John Neumann

John Neumann nsIn existographies, John von Neumann (1903-1957) (IQ:190|#40) [RGM:234|1,500+] (Gottlieb 1000:959) (Becker 160:142) (Odueny 100:54) (RE:53) [CR:219], born “Janos” (anglicized to John), called the diminutive "Jancsi" in Hungary, which became “Johnny” in America, the "von" title purchased by his Father in 1913, was a Hungarian-born American mathematician, chemical engineer, economist, physicist, and computer scientist, a distinguished polymath, magnitude genius (see: genius), and "last" known of the supposed dying bread of "last universal geniuses" (see: hydraism), noted for his work in quantum thermodynamics (1927, 1932), human free energy theory of economic thermodynamics (1934-1944), for his ill-fated 1940 suggestion to American electrical engineer Claude Shannon to call information by the name “entropy” as in "information entropy" (see: Neumann-Shannon anecdote), for his computer system architecture designs (von Neumann architecture, 1945), and for his ideas on automaton self-assembly (see: Neumann automaton) (1948,1951), and for his work in computer science thermodynamics (1949). [1]

Quantum thermodynamics
See main: Quantum thermodynamics
In 1927, Neumann published "Thermodynamics of Quantum Mechanical Assemblies", in which he extended Austrian physicist Ludwig Boltzmann’s notion of entropy to quantum systems. [9] This was later expanded into his 1932 book Mathematical Foundations of Quantum Mechanics, wherein he included the earlier 1922 work of his friend Hungarian physicist Leo Szilard on Maxwell's demon and the newly conceived uncertainty principle conceived in 1927 by Werner Heisenberg. [10]

Game theory
In 1928, Neumann published “On the Theory of Games” in which he set out to answer the following question: [24]

“n players, S1, S2, … Sn, are playing a given game of strategy. How must one of the participants, Sm, play in order to achieve a most advantageous result?”

In the decades to follow, Neumann would go onto attempt to incorporate game theory into economics intermixed with thermodynamics.

Economics
In 1932, Neumann read his “A Model of General Economic Equilibrium”, in the winter at the mathematical seminar of Princeton University, in which derives a function φ (X, Y) related to the production of goods, based on the model of thermodynamic potentials, which he abstracts as follows: [6]

“A direct interpretation of the function Φ(X,Y) would be highly desirable. Its role appears to be similar to that of thermodynamic potentials in phenomenological thermodynamics; it can be surmised that the similarity will persist in its full phenomenological generality (independent of our restrictive idealisations).”

Because of this paper, it is often said that Neumann was one of the first to apply the classical thermodynamic approach to economics; though, of course, the subject dates back much earlier. On this publication, American economist Paul Samuelson, in his review, comments: [29]

“Of the attempts to find analogies between thermodynamics and economies there is alas no end. But the mathematical genius John von Neumann has earned the right to command our investigation when he suggests that his growth model (1945; earlier 1932 and 1937) defines a function:

Value of Inputst / Value of Outputst+1 = ∑injmPiAijXj/∑injmPiBijXj = ϕ(P,X)

whose “role seems to be similar to that of thermodynamic potentials in phenomenological thermodynamics.” (1945, p.1)

In 1934, Abraham Flexner, the director of the Institute for Advanced Study at Princeton, sent Georges Guillaume's economic thermodynamics PhD dissertation (turned book) On the Fundamentals of the Economy with Rational Forecasting Techniques—in which, critical of the Lausanne school, he applies the formalisms of thermodynamics to the theory of value—to Neumann to read. After reading the book, a month latter, Neumann commented back: [25]

“I think that the basic intention of the authors, to analyze the economic world, by constructing an analogous fictitious ‘model’, which is sufficiently simplified, so as to allows an absolutely mathematical treatment, is—although not new—sound, and in the spirit of exact sciences. I do not think, however, that the authors have a sufficient amount of mathematical routine and technique, to carry out program out.
Neumann economic thermodynamic variables table
Neumann's 1934 human thermodynamics variables table, based on his review of Georges Guillaume's 1934 On the Fundamentals of the Economy with Rational Forecasting Techniques. [25]

I have the impression that the subject is not yet ripe (I mean that it is not yet fully enough understood, which of its features are the essential ones) to be reduced to a small number of fundamental postulates—like geometry, or mechanics (cf. pgs. 77-78). The analogies with thermodynamics are probably misleading (cf. pgs. 69, 85). The authors think that the ‘amortization’ is analogous to ‘entropy’. It seems to me, that if this analogy can be worked out at all, the analogon of ‘entropy’ must be sought in the direction of ‘liquidity’. To be more specific: if the analogon of ‘energy’ is ‘value’ of the estate of an economical subject, then analogon of its thermodynamic ‘free energy’ should be its ‘cash value’.

