Tuesday, 21 July 2015

Benoît Mandelbrot's contribution to Finance

  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 'poet of randomness'. Before Nassim Taleb ,Black Swans were dealt by him in a philosophical and aesthetic way.Mandelbrot was initially a probability guy but later went into other fields of maths and made his name in other fields.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 Louis Bachelier's model.Mandelbrot linked randomness to geometry and made randomness a more natural science.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. 

Benoît Mandelbrot , the late Sterling Professor of Mathematical Sciences at Yale University

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.

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. 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 .

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’.”

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.

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. 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.

Mandelbrot used his fractal theory to explain the presence of extreme events in Wall Street.

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 Mandelbrot also knew the pitfalls in Bachelier's model.For this reason many call him as the "father of Quantitative Finance".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.

Mandelbrot's contribution in finance fall into three main stages:

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.

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.

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.

Saturday, 4 January 2014

John von Neumann

Man and Machine

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. He was a polymath who possessed fearsome technical prowess and is considered "the last of the great mathematicians".A mathematician obsessed with making contribution to every branch of mathematics. 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.

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.

Early Life and Education in Budapest

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.

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.

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  Journal of the German Mathematical Society, dealing with the zeros of certain minimal polynomials.

University — Berlin, Zurich and Budapest

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.

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.

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.
This von Neumann stamp, issued in Hungary in 1992, honors his contributions to mathematics and computing.

 Game Theory

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. Von Neumann loved games and toys, which probably contributed in great part to his work in Game Theory.

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 Prisoner's Dilemma, "an intersection in Princeton was nicknamed "Von Neumann Corner" for all the auto accidents he had there."

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 Prisoner's Dilemma, 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.

For Von Neumann, the inspiration for game theory was poker, a game he played occasionally and not terribly well. 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.

Their book, Theory of Games and Economic Behavior, 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.
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.

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.

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.

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 Prisoner's Dilemma 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.

Quantum Mechanics

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.

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.

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.

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.


Marriages and America

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.

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.
Neumann at Princeton

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.

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.

Von Neumann’s Role in Nuclear Development

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.

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.

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.

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.

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.

Development of Modern Computing
Von Neumann with first super Computer

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.

A von Neumann language 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.

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.

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.

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.

Untimely End

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.

Saturday, 26 October 2013

Logic:Ludwig Wittgenstein & Bertrand Russell

Ludwig Wittgenstein was born in Vienna in 1889 and died in Cambridge in 1951.Wittgenstein found the final solution to philosophy, a man who put end to philosophy by changing it into logic. 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) Principia Mathematica (1910-1913), a monumental treatise which bases mathematics on logic. 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.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. Later he went on to know the meaning of life, which is knowing the world and knowing God. Later he wrote “Tractus Logico Philosophicus” 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.

"He was perhaps the most perfect example I have known of genius as traditionally conceived, passionate, profound, intense, and dominating".
Bertrand Russell on Wittgenstein

Ludwig Wittgenstein, 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 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). 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 work on logic. They wanted him to explain the Tractatus, 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, Philosophical Investigations. Before its publication, direct acquaintance with his new ideas had been confined to those who attended his lectures and seminars in Cambridge.
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.
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.

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.

2. The General Character of Wittgenstein's Philosophy
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.

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.
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.

3. Wittgenstein's Early Philosophy
In spite of the wider appeal of his later work, Wittgenstein was without a doubt a philosopher's philosopher. In the Tractatus he developed a theory of language that was designed to explain something that Russell had left unexplained in Principia Mathematica, 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.

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.
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.
The pictorial character of propositions is a theme with many developments in the Tractatus. 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.

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 Tractatus 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 Tractatus 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.
"The world is the totality of facts,not of things" Tractatus Logico Philosophicus

“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.”
Ludwig Wittgenstein,  Tractatus .
4. Propositions as Pictures
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.
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.
What can be said at all can be said clearly, and what we cannot talk about we must pass over in silence.
—Ludwig Wittgenstein in Tractatus.

5. Transition
The first modification of the system of the Tractatus 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 Tractatus. He therefore dropped the requirement.

There are two things that make this change of mind important. Firstly, though the Tractatus 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. The atomism of the Tractatus 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. 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.

6. His Later Philosophy: The Blue Book
The most accessible exposition of the leading ideas of Wittgenstein's later philosophy is to be found in the Blue Book (1958), 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.
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.
There are, in fact, two discussions in the Blue Book 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.
The discussion of meaning is a development of a point made in the Tractatus: '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.

The discussion of meaning in the Blue Book develops the isolated remark in the Tractatus 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 Philosophical Investigations (1953).
The treatment of the self in the Blue Book is very clear and strongly argued. As in the Notebooks and the Tractatus, 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.

7. Philosophical Investigations: the Private Language Argument
The so-called 'private language argument' of Philosophical Investigations 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 distinguish 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.

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 Philosophical Investigations 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.

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 sense datum theory which he himself adopted in the Tractatus (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 Tractatus' (Malcolm, 1986, ch. 4). But a brief review of the development of his philosophy of mind will show that these suggestions are mistaken.

Anyone who compares what Wittgenstein said about simple objects in the Notebooks and in the Tractatus 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 Tractatus, 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.

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.
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 Tractatus. 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 Tractatus, 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.
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 expression of the sensory type and seldom a description of it. This move was the beginning of a reconstruction of the situation, designed to lead to a better account of sensation-language.

             The destructive move is made most perspicuously in Philosophical Investigations. 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).
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 Logical Structure of the World (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.
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.

8. Philosophical Investigations: Meaning and Rules
Another, similar-sounding, but in fact very different, question is discussed in Philosophical Investigations. 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 Tractatus in order to see why it is important.
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 Tractatus 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.

             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 Philosophical Investigations. 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 Philosophical Investigations. 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.
                                                 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 Philosophical Investigations 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 Philosophical Investigations or in Remarks on the Foundations of Mathematics, he does not re-assert it either. There seems to be some ambivalence.
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.
Ludwig Wittgenstein, Professor of Philosophy, Cambridge

9. The Authority of Rules
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.
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.

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.

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.

10. Wittgenstein's Philosophy of Mathematics
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).
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.

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.

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.

Bertrand Russell 

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. 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.”
"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....
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.

Russell’s early life was marred by tragedy and bereavement. 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.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.

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 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 An Essay on the Foundations of Geometry, a revised version of which was published as his first philosophical book in 1897. Following Kant’s Critique of Pure Reason (1781, 1787), this work presented a sophisticated idealist theory that viewed geometry as a description of the structure of spatial intuition.

In 1896 Russell published his first political work, German Social Democracy. 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.

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.

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 logicism—was stated at length in The Principles of Mathematics(1903). 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 The Principles of Mathematics, Russell discovered that he had been anticipated in his logicist philosophy of mathematics by the German mathematician Gottlob Frege, whose book The Foundations of Arithmetic (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.

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 The Principles of Mathematics as a “retreat from Pythagoras.” The first step in this retreat was his discovery of a contradiction—now known as Russell’s Paradox—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.
The Pioneering book on Logic

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 The Principles of Mathematics. 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 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, Alfred North Whitehead, had finished the three volumes of Principia Mathematica (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.
Russell while a student at Cambridge

Principia Mathematica is a herculean attempt to demonstrate mathematically what The Principles of Mathematics 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 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. Principia Mathematica 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.

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.
In philosophy the greatest impact of Principia Mathematica has been through its so-called theory of descriptions. This method of analysis, first introduced by Russell in his article “On Denoting” (1905), 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.
Bertrand Russell, 3rd Earl Russell

Russell later said that his mind never fully recovered from the strain of writing Principia Mathematica, and he never again worked on logic with quite the same intensity. In 1918 he wrote An Introduction to Mathematical Philosophy, which was intended as a popularization of Principia, but, apart from this, his philosophical work tended to be on epistemology rather than logic. In 1914, in Our Knowledge of the External World, Russell argued that the world is “constructed” out of sense-data, an idea that he refined in The Philosophy of Logical Atomism (1918–19). In The Analysis of Mind (1921) and The Analysis of Matter (1927), he abandoned this notion in favour of what he called 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.

Russell’s intellectual fame was firmly established when he coauthored one of the most influential and widely discussed books on the subject of mathematics, Principia Mathematica, which theorized that mathematics and logic are identical and that mathematical principles can be deduced by a handful of logical ideas. Principia Mathematica 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 The Problems of Philosophy, 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, The History of Western Philosophy (1945).
One of the best book ever written

Russell felt that all organized religions were relics from humanity’s barbaric history. Despite his ultimate rejection of God, however, he admitted in The Elements of Ethics (1910) that obedience to rules like the Ten Commandments “will in almost all cases have better consequences than disobedience.” In his 1927 work, Why I Am Not a Christian, Russell confessed his agnostic and anti-Christian views.
It comes as no surprise, therefore, that in 1929, in Marriage and Morality, 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 Marriage and Morality, 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.

Connected with the change in his intellectual direction after the completion of Principia 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 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 The Problems of Philosophy (1911), 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 Principia Mathematica.
In the same year that he began his affair with Morrell, Russell met 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. However, Wittgenstein’s own work, eventually published in 1921 as Logisch-philosophische Abhandlung ( Tractatus Logico-Philosophicus , 1922), undermined the entire approach to logic that had inspired Russell’s great contributions to the philosophy of mathematics. 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.

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 Power, 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.

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 Power, 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

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 Principles of Social Reconstruction (1916),Roads to Freedom (1918), and The Prospects of Industrial Civilization (1923). 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 The Practice and Theory of Bolshevism (1920).
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 On Education (1926), Marriage and Morals (1929), and The Conquest of Happiness (1930)—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.
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, Albert C. Barnes, and lost his job, Russell was able to turn the lectures he delivered at the foundation into a book, A History of Western Philosophy (1945), which proved to be a best-seller and was for many years his main source of income.

In 1944 Russell returned to Trinity College, where he lectured on the ideas that formed his last major contribution to philosophy, Human Knowledge: Its Scope and Limits (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.

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 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.

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.

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.
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.
Russell in a rally

Russell devoted the final two decades of his life primarily to combating nuclear proliferation. 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 Russell-Einstein Manifesto. 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.

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.

 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.”

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. Russell claimed that Socrates was more worthy of reverence than Jesus Christ, so found his only religious satisfaction in an evolving idealistic philosophy.