See main: Quantum thermodynamicsIn 1927, Neumann published "Thermodynamics of Quantum Mechanical Assemblies", in which he extended Austrian physicist Ludwig Boltzmann’s notion of entropy to quantum systems.  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. 
“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?”
“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).”
“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 = ∑in∑jmPiAijXj/∑in∑jmPiBijXj = ϕ(P,X)
whose “role seems to be similar to that of thermodynamic potentials in phenomenological thermodynamics.” (1945, p.1)
“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'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. |
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.”
“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 drawing analogies between economics and physics, von Neumann and Morgenstern talked a lot about the theory of heat.”
“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.”
|Left: Neumann (age 25) in 1928. . Center: Neumann in 1948. Right: Neumann at Princeton. |
See main: Neumann-Shannon anecdote; Information theoryIn 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. 
See main: Neumann automaton theoryIn 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).  The important point about Neumann’s automaton, however, is that it would require a source of free energy in order to function.  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). 
|Left: Neumann age 7.  Right: Neumann with one of his brothers and cousin.  See also: John von Neumann as Seen by his Brother by Nicholas Von Neumann. |
“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.”
“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.”
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.”
See main: Neumann on godIn 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.” 
“When the judge his seat hath taken .. what shall wretched I then plead? Who for me shall intercede when the righteous scarce is freed?
“So long as there is a the possibility of eternal damnation for nonbelievers it is more logical to be a believer at the end.”
“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) 
“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.”
“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 
“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 
“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) 
|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. |