The technique of the authors to set up and deal with equations is rather primitive, the way, for instance, in which they discuss the fundamental equations (1) and (2) on page 81-85 is incomplete, as they omit to prove that 1: the resulting prices are all positive (or zero), 2: that there is only one such solution. A correct treatment of this particular question, however, exists in the literature. Various other technical details in the setting up of their equations and in their interpretations could be criticized, too. I do not think that their discussion of the ‘stability of solutions’, which is the only satisfactory way to build up a mathematical theory of economic cycles and of crises, is mathematically satisfactory.

The emphasis the authors put on the possibility of states of equilibrium in economics (cf. pgs. 68-69) seems to me to entirely justified. I think that the importance of this point has not always been duly acknowledged. I cannot judge the value of their statistical methods, as they are given in the last part of the book for practical purposes. Their aim is to diagnose the present status of economics, and to lead to forecasts. But I think that the theoretical deduction, which lead to them is weak and incomplete.”

This incident is recounted in greater detail by Canadian economics historian Robert Leonard’s 1995 article “From Parlor Games to Social Science”. In this article, Leonard summarizes that Guillaume's book is an effort to construct a mathematical economics in the manner of Leon Walrus and Vilfredo Pareto, treating not just exchange but also global production, along the way drawing parallels between economics and physics. [4]

Game theory biographers Giorgio Israel and Ana Gasca, to note, give their opinion on the above passage that “here there is no outright rejection of the attempt to construct concepts similar to physics”, and go on to give the rather dubious statement that “Neumann was later to claim that a worthwhile foundation of mathematical economics should be based on the rejection of mechanical and physical reductionism and on the creation of a new kind of mathematics better suited to the specific nature of socio-economic phenomena.” [28] The assertion here that Neumann in later years became anti-reductionist in view, contradicts his 1938 basing of economics on thermodynamic potentials, his 1948 floating parts automaton theory, and his 1956 mind as computer treatise.

American physicist and econophysicist Joseph McCauley comments the following on Neumann here: [26]

“Attempts at neo-classical equilibrium economic analogies with thermodynamics go back to Guilluame and Samuelson. Von Neumann apparently believed that thermodynamic formalism could potentially be useful in computer theory, for formulating a description of intelligence, and was interested in the possibility of a thermodynamics of economics. But presented with Guillaume’s work, he criticized it on the basis of the misidentification of a quantity as entropy.”

In 1944, Neumann, together with German-born Austrian economist Oskar Morgenstern, publishes Theory of Games and Economic Behavior, in which they employ physics models and discuss physics (26+ pages), the theory of heat (6+ pages), and thermodynamics on one page. [24] As American science writer Tom Siegfried summarizes: [15]

“In drawing analogies between economics and physics, von Neumann and Morgenstern talked a lot about the theory of heat.”

The one page where the term thermodynamics is an effort to speculate on a type of utility thermometer:

“Given a physical quantity, the system of transformations up to which it is described by numbers may vary in time, i.e. with the stage of development of the subject. Thus temperature was originally a number only up to any monotone transformation. With the development of thermometry particularly of the concordant ideal gas thermometry the transformations were restricted to the linear ones, i.e. only the absolute zero and the absolute unit were missing. Subsequent developments of thermodynamics even fixed the absolute zero so that the transformation system in thermodynamics consists only of the multiplication by constants. Examples could be multiplied but there seems to be no need to go into this subject further. For utility the situation seems to be of a similar nature.”

This quote, comparing temperature measurement with utility, to note, seems to be a rehash of Henri Poincare’s circa 1807 commentary to Leon Walras about the possibility of measuring satisfaction, as found in Walras’ 1909 “Economics and Mechanics” or Francis Edgeworth’s 1915 “Recent Contributions to Mathematical Economics”. [27] As Morgenstern was a Edgeworth aficionado, this add possibly could have come from him.

The majority of mentions of "heat" in the book revolve around using it as a comparative example to help elucidate the objects raised about utility not being a measurable quantity.
Neuman photos (adult)
Left: Neumann (age 25) in 1928. [16]. Center: Neumann in 1948. Right: Neumann at Princeton. [16]

Information theory | Entropy
See main: Neumann-Shannon anecdote; Information theory
In 1940, Neumann suggested the idea to American electrical engineer Claude Shannon that he should call his new equation for the transmission of information by the namesake "entropy" (instead of either uncertainty or information, which Shannon was leaning towards), since, according to Neumann, Shannon's logarithm equation had the same mathematical form or isomorphism form to both his equation for quantum entropy and his associated Hungarian physicist Leo Szilard's equation for Maxwell's demon mental entropy generation, after which in 1945 Shannon began to adopt the view that "information is negative entropy". Likewise, sometime between 1937 and 1947, Neumann also consulted American mathematician Norbert Wiener about his information communication equations, after which in 1945, Wiener began to tout the view that "information is entropy". The result, in the centuries to follow, has been the adoption of the misaligned viewpoint, by many, that information theory is the backbone of thermodynamics, which is not the case. [11]

Automaton free energy theory
See main: Neumann automaton theory
In a 1948 Hixton Symposium, organized by American chemical engineer Linus Pauling, Neumann invented a famous thought experiment which illustrates the role which free energy plays in creating statistically unlikely configurations of matter. Neumann imagined a robot or automaton, made of wires, electrical motors, batteries, etc., constructed in such a way that when floating on a lake stoked with component parts, it will reproduce itself (self-replicate). [2] The important point about Neumann’s automaton, however, is that it would require a source of free energy in order to function. [1] Neumann gave an outline of his theory in 1948, and an expanded chapter version of the same theory in 1951, but did not go on to complete and publish any further work on this topic. Most of what is written on his automaton free energy theory has been published post humorously (Ed. A. W. Burks, 1966). [4]
Neumann (young)
Left: Neumann age 7. [16] Right: Neumann with one of his brothers and cousin. [16] See also: John von Neumann as Seen by his Brother by Nicholas Von Neumann. [17]

Education
Neumann was a child prodigy born with a photographic memory, able to divide eight digit numbers in his head, exchange jokes in classical Greek (with his father), and to memorize the names, numbers, and addresses in phone books (displayed as a game to guests), all by the age of six. [3]

Regarding phone book memorization, a visiting guest would select a page and column of the phone book at random, then let Johnny read the columm over a few times, then hand the page back to the guest, after which he could answer any question, such as who has number such and such? He could recite names, addresses, and numbers in order, and so on. [19]

By the age of 8, he was familiar with differential and integral calculus. [20]

Neumann's early mathematical ability is speculated to have been a product of his intrigue of his grandfather’s ability to rapidly perform complex mathematical calculations. [8]

By age 10, the family library had become one of Johnny’s favorite spots. As American writer William Poundstone (Ѻ) summarizes: [19]

“The Neumann household was a congenial environment for a child prodigy’s intellectual development. Max Neumann bought a library in an estate sale, cleared one room of furniture to house it, and commissioned a cabinetmaker to fit the room with floor-to-ceiling bookcases. Johnny spent many hours reading books from the library. One was the [44-volume] encyclopedic history of the world edited by the once-fashionable German historian Wilhelm Oncken. Von Neumann read it volume by volume. He would balk at getting a haircut unless his mother let him take a volume of Oncken along. By the outbreak of World War I, Johnny had read the entire set and could draw analogies between current events and historical ones, and discuss both in relation to theories of military and political strategy. He was ten years old.”

From age 8 to 18, 1911-1921, Johnny attended the Lutheran Gymnasium, a high school for boys with a strong academic reputation, in Budapest. Neumann’s mathematical prowess was said to have been discovered by Hungarian mathematician Laszlo Racz, who recognized John’s math talent, gave him special tutelage, and made recommendations for a special program to his father Max Neumann. [18]

In 21 Mar 1919, when Johnny was 15, Jewish Hungarian utopian socialist Bela Kun seized power of Hungary, as head of the workers and peasants’ state, and began to implement the prescriptions of Karl Marx and Vladimir Lenin, such as issuing a decree transferring ownership of land, businesses, and means of production to the proletariat. In Aug 1919, Kun’s government collapsed, after which Admiral Miklos Horthy seized power, initiating fascism. Kun’s collapse was blamed on the Jews, after which lynch mobs began to operate, some 5,000 people were killed, and 100,000 people fled Hungary. Neumann, when he turned 18, left first to Germany then to America.

In 1926, at the age of 23, Neumann simultaneously competed a BS degree in chemical engineering, from the Technische Hochschule Zurich, and a PhD in mathematics (with minors in experimental physics and chemistry), with a thesis on set theory, from Pázmány Péter University in Budapest. Biographer William Poundstone elaborates on this as follows: [19]

“Von Neumann’s complex college career spanned three nations. In 1921, he enrolled in the University of Budapest but did not attend classes. He showed up only to ace the exams. Simultaneously, he enrolled at the University of Berlin, where he studied chemistry through 1923. After Berlin, his academic grand tour took him to the Swiss Federal Institute Technology of Zurich. There he studied chemical engineering, earning a degree in 1925. Finally, he received his PhD in mathematics—with minors in physics and chemistry—from the University of Budapest in 1926. He was then named Privat dozent (assistant professor) at the University of Berlin, reportedly the youngest man ever to hold that position.”

Neumann if frequently known for his quick thinking. An oft cited example is how he famously solved the "fly problem" in a matter of seconds at a cocktail party: [12]

Fly puzzle: “Two bicyclists are 20 miles apart and head toward each other at 10 miles per hour each. At the same time a fly traveling at a steady 15 miles per hour starts form the front wheel of the northbound bicycle. It lands on the front wheel of the southbound bicycle, and then instantly turns around and flies back, after landing instantly flies north again. What total distance did the fly cover before it was crushed between the two front wheels?”

When the question was put to Neumann he danced around and answered immediately: “15 miles”. When someone exclaimed: “Oh, you’ve heard that one before?” The puzzled Johnny replied: “I simply summed the infinite series.”

(add)

Religion
Neumann was born into a Jewish family with “ambivalent” religious attitudes; his family was so ecumenical that his family put up a Christmas tree, exchanged gives, and sang Christmas carols each year with their German governess, while also maintaining equal secular observance of the major Jewish holidays. Once Johnny’s brother Michael asked his father Max Neumann why the family considered itself Jewish when it did not observe the religion seriously? Max’s answer was: “tradition”. This religious confusion, according to William Poundstone, “would follow von Neumann throughout his life”, inclusive of a nominal conversion to Catholicism at the time of his first marriage, stitched together with an essentially agnostic belief system stance [19]

Neumann’s friend Edward Teller commented that whenever Neumann was tempted to curse, he would refrain himself and joke, “Now I will have to spend two hundred fewer years in purgatory.” [19]

While Neumann remained overtly secular and religiously-neutral or agnostic throughout his existence, when he went into his 18-month death bed, things changed (see below).

IQ
IQ guesstimate of 200 (Ѻ); IQ ranked (2016) by AI zealot / Christopher Langan fan, at 225 (Ѻ).

Death | Reaction end
See main: Neumann on god
In the spring of 1956, at the hospital, while in his last 18-months dereacting (dying) from cancer, Neumann invited Anselm Strittmatter, a well-educated Roman Catholic priest who could discuss classical Rome and Greece, to visit him for consultation, who thereafter he began to see regularly. During these visits, Neumann expressed great fear of death. To his visitors, he despaired that “he could not visualize a world which did not include himself thinking within it.” [22]

Neumann recited in Old Latin passages about judgment, right and wrong, and freedom:

“When the judge his seat hath taken .. what shall wretched I then plead? Who for me shall intercede when the righteous scarce is freed?

Neumann told the priest that Blaise Pascal, in section 233 of Pensees, had a point, referring to Pascal's wager, commenting something to the effect of: [22]

“So long as there is a the possibility of eternal damnation for nonbelievers it is more logical to be a believer at the end.”

Neumann, in short, he sided with Pascal about betting one's afterlife on belief or nonbelief in the existence of god.

Subsequently, so to not lose in the wager, Father Strittmatter administered the last sacraments to him. [20] To his mother, who was also dying from cancer during this period, he expressed the following similar view:

“There probably has to be a God, because it is more difficult to explain if there is than if there isn't.”
— John Neumann (c.1956), said to his mother late in life (reaction existence) [12]

Some of Von Neumann's friends, having always known him as "completely agnostic", believed that his religious conversion was not genuine since it did not reflect his attitudes and thoughts when he was healthy. As Dutch-born American physicist and science historian Abraham Pais reports: [23]

“He had been completely agnostic for as long as I had known him. As far as I could see this act did not agree with the attitudes and thoughts he had harbored for nearly all his life.”

Even after his conversion, Father Strittmatter recalled that von Neumann did not receive much peace or comfort from it as he still remained terrified of death.

Quotes | On
The following are related quotes on or about Neumann:

“I have had the privilege of hearing Dr. von Neumann speak on various occasions, and I always find myself in the delightful but difficult role of hanging on to the tail of a kite. While I follow him, I can’t do much creative thinking as we go along.”
— DH. Gerard, Neumann “Fourth Lecture” discussions (1948)

“It isn't often that the human race produces a polymath like von Neumann.”
— Howard Rheingold (2000), Tools for Thought [14]

“If any one person in the previous century personified the word polymath, it was von Neumann.”
Tom Siegfried (2006), A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature [15]

Quotes | By
The following are noted and or relevant quotes by Neumann:

“It is just as foolish to complain that people are selfish and treacherous as it is to complain that the magnetic field does not increase unless the electric field has a curl. Both are laws of nature.”
— John Neumann (date) [13]
John von Neumann (1992)
Neumann, a multi-cited universal genius, possibly the last known descendent of "last universal geniuses", hailed for his diverse achievements in a number of deep fields: thermodynamics, quantum mechanics, computer technology, artificial life theory, economics, etc. [12]

References
1. Avery, John. (2003). Information Theory and Evolution (pg. 89, 93). London: World Scientific.
2. (a) Neumann, John von. (1963). "Probabilistic Logic and the Synthesis of Reliable Organisms from Unreliable Components", in Collected Works (A. Taub editor), Vol. 5, pgs. 341-47. MacMillian, New York.
(b) Neumann, John von. (1966). Theory of Self-Replicating Automata, A. Burks, ed. University of Illinois Press.
3. (a) John von Neumann (1903-1957) – AtomicArchive.com.
(b) John Von Neumann (biography) – MacTutor History of Mathematics Archives.
4. Wiener, Norbert and Schade, J.P. (1965). Cybernetics of the Nervous System (pg. 29). Elsevier Pub. Co.
6. (a) Neumann, John. (1932). “A Model of General Economic Equilibrium”, Read at Mathematical Seminar Princeton University, Winter; first published in German under the title Ube rein Okonomisches Gleichungssystem und ein Verallgemeinerung des Brouwerschen Fixpunktsatzes in the volume entitled Ergebuisse eines Mathematischen Seminars (editor: K. Menger). Vienna, 1938; translated into English by Oskar Morgenstein; in: Collected Works (ed. J.H. Taub) (Vol VI) London: Pergamon.
(b) Neumann, John. (1945-46). “A Model of General Economic Equilibrium.” Review of Economic Studies, 13: 1-9.
(c) Burley, Peter and Foster, John. (1994). Economics and Thermodynamics: New Perspectives on Economic Analysis (pgs. 17, etc.). Kluwer Academic Publishers.
7. (a) Leonard, Robert J. (1995). “From Parlor Games to Social Science: von Neumann, Morgenstern, and the Creation of Game Theory, 1928-1944” (abs), Journal of Economic Literature, 33(2): 730-761.
(b) Leonard, Robert J. (2010). Von Neumann, Morgenstern, and the Creation of Game Theory (pg. i). Cambridge University Press.
(c) Neumann, John. (1934). “Letter to Abraham Flexner”, May 25, Faculty Files, Folder 1933-35. VNIAS.
8. Hersh, Reuben and John-Steiner, Vera. (2010). Loving and Hating Mathematics (pg. 54). Princeton University Press.
9. Neumann, John. (1927). “Thermodynamics of Quantum Mechanical Assemblies” (“Thermodynamik quantummechanischer Gesamheiten”), Gott. Nach. 1:273-91.
10. Neumann, John. (1932). Mathematical Foundations of Quantum Mechanics (Mathematische Grundlagen der Quantenmechanik) (translator: R.T. Beyer) (§2: Thermodynamical Considerations, pgs. 359-). Princeton University Press, 1955.
11. Thims, Libb. (2012). “Thermodynamics ≠ Information Theory: Science’s Greatest Sokal Affair” (url), Journal of Human Thermodynamics, 8(1): 1-120, Dec 19.
12. MacRae, Norman. (1992). John Von Neumann: the Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More (fly puzzle, pgs. 10-11; God, pg. 43; deathbed, pgs. 378-79). American Mathematical Society.
13. John von Neumann – Wikiquote.
14. Rheingold, Howard. (2000). Tools for Thought: the History and Future of Mind-Expanding Technology (pg. 66). MIT Press.
15. Siegfried, Tom. (2006). A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature (polymath, pg. 28; analogies, pg. 39). National Academies Press.
16. Anon. (2004). “Student Research: Why Not Bomb Them Today” (GermanEnglish), Nov 18.
17. Vonneuman, Nicholas A. (1987). John von Neumann as Seen by his Brother (Goethe, pg. 53). Publisher.
18. (a) Vonneuman, Nicholas A. (1987). John von Neumann as Seen by his Brother (prodigy, pg. 26). Publisher.
(b) Author. (1983). “Article”, Hungarian Digest (pg. 37). Lapkiado Publishing House.
(c) Poundstone, William. (2011). Prisoner’s Dilemma (§: The Child Prodigy, pgs. 12-). Random House LLC.
19. Poundstone, William. (2011). Prisoner’s Dilemma (§: The Child Prodigy, pgs. 12-). Random House LLC.
20. Halmos, P.R. (1973). “The Legend of von Neumann”, The American Mathematical Monthly, 80(4):382-94.
21. Blair, Clay, Jr. (1957). Passing of a Great Mind (pgs. 89-104), Life Magazine, 25 Feb.
22. MacRae, Norman. (1992). John Von Neumann: the Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More (deathbed, pgs. 378-79). American Mathematical Society.
23. Pais, Abraham. (2006). J. Robert Oppenheimer: A Life (pg. 109). Oxford University Press.
24. Neumann, John and Morgenstern, Oskar. (1944). Theory of Games and Economic Behavior (pdf) (physics, 23+ pgs; heat, 6+ pgs; thermodynamics, pg. 23). Princeton University Press, 2007.
25. (a) Neumann, John. (1934). “Letter to Abraham Flexner”, May 25, Faculty Files, Folder 1933-35. VNIAS.
(b) Leonard, Robert J. (1995). “From Parlor Games to Social Science: von Neumann, Morgenstern, and the Creation of Game Theory, 1928-1944” (abs), Journal of Economic Literature, 33(2): 730-761.
(c) Mirowski, Philip. (2002). Machine Dreams: Economics becomes a Cyborg Science (pg. 104). Cambridge University Press.
26. McCauley, Joseph L. (2003). “Thermodynamic Analogies in Economics and Finance: Instability of Markets.” Physica A, 329 (2003): pp. 199-212.
27. (a) Walras, Leon. (1909). “Economics and Mechanics” ("Economie et Mecanique"), Bussetin de la societe vaudoise des sciences naturelles, 5th series, 45 (166): 313-27.
(b) Edgeworth, Francis. (1915). “Recent Contributions to Mathematical Economics” (abs) (quote, pgs. 57-58), Economic Journal, 25(97):36-63.
28. Israel, Giorgio and Gasca, Ana M. (1999). The World as a Mathematical Game: John von Neumann and Twentieth Century Science (pg. 127). Springer.
29. Samuelson, Paul. (1992). “Economics and Thermodynamics: von Neumann’s Problematic Conjecture”, in: Rational Interaction, Essays in Honor of John C. Harsanyi (editor: Paul Selten) (pgs. 377-). Springer-Verlag, 1992.

Further reading
● Neumann, John. (1956). The Computer and the Brain (preface: Klara Neumann; forward: Paul Churchland and Patricia Churchland). Yale University Press, 2000.

External links
John von Neumann – Wikipedia.

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