"Father of the Hydrogen Bomb" Ulam: My Friend John von Neumann | Commemorating the 120th Anniversary of John von Neumann's Birth (Part 1)

December 28, 2023 is the 120th birthday of John von Neumann, a Hungarian mathematician, physicist, computer scientist, and engineer. He made so many contributions throughout his legendary life that even just listing and briefly explaining his important work is beyond the ability of ordinary professionals. When von Neumann died, the Bulletin of the American Mathematical Society launched a commemorative special issue, in which the famous mathematician and "father of the hydrogen bomb" Ulam wrote a long article to introduce von Neumann's life and work in chronological order. We have translated the full text (in two parts), and perhaps from Ulam's story we can understand why von Neumann made so many contributions. This article is dedicated to this great all-round scholar.

By Stanisław Ulam

Translation | Yuanyuan

On February 8, 1957, John von Neumann died. Mathematics lost one of its most original, insightful, and versatile minds; science lost a universal genius and a unique interpreter of mathematics. He was able to bring the latest (and develop potential) methods to apply to physics, astronomy, biology, and new technologies. Many outstanding people have recounted and praised his contributions. The purpose of this article is to give a brief account of his life and work in the context of our friendship, which lasted for 25 years. (Editor's note: The paper numbers introduced in the article are the numbers of the author's list of appendices, which we list in the form of annotations at the end of the article.)

Brief biography

John von Neumann (nicknamed "Johnny," a household name in the United States) was born on December 28, 1903, in Budapest, Hungary (then part of the Austro-Hungarian Empire), the eldest of three boys. His family was wealthy; his father, Max von Neumann, was a banker. Johnny was privately educated at a young age. In 1914, when World War I broke out, he was ten years old and entered a Lutheran high school.

In the two decades before and after World War I, Budapest proved to be a super-fertile breeding ground for scientific talent. It is left to historians of science to discover and explain why so many outstanding people were born here (their names are everywhere in the annals of mathematics and physics today; editor's note: see "This inconspicuous small country has produced the most outstanding group of people in the history of science"). Johnny was probably the brightest star of this group of scientists. When asked what caused this statistically unlikely phenomenon, he would say that it was a coincidence of cultural factors that he could not explain precisely: external pressures on the entire society of Central Europe, deep insecurity in the subconscious of individuals, and the necessity to produce something extraordinary or face extinction. World War I broke the original economic and social patterns. Budapest, once the second capital of the Austro-Hungarian Empire, was now the main city of a small country. For many scientists, they had to emigrate and make a living in other less restricted and remote places.

According to his classmate Fellner,1 Johnny's unusual abilities attracted the attention of an early teacher, László Rátz, who suggested to Johnny's father that it was pointless to teach Johnny mathematics in the traditional way at school and that he should receive private math tutoring. So, under the guidance of Professor József Kürschak and with Michael Fekete, who was then an assistant professor at the University of Budapest, Johnny learned various mathematical problems.

By the time he passed the "matura" in 1921, Johnny was already a recognized professional mathematician. His first paper was a collaboration with Fekete, completed before he was 18. For the next four years, Johnny was enrolled as a mathematics student at the University of Budapest, but spent most of his time at ETH Zurich in Switzerland, where he earned his undergraduate degree as a "Diplomingenieur in Chemie," and in Berlin.

At the end of each semester, he returned to the University of Budapest to pass his course exams (without attending lectures, which was somewhat irregular). He obtained a doctorate in mathematics in Budapest and a degree in chemistry in Zurich at the same time. While in Zurich, he spent a lot of his spare time on mathematical problems, writing articles and corresponding with mathematicians. At that time, Hermann Weyl and George Pólya were both in Zurich, and Johnny had contact with them. Once, when Weyl was away from Zurich for a short time, Johnny took his place in the class.

It is worth noting that, in general, it is not uncommon for young prodigies to produce original mathematical work in Europe. There seems to be at least a two or three year gap in professional education compared to the United States, which may be due to the more intensive education system (preparatory courses) between high school and college in the United States. However, even among child prodigies, Johnny was exceptional. He began his original work while still a student. In 1927, he became a private lecturer (Privatdozent) at the University of Berlin, and worked in this capacity for nearly 3 years. During that time, he became known to mathematicians all over the world because of his papers on set theory, algebra, and quantum theory. I remember that in 1927, when he came to Lwów (then part of Poland) to attend a mathematician's congress, his work on the foundations of mathematics and set theory was already well known. We, the students, used his results as examples of the work of young geniuses.

In 1929 he came to the University of Hamburg, still as a private lecturer. In 1930 he came to the United States for the first time, as a visiting lecturer at Princeton. I remember Johnny telling me that even though there were only a few existing and future vacancies in German universities, there were still forty or sixty lecturers who were eager to become professors in the near future. Johnny calculated, in his typically rational way, that the number of professorial appointments expected "within three years" was three, while there were forty (candidate) lecturers! He also felt that the upcoming political events would make intellectual work very difficult.

In 1930, he accepted a visiting professorship at Princeton University, lecturing for part of the academic year and returning to Europe in the summer. He became a permanent professor at Princeton in 1931. In 1933, he was invited to join the Institute for Advanced Study (IAS) at Princeton as a professor, becoming the youngest tenured member of the Institute.

Johnny married Marietta Kovesi in 1930. Their daughter Marina was born in Princeton in 1935. In the Institute's early years, visiting scholars from Europe found it to be a very informal place, but with a strong scientific atmosphere. The Institute's professors had their offices in Fine Hall (part of Princeton University), and the Institute and the school's departments were full of celebrities. At any given time, it was probably one of the most talented places in the fields of mathematics and physics.

I first came to the United States in late 1935, at Johnny's invitation. Professor Oswald Veblen and his wife arranged a delightful social life, and I found that von Neumann's house [and that of James Waddell Alexander] became almost the center of the party. It was a time of depression, but the Institute managed to keep a considerable number of local and visiting mathematicians relatively carefree.

Johnny’s first marriage ended in divorce. He remarried during a trip to Budapest in the summer of 1938 and brought his second wife, Klára Dan, back to Princeton. His home remained a gathering place for scientists. His friends remembered his hospitality and the atmosphere of intelligence and wit. Klára later became one of the first programmers to write mathematical problems for electronic computers, and she developed some of the early techniques of this art.

With the start of the war in Europe, Johnny's activities outside the Institute began to multiply. A list of his positions, organizational memberships, etc., is provided at the end of this article, but this list alone gives us an idea of ​​the amount of work Johnny did for various scientific projects both within and outside the government.

In October 1954, he was appointed by the President to the Atomic Energy Commission. He took a leave of absence from Princeton and resigned from all positions except that of Chairman of the ICBM Committee. Admiral Lewis Strauss, the Chairman of the Commission and a long-time friend of Johnny, immediately recommended Johnny for nomination upon learning of the vacancy on the Commission. Of Johnny's brief service on the Commission, he wrote:

"Johnny was extremely useful between the time of his appointment and the late fall of 1955. He had an invaluable ability to break down the most difficult problems into their components so that they became very simple. Everyone was wondering why we couldn't see the answer as clearly as he did. In this way he greatly facilitated the work of the Atomic Energy Commission."

Johnny's health had always been good, but from 1954 he looked very tired. In the summer of 1955, he discovered the first signs of a fatal disease through X-rays. A long and cruel illness gradually put an end to all his activities. Finally he died at Walter Reed Hospital in Washington at the age of 53.

John von Neumann in the Eyes of His Friends

Johnny's friends remember him as standing in front of a blackboard or discussing problems at home with a characteristic posture. Somehow, his gestures, smiles and eye contact always reflected his thoughts or the essence of the problem he was discussing. He was of medium height, quite slim when he was young, but later became fatter; he walked with small steps, never very fast, but with quite random acceleration. Whenever a problem showed the characteristics of a logical or mathematical paradox, a smile would flash across his face. In addition to his preference for abstract wisdom, he also greatly appreciated (even hungry for) more down-to-earth comedy and humor.

His mind seems to have brought together a number of abilities which, if not contradictory, were at least independent - each requiring powers of concentration and memory so great that they are rarely found in the same person. These abilities are: a sense of mathematical thought in a set-theoretic manner, formally based on algebraic forms; a knowledge and understanding of the essential content of classical mathematical analysis and geometry; and a keen sense of the potential application of modern mathematical methods to existing and new problems in theoretical physics. All this is concretely demonstrated by his outstanding original work, which covers a very wide range of contemporary scientific thought.

His conversations with friends on scientific matters could last for hours, and he was never short of topics, even if they were not mathematical.

Johnny had a keen interest in people and loved gossip. One often got the sense that he was collecting from memory various traits of people, as if preparing a statistical study. He was also concerned with the changes that occurred over time. He mentioned to me several times in his youth that he believed that creative mathematical ability declined after about the age of 26, but that a certain more prosaic experience and wit that developed with experience compensated for the gradual loss of ability, at least for a while. Later, he gradually raised this limit.

He occasionally made comments about other scientists in conversation, and in general his views were quite tolerant, but he also often praised and criticized. In fact, he was very cautious in expressing his judgments, and he was reluctant to make any final opinions about others: "Let Rhadamanthys and Minos... judge..." Once he was asked about this, and he said that he considered Erhard Schmidt and Weyl to be mathematicians who had a great influence on him, especially in the technical aspects of his early work.

Johnny was considered by many to be an excellent committee chairman (a peculiarly modern activity), someone who would forcefully assert his technical opinions but easily deferential on personal or organizational matters.

Despite his great abilities and his full awareness of them, he lacked a certain self-confidence. Johnny greatly admired several mathematicians and physicists, regarding them as possessing qualities that he himself could not attain. I think the qualities that made him feel this way were an intuition for new truths, a relatively simple mental ability, or a kind of genius - a seemingly irrational insight into the statement or proof of a new theorem.

He was well aware that the criteria for the value of mathematical work were to some extent purely aesthetic. He once expressed the fear that in our present civilization the value of abstract scientific achievements might diminish, that “human interests might change, that present scientific curiosity might cease, that human thought might be entirely different in the future.” At one point in the conversation, the accelerating pace of technological progress and changes in the way humans live made it seem as if we were approaching some fundamental singularity in human history, beyond which human affairs as we know them could no longer continue.

Johnny's friends loved his wonderful sense of humor. Among scientific colleagues, he could make illuminating (often sarcastic) comments on historical or social phenomena in a mathematician's way, showing the kind of inherent humor that only propositions in the empty set are correct. This was usually appreciated by mathematicians. Of course, he did not regard mathematics as inviolable. I remember a discussion about a physics problem in Los Alamos, in which the mathematical argument used ergodic transformations and the existence of fixed points. He suddenly laughed and said, "Modern mathematics can be applied after all! We don't know it a priori, right, but it may be..."

His main interest outside of science was in the study of history, and his knowledge of ancient history was incredibly detailed. For example, he could remember all the anecdotes from Edward Gibbon's The History of the Decline and Fall of the Roman Empire, and loved to engage in historical discussions after dinner. On a trip south to Duke University to attend a meeting of the American Mathematical Society (AMS), we passed near the battlefields of the Civil War and were struck by his familiarity with the most minute details of the battles. This encyclopedic knowledge shaped his view of the future course of events through a kind of "analytic continuation." I can testify that most of his guesses were surprisingly correct in his predictions of the political events leading up to World War II and the military events during the war. However, after the war, he thought it was highly likely that disaster would occur immediately, and fortunately, his concerns were proven to be wrong. Perhaps he had a tendency to take too purely rational a view of historical events, and this tendency may have been caused by an overly formalized game-theoretic approach.

Among other accomplishments, Johnny was an excellent linguist. He remembered very well the Latin and Greek he had learned in school. In addition to English, he spoke fluent German and French. His speeches in the United States were known for their literary quality (only a few trademark mispronunciations that his friends loved to hear, such as "integhers"; Translator's note: integers should be "integers"). During his frequent trips between Los Alamos and Santa Fe (New Mexico), his knowledge of Spanish was less than perfect, and during his trips to Mexico he tried to express himself by using "New-Castilian", a language of his own creation - English words with the prefix "el" and the appropriate Spanish endings.

Before the war, Johnny would spend his summers in Europe and give lectures (at Cambridge University in 1935, at the Institut Henri Poincare in Paris in 1936). He often mentioned that he found it almost impossible to do scientific work there because of the tense political atmosphere, and after the war he travelled abroad only when he had to.

Since arriving in the United States, he has appreciated the opportunities here and has high hopes for the future of scientific work here.

Von Neumann's Great Achievements

Our chronological review of von Neumann's interests and achievements is largely a review of the development of science as a whole over the past 30 years. In his young work, he focused not only on mathematical logic and axiomatic set theory, but also on the substance of set theory itself, and obtained interesting results in measure theory and the theory of real numbers.

It was during this period that he also began his representative work in quantum theory, namely on measurement in quantum mechanics and the mathematical foundations of the new statistical mechanics. His in-depth study of operators on Hilbert space can also be traced back to this period. His research went far beyond the direct needs of physical theory, for example, he pioneered a detailed study of operator rings (translator's note: von Neumann algebra) with independent mathematical significance; research on continuous geometry also began during this period.

Von Neumann’s awareness of the results obtained by other mathematicians and the potential they implied was astonishing. In his early work, Émile Borel’s paper on the minimax property inspired von Neumann’s paper “The Theory of Social Games” [17], ideas that would later culminate in one of his most original masterpieces, the theory of games. Bernard Koopman’s ideas about the possibility of treating problems in classical mechanics through operators on function spaces inspired him to give the first rigorous proof of the ergodic theorem in mathematics. Alfréd Haar’s construction of measures in groups inspired his ingenious partial solution of Hilbert’s fifth problem, showing the possibility of introducing analytic parameters in compact groups.

In the mid-1930s, Johnny was fascinated by the problem of turbulence in hydrodynamics and he realized the mystery behind nonlinear partial differential equations. Since World War II, his work has involved the study of hydrodynamic equations and shock theory. The phenomena described by these nonlinear equations cannot be solved analytically, and even a qualitative understanding is impossible with current methods. In his opinion, numerical calculations seemed to be the most promising way to understand the behavior of such systems. This prompted him to investigate new possibilities for calculations on "electronic machines" from the very beginning. He began to study the theory of computation and started work on the theory of automata, which is still under development today. It was during these studies that he became interested in the workings of the nervous system and the systematic nature of living organisms, to which he devoted much energy.

This journey through the many fields of mathematical science is not the result of restlessness. It is neither the pursuit of novelty nor the desire to apply a small number of general methods to many different special cases. Unlike theoretical physics, mathematics is not limited to a few core problems. Von Neumann believed that the pursuit of unity, if based on a purely formal basis, is doomed to fail. This broad curiosity is based on some metamathematical motivations and is strongly influenced by the physical real world - those physical phenomena may not be formalized for a long time to come. (Editor's note: See "Yang Zhenning Comments on the Axiomatization of Physics".)

Mathematicians often face two conflicting motivations when they embark on creative work: the first is to add to the existing edifice - people can quickly gain recognition by solving existing problems; the second is the desire to blaze new trails, integrating existing cognition to create new fields. The latter approach is a riskier undertaking, and the final judgment of its value or success will only come in the future. In his early work, Johnny chose the first. In his later years, he felt confident enough in himself that he freely but also painstakingly created a possible new mathematical discipline - the combinatorial theory of automata and organisms. (Editor's note: See "The Legacy of Turing and von Neumann: The Architecture of Living Computers") But illness and early death prevented him from making only a start.

In his constant search for applicability, and his instinct for general mathematics for all the exact sciences, he reminds one of Euler, Poincare, or, more recently, perhaps Hermann Weyl. One should remember that the variety and complexity of contemporary problems greatly exceed those faced by the first two. Johnny regretfully observed in his last article that there is probably no brain now capable of learning more than one-third of the field of pure mathematics.

Early work: set theory and algebra

Von Neumann's first paper, written in collaboration with Fechter, dealt with the zeros of certain minimal polynomials. It was a generalization of Fechter's theorem on the location of the roots of Chebyshev polynomials, and was completed in 1922, when von Neumann was less than 18 years old.

Another teenage work was a paper on uniformly dense sequences (written in Hungarian, with an abstract in German), which proved that reordering a dense sequence yields a uniformly dense sequence. This work did not yet reveal the depth of his mathematical ideas nor the technical difficulties that would come, but the choice of topic and the simplicity of the techniques used in the proof foreshadowed future research that would combine set-theoretic intuition with algebraic techniques.

The era was marked by the attention paid by a large number of young mathematicians to set theory. The great ideas of George Cantor were eventually reflected in the theory of functions of the real variable, in topology, and later in analysis, through the work of the great Frenchmen René-Louis Baire, Borel, Henri Lebesgue, and others. At the turn of the century, these things were not part of the basic intuition of young mathematicians. After the end of World War I, it was noticed that these ideas became instinctive for the new generation of mathematicians.

The paper on transfinite ordinals [2]3 already demonstrated von Neumann's unique approach and style to set theory using algebra. The first sentence of the paper states frankly: "The purpose of this work is to formulate Cantor's concept of ordinals in a concrete and precise manner." As the preface states, Cantor's own somewhat vague formulation is replaced by the definition given in Zermelo's axiom system. In addition, he outlines a rigorous basis for the definition by transfinite induction. The introduction emphasizes the rigorous formalist approach, and von Neumann even points out with some pride that the symbol ... [for et cetera] and similar expressions have never been used before. This treatment of ordinals - later considered by Kazimierz Kuratowski - is the best way to introduce this idea so far, which is very important for "construction" in abstract set theory. According to von Neumann's definition, every ordinal is the set of all smaller ordinals, which makes the theory of ordinals very elegant and avoids the concept of ordertype, which is somewhat vague because the set of all orders in axiomatic set theory that are isomorphic to a given order does not constitute a set (does not exist).

The paper on Priifer's theory of ideal algebraic numbers [5]4 hints at the breadth of his future research interests. This paper deals with set-theoretic problems and the enumeration of branches of relatively prime ideals. Heinz Prüfer introduces ideal numbers as "ideal solutions of infinitely generated congruence relations". The technique used by von Neumann in this paper is similar to the work of Kürschak and Mihály Bauer on Hensel's p-adic numbers. Here he again shows the effectiveness of generalizing constructions on finite algebras to the infinite domain (the case of infinite countables and the continuum) - a method that became very common in mathematical research in the following decades. Another indication of his interest in algebra is a short note on Minkowski's theory of linear functionals [39]5.

Much of von Neumann's early work reflects his axiomatic vision, in the sense that his ideas are more formal and more precise than those originally envisioned by logicians in the early 20th century. Most of von Neumann's papers from 1925 to around 1929 attempt to spread the axiomatic spirit, even for physical theory. He was not satisfied with the existing formulation, even for set theory itself, as he again stated in the first sentence of his paper on the axiomatization of set theory [3]6: "The aim of the present work is to give a logically incontrovertible axiomatization of set theory"; the next sentence reads: "I shall begin by describing some of the difficulties of axiomatization which will make the present paper useful."

The last sentence of this 1925 paper is the most interesting. Von Neumann points out the limitations of any formalization of an axiomatic system. Here is perhaps a vague anticipation of Kurt Gödel's statement about the existence of undecidable propositions in formal systems. The last sentence is: "For the present, we can do no more than to state that there are objections to set theory itself and that there is no way to avoid these difficulties at present." [Perhaps here we are thinking of a similar statement from a completely different field of science: Wolfgang Pauli's assessment of the status of relativistic quantum theory in his article in the Handbuch der Physik; infinities and divergences still play a mysterious role in field theory.]

His second paper on the subject[18]8 was titled “Die Axiomatisierung der Mengenlehre” (1925).

The simplicity of the axiom system is astonishing, with the introduction of first-order and second-order objects corresponding to sets and properties of sets in naive set theory, respectively; these axioms take little more than a page to print, yet are sufficient to establish almost all of naive set theory, and with it all of modern mathematics. To this day, this is one of the best foundations for set-theoretic mathematics. Gödel used a system inspired by this approach in his great work on the independence of the axiom of choice and the continuum hypothesis. Significantly, in von Neumann's first paper on the axiomatization of set theory, he explicitly recognized the two fundamentally different directions that mathematicians had taken in order to avoid Burali-Forti's paradox, Richard's paradox, and Russell's paradox. The group consisting of Bertrand Russell, Julius König, LEJ Brouwer, and Weyl took the more radical view that the entire logical foundation of the exact sciences had to be restricted to prevent paradoxes of the type mentioned above. "The general impression of their work is almost crushing," said von Neumann. He objected to Russell's basing the whole of mathematics on the dubious axiom of reducibility; he also objected to Weyl and Brouwer's rejection of what he considered to be much of the more meaningful aspects of mathematics and set theory.

He was more understanding of the second, less radical group, which included Ernst Zermelo, Abraham Fraenkel, and Arthur Moritz Schoenflies. Von Neumann knew that their work (and his own) was far from complete, and he made it clear that the axioms seemed somewhat arbitrary. He said that their axiomatization could not prove that the formal system could not lead to contradictions, but even if naive set theory could not be taken seriously in this sense, at least most of what it contained could be restated as proofs in the formal system, and these "formal" contents could be clearly defined.

Von Neumann gave the first finite axiomatization of the foundations of set theory, with axioms that had a simple logical structure like those of elementary geometry. The simplicity of the axiom system and the formal features of the reasoning seemed to achieve Hilbert's goal of treating mathematics as a finite game. Here we can see the roots of von Neumann's future interest in computers and the "mechanization" of proofs.

Starting from these axioms, most of the important concepts of set theory are derived with amazing algebraic efficiency; this economy of treatment seems to indicate that essential simplicity is more of a concern than skill for simplicity's sake. This provides the basis for studying the limitations of finite formal systems in terms of the concept of "machine" or "automaton".

What I find odd is that in many mathematical discussions of topics in set theory and related fields, von Neumann also seems to think formally. Most mathematicians seem to have an intuitive framework in mind for discussing problems in these fields—based on geometry or on diagrams, arrows, and the like that represent abstract sets. Von Neumann gives the impression that he thought in a purely formal, sequential manner. I would say that his intuitive basis—like other more “direct” intuitions—can lead to new theorems and proofs, and his type of intuition seems to be quite rare. If, as Poincare said, mathematicians are divided into two categories—visual intuition and auditory intuition, Johnny probably belongs to the latter. And in his case, the “auditory intuition” can be quite abstract. More precisely, it involves the connection and transformation between the game of proofs in formal languages ​​and the mathematical interpretation of those symbols. It’s a bit like the difference between recording chess moves in algebraic notation and picturing the actual chess board in your mind.

In some recent discussions about the state of the foundations of mathematics, von Neumann seems to suggest that, in his view, the story is far from over. Rather than marking the end of the story, Gödel’s discovery calls for a new way to understand the role of formalism in mathematics.

His paper [16]10 gave a rigorous axiomatic treatment of the informal discussion in [2]. The first part of the paper introduces the basic operations in set theory, the foundations of the theory of equivalence, similarity, and well-ordering, and finally, based on the treatment of ordinals, proves the possibility of finite or transfinite inductive definitions. At the end of the introduction to the paper, von Neumann correctly pointed out that transfinite induction had not been rigorously introduced in any previous axiomatic or non-axiomatic system of set theory.

Perhaps the most interesting of von Neumann's papers on set theory axioms is [23]11. This paper discusses the necessary and sufficient condition for all sets satisfying a certain property to form a new set. The condition is that there does not exist an injection from the class of all sets to the class of sets satisfying the property. This existence principle of sets was used by von Neumann as axiom 12, and some of the axioms assumed in other systems, in particular the axiom of choice, follow from it. We have now shown that the reverse is also true, that is, these other axioms also lead to this von Neumann axiom. Therefore, if the usual axioms are consistent, then this axiom is also consistent.

His great paper in the Mathematische Zeitschrift[12]13, “Zur Hilbertschen Beweistheorie” (On Hilbert’s Theory of Proof) was devoted to the problem of avoiding contradictions in mathematics. This classic study sets out the original ideas behind the formalism of mathematics in general. The original paper emphasizes that this complex problem, initiated and developed by Hilbert and worked on by others such as Paul Bernays and Wilhelm Ackermann, has not yet been satisfactorily solved. In particular, von Neumann pointed out that Ackermann’s proof of consistency cannot be applied to classical analysis, and that we can only prove the consistency of a subsystem of it by strictly finite methods. In fact, von Neumann proved (although he did not state this explicitly) that the logical theory of quantifiers and propositional conjunctions about finite (i.e., decidable) relations is consistent. This is not far from the limit of what Hilbert’s original plan, which was to use strictly finite methods, could achieve. But von Neumann speculated at the time that the consistency of all analysis could be proved in the same way. At present, one has the impression that the ideas elaborated by the work of Hilbert and his school, developed in such a precise way and then completely revolutionized by Gödel, have not yet come to an end. Perhaps we are in the midst of another great process: the "naive" treatment of set theory and the attempts to formalize the metamathematics derived from our intuition of infinity are turning to the future "super-set theory". It is not uncommon in the history of mathematics that the intuition of leading mathematicians about existing scientific problems, or more precisely, a shared vague feeling, is then formally incorporated into a "super-system" that involves the essence of the original system.

Von Neumann’s interest in the foundations of mathematics continued until the end of his life, and 25 years after the publication of these papers, one can find traces of that work in the logic of computers he constructed.

While von Neumann was working on the foundations of mathematics, he also made unique advances in set theory itself, and in the theory of real variables and algebras motivated by problems in set theory. For example, von Neumann constructed a subset of the real numbers that is equiminal to the continuum, such that any finite number of elements in it is algebraically independent. This proof did not use the axiom of choice. In a paper published in the same year in the Foundations of Mathematics[14]14, he gave a method for decomposing an interval into countably disjoint and congruent subsets. This method solved a problem of Hugo Steinhaus, who needed a special construction to perform such a decomposition on an interval. The corresponding decomposition of the circle by Felix Hausdorff was much easier. (This is because the circumference is a group manifold.)

In his paper on general measure theory [28]15, von Neumann solved the problem of finite additive measures for subsets of groups. He generalized Hausdorff's paradox of decomposition of the sphere, and the related theory of the wonderful decomposition of the three-dimensional sphere by Stefan Banach and Alfred Tarski, from Euclidean space to general non-Abelian groups (Editor's note: see "Magic of Mathematicians: One Thing Becomes Two"); Banach's positive result on the existence of additive measures for all subsets of the plane was generalized to subsets of general commutative groups. Johnny finally concluded that all solvable groups are "measurable" (i.e., such measures can be introduced into them).

This paper generalized the results of set theory on Euclidean space to more general topological and algebraic structures, one of the first examples of similar problems and methods that have become increasingly prominent. The "congruence" of two sets is more generally understood as equivalence under a given group of transformations; the measure is a generalized additive function. Similarly, this formulation of the problem foreshadows the work of Haar and the study of the decomposition of the Hausdorff-Banach-Tarski paradox.16

In the "miracle year" of 1928, von Neumann also wrote about game theory. This was his first work in what became an important area in the combinatorial field, which is now flourishing and has many applications. It is hard to believe that, starting in 1927, while completing the above work, he was also able to publish a large number of papers on the mathematical foundations of quantum theory, probability problems in quantum statistical theory, and also obtained important results in the representation of continuous groups!

Real variable function theory, measure theory, topology, continuous groups

Professor Paul Halmos's article describes von Neumann's important contributions to measure theory. We briefly introduce his work in this area against the backdrop of his other contributions.

The paper [35]17 solved a problem raised by Haar, regarding the selection of representative elements of a function class, considering the product of powers of a finite system and the linear manifolds defined on it; if two functions on a linear manifold are equal outside a set of zero measure, then they are defined to be equivalent. This problem was generalized to measures other than the Lebesgue measure, and a similar problem was explicitly solved.

The paper [45]18 proved an important result in measure theory: any measure-preserving Boolean mapping between two measurable sets (in two measure spaces) is generated by a measure-preserving point transformation. This result is very important for proving that a more general completely separable measure space is equivalent to a Euclidean space with Lebesgue measure, so that we can simplify the Boolean algebra of all measurable sets to the usual Lebesgue measure for study.

In [51]19, von Neumann proved the uniqueness of the Haar measure constructed by Haar (see Ann. of Math. vol. 34, pp. 147-169), which requires that the (Lebesgue-type) measure be invariant under left- or right-multiplication by the group. The uniqueness of the Haar measure had already been proved for compact groups. Von Neumann introduced a different construction from Haar's in his proof. This paper preceded the construction of a general theory of almost periodic functions on separable topological groups and was compatible with their orthogonal representation theory.

In the paper [54]20, von Neumann extended the concept of completeness, which had previously been defined only for metric spaces, to linear topological spaces, and obtained interesting examples of spaces that are not metric spaces but are complete. Of course, this case involves inseparable spaces. The paper also contains novel constructions of pseudo-metric and convex spaces.

In a paper co-authored with Pascual Jordan [59]21, they solved the problem of characterizing generalized Hilbert spaces in linear metric spaces proposed by René Maurice Fréchet. This paper strengthened Fréchet's result and obtained a necessary and sufficient condition: a linear metric space L is isomorphic to Hilbert space if and only if every 2-dimensional linear subspace is isomorphic to Euclidean space.

The results in [35] were generalized in a paper with Marshall Harvey Stone [60]22. The paper [35] dealt with the problem of selecting representatives from the remainder class after subtracting a given bilateral ideal from an abstract ring. The paper contains a number of results on the representation theory of Boolean rings subtracted from an ideal.

In a paper in the Russian journal Sbornik [64]23, von Neumann again examined the uniqueness of the Haar measure. Previous proofs of uniqueness had been made by methods other than the Haar construction, which did not contain arbitrary elements and automatically derived the uniqueness of the measure. This paper gives an independent method for the uniqueness of doubly invariant outer measures for locally compactly separable groups. [At the same time, André Wei gave a different proof.]

In a paper co-authored with Kuratowski [69]24, they obtained some precise and powerful results on the projective properties of certain sets of real numbers defined by transfinite induction. The famous Lebesgue set25, previously shown by Kuratowski to belong to projective class 3, was shown to be the difference of two analytic sets and thus to belong to projective class 2. With the help of some more general constructions, they obtained more general theorems on the analytic character of sets (in the Hausdorff sense). This result seems to have important implications for the currently incomplete theory of projective sets.

The review paper On infinite direct products [75]26, published in Compositio Mathematica, covers the algebraic theory of operators and the measure theory of this system, which is very important in modern abstract analysis. Von Neumann summarizes some previous work on the algebra of functional operators and the topology of operator rings, including non-separable hyper-Hilbert spaces. From a methodological point of view and from a practical construction, this paper contains groundbreaking content in the study of algebra at the time and is also an excellent introductory paper. Starting with vector spaces, the paper first treats their products, then the linear operators on these structures, and finally the classes of these operators, and again from the "first level" to examine the algebraic properties of these operators as vector spaces. Von Neumann intended to draw an analogy between this elegant system and hyperquantization in quantum theory, and he saw the paper in particular as a mathematical preparation for non-countable products.

The paper [24]27 studied a series of questions derived from Hilbert's fifth problem: whether it is possible to make group operations analytic by parameter transformation of continuous groups. In my opinion, this paper is the first important result in this field. It deals with subgroups of linear transformation groups of n-dimensional space and obtains a positive result: every such continuous group has a normal subgroup, and there exists an analytic parameter representation locally, and such parameter representation corresponds one-to-one with a finite number of parameters.

These theorems showed for the first time that group properties prevented the common "pathological" possibilities in the theory of functions of a real variable. The paper revealed in detail the structure of these groups by expressing the elements as products of exponential operators, and the results were later generalized by Élie Cartan to subgroups of general Lie groups and simplified. These results show that a linear manifold is an infinitesimal group of the whole group G if it satisfies the property that if it contains the matrices U, V, it also contains the commutator UV-VU. This paper is very important because it was written before Cartan and after Ado. Of course, von Neumann's own paper [48]28 solved Hilbert's fifth problem for compact groups.

This beautiful result is based on Haar's paper (published in the same issue), in which Haar introduced invariant measure functions in continuous groups. Inspired by this, von Neumann adopted a theorem similar to Peter-Weyl integration on groups, and used linear combinations of finite characteristic functions of integral operators to approximate functions (which was proposed in Schmidt's doctoral thesis), as well as cleverly using Brouwer's theorem of regional invariance in n-dimensional Euclidean space, and finally proved that compact n-dimensional topological groups are continuously isomorphic to closed groups composed of unitary matrices in finite-dimensional space.

The method of this paper allows us to represent more general groups (not necessarily n-dimensional) as subgroups of infinite products of such n-dimensional groups. In the second part of the paper an example is given: a finite-dimensional noncompact group consisting of transformations acting on Euclidean space, where no matter how the parameters are changed, these transformations cannot be made analytic. This was almost 20 years before the complete solution of Hilbert's fifth problem, including the work of Deane Montgomery and Andrew Gleason on the "open" (i.e. noncompact) n-dimensional case. Von Neumann's achievement required a combination of a deep knowledge of set theory, real variable techniques, an intuition for Brouwer's topological ideas, and a real understanding of integral equation techniques and matrix calculus.

In a paper with Jordan and Eugene Wigner [50]29, we can find von Neumann combining abstract algebraic ideas with analytical techniques. This paper is about the algebraic generalization of the formalization of quantum mechanics, which is considered to be a possible starting point for future generalizations of the theory of quantum mechanics and deals with commutative but non-associative hypercomplex algebras. The basic result is that all these formally real finite, commutative r-number systems are just matrix algebras, with one exception. However, this exception seems too narrow for the generalizations required by quantum theory.

In an unpublished abstract submitted to the Bulletin of the American Mathematical Society (Appendix 2[14]30), von Neumann proposed a unity theorem for the group of all homeomorphisms on the 3-sphere. The actual theorem is: given any two homeomorphisms A and B (neither of which is the identity mapping), there are a finite number (23 is enough) of conjugates of A such that their product equals B.

Hilbert Space, Operator Theory and Operator Rings

Von Neumann's fundamental and comprehensive research on Hilbert spaces, operator theory, and operator rings can be found in his papers with Professors Francis Joseph Murray and Richard V. Kadison. His initial interest in the subject arose from the rigorous mathematical formulation of quantum theory.

In 1954, von Neumann told a National Academy of Sciences questionnaire that he considered this work one of his three most important mathematical contributions. In terms of length alone, papers on these topics account for about a third of his published works. They include very detailed analyses of the properties of linear operators and algebraic studies of classes of operators (operator rings) in infinite-dimensional spaces. These results achieved the purpose he stated in his book Mathematical Foundations of Quantum Mechanics (Mathematical Foundations of Quantum Mechanics), namely to prove that the mathematical ideas first proposed by Hilbert can provide a sufficient foundation for quantum physics, and that there is no need to introduce a new mathematical system for this physical theory.

Von Neumann's incredibly detailed classification of the linear properties of unitary space solved many problems involving unbounded operators. It gave a complete theory of hypermaximal transformations, bringing Hilbert space almost as completely within the grasp of mathematicians as finite-dimensional Euclidean space.

Throughout his career, von Neumann remained interested in this subject. Even until the end, while pursuing other research, he obtained and published results on the properties of operators and on spectral theory. The paper [106]31 was published in 1950 in celebration of Schmidt's 75th birthday (it was Schmidt who introduced him to the fascination of this subject). No one has done more than von Neumann to explore the mystery of noncompactness, at least in the unitary case and its linear transformations. Work in this direction will build on his results for a long time to come. This work is now being vigorously carried forward by his collaborators and former students (especially Murray) and others, and we can fully expect that they will provide more valuable insights into the properties of linear operators.

Lattice theory and continuous geometry

Garrett Birkhoff's article "Von Neumann and lattice theory" documents Johnny's work on lattice theory and continuous geometry. Von Neumann's interest in these theories was also based on the potential applications of these new combinatorial and algebraic structures to quantum theory.

Around 1935, Birkhoff developed and generalized lattice theory from Richard Dedekind's original formulation. At about the same time, Marshall Stone systematically formulated the properties of Boolean algebra in algebraic theory and set theory. I remember that in the summer of 1935, Birkhoff, Stone, and von Neumann stopped in Warsaw on their way back from a mathematical conference in Moscow and gave a brief lecture at a meeting of the Warsaw Mathematical Society on the new developments in these fields, including the new formulation of the logic of quantum theory. The discussion that followed gave rise to expectations about the potential applications of describing quantum theory in the language of general Boolean algebra and lattice theory. Von Neumann later returned to these attempts several times, but most of his ideas in this direction are recorded only in unpublished notes.32

His work on continuous and pointless geometry was based on the conviction that the original concepts of quantum theory were concerned with such objects; the "universe of discourse" apparently consisted of classes or linear manifolds of equivalent points in Hilbert space. (Dirac made this point explicitly in his book.)

Some of this related work was presented in symposiums and the content is included in the Princeton Institute Lectures; some of it survives in manuscript form. In discussing these issues with von Neumann, my impression was that, starting around 1938, he felt that existing problems in nuclear physics and new discoveries gave rise to completely different types of new problems, and that insisting on a mathematically perfect system to explain quantum theory became less important. After the war, he expressed some views similar to those of Einstein, believing that the richness of nuclear physics and elementary particle physics was bewildering, so any attempt to establish a general quantum field theory was premature, at least at that time. (To be continued)

Notes

1. This information was conveyed by Fellner in a letter recalling Johnny's early studies.

2. [17]Zur Theorie der Gesellschaftsspiele, Math. Ann. vol. 100 (1928) pp. 295-320.

3. [2]Zur Einfiihrung der transfiniten Ordnungszahlen, Acta Univ. Szeged vol.1 (1923) pp. 199-208.

4. [5]Zur Priiferschen Theorie der idealen Zahlen, Acta Univ. Szeged vol. 2 (1926) pp. 193-227.

5. [39]Zum Beweise des Minkowskischen Satzes über Linearformen, Math. Zeit. vol. 30 (1932) pp. 1-2.

6. [3]Eine Axiomatisierung der Mengenlehre, J. Reine Angew. Math. vol. 154 (1925) pp. 219-240.

7. In connection with this article, Professor Fraenkel of the Hebrew University of Jerusalem wrote to me the following: "Around 1922-1923, when I was a professor at the University of Marburg, I received from Professor Erhard Schmidt in Berlin (on behalf of the editorial office of the Mathematicische Zeitschrift) a very long manuscript by an unknown author, signed by Johann von Neumann, entitled Die Axiomatisierung der Mengenlehre, which was his final doctoral thesis but was not published until 1928 in the Mathematische Zeitschrift (Vol. 27). I was asked for my opinion because the article seemed difficult to understand. I do not claim to understand everything, but I saw enough to see that it was a masterpiece, and I recognized the "lion's paw" (ex ungue To answer these questions, I invited the young scholar to visit Marburg, discussed with him, and strongly advised him to prepare an informal paper explaining this highly technical article, emphasizing the new approach to the problem and its fundamental results. He wrote an article entitled "Mengele's Axioms" (Eine Axiomatisierung der Mengenlehre), which I later published in 1925 in the Journal für die reine und angewandte Mathematik (Volume 154), of which I was then associate editor.

8. [18]Die Axiomatisierung der Mengenlehre, Math. Zeit. vol. 27 (1928) pp.669-752.

9. This, of course, is exactly what Leibniz had in mind.

10. [16] Über die Definition durch trans finite Induktion, und verwandte Fragen der allgemeinen Mengenlehre, Math. Ann. vol. 99 (1928) pp. 373-391.

11. [23] Über eine Widerspruchfreiheitsfