then the second statement is an empirical law.<br />Poincaré’s attitude towards conventionalism is illustrated by the following statement, which concluded his analysis on classical mechanics in <i>Science and Hypothesis</i>:<br /><blockquote>Are the laws of acceleration and composition of forces nothing but arbitrary conventions? Conventions, yes; arbitrary, no; they would seem arbitrary if we forgot the experiences which guided the founders of science to their adoption and which are, although imperfect, sufficient to justify them. Sometimes it is useful to turn our attention to the experimental origin of these conventions.</blockquote><br /><b> <span style="color: red;">5. Science and Hypothesis</span></b><br />According to Poincaré, although scientific theories originate from experience, they are neither verifiable nor falsifiable by means of the experience alone. For example, look at the problem of finding a mathematical law that describes a given series of observations.In this case, representative points are plotted in a graph, and then a simple curve is interpolated.The curve chosen will depend both on the experience which determines the representative points and on the desired smoothness of the curve even though the smoother the curve the more that some points will miss the curve. Therefore, the interpolated curve — and thus the tentative law — is not a direct generalization of the experience, for it ‘corrects’ the experience.The discrepancy between observed and calculated values is thus not regarded as a falsification of the law, but as a correction that the law imposes on our observations. In this sense, there is always a necessary difference between facts and theories, and therefore a scientific theory is not directly falsifiable by the experience.<br /><br />For Poincaré, the aim of the science is to prediction. To accomplish this task, science makes use of generalizations that go beyond the experience.In fact, scientific theories are hypotheses. But every hypothesis has to be continually tested. And when it fails in an empirical test, it must be given up. According to Poincaré, a scientific hypothesis which was proved untenable can still be very useful. If a hypothesis does not pass an empirical test, then this fact means that we have neglected some important and meaningful element; thus the hypothesis gives us the opportunity to discover the existence of an unforeseen aspect of reality. As a consequence of this point of view about the nature of scientific theories, Poincaré suggests that a scientist must utilize few hypotheses, for it is very difficult to find the wrong hypothesis in a theory which makes use of many hypotheses.<br />For Poincaré, there are many kinds of hypotheses:<br /><ul><li>Hypotheses which have the maximum scope, and which are common to all scientific theories (for example, the hypothesis according to which the influence of remote bodies is negligible). Such hypotheses are the last to be changed.</li><li>Indifferent hypotheses that, in spite of their auxiliary role in scientific theories, have no objective content (for example, the hypothesis that unseen atoms exist).</li><li>Generalizations, which are subjected to empirical control; they are the true scientific hypotheses.</li></ul>Regarding Poincaré’s point of view about scientific theories, the following have the most lasting value:<br /><ul><li>Every scientific theory is a hypothesis that had to be tested.</li><li>Experience suggests scientific theories; but experience does not justify them.</li><li>Experience alone is unable to falsify a theory, for the theory often corrects the experience.</li><li>A central aim of science is prediction.</li><li>The role of a falsified hypothesis is very important, for it throws light on unforeseen conditions.</li><li>Experience is judged according to a theory. </li></ul><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-oglHZbr7tSk/VYt61Gn8t7I/AAAAAAAAYaU/iqFIJxeG5ME/s1600/41b38oZf34L._UY250_.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-oglHZbr7tSk/VYt61Gn8t7I/AAAAAAAAYaU/iqFIJxeG5ME/s1600/41b38oZf34L._UY250_.jpg" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"></div><br /><span style="color: red;"><b> 6.Randomness</b></span><span class="Apple-style-span" style="font-family: Calibri,sans-serif; font-size: 15px; line-height: 17px;"><span style="color: red;">.</span><span class="Apple-style-span" style="color: white;"> </span></span><span class="Apple-style-span" style="font-size: 15px; line-height: 17px;"><span class="Apple-style-span" style="font-family: inherit;"> </span></span><br /><span class="Apple-style-span" style="font-size: 15px; line-height: 17px;"><span class="Apple-style-span" style="font-family: inherit;">Poincre was first big gun to understand mathematical techniques and limits involved in randomness and hence forecasting limits</span><b style="font-family: Calibri, sans-serif;">.</b></span>Poincare was the first thinker to go against Gaussian or Normal Bell curve and suggested that such curve is useless for big and rare events. Many claim that Poincare was the first one to give idea of relativity but he never made it big to get prominence.Poincare also started the study of fractals and their use in physics,math and randomness etc,Mandlbrolt later continued his work 100 years after Poincre was first big gun to understand mathematical techniques and limits involved in randomness and hence forecasting limits. Poincare's research on solar system got a Prize which was highest at that time. <span style="color: #3d85c6;">Poincare was the advisor of Louis Bachelier, the pioneer of Financial Maths</span><span class="Apple-style-span" style="font-family: inherit;"><span style="color: #3d85c6;">.</span><span class="Apple-style-span" style="line-height: 19px;">Poincaré saw mathematical work in </span><span class="Apple-style-span" style="line-height: 19px;">economics</span><span class="Apple-style-span" style="line-height: 19px;"> and finance as peripheral. In 1900 Poincaré commented on </span><span class="Apple-style-span" style="line-height: 19px;">Louis Bachelier</span><span class="Apple-style-span" style="line-height: 19px;">'s thesis "The Theory of Speculation", saying: "M. Bachelier has evidenced an original and precise mind [but] the subject is somewhat remote from those our other candidates are in the habit of treating." (Bernstein, 1996, pp. 199–200) Bachelier's work explained what was then the French government's pricing options on French Bonds and anticipated many of the pricing theories in financial markets used today</span> .</span><span style="color: #3d85c6;">Poincare suggested that as you project in the future you may need an increasing amount of precision ,near precision is not possible .</span> Think of forecasting as in terms of tree branches, this grows in multiple ways and doubling every time so such increasing amount requires a lot of precision.Poincare suddenly died in July, 1912 and left his works for generations.<br /><br /><span class="Apple-style-span" style="color: #cc0000;"><b>According to famous French mathematician Jean Leray "very few men were able to follow Poincare's thoughts , he had no students.After one century of mathematical work and advancement we are able to understand his works and thoughts more easily."</b></span><br /><span class="Apple-style-span" style="color: #cc0000;"><b><br /></b></span><span class="Apple-style-span" style="color: #cc0000;"><b><br /></b></span> <b> Bibliography</b><br />COLLECTED SCIENTIFIC WORKS (in French).<br /><ul><li>Oeuvres, 11 volumes, Paris : Gauthier-Villars, 1916-1956 <br /><br />PHILOSOPHICAL WORKS.</li><li>1902 <i>La science et l’hypothèse</i>, Paris : Flammarion (<i>Science and hypothesis</i>, 1905)</li><li>1905 <i>La valeur de la science</i>, Paris : Flammarion (<i>The value of science</i>, 1907)</li><li>1908 <i>Science and méthode</i>, Paris : Flammarion (<i>Science and method</i>, 1914)</li><li>1913 <i>Dernières pensées</i>, Paris : Flammarion (<i>Mathematics and science: last essays</i>, 1963)</li><li>The first three works are translated in <i>The foundations of science</i>, Washington, D.C. : University Press of America, 1982 (first edition 1946).</li></ul>Nassif Nabeal (Nassif Medici)http://www.blogger.com/profile/13708191074451322062noreply@blogger.com0tag:blogger.com,1999:blog-5761410253972037435.post-62292083085376201702013-10-21T06:57:00.001-07:002015-07-10T20:44:31.671-07:00Benoît Mandelbrot, the modern Leonardo Da Vinci<span style="color: #0b5394;">MATHEMATICS </span>is a curious subject. Though often classed as one, it is not really a science. That scientists use it to describe their interpretation of reality is not quite the same thing. Nor, though, is it an art—not, at any rate, in the modern meaning of that word. The aesthetics of the subject, which any mathematician will tell you are the driving force behind his passion, are not obvious to the senses in the way that those of a painting, a symphony or a play are. Yet <b><span style="color: red;">Benoît Mandelbrot</span></b><span class="Apple-style-span" style="color: red;">’s</span> celebrity beyond the academy is largely due to art in its modern, sensuous, sense. For the “set” to which he gave his name, when computed, drawn on a complex plane and suitably tinted, appealed greatly to the senses—as a million posters, greetings cards and T-shirts, bought by people who had not the faintest idea what it was, attest. <br />The Mandelbrot set is a collection of points in the complex-number plane. The formula for calculating these numbers is <i>z<sub>n+1</sub> = z<sub>n</sub><sup>2</sup> + c</i>, where <i>c</i> is a complex number and <i>n</i> (representing the digits 1 to infinity) counts the number of times the calculation has been performed. <i>Z</i> starts as any number you like, and changes with each calculation, the value of <i>z<sub>n+1</sub></i> being used as <i>z<sub>n</sub></i> the next time round. Sometimes the value of <i>z</i> remains finite, no matter how large <i>n</i> gets. In that case, <i>c</i> is part of the Mandelbrot set. Sometimes <i>z</i> shoots off to infinity. In that case, <i>c</i> is not part of the set. The boundary between the two is the swirling fractal line that so appeals to the eye, and the colours of the points outside the set indicate how long the calculation takes to start shooting off to infinity. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/-WOrEkzMezHY/T4szqDodkDI/AAAAAAAALjw/MV5K2aXq9gE/s1600/118_dyson-mandelbrot.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="400" src="http://3.bp.blogspot.com/-WOrEkzMezHY/T4szqDodkDI/AAAAAAAALjw/MV5K2aXq9gE/s400/118_dyson-mandelbrot.jpg" width="298" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Benoît Mandelbrot</td></tr></tbody></table><span style="color: #0b5394; font-size: small;"><span style="font-family: inherit;"><br /></span></span><br /><div class="MsoNormal"><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="color: #0b5394; font-size: small;"><span class="Apple-style-span"><b>Probability and Randomness</b></span></span></span><span style="font-size: 12pt; line-height: 115%;"><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="color: #0b5394; font-size: small;"><b>:</b></span></span> Other mathematicians of probability like Kolmogorov may be more academic or progressive but Mandelbrot was unique he proved that mathematicians actually understand randomness</span><span style="font-size: 12pt; line-height: 115%;">.In fact Nassim Nicholas Taleb's "Black Swan Theory" is inspired by work of Mandelbrot as Mandelbrot was much concerned about high-risk rare events(Black Swans).Nassim and Mandelbrot collaborated in research tasks related to risk management and Mandelbrot was later called by Nassim as "poet of randomness".Black Swans were dealt by him in a philosophical and aesthetic way.</span><span class="Apple-style-span" style="font-size: 16px; line-height: 18px;">Mandelbrot was initially a probability guy but later went into other fields of maths and made his name in other fields.</span><span style="font-size: 12pt; line-height: 18px;"><span class="Apple-style-span" style="font-family: inherit;">In 1960s Mandelbrot presented his ideas on prices of commodity and stock prices and made a contribution on mathematics of randomness in economic theory.Mandelbrot also knew the pitfalls in Bachelier's model.</span></span><span class="Apple-style-span" style="font-size: 16px; line-height: 18px;">Mandelbrot linked randomness to geometry and made randomness a more natural science.</span><span class="Apple-style-span" style="font-size: 16px; line-height: 18px;">If stock markets were Gaussian then stock market crashed would have happen once in a Billion years. Mandelbrot's randomness methods make the statistics methods look useless. </span></div><div class="MsoNormal"><span style="font-size: 12pt; line-height: 115%;"><br /></span></div><div class="MsoNormal"><span style="font-size: 12pt; line-height: 115%;"><span style="color: #cc0000;">Fractals are linked with power laws, Mandelbrot worked on it and applied it to randomness. Mandelbrot designed the mathematical object called "Mandelbrot set" and later worked on shapes and fractals of maths and also worked on Chaos Theory.</span> These objects play an important role on aesthetics , music , architecture , poetry , gestures and tones are derived from fractals . Mandelbrot's book "Fractal Geometry of Nature" it made a fame in arts , visual arts and every artistic circle. He was later offered a position in Medicine <b style="mso-bidi-font-weight: normal;">,<span style="color: #0b5394;"> </span></b><span style="color: #0b5394;"><span style="color: #cc0000;">all artists used to call Mandelbrot "The Rock Star of Mathematics"</span></span><span style="color: #cc0000;">.</span> Mandelbrot became famous because of the number of applications of mathematics in our society. Mandelbrot was initially a probability guy but later went into other fields of maths and made his name in other fields.</span><br /><br /></div><span style="font-size: 12pt; line-height: 115%;"><span class="Apple-style-span" style="font-family: inherit;">In 1960s Mandelbrot presented his ideas on prices of commodity and stock prices and made a contribution on mathematics of randomness in economic theory. Just is it as easy to reject innocence than accept it, it is easier to reject a Bell curve than accept it .It's difficult to reject Fractal than accept it because a single event can cause a disaster.</span></span><br /><span class="Apple-style-span" style="font-family: Georgia, 'Times New Roman', serif;"><br /></span><span class="Apple-style-span" style="font-family: Georgia, 'Times New Roman', serif;"><b><span class="Apple-style-span" style="color: #0b5394;"><br /></span></b></span><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span class="Apple-style-span"><span class="Apple-style-span" style="color: #0b5394; font-weight: bold;">Mandelbrot's Contribution in Finance.</span></span></span><br /><br /><br />The first formal model for security price changes was put forward by Bachelier (1900). His price difference process in essence sets out the mathematics of Brownian Motion before Einstein and Wiener rediscovered his results in 1905 and 1923 in the context of physical particles, and in particular generates a Normal (i.e. Gaussian) distribution where variance increases proportionally with time. A crucial assumption of Bachelier’s approach is that successive price changes are independent. His dissertation, which was awarded only a “mention honorable” rather than the “mention très honorable” that was essential for recognition in the academic world, remained unknown to the financial world until Osborne (1959), who made no reference to Bachelier’s work, rediscovered Brownian Motion as a plausible model for security price changes.<br /><br /><span style="color: #cc0000;">But in 1963 the famous mathematician Mandelbrot produced a paper pointing out that the tails of security price distributions are far fatter than those of normal distributions (what he called the “Noah effect” in reference to the deluge in biblical times) and recommending instead a class of independent and identically distributed “alpha-stable” Paretian distributions with infinite variance. </span>Towards the end of the paper Mandelbrot observes that the independence assumption in his suggested model does not fully reflect reality in that “on closer inspection … large changes tend to be followed by large changes – of either sign – and small changes tend to be followed by small changes.” Mandelbrot later called this the “Joseph effect” inreference to the biblical account of seven years of plentiful harvests in Egypt followed by seven years of famine. Such a sequence of events would have had an exceptionally low probability of taking place if harvest yields in successive years were independent. While considering how best to model this dependence effect, Mandelbrot came across the work of Hurst (1951, 1955) which dealt with a very strong dependence in natural events such as river flows (particularly in the case of the Nile) from one year to another and developed the Hurst exponent H as a robust statistical measure of dependence. Mandelbrot’s new model ofFractional Brownian Motion, which is described in detail in Mandelbrot & van Ness (1968), is defined by an equation which incorporates the Hurst exponent H. Many financial economists, particularly Cootner (1964), were highly critical of Mandelbrot’s work, mainly because – if he was correct about normal distributions being seriously inconsistent with reality – most of their earlier statistical work, particularly in tests of the Capital Asset Pricing Model and the Efficient Market Hypothesis, would be invalid. Indeed, in his seminal review work on stockmarket efficiency, Fama (1970) describes how non-normal stable distributions of precisely the type advocated by Mandelbrot are more realistic than standard distributions .<br /><br />“Economists have, however, been reluctant to accept these results, primarily because of the wealth of statistical techniques available for dealing with normal variables and the relative paucity of such techniques for non-normal stable variables.”<br /><br />Partly because of estimation problems with alpha-stable Paretian distributions and the mathematical complexity of Fractional Brownian Motion, and partly because of the conclusion in Lo (1991) that standard distributions might give an adequate representation of reality, Mandelbrot’s two suggested new models failed to make a major impact on finance theory, and he essentially left the financial scene to pursue other interests such as fractal geometry. However, in his “Fractal Geometry of Nature”, Mandelbrot (1982) commented on what he regarded as the “suicidal” statistical methodologies that were standard in financetheory: “Faced with a statistical test that rejects the Brownian hypothesis that price changes are Gaussian, the economist can try one modification after another until the test is fooled. A popular fix is censorship, hypocritically called ‘rejection of outliers’. One distinguishes the ordinary ‘small’ price changes from the large changes that defeat Alexander’s filters. The former are viewed as random and Gaussian, and treasures of ingenuity are devoted to them … The latter are handled separately, as ‘nonstochastic’.”<br /><br />Shortly after the “Noah effect” manifested itself with extreme severity in the collapse of Long-Term Capital Management, Mandelbrot (1999) produced a brief article, the cover story of the February 1999 issue of “Scientific American”, in which he used nautical analogies to highlight the foolhardy nature of standard risk models that assumed independent normal distributions. He also pointed out that a more realistic depiction of market fluctuations, namely Fractional Brownian Motion in multifractal trading time, already existed.<br /><br /><span class="Apple-style-span" style="font-family: Georgia, 'Times New Roman', serif;"><br /></span><span style="font-family: Georgia,"Times New Roman",serif;"><span style="font-size: small;"><span style="color: #3d85c6;"><span class="Apple-style-span" style="line-height: 18px;">In fact he was one of the pioneers in studying the variation of financial prices even before Bchelier's Brownian model became widely accepted in academia and </span><span class="Apple-style-span" style="line-height: 18px;"><span class="Apple-style-span">Mandelbrot also knew the pitfalls in Bachelier's model.</span></span><span class="Apple-style-span" style="line-height: 18px;">For this reason many call him as the "father of Quantitative Finance"</span> </span></span></span><br /><br /><br /><span style="color: #0b5394;"><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span class="Apple-style-span"><br /></span></span></span><span style="color: #0b5394;"><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span class="Apple-style-span"><b>Mandelbrot's contribution in finance fall into three main stages:</b></span></span></span><br /><span class="Apple-style-span" style="color: #3d85c6;"><b><br /></b></span>He was the first to stress the essential importance, even in a first approximation, of large variations that may occur as sudden price discontinuities. The Brownian model is unjustified in neglecting them. They are not “outliers” one can safely disregard or study separately. To the contrary, their distribution is much more important than that of the "background noise" constituted by the small changes of Brownian motion. He followed this critique in by showing in 1963 that the big discontinuities and the small "noise" fall on a single power-law distribution and represented them by a scenario based on Levy stable distributions. He and Taylor introduced in 1967 the new notion of intrinsic "trading time." In recent years, fractal trading time and his 1963 model have gained wide acceptance.<br /><br />Secondly, Mandelbrot tackled the fact that the “background noise” of small price changes is of variable “volatility.” This feature was ordinarily viewed as a symptom of non-stationality that must be studied separately. To the contrary, Mandelbrot interpreted this variability as indicating that price changes are far from being statistically independent. In fact, for all practical purposes, their interdependence should be viewed as continuing to an infinitely long term. In particular, it is not limited to the short term that is studied by Markov processes and more recently ARCH and its variants. In fact, it too follows a power-law side of dependence. He followed this critique and illustrated long-dependence by introducing in 1965 a process called fractional Brownian motion which has become very widely used.<br /><br />Thirdly, he introduced the new notion of multifractality that combines long power-law tails and long power-law dependence. Early on, his work was motivated by the context of turbulence, but he immediately observed and pointed out that in 1972 the same ideas also apply to finance. After a long hiatus while he was developing other aspects of fractal geometry, he returned to finance in the mid-1990s and developed the multifractal scenario theory in detail in his 1997 book "Fractals and Scaling in Finance". The concept of scaling invariance used by Mandelbrot started by being perceived as suspect, because at that time other fields did not use it. However the period after 1972 also saw the growth of a new subfield of statistical physics concerned with “criticality.” The concepts used in that field are similar to those Mandelbrot had been using in finance.In "The Misbehavior of Markets", another popular book by Mandelbrot,he argues that the Gaussian models for financial risk used by economists like William Sharpe and Harry Markowitz should be discarded, since these models do not reflect reality. Mandelbrot argues that fractal techniques may provide a more powerful way to analyze risk. <br /><span class="Apple-style-span" style="font-family: Georgia, 'Times New Roman', serif;"><span class="Apple-style-span" style="color: #0b5394;"><b><br /></b></span></span><span class="Apple-style-span" style="font-family: Georgia, 'Times New Roman', serif;"><span class="Apple-style-span" style="color: #0b5394;"><b><br /></b></span></span><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="color: blue;"><span style="color: #0b5394;"><b>Fractal Geometry</b></span>.</span> </span>Complex-number plane. Lewis Carroll, no mean mathematician himself, asked Alice to believe as many as six impossible things before breakfast. These could easily have been two of them.First, then, the complex plane. This is the space on which all numbers, real, imaginary and combinations of the two, can be plotted. A real number is the familiar sort from normal arithmetic. An imaginary one is a multiple of the square root of -1.<br /><br />Mathematicians struggled for centuries with the question of what, multiplied by itself, gives the answer -1 before one of them, Leonhard Euler, suggested that the best way to deal with the problem was to invent a new symbol (he chose <i>i</i>) and live with the consequences. It works. And, as Euler’s successor, Carl Friedrich Gauss, was to discover, if you plot real numbers on one axis of a graph and imaginary ones on the other, you create a plane that represents both sorts of numbers. Complex numbers, which have a real and an imaginary part added together, are the points on this plane that do not lie on either axis.The invention of complex numbers was a watershed in mathematics. It also marked the moment when maths began to slip away from being part of the armamentarium of any educated person and towards the dizzyingly abstruse field it has become today. But a fractal is something a ten-year-old child might hit on.<br /><br /><span class="Apple-style-span" style="font-family: inherit;">In 1975 he invented the word fractal to describe his discoveries. But the breakthrough that made them famous was the ability of computers to plot them in a way that is easy on the eye. Thus were launched the posters, the cards and the T-shirts.While the ideas behind fractals, iteration and self-similarity are ancient, it took the coining of the term "fractal geometry" in 1975 and the publication of <span style="color: #3d85c6;">The Fractal Geometry of Nature</span> in French in the same year to give the quest an identity. As Mandelbrot put it, "to have a name is to be" — and the field exploded.</span>Extending fractals into the plane of complex numbers followed in 1979.<br /><span class="Apple-style-span" style="font-family: inherit;"><br /></span><br /><div><span class="Apple-style-span" style="font-family: inherit;">Fractal geometry is now being used in work with marine organisms, vegetative ecosystems, earthquake data, the behaviour of density-dependent populations, percolation and aggregation in oil research, and in the formation of lightning. Lightning resembles the diffusion patterns left by water as it permeates soft rock such as sandstone: computer simulations of this effect look exactly like the real thing.</span><br /><span class="Apple-style-span" style="font-family: inherit;">Fractals hold a promise for building better roads, for video compression and even for designing ships that are less likely to capsize. The geometry is already being successfully applied in medical imaging, and the forms generated by the discipline are a source of pleasure in their own right, adding to our aesthetic awareness as we observe fractals everywhere in nature.</span><br /><span class="Apple-style-span" style="font-family: inherit;"><br /></span></div><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-giqBbazDuYM/T8SX09zWdHI/AAAAAAAAM9U/gZi9qKvNZcQ/s1600/carbajo-fr-gno-cells_alive.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="300" src="http://1.bp.blogspot.com/-giqBbazDuYM/T8SX09zWdHI/AAAAAAAAM9U/gZi9qKvNZcQ/s400/carbajo-fr-gno-cells_alive.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">This fractal, <i>Cells Alive</i>, is a winner at <span class="pegado-blanco">“<a href="http://www.fractalartcontests.com/2009">Benoît Mandelbrot Fractal Art Contest 2009</a>”</span>. </td></tr></tbody></table><br />What is the length of a country’s coastline? Any encyclopedia will give you a figure. Yet stand by the sea and watch the irregularity of its edge, and you begin to doubt. It is not just a matter of tide and waves. Even measuring the boundary of a static body of water is no mean feat. The closer you look, the more irregular the line. That, at bottom, is what describes a fractal. When you magnify it, it rushes away from you and becomes a simulacrum of its larger self, eventually infinitely long. <br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://3.bp.blogspot.com/-uIbiamXPF4s/T4sz0o6u3hI/AAAAAAAALj4/kCRhzuisWKc/s1600/mandelbrot.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://3.bp.blogspot.com/-uIbiamXPF4s/T4sz0o6u3hI/AAAAAAAALj4/kCRhzuisWKc/s1600/mandelbrot.jpg" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Benoît Mandelbrot</td></tr></tbody></table><br /><br /><br /><div style="background-repeat: no-repeat no-repeat; border-collapse: collapse; margin-bottom: 13px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="color: #0b5394;"><b>Computers and IBM</b>.</span></span><span class="Apple-style-span" style="font-family: inherit;">His interest in computers was immediate, and his use of the new resource grew rapidly. He returned to France, married Aliette Kagan and became a professor at the University of Lille and then at the Centre National de la Recherche Scientifique in Paris. His academic future looked assured. But he felt uncomfortable in that environment, and in 1958 he spent the summer at IBM as a faculty visitor.</span></div><div style="background-repeat: no-repeat no-repeat; border-collapse: collapse; margin-bottom: 13px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span class="Apple-style-span" style="font-family: inherit;">The company asked him to work on eliminating the apparently random noise in signal transmissions between computer terminals. The errors were not in fact completely random – they tended to come in bunches. Mandelbrot observed that the degree of bunching remained constant whether he plotted them by the month, the week or by the day. This was another step towards his fractal revelation.</span></div><div style="background-repeat: no-repeat no-repeat; border-collapse: collapse; margin-bottom: 13px; margin-left: 0px; margin-right: 0px; margin-top: 0px; padding-bottom: 0px; padding-left: 0px; padding-right: 0px; padding-top: 0px;"><span class="Apple-style-span" style="font-family: inherit;">During the 1960s Mandelbrot's quest led him to study galaxy clusters, applying his ideas on scaling to the structure of the universe itself. He scoured through forgotten and obscure journals. He found the clue he was looking for in the work of the mathematician and meteorologist Lewis Fry Richardson: he took a photocopy, and when he returned to consult the volume further, found it had gone to be pulped. Nonetheless, he knew he had struck a rich seam.<span style="font-family: inherit;"><span style="font-family: inherit;">Mandelbrot</span> also tested the Fractal Theory in financial markets and certain d<span style="font-family: inherit;">istribution<span style="font-family: inherit;">s and methods for High Impact and Rare events(Black Swans) using computers and Monte Carlo Methods.</span></span></span></span></div>Before all this Dr Mandelbrot worked in the obscurity that modern mathematicians have resigned themselves to. He had followed, albeit belatedly, a path familiar to Jewish intellectuals driven from eastern Europe by the rise of the Nazis. His family fled Poland for France before the second world war and, though they stayed there for the duration, the young Benoît afterwards oscillated between France and the United States before settling for America in 1958. Once there, he worked for IBM. Among other things, he modelled electrical noise. Which, it turns out, is fractal. That it was the transmogrification of his formula by computers which brought him fame is thus appropriate.<br /><span style="color: blue;"><br /></span><span style="color: blue; font-family: Georgia, 'Times New Roman', serif;"><a href="http://www.blogger.com/blogger.g?blogID=3055784350013401414" name="game,_set_and_match"></a> <span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="color: #0b5394;"><b>Game, set and match?</b></span></span></span>For a time, fractals seemed the answer to everything: the shape of clouds, the growth of organisms, even why the night sky is dark. Then the world lost interest. <br />Perhaps it should not have. For among Dr Mandelbrot’s beliefs was a conviction that financial-market movements, too, have fractal forms, rather than the familiar bell shapes of “normal” distribution that Gauss also described. If Dr Mandelbrot’s belief was correct, trading models based on Gauss’s distribution are wrong. <br />That markets are not Gaussian has now been accepted. Dr Mandelbrot’s interpretation, however, has not. Even if it had been, the bankers might not have noticed. They preferred algorithms to geometry.<br /><br />Artists have been fascinated by geometry for as long as mathematicians have. The studies of Euclid are reflected in the regularities of classical and Renaissance architecture, from the Pantheon in Rome to the Duomo in Florence. But artists and architects were also thinking centuries ago about non-regular, curving geometries. You could argue that fractals give us the mathematics of the Baroque – they were anticipated by Borromini and Bach. I have a facsimile, given away by an Italian newspaper, of part of Leonardo da Vinci's Atlantic Codex, which contains page after page of his attempts to analyse the geometry of twisted, curving shapes.<br /><br /><span style="color: #cc0000;"><b>Mandelbrot was a modern Leonardo, a man who showed the beauty in nature and a man who worked in many fields of mathematics. He was a prophet of the curving universe and gave us, in the endlessly playful geometry of fractals, a visual lexicon for our complex world.</b></span><br /><div class="separator" style="clear: both; text-align: center;"></div><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://4.bp.blogspot.com/-a_Y0_Yz-ugs/T3xCgBFfWfI/AAAAAAAAK6o/KFM643hscGA/s1600/800px-Mandel_zoom_08_satellite_antenna.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="240" src="http://4.bp.blogspot.com/-a_Y0_Yz-ugs/T3xCgBFfWfI/AAAAAAAAK6o/KFM643hscGA/s320/800px-Mandel_zoom_08_satellite_antenna.jpg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"> Fractal Geometry</td></tr></tbody></table><br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/-b0Ys3Jiwg4s/T3xCkj9sF3I/AAAAAAAAK64/frPGr6gW31w/s1600/Julia_and_Mandelbrot.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="238" src="http://2.bp.blogspot.com/-b0Ys3Jiwg4s/T3xCkj9sF3I/AAAAAAAAK64/frPGr6gW31w/s320/Julia_and_Mandelbrot.jpg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"> Fractal Geometry</td></tr></tbody></table><div class="separator" style="clear: both; text-align: center;"></div><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://4.bp.blogspot.com/-WJm30-3qCDI/T3xCrKQkgEI/AAAAAAAAK7I/PPxzgQGZoIE/s1600/fractal3.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="240" src="http://4.bp.blogspot.com/-WJm30-3qCDI/T3xCrKQkgEI/AAAAAAAAK7I/PPxzgQGZoIE/s320/fractal3.jpg" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"> Fractal Geometry</td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-EjaBVfGp7hY/T3xCu6dfBjI/AAAAAAAAK7Q/CJrqFbbqWdQ/s1600/image1%5B1%5D.png" style="margin-left: auto; margin-right: auto;"><img border="0" height="241" src="http://1.bp.blogspot.com/-EjaBVfGp7hY/T3xCu6dfBjI/AAAAAAAAK7Q/CJrqFbbqWdQ/s320/image1%5B1%5D.png" width="320" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Fractal Geometry</td></tr></tbody></table>Nassif Nabeal (Nassif Medici)http://www.blogger.com/profile/13708191074451322062noreply@blogger.com0tag:blogger.com,1999:blog-5761410253972037435.post-44105552182412894402013-10-21T06:47:00.000-07:002013-10-21T06:47:05.013-07:00Mathematical Research in France<span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"><b><span style="color: #3d85c6;">" More than any other city on the planet, Paris is the world’s center for mathematics...”</span></b></span></span><br /><span style="font-family: Arial,Helvetica,sans-serif;"><span style="font-size: small;"><b><span style="color: #3d85c6;">Marcel Berger’s assertion was confirmed by the granting in 2010<span style="font-size: small;"> </span>of two more Fields medals to Parisian mathematicians".</span></b></span></span><br /><br />This statement by mathematician Marcel Berger, an internationally renowned specialist in differential geometry who has worked in Japan and the United States, as well as in France, is not an idle one − it was confirmed in a study conducted by the American periodical ScienceWatch in 2005. Paris indeed has the world’s highest concentration of mathematicians, many at IHES or Pierre et Marie Curie<br />(Paris 6), Paris Diderot (Paris 7) , and Paris-Sud 11.<span class="st">École Normale Supérieure ,Paris Dauphine, Ecole Polytechnique</span> ,Toulouse, Strasbourg, and Grenoble are pioneers of their fields. In addition, one can cite the 40 joint research units maintained by CNRS and INRIA with universities in Bordeaux, Lyon, Lille, and Rennes…<br /><br /><span style="font-family: Verdana,sans-serif;"><span style="color: #e06666;"><b><span style="font-size: small;">A strong mathematical tradition</span></b></span></span><br /><br />The century of Louis XIV was also that of Descartes, Fermat, and Pascal. At the time of the Revolution, Laplace, Lagrange, Legendre, Condorcet, d’Alembert, and Monge commanded center stage in mathematics. They, in turn, were followed by Fourier, Cauchy, Galois, Poncelet, and Chasles − a line of succession just as impressive, if less often invoked, as that linking France’s writers. We forget that at the outset of the 19th century more foreign scholars arrived in Paris for its scientific culture than for its literary dazzle. By the end of the century and into the early 20th century, the capital played host to the prominent personalities of Jordan, Poincaré, Borel, Lebesgue, and<br />Lévy, among others.<br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://4.bp.blogspot.com/-Wx_BV2u71LQ/UQ4T9RGhGoI/AAAAAAAAVgk/YqhEfvBI320/s1600/Pierre-Simon_Laplace.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="http://4.bp.blogspot.com/-Wx_BV2u71LQ/UQ4T9RGhGoI/AAAAAAAAVgk/YqhEfvBI320/s320/Pierre-Simon_Laplace.jpg" width="240" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;"><span style="color: #3d85c6;">Pierre-Simon Laplace, also known as Newton of France.</span></span></td></tr></tbody></table><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-S7gas7Zj8uM/UJYIxHAXRsI/AAAAAAAAU_w/NHTjJ9JmCM4/s1600/bourbaki+2.png" style="margin-left: auto; margin-right: auto;"><img border="0" height="287" src="http://1.bp.blogspot.com/-S7gas7Zj8uM/UJYIxHAXRsI/AAAAAAAAU_w/NHTjJ9JmCM4/s400/bourbaki+2.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="color: #3d85c6;"><span style="font-family: Times,"Times New Roman",serif;"><span style="font-size: small;">Here is a picture from 1951 of the Bourbaki group meeting at some pleasant retreat. From left to right, we have Jacques Dixmier, Jean Dieudonné taking notes, Pierre Samuel lighting a cigarette in a holder, André Weil wearing shorts, Jean Delsarte leaning back in a chair and Laurent Schwartz under the parasol, also busy scribbling away.</span></span></span></td></tr></tbody></table><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><span style="color: #3d85c6; font-size: small;"><a href="http://2.bp.blogspot.com/-hzXRMRivlXc/UJ3dfAAuVrI/AAAAAAAAVEA/XspxT2V8Qek/s1600/Henri-Poincar--008.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="240" src="http://2.bp.blogspot.com/-hzXRMRivlXc/UJ3dfAAuVrI/AAAAAAAAVEA/XspxT2V8Qek/s400/Henri-Poincar--008.jpg" width="400" /></a></span></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="color: #3d85c6; font-size: small;"><span style="font-family: "Trebuchet MS",sans-serif;"><b>Henri Poincaré</b> (<span class="mw-formatted-date" title="1854-04-29">29 April 1854</span> – <span class="mw-formatted-date" title="1912-07-17">17 July 1912</span>) was France's greatest mathematician<span style="font-size: small;"><span style="font-size: small;">, he published in every branch of mathematics and was also a world<span style="font-size: small;">-renowned </span></span></span></span></span><span style="color: #3d85c6; font-size: small;"><span style="font-family: "Trebuchet MS",sans-serif;"><span style="font-size: small;"><span style="font-size: small;"><span style="font-size: small;"><span style="font-size: small;">t</span>heoretical physic<span style="font-size: small;">ist, engineer and philosopher of science.</span> </span></span></span></span></span></td></tr></tbody></table>The 1930s saw the founding of the Bourbaki group,a group founded by Mathematics professors based at <span class="st">École Normale Supérieure</span> which revolutionized Mathematics, preparing the way for the prodigious expansion of the 1950s and beyond. The reasons for that expansion are many: an increase in the theoretical research that underpins practical applications in every economic sector, in parallel with the explosion of computer science and robotics; the “mathematicization” of economic analysis;foundation of Financial mathematics, the flexibility and diversity of the system of mathematical research, which had been freed from some of the constraints of the university system by the emergence of other sources of financing; the autonomy of mathematical researchers, who are less dependent on large budgets than researchers in some other disciplines; the arrival in France of Russian mathematicians; the prestige in France of pure intellectual research; and the commitment of great mathematicians to the freedom of thought and criticism.Today many students from all over the world come to France to study maths with <span class="st">Université Paris-Sud and </span><span class="st">Université Pierre-et-Marie-Curie as top centers of mathematical research.French mathematicians were pioneers of Financial Math and still are the leading academics in Financial Math, </span><span class="st">Nicole El Karoui, Marc Yor and </span><span class="st"><span class="st">Hélyette Geman<b> </b></span>are world's best financial math academics.</span><br /><br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/-e_vPnvvJJBs/UQ4Rj0KPGiI/AAAAAAAAVfI/bhsDtxn0bVU/s1600/Nicole_El_Karoui_2008.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="266" src="http://2.bp.blogspot.com/-e_vPnvvJJBs/UQ4Rj0KPGiI/AAAAAAAAVfI/bhsDtxn0bVU/s400/Nicole_El_Karoui_2008.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="color: #3d85c6;"><span style="font-size: small;">Professor Nicole El Karoui </span></span><span style="color: #3d85c6;"><span style="font-size: small;">,French mathematician<span style="font-size: small;"> </span>and pioneer in the development of Mathematical Finance."DEA El Karoui",a graduate program<span style="font-size: small;">m</span>e run by El Karoui is one of the most prestigious program in quantitative finance in the world.</span></span></td></tr></tbody></table><span class="st"><br /></span><br /><span style="font-family: Arial,Helvetica,sans-serif;"><span style="color: #e06666;"><b><span style="font-size: small;">Fields Medal: 11 of 52 recipients have come from French<span style="font-size: small;"> </span>institutions.</span></b></span></span><br /><br />Since 1936, Fields medals have been awarded every four years to mathematicians under the age of 40. The first French winner was Laurent Schwartz, a graduate of the École Normale Supérieure (ENS) and professor at the École polytechnique. More recent demonstrations of the tradition of French excellence in mathematics came with award of Fields medals to Laurent Lafforgue (2002), a graduate of École Normale Supérieure and professor at the Institut des Hautes Études Scientifiques (IHES); to Wendelin Werner (2006), professor at Université Paris-Sud 11 and École Normale Supérieure; and, in 2010, to Cédric Villani, director of the Institut Henri Poincaré in Paris (Université Pierre et Marie Curie and Centre National de la Recherche Scientifique) and professor at the École Normale Supérieure de Lyon, and Ngô Bảo Châu, professor at Université Paris-Sud.<br /><br /><span style="font-family: Arial,Helvetica,sans-serif;"><span style="color: #e06666; font-size: small;"><b>The Abel Prize, established in 2003 has already had 3 French<br />recipients:</b></span></span>Jean-Pierre Serre of the Collège de France (2003), Jacques Tits of the Collège de France (2008, jointly with American John Griggs Thompson), and Franco- Russian Mikhaïl Leonidovich Gromov, of IHES (2009).Nassif Nabeal (Nassif Medici)http://www.blogger.com/profile/13708191074451322062noreply@blogger.com0tag:blogger.com,1999:blog-5761410253972037435.post-91890172006207979812013-10-21T06:42:00.001-07:002015-06-09T13:47:27.084-07:00Karl Pearson and the Origins of Modern Statistics<div><b> <span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="color: red; font-size: small;">Introduction</span></span></b><br /><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><b><span style="color: red;"> </span></b></span> <br />A Renaissance scientist in Victorian London, Karl Pearson (1857-1936), was a prodigious and consummate literary polymath whose quest for philosophical, spiritual and numerical truth was his lifelong odyssey. <sup><a href="http://www.rutherfordjournal.org/article010107.html#sdendnote1sym" id="sdendnote1anc" name="sdendnote1anc"></a></sup> As a student of the Cambridge Mathematics Tripos system, Pearson learned to use applied mathematics as a pedagogical tool for determining the truth (i.e., ‘one that provided the standards and the means of producing reliable knowledge’ <sup><a href="http://www.rutherfordjournal.org/article010107.html#sdendnote2sym" id="sdendnote2anc" name="sdendnote2anc"></a></sup>). After graduating from Cambridge, Pearson spent a year studying in Heidelberg and Berlin in pursuit of the truth, which led him to write a passion play, some poetry, a romantic novel and a German book on aspects of the history of art. His passionate Germanic interests, which underscored his desire to find the truth, were pursued whilst he was writing papers and books on elasticity, engineering, mechanics, philosophy and physics, but the truth eluded him in these scientific fields: mathematical statistics was to become another means to that truth.<br /><br />Pearson’s legacy of establishing the foundations to contemporary mathematical statistics helped to create the modern world view, for his statistical methodology not only transformed our vision of nature, but it also gave scientists a set of quantitative tools to conduct research, accompanied with a universal scientific language that standardised scientific writing in the twentieth century. Despite his enduring contribution to statistics, Pearson had no preconceived ideas of becoming a statistician when he graduated from Cambridge in 1879. By the time he began to look for work in the 1880s, there were several options available to him: he could have become a mathematical physicist, an engineer, a lawyer, a specialist of medieval German literature, a philosopher or a mathematics teacher. The idea of creating a new mathematical discipline was not even on the horizon. Whilst I have been challenging existing accounts of Pearson for the last ten years and, in particular, redressing Weldon’s influential role in Pearson’s life, <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote3sym" id="sdendnote3anc" name="sdendnote3anc"></a> this paper will examine those factors that led Pearson to become a statistician and, in the process, to go on to create the foundations of modern mathematical statistics, having started his career initially as an elastician (that is, someone who devised mathematical equations for elastic properties of matter). <br /><br /><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="color: red; font-size: small;"><b>Historiography of Pearsonian Statistics</b></span></span><br /><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="color: red;"><b> </b></span></span> <br />Much of the scholarship on Pearson has been shaped by mono-causal and uni-dimensional accounts; thus, less effort has been made to produce a more nuanced and more balanced account of Pearson’s professional life. Pearson has long been erroneously viewed as a disciple of Francis Galton who, it is believed, followed in his footsteps and merely expanded what Galton started. Consequently, it has been falsely assumed by various scholars that Pearson's motive to create a new statistical methodology arose from problems of eugenics. <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote4sym" id="sdendnote4anc" name="sdendnote4anc"></a>As I have already argued, Pearson not only managed the Drapers’ Biometric and the Galton Eugenics laboratories separately, which occupied separate physical spaces, but he maintained separate financial accounts, established very different journals and created two completely different methodologies. Moreover, he took on his work in the Eugenics Laboratory reluctantly and primarily as a personal favour to Galton. Pearson thus emphasised to Galton that the sort of sociological problems that he was interested in pursuing for his eugenics programme were markedly different from the research that was conducted in the Biometric Laboratory. <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote5sym" id="sdendnote5anc" name="sdendnote5anc"></a>Conversely, many statisticians who have written on Pearson have invariably assumed that the impetus to Pearson’s statistics came from his reading of Galton’s<i>Natural Inheritance.</i> <i><a href="http://www.rutherfordjournal.org/article010107.html#sdendnote6sym" id="sdendnote6anc" name="sdendnote6anc"></a></i> This view, however, fails to take into account that Pearson’s initial reaction to Galton’s book in March 1889 was actually quite cautious. It was not until 1934, almost half a century later, when Pearson was 78 years old, that he reinterpreted the impact Galton’s book had on his statistical work in a more favourable light — long after Pearson had established the foundations to modern statistics. <br />Additionally, nearly all historians of science have failed to take into account that Pearson and Galton’s ideas, methods and outlook on statistics were as different from each other as two people could have possibly been. Whilst Pearson’s main focus was goodness of fit testing, Galton’s emphasis was correlation (though Galton never even used Pearson’s product-moment correlation); Pearson made higher level mathematics (i.e., determinantal algebra) a requisite for doing statistics and his work was thus more mathematically complex than Galton’s; Pearson was interested in very large data sets (more than 1000) whereas Galton was more concerned with smaller data sets of around 100 (owing to the explanatory power of percentages) and Pearson undertook long term projects over several years, whilst Galton wanted faster results. Moreover, Galton thought all data had to conform to the normal distribution, whereas Pearson emphasised that empirical distributions could take on any number of shapes.<br /><br />The idea that all data had to conform to the normal distribution was rooted in the philosophical ideologies of determinism and Aristotelian essentialism. Galton’s belief in essentialism, which was the dominant thinking of the taxonomists, typologists and morphologists until the end of the nineteenth century and gave rise to the morphological concept of species, implied that species regressed back to the mean value. Galton was, therefore, convinced that all biological data could only be normally distributed. He was, in fact, so committed to his idea of a ubiquitous normal curve, that he created a mechanical device, a modified pantograph, to stretch or squeeze any figure in two directions until it was normally distributed. The Belgian statistician, Adolphe Quetelet (1796-1874) attached so much significance to the normal curve because of his belief in determinism, which meant that there was an ideal statistical mean value and that the normal curve was the ideal curve, since it followed the law of errors; hence, all the variation around the mean had to conform to this curve. That Pearson would go on to create a new type of statistics was to a large extent in response to the unshakable conviction, held by so many vital statisticians, mathematicians and philosophers, that the normal distribution was the only feasible distribution for the analysis and interpretation of statistical data. Such was the tyranny of the normal curve, that by the end of the nineteenth century, most statisticians assumed that no other curve could be used to describe data, but this monolithic view was challenged by Pearson in the last decade of the nineteenth century.<br /><br />There has also been a tendency to overemphasise the role that Pearson’s iconoclastic and positivistic book, the <i>Grammar of Science, </i>played in the development of Pearson’s statistical work, whilst neglecting other influential factors in his life. <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote7sym" id="sdendnote7anc" name="sdendnote7anc"></a> This book, which contains Pearson’s first eight Gresham lectures, written when he was 34 years old represents his philosophy of science as a young adult, and does not reveal everything about Pearson’s thinking and ideas, especially those in connection with his development of the modern theory of mathematical statistics. Thus, it is not helpful to see this particular book as an account of what Pearson was to do throughout the remaining 42 years of his working life. This uni-dimensional account of Pearson’s work belies the complexity of Pearson, who was a far more multifaceted person than has been conveyed by a number of historians and philosophers of science.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://1.bp.blogspot.com/-iXvTaKob-_s/UmVX7YQwPpI/AAAAAAAAV3U/7Qa7-L1lmPc/s1600/pearson.jpg" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" height="400" src="http://1.bp.blogspot.com/-iXvTaKob-_s/UmVX7YQwPpI/AAAAAAAAV3U/7Qa7-L1lmPc/s400/pearson.jpg" width="342" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;"><span style="font-size: small;">Karl Pearson, applied mathematician, philosopher of science, biometrician, statistician, eugenist, contributor to “the woman’s question” and founder of the world's first Statistics department at University College London.</span></td></tr></tbody></table><div class="separator" style="clear: both; text-align: center;"></div><span style="color: red;"></span><br /><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: red;"><br /></span></span><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="color: red;"><b>W.F.R. Weldon and Darwinian Variation</b></span></span><br /><span style="color: red;"><b> </b></span> <br />The person who was, undoubtedly, the most influential individual in Pearson’s life was the Cambridge-trained Darwinian zoologist, Walter Frank Raphael Weldon (1860-1906), known as ‘Raphael’ to Pearson, who provided Pearson with the much-needed impetus, data and indefatigable support that enabled Pearson to change careers from being an elastician to becoming a biometrician and, consequently, to construct a new statistical methodology. <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote8sym" id="sdendnote8anc" name="sdendnote8anc"></a>Whilst Galton did, indeed, play a role in the development of some of Pearson’s statistical methods, mainly for correlation and regression, his role was not as significant as that of Weldon. Moreover, it was Weldon who introduced Pearson to Galton in 1894, three years after Pearson and Weldon first met. I will argue that the reasons Pearson was able to help Weldon in 1893, which led to Pearson’s creation of the new discipline of mathematical statistics, were due to the following five factors: (1) Pearson’s training for the Cambridge Mathematics Tripos examination, where he learned to use mathematics as a pedagogical tool for finding the truth, which set him on a life-mission to find this truth; (2) his disillusionment with physics in the late 1880s; (3) his belief that his earliest statistical ideas and methods that he devised in his Gresham Lectures from 1891 to 1894 could be applied to problems of evolution in 1892; (4) the fortuitous timing of Weldon’s statistical questions to Pearson in 1892 – just when Pearson was beginning to devise new statistical methods and (5) Pearson’s quest to achieve immortality by leaving a legacy that would survive him. <br />Weldon occupied a central position in Pearson’s life, which can be easily documented from one of the most extensive sets of letters in Pearson’s archives, which consist of nearly 1,000 letters of Pearson, Weldon and his wife Florence. Pearson and Weldon’s letters tell the story of an intellectual love affair that was in full bloom throughout the 1890s, when Charles Darwin’s ideas had kindled their intellectual partnership, when they began to look for empirical evidence of natural selection and a way to understand the circumstances needed for the formation of new species (or speciation). Once Darwin introduced the idea of continuous variation into biological discourse, his cousin, Francis Galton, began to devise statistical methods to measure this variation. Galton’s work captured the interest of Weldon.<br /><br />Pearson recognized the fundamental statistical concepts in Darwin’s work, as ‘every idea of Darwin, from variation, natural selection, inheritance to reversion, seemed to demand statistical analyses’. <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote9sym" id="sdendnote9anc" name="sdendnote9anc"></a>Darwin had not only shown that variation was measurable and meaningful by emphasising statistical populations rather than focusing on one type or essence, but he also discussed various types of correlation that could be used to explain natural selection. As the evolutionary biologist, Sewall Wright (1899-1988) remarked in 1931, ‘Darwin was the first person to effectively view evolution as primarily a statistical process in which random heredity variation merely furnished the raw material.’ <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote10sym" id="sdendnote10anc" name="sdendnote10anc"></a>Pearsonian mathematical statistics was thus built upon Charles Darwin’s recognition that species comprised different sets of ‘statistical populations’ underpinned by individual <i>variation</i>. Biological Darwinism not only rejected the essentialistic concept of species, but it precipitated a new way of thinking that led Pearson and Weldon to create a new methodology. <br />Weldon first met Pearson in the autumn of 1891 at the Association for Promoting a Professorial University for London. From 1891 until Weldon took up his post at Oxford in 1899, they saw each other almost daily and often several times a day. Pearson not only regarded Weldon as ‘one of the closest friends he ever had’ and valued his opinions more highly than those of anyone else, but by the time Pearson met Weldon, he found someone whose ideas meshed with those he had been developing in his Gresham lectures. Because Weldon’s impact on Pearson led him to change the direction of his career, Pearson acknowledged Weldon as the person ‘who changed the whole drift of my work and left a far deeper impression on my life’ than anyone else. <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote11sym" id="sdendnote11anc" name="sdendnote11anc"></a> Tragically, this intellectual love affair was cut short by Weldon’s death in 1906 and left Pearson inconsolably bereft for the remaining 30 years of his life. To gain an understanding of the multifaceted development of Pearson, his early life and education, including his time in Cambridge and Germany, are addressed in the following sections. <br /><br /><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="color: red;"><b>Pearson’s Early Life and Education</b></span></span><br /><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: red;"><b> </b></span></span> <br />Carl was the younger son and second of three children; born in London, he was of Yorkshire descent, as most of his ancestors came from the North Riding. (The University of Heidelberg changed the spelling of his name in 1879 when they enrolled him as ‘Karl Pearson’; he used both variants of his name interchangeably until 1884 when he finally adopted Karl, eventually becoming known as ‘KP’.) His mother, Fanny Smith, came from a family of master mariners who sailed their own ships from Hull; his father William, who read law at Edinburgh, was a successful barrister and a Queen’s Counsel (QC) at the Inner Temple of the Royal Courts of Justice. They were a family of dissenters and of Quaker stock; Carl’s maternal grandfather was a Unitarian minister. By the time Carl was 22 he had rejected Christianity and adopted ‘Freethought’ as a nonreligious faith that was grounded in science, though he distinguished his views from a ‘Freethinker’ (i.e., a person who forms opinions about religion on the basis of reason without recourse to authority or established beliefs).<br /><br /><a href="http://www.rutherfordjournal.org/article010107.html#sdendnote12sym" id="sdendnote12anc" name="sdendnote12anc"></a> <br />His father was a very hard-working and taciturn man who was never home before seven in the evening. Once he was home he worked till midnight and was up at four in the morning reviewing his briefs. The only time he spent any time with his children was during the holidays. But to be home with their father was ‘simply purgatory’ because he never spoke a word to anyone. As a child, Carl was rather frail, often ill and prone to depression. Both Carl and his elder brother, Arthur found their father’s attitude to be oppressive and they worried continually about their mother’s well being.When Carl was four, he had French lessons at their home on the Camden Road, opposite Holloway Gaol. After the family moved to Northwick, hear Harrow-on-the-Hill in 1863 he and his brother, Arthur, attended a small school with 15 pupils in Harrow, established by a William Penn, who also provided home tuition for Carl in 1866. When the family moved to Mecklenburg Square, Bloomsbury later that year, the boys settled very happily into University College London School, then on Gower Street. Carl stayed for seven years, until he was 16 years old.<br /><br />From the beginning, both parents wanted their sons to attend Cambridge, and at least one of them was expected to read mathematics. Since Arthur had received the Marlowe scholarship to read Classics at Trinity Hall, Cambridge, it was left to Carl to study mathematics. When Carl was 15 years old, his father began to look for a good Cambridge Wrangler to prepare him for the Mathematics Tripos examination. William’s search was motivated because he was expecting far more from Carl than he was from Arthur, for he had thought that Carl was destined for a distinguished university career.<a href="http://www.rutherfordjournal.org/article010107.html#sdendnote13sym" id="sdendnote13anc" name="sdendnote13anc"></a> At 16, Carl went up to Hitchin, 25 miles south-west of Cambridge, where he stayed for five months receiving tuition in mathematics. Very unhappy there, he left in July 1874 to go to Merton Hall, Cambridge to be coached in mathematics under a number of tutors including the great mathematics tutor, John Edward Routh, whom his father recommended to his son, since Routh ‘had coached more Senior Wranglers than any other man’.<a href="http://www.rutherfordjournal.org/article010107.html#sdendnote14sym" id="sdendnote14anc" name="sdendnote14anc"></a>Routh introduced Carl to the mathematical theory of elasticity, and started his hour-long tutorial sessions with him, at seven in the morning. Carl stayed at Merton Hall from mid July 1874 until 15 April 1875 when he took his entrance examination at various Cambridge colleges. His first choice was Trinity College, where he failed the entrance exam choice; his second choice was King’s College from whom he received an Open Fellowship. He began to study at King’s on 9 October.<br /><br />Being away from a depressing family life, Carl found the highly competitive and intellectually demanding requirements of the Mathematics Tripos to be the tonic he needed. He came to life in this environment and his health improved. In addition to studying, students of the Maths Tripos were expected to take regular exercise as a means of preserving a robust constitution and regulating the working day.<a href="http://www.rutherfordjournal.org/article010107.html#sdendnote15sym" id="sdendnote15anc" name="sdendnote15anc"></a> Pearson carefully balanced hard mathematical study against such physical activities as walking, ice-skating, ice hockey and lawn tennis. When the weather was vile, for exercise he worked out indoors using dumbbells and played billiards. Success in competitive sport became a hallmark of the rational body whilst the hard study of mathematics was a manly pursuit.<br /><br />As a diversion from mathematics, in Pearson’s second year, he began to attend lectures on Dante and Spinoza. His mathematics tutor at Kings, Oscar Browning (1837-1923) recommended that Pearson read such Romantic works as Goethe’s <i>Faust </i>and <i>Wilhelm Meister </i>and Percy Shelley’s <i>Peter Bell the Third. </i>Inspired by Goethe’s <i>The Sorrows of Young Werther</i>, perhaps the first European cult novel, Pearson ‘was determined to write a book in the genuine gush style’ and wrote his first book, a romantic novel, <i>The New Werther</i>. <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote16sym" id="sdendnote16anc" name="sdendnote16anc"></a> This was a literary work on idealism and materialism, written in the form of letters to his fiancée, Ethel, from a young man, Arthur, wandering in Germany and who, like Pearson, was searching for a creed of life and turned to philosophy, religion and history in the hopes of finding some underlying principle to life. Pearson graduated with honours in 1879, being the Third Wrangler in the Mathematics Tripos. He subsequently received a Fellowship from King’s, which gave him financial independence for seven years. (He was made an Honorary Fellow of King’s in 1903.) For the Victorians, being placed a high wrangler was a mark of enormous intellectual and social distinction. <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote17sym" id="sdendnote17anc" name="sdendnote17anc"></a> This was, in fact, such a high distinction that the names and the photographs of the top three Wranglers were published in all the national newspapers. <br /><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: red;"><br /></span></span><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="color: red;"><b>Pearson’s German Wanderlust </b></span></span><br /><span style="color: red;"><b> </b></span> <br />Immediately after finishing the Tripos, Pearson began to plan his trip to Germany whilst pursuing other activities that interested him. Having already considered the possibility of becoming an elastician, Pearson began to spend three hours a day in Professor James Stuart's engineering workshop, whilst studying medieval languages, and reading philosophy – with the hopes of becoming a philosopher. He left for the continent in April 1879 to improve his German and to study physics and metaphysics. When Pearson was in Heidelberg, he read the works of Berkeley, Locke, Kant and Spinoza, but found his ‘faith in reason has been shattered by the merely negative results which he found in these great philosophers that he despaired his little reason leading him to anything’. <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote18sym" id="sdendnote18anc" name="sdendnote18anc"></a> He subsequently abandoned philosophy because ‘it made him miserable and would have led him to inevitably short-cut his career’. <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote19sym" id="sdendnote19anc" name="sdendnote19anc"></a> Later that summer Pearson was at such a ‘low ebb of despair’, in his search for the truth, that he was tempted to become a Roman Catholic. Whilst he wanted to know ‘what is the truth?’ he also realised that ‘what is the truth for one, may not be the truth to another’ man. <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote20sym" id="sdendnote20anc" name="sdendnote20anc"></a>Angst-ridden, he wanted someone to tell him what his duty was, since he felt he had none. <br />This quest for the truth, which was of paramount importance to a Cambridge trained mathematician, became a template for the work he pursued and ultimately shaped the direction of his career. His time in Germany became a period of self-discovery: the romanticist and the idealist discovered positivism. Pearson thus adopted and coalesced two different philosophical traditions to fulfil two different needs, for idealism was concerned with nature and personal feelings, whereas positivism dealt with science and professional goals. <br /><br />Having relinquished philosophy, Pearson soon realised he would never be a great mathematician or physicist, like James Clerk Maxwell, William Thomson or Hermann von Helmholtz, because, as he put it, he ‘was not a born genius’. His very good friend from King’s College, Robert Parker, did not, however, accept Pearson’s self-appraisal. Parker did not believe in born geniuses and admonished Pearson because ‘for someone who had studied mathematics and philosophy you would have had as much a chance as most men of turning into a scientific God to be stuck on the mantelpiece of future generations of seedy undergraduates’. <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote21sym" id="sdendnote21anc" name="sdendnote21anc"></a>With some reluctance, Pearson decided to study Roman international law in Berlin in June 1879, even though he was still scribbling problems of mathematical physics. In fact, he was ‘more or less distracted by mathematics during the whole of his time’ in Heidelberg and Berlin. <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote22sym" id="sdendnote22anc" name="sdendnote22anc"></a>Whilst he had initially considered doing the Law Tripos at Cambridge, when he returned to London in June 1880 he decided instead to read law at Lincoln’s Inn and took up rooms with Parker. Though Pearson had made this decision, admittedly under duress from his father, he was ‘quite determined not to even open up a law book and still less to enter the law circuit’ as his father proposed he should do, for he intended to ‘plod on at old German’ instead. <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote23sym" id="sdendnote23anc" name="sdendnote23anc"></a> <br />By October 1881, Pearson’s father was becoming deeply worried about the unsettled life of his son, who occasionally needed money from him. William castigated Carl that it was ‘high time you did something [with your life] as three years have already gone’ and that ’you must take to reading law and only reading it’. <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote24sym" id="sdendnote24anc" name="sdendnote24anc"></a>Pearson, however, wanted <i>another</i> six years before he did anything at the bar. Nevertheless, a month later he was called to the Bar, but after setting up a partnership deed between two turnip-top sellers in Covent Garden Market, which took him three agonising days to complete, he realised he ‘hated the law’.<a href="http://www.rutherfordjournal.org/article010107.html#sdendnote25sym" id="sdendnote25anc" name="sdendnote25anc"></a> His mother was not surprised when Carl gave up the law, as she did not think that it was ‘quite what his taste and inclination would really dictate’.<a href="http://www.rutherfordjournal.org/article010107.html#sdendnote26sym" id="sdendnote26anc" name="sdendnote26anc"></a> Looking for other opportunities, Pearson thought ‘success might be a possible option on the road to science’.<a href="http://www.rutherfordjournal.org/article010107.html#sdendnote27sym" id="sdendnote27anc" name="sdendnote27anc"></a> With such a constant stream of activities and moving from one topic to another, Pearson confessed to his brother, Arthur, that ‘I hardly know what to expect from myself, as I have so many different impulses which lead me in such opposite directions’.<br /><br /><a href="http://www.rutherfordjournal.org/article010107.html#sdendnote28sym" id="sdendnote28anc" name="sdendnote28anc"></a> <br />Pearson’s interests soon turned to German folklore and literature, the history of the Reformation and the German humanists. From 1882 to 1884 he lectured to working men’s clubs around London. He lectured on heat in Barnes, on Martin Luther in Hampstead and on Karl Marx and Ferdinand Lassalle at Revolutionary Clubs around Soho. His lectures were, indeed, very popular and were often sold out Many of his lectures on humanism and intellectual welfare in Germany were about his search for truth amongst religious systems, and he surmised that ‘all systems of religion are of necessity half-truths’.<a href="http://www.rutherfordjournal.org/article010107.html#sdendnote29sym" id="sdendnote29anc" name="sdendnote29anc"></a> As a philological scholar of medieval German folklore, literature and its language, he was short-listed for the newly created readership in German, at Cambridge, in April 1884, except he ‘longed to be working with symbols instead of words’.<a href="http://www.rutherfordjournal.org/article010107.html#sdendnote30sym" id="sdendnote30anc" name="sdendnote30anc"></a>By then he had already established himself as a respectable elastician and he began to write some papers on the theory of elastic solids and fluids, as well as some mathematical physics papers on optics and ether squirts.<br /><a href="http://www.rutherfordjournal.org/article010107.html#sdendnote31sym" id="sdendnote31anc" name="sdendnote31anc"></a> <br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-vz2P_YP4ctI/T5koEH910jI/AAAAAAAAL04/kkSwpkK0JpE/s1600/grammar-science-karl-pearson-hardcover-cover-art.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://2.bp.blogspot.com/-vz2P_YP4ctI/T5koEH910jI/AAAAAAAAL04/kkSwpkK0JpE/s1600/grammar-science-karl-pearson-hardcover-cover-art.jpg" /></a></div><span style="font-family: Georgia,"Times New Roman",serif;"><br /></span><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="color: red;"><b>Mathematical Physics and the Quest for the Truth</b></span></span><br /><span style="color: red;"><b> </b></span> <br />As an elastician, some of Pearson’s early work on elasticity involved, for example, determining the bending moments of a bridge span and calculating stresses on masonry dams. Since elasticity dealt with practical problems and determining the geometry of space by examining surfaces within the body, Pearson’s vision of the truth would have eluded him. Since an elastician cannot really see what is going on, except in a limited manner (e.g., at the point of rupture), any notion of the ‘truth’ would not have been physically obvious.Ether, as Pearson explained, had been conceived by physicists to have been ‘the medium that could fill up the interstices between bodies and between the atoms of bodies’ in outer space.<a href="http://www.rutherfordjournal.org/article010107.html#sdendnote32sym" id="sdendnote32anc" name="sdendnote32anc"></a> Pearson’s theory of ether squirts was the final product from his theory of electromagnetism and atomism that he had been working on in the late 1880s. To Pearson, matter was geometry in motion, for it represented the changing shape of space. Pearson’s ideas on ether squirts were influenced by the work of the philosopher and mathematician, William Kingdon Clifford (1845-1879), who presented the idea that matter and energy are simply different types of curvature of space and whose book, the <i>Common Sense of the Exact Sciences, </i>Pearson was asked to finish after Clifford’s early death. Pearson developed his theory by combining two strands of ideas, one on pulsating spheres of ether and the other on Clifford’s twists (or space curvatures) which Pearson regarded as comparable to magnetic induction. However, by the late 1880s, Pearson had become disillusioned with his idea of ether squirts, due largely to the lack of support he received from British physicists. His paper was eventually published in the<i>American Journal of Mathematics.</i><a href="http://www.rutherfordjournal.org/article010107.html#sdendnote33sym" id="sdendnote33anc" name="sdendnote33anc"></a> <br />Pearson thus concluded that it was not possible for the human mind to have knowledge of ultimate reality. The initial excitement that ether squirts could have provided him with the means to find that Victorian Cambridge idea of the truth, diminished when he realised he could not measure or weigh the ether squirts in outer space. Pearson abandoned this theoretical work and his interests returned to elasticity, if only to finish edit Isaac Todhunter’s <i>History of the Theory of Elasticity.</i><br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-yH1x97H52d0/T6dwW4JrUkI/AAAAAAAAMaM/T22zbArNUJc/s1600/a-history-theory-elasticity-strength-materials-form-galilei-karl-pearson-hardcover-cover-art.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://1.bp.blogspot.com/-yH1x97H52d0/T6dwW4JrUkI/AAAAAAAAMaM/T22zbArNUJc/s1600/a-history-theory-elasticity-strength-materials-form-galilei-karl-pearson-hardcover-cover-art.jpg" /></a></div><i> </i><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"> </span><br /><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><br /></span><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="color: red;"><b>Applied Mathematics and University College London</b></span></span><br /><span style="color: red;"><b> </b></span> <br />Between 1879 and 1884 Pearson applied for more than six mathematical posts in Dundee, Leeds, London, Manchester and Sheffield. Having no luck in finding a job that satisfied him, he thought of taking up a secretaryship in a hospital, becoming a school master, immigrating to North America or even returning to the law. Pearson took on a temporary job teaching mathematics at King’s College, London in 1883. He accepted the Goldsmid Chair of ‘Mechanism and Applied Mathematics’ in October 1884, at University College London succeeding the German mathematician, Olaus Henrici (1840-1918). <b> </b><span style="color: #6fa8dc;">Pearson played a pivotal role in the institutional changes at University College London, as he created the Department of Structural (now Civil) Engineering in 1892, established a Department of Astronomy in 1904 with two observatories (the Transit and the Equatorial Houses) and founded the Biometric School in 1893, which was incorporated into the Drapers’ Biometric Laboratory in 1903 and became the Department of Applied Statistics in 1911 (now the Department of Statistical Sciences,world's first department of Statistics)<b>.</b> </span>He also helped to establish the departments of anthropology and genetics. Over a period of 28 years, he founded and edited seven academic journals of which <i>Biometrika </i>is the best known periodical.Though Pearson had finally landed a permanent job, he was unhappy a month after he started to teach. He lamented to Robert Parker that ‘if I only had a spark of originality or was a genius, I would have <i>never</i> have settled down to the life of a teacher, but instead would have wandered through life in the hope of producing something that might survive me’. <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote34sym" id="sdendnote34anc" name="sdendnote34anc"></a> But Parker could see all that Pearson wanted out of life was to be comfortable, having a little work and plenty of cash to make him financially independent. Nevertheless, Pearson captivated the interest of his many engineering students; he lectured to groups, ranging from 80 to 100 students, for 11 hours a week. He felt there was a sense of power and inspiration in maintaining order among so many high-spirited young men, especially when failure would have meant riots. His colleague, the eminent chemist, Sir William Ramsay (1852-1916), once remarked to Pearson that they ‘were the only men who [could] hold big classes in complete silence in the College’ .<br /><br /><a href="http://www.rutherfordjournal.org/article010107.html#sdendnote35sym" id="sdendnote35anc" name="sdendnote35anc"></a> <br />Pearson continued to inspire his students in the twentieth century after he had established the discipline of mathematical statistics. According to a number of these students, they thought Pearson had the rare gift of complete clarity, coupled with an understanding and appreciation of what the student was going through. Moreover, Pearson showed a willingness to take the time to explain an idea so completely by numerical example that anyone could understand the lessons so long as they were willing to do some hard thinking.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-nXrJAfJNY_k/T5knrUK64bI/AAAAAAAAL0w/OIfPDlLrU_Y/s1600/k7786.gif" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://2.bp.blogspot.com/-nXrJAfJNY_k/T5knrUK64bI/AAAAAAAAL0w/OIfPDlLrU_Y/s320/k7786.gif" width="221" /></a></div><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><br /></span><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="color: red;"><b>The Gresham Lectures of Statistics</b></span></span><br /><span style="font-family: Georgia,"Times New Roman",serif;"><span style="color: red;"><b> </b></span></span> <br />Pearson was, however, an ambitious man: shortly after his marriage in 1890 he took up the Gresham Chair of Geometry at Gresham College in the City of London. When Sir Thomas Gresham (1519-1579) founded his College, he established seven professorships on the lines of an old mediaeval university in which all knowledge fell into one of the seven divisions: divinity, astronomy, geometry, music, law, physic and rhetoric. The early occupants of the Gresham Chairs in Geometry and Astronomy, such as Christopher Wren, Robert Hooke, Robert Boyle and William Petty, were among the most distinguished scientific men of their time. <br />Pearson delivered a total of 38 lectures beginning in the spring of 1891 and ending in the summer of 1894; there were also five guest speakers (including Weldon) who delivered lectures when Pearson was ill. He eventually had to give up this post because his doctor recommended that he cut back on the amount of work he was doing Pearson held this post for three years, concurrently with his post at UCL. These lectures were aimed at a very different audience from his engineering students at UCL. With the increasing development of the specialisation of academic disciplines in Victorian universities, there was a concomitant development of vocational education, which provided practical education and training to the working class through such organisations as the mechanics institutes, mining schools and agricultural programmes. Thus, students who attended Pearson’s Gresham lectures consisted of the industrial class, artisans, clerks and others who worked during the day in the financial district in the City of London. <br /><br />Pearson wanted to introduce these students to a way of thinking that would influence how they made sense of the physical world. Whilst his first eight lectures formed the bases of the <i>Grammar of Science,</i> the remaining 30 lectures dealt with statistics, which represent Pearson’s earliest ideas about mathematical statistics. As he could not lead these students through the ‘mazy paths of mathematical theory’, he had to find a way to present material that would be accessible to this audience. He chose statistics as a topic for these students for he thought they would understand insurance, commerce and trade statistics and could relate to such games of chance involving Monte Carlo Roulette, lotteries, dice and coins. He appealed to them by using graphs, geometric figures and illustrations to teach statistics and deduction by easy arithmetic.Pearson explained to these students, that ‘the geometry of statistics was not just about the graphical representation of data, but it also was a fundamental process of statistical inquiry’. For Pearson, ‘geometry was a mode of ascertaining the numerical truth and a means of statistical research’.<a href="http://www.rutherfordjournal.org/article010107.html#sdendnote36sym" id="sdendnote36anc" name="sdendnote36anc"></a> Furthermore, he proceeded to redefine statistics to these students because statistics did not have to be confined to vital statistics or the measurement of social phenomena, as Pearson argued that statistics was not a branch of sociology, but rather an abstract science in its own right. Thus, it seemed to him that it is clear that a great future awaits our present statistics and we may reasonably anticipate that the combination of statistics and analysis will create a science which will excel every other branch of mathematics, including astronomy, mechanics and physics. <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote37sym" id="sdendnote37anc" name="sdendnote37anc"></a> <br /><br />He recognised that statistics could be useful for problems in biology, to measure variation, Darwinian natural selection and for problems of heredity.Thus, we can learn far more about Pearson by looking at <i>all</i> of his Gresham lectures, rather than simply looking only at the first eight published lectures, which formed the basis of the <i>Grammar of Science.</i> His 30 statistical Gresham lectures not only provided Pearson with the opportunity to create a new statistical methodology, but they signified a turning point in Pearson’s career, for he was able to pursue his goal of finding the numerical truth when he began to teach the geometry of statistics. <br /><br />Though Pearson was to find himself on the threshold of creating a new kind of statistics at the end of 1891, it was not until Weldon asked him for his advice on the data from his Naples crabs in 1892 that Pearson’s early statistical ideas came to fruition. When Weldon was a student at UCL, before going up to Cambridge to read zoology, he acquired a respectable knowledge of mathematics from Pearson’s predecessor, Olaus Henrici, whose emphasis on the use of graphical methods to teach mathematics, may have shaped Weldon’s use of graphical procedures when analysing some of his statistical data in the early 1890s. This also may have helped foster his symbiotic relationship with Pearson who used geometry as a heuristic device to teach statistics. From 1892 to 1897, Weldon and his wife Florence travelled during the summer holidays to Guernsey, Rome, the south of France and the Bahamas to collect marine biological data. Before moving to London from Cambridge, in the autumn of 1889 to take up his new post in the Jodrell Chair of Zoology at UCL, Weldon went to Plymouth that summer to collect data on marine organisms and he began to use Galton’s statistical methods after he read <i>Natural Inheritance</i>. Weldon had first met Galton in Swansea some nine years earlier at the annual meeting of British Association for the Advancement of Science. <br />Pearson had just devised the standard deviation in 1892 when Weldon approached him for assistance because he found that one of his distributions of data was bimodal, while the rest of his data were normally distributed. Weldon’s attemptWeldon’s attempt to break up his double humped curve into two normal components was derived from Galton’s belief that all distributions should be normally distributed. Weldon then concluded that either the crabs from Naples were two distinct races or they were in the process of creating a new species.He also seems to have been exploring Galton’s claim that a new species could be established only by a sport, jump or saltation producing a new type (i.e., instantaneous speciation). Pearson wanted to find another way to interpret the data without trying to normalise it, as Galton and Quetelet had done. Pearson and Weldon thought it was important to make sense of the shape of the curve without distorting its original shape, as it might have revealed something about the creation of new species. Thus, for him the truth could not be found by forcing data to conform to the normal curve.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/-nlkvyAJ1pjY/T4L-zeu81wI/AAAAAAAALRM/mkBWuUC4Eck/s1600/aged-87-with-karl-pearson.gif" style="margin-left: auto; margin-right: auto;"><img border="0" height="320" src="http://2.bp.blogspot.com/-nlkvyAJ1pjY/T4L-zeu81wI/AAAAAAAALRM/mkBWuUC4Eck/s320/aged-87-with-karl-pearson.gif" width="275" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Pearson with Francis Galton</td></tr></tbody></table><br />Weldon wanted to measure the variation in an attempt to gain some understanding of evolution, to determine how a new species emerged and then to detect empirical evidence of natural selection. He needed a statistical system that could measure the variation of his crabs and one that could systematically handle large amounts of data, since he had collected thousands of crab measurements. Pearson realised that a very large sample is essential when trying to show empirical evidence of natural selection. Since Galton’s samples were usually not larger than 100, his statistical methods were not amenable to Weldon’s data. To help Weldon, Pearson had to create a formalised system of frequency distributions that could handle large sample sizes (e.g., more than 1,000 female crabs) and to develop a system that did not rely on the normal distribution. <br />After Pearson helped Weldon, it then became possible to make comparisons or generalisations with other data sets that had been previously impossible to make. Pearson had already introduced the ‘histogram’ on 18 November 1891, a term he coined to designate a ‘time-diagram’ in his lecture on ‘Maps and Chartograms’. He explained that the <i>histogram</i> could be used for <i>historical </i>purposes to create blocks of time of ‘charts about reigns or sovereigns or periods of different prime ministers’. <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote38sym" id="sdendnote38anc" name="sdendnote38anc"></a>The histogram is similar to a bar chart, except the histogram is contiguous; it has no gaps, and is more faithful to exact data, whereas a bar chart has gaps and uses nominal data. By November 1893, Pearson realised that scientists were approaching the question of heredity and evolution from a new standpoint. Thus, he pronounced that ‘for the first time in biology, there is a chance of the science of life being an exact, a mathematical science’ and that it was ‘largely to Weldon that we owe this attempt to give an exact aspect to the problem of evolution: his intensely laborious and careful measurement on the organs of shrimp and crab are the fist step in the right direction’. <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote39sym" id="sdendnote39anc" name="sdendnote39anc"></a> <br /><br /><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="color: red;"><b>Goodness of Fit Testing</b></span></span><br /><span style="font-family: "Helvetica Neue",Arial,Helvetica,sans-serif;"><span style="color: red;"><b> </b></span></span> <br />Once Pearson began to help Weldon with his data, he began to import Cambridge maths into his statistical methods and theory. <b> </b>He adapted the mathematics of mechanics, using the <i>method of moments</i>, to construct a new statistical system to interpret Weldon’s data that produced asymmetrical curves, since no such system existed at the time. The term ‘moment’ originated from mechanics and is a force about a point of rotation, whilst moments in statistics are averages. From the method of moments, Pearson established four parameters for curve fitting to show how the data clustered (the mean), how it spread (the standard deviation), if there were a loss of symmetry (skewness) and if the shape of the distribution were peaked or flat (kurtosis). These four parameters describe the essential characteristics of any distribution: the system is parsimonious and elegant. The method of moments, which enabled Pearson to create the infrastructure of his statistical methodology, are <i>still</i> essential for interpreting any set of statistical data, whatever shape the distribution takes. Hence, this system allowed Pearson to analyse data that resulted in various shaped distributions, and enabled him to move beyond the limitations of the normal curve. <br />After Pearson examined Weldon’s asymmetric curves that had been derived from his crab data in Naples, Pearson decided that an objective method of measuring the goodness of fit was a desideratum for distributions that did not conform to the normal curve. Pearson’s earliest consideration of determining a measure of the goodness of fit test came out of his lecture on 21 November 1893, when he asked his students, ‘Can you always fit a normal curve to a set of data?’ The answer was ‘not always’ since there were many types of data that could produce asymmetric curves and thus would not fit into a normal curve.<a href="http://www.rutherfordjournal.org/article010107.html#sdendnote40sym" id="sdendnote40anc" name="sdendnote40anc"></a> Pearson went on the devise his first goodness of fit test for asymmetrical distributions, using the sixth moment from the method of moments in 1892.<br /><br />Quetelet had, in fact, made one of the earliest attempts to fit a set of observational data to a normal curve in 1840, which Francis Galton began to use in 1863. Wilhelm Lexis devised the <i>Lexican Ratio L</i> as a goodness of fit test to determine if an empirical distribution conformed to the normal distribution, whilst Francis Ysidro Edgeworth provided a goodness of fit test in 1887 that was based on a normal approximation to the binomial distribution.<a href="http://www.rutherfordjournal.org/article010107.html#sdendnote41sym" id="sdendnote41anc" name="sdendnote41anc"></a> Though many other nineteenth century scientists attempted to find a goodness of fit test, such as the American statistician, Erasmus Lyman de Forest, and the Italian mathematician, Luigi Perozzo, they did not give any underlying theoretical basis for their formulas, which Pearson managed to do. <br /><br />By the time Pearson had finished his 30 statistical Gresham lectures in May 1894, he had provided the underpinnings of his statistical methodology and he was in the process of creating a new academic discipline. With such well-attended lectures, his public life was well developed by the time he began to each statistics at UCL. He was thus able to bring to UCL what he gained at Gresham. In October, he began to offer a set of lectures at UCL on the ‘theory and Practice of Statistics’ for one hour a week for those ‘desiring to study Animal Variation, to deal with Errors of Physical Observations or to become Actuaries’. <a href="http://www.rutherfordjournal.org/article010107.html#sdendnote42sym" id="sdendnote42anc" name="sdendnote42anc"></a>These lectures were, for Pearson, ‘not a part of his regular duty, but solely instituted because [he] was interested in developing a modern theory of statistics’.<br /><br /><a href="http://www.rutherfordjournal.org/article010107.html#sdendnote43sym" id="sdendnote43anc" name="sdendnote43anc"></a> <br />In the last set of his Gresham lectures in May 1894, Pearson discussed various procedures for goodness of fit testing for asymmetrical curves. He introduced his second measure of a goodness of fit test at UCL in the spring of 1894, which he thought provided a ‘fairly reasonable measure of a goodness of fit’. He continued to work on improving this method throughout the 1890s until he devised his chi-square goodness of fit test in 1900. However, before he reached this solution in 1900, his work was interrupted by Francis Galton who needed some help with correlation. The interruption proved beneficial, as Pearson was able to expand the corpus of his statistical methods; he went on to devise 22 methods of correlation of which 11 continue to be used today.<br /><br />On Christmas Day in 1896, Pearson wrote to Galton that he wanted to develop a goodness of fit test for asymmetrical distributions for biologists and economists, which meant that Pearson also needed to create a corresponding probability distribution that was asymmetrical in shape. Pearson’s ongoing work on curve fitting signified that he needed a criterion to determine how good the fit was, which led him to devise different goodness of fit tests. This work underpinned the infrastructure to his statistical theory and encompassed his entire working life as a statistician; this began in 1892 when he introduced the sixth moment as a measure of goodness of fit for Weldon’s data, continued throughout the 1890s, culminated in 1900 when he found the exact chi-square distribution from the family of Gamma distributions and devised his <i>chi-square </i>(Χ<sup>2</sup>, P) <i>goodness of fit test </i>and ended with the last paper he wrote when he was 79 years old. Indeed, the chi-square goodness of fit test represented Pearson’s single most important contribution to the modern theory of statistics, for it raised substantially the practice of mathematical statistics.<br /><table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody><tr><td style="text-align: center;"><a href="http://2.bp.blogspot.com/-uhPCEQZEIR8/T6dvZ0_5WgI/AAAAAAAAMaE/Nwu2h1ECMJQ/s1600/chisqreject2.jpg" style="margin-left: auto; margin-right: auto;"><img border="0" height="213" src="http://2.bp.blogspot.com/-uhPCEQZEIR8/T6dvZ0_5WgI/AAAAAAAAMaE/Nwu2h1ECMJQ/s400/chisqreject2.jpg" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">Pearson created Chi-squared distribution and its goodness of fit test.</td></tr></tbody></table><br /><br /><br /> The overriding significance of the chi-square distribution and its goodness of fit test meant that statisticians could use statistical methods that did not depend on the normal distribution to interpret their findings. <br />Pearson also established the professional accoutrements necessary to establish and to institutionalise the new discipline of mathematical statistics: with Weldon he founded <i>Biometrika</i> in November 1900, he established the Drapers’ Biometric Laboratory in 1903, and set up the first-ever degree course in mathematical statistics in 1917. Largely due to Weldon assistance, and later that of his many students, including George Udny Yule and William Sealy Gosset (or Student) as well as Francis Galton, Pearson was able to firmly establish the new discipline of mathematical statistics in the early years of the twentieth century, which provided the foundations for such statisticians as Ronald A. Fisher to make further advancements in the development of the modern theory of mathematic statistics.<br /><br /></div>Nassif Nabeal (Nassif Medici)http://www.blogger.com/profile/13708191074451322062noreply@blogger.com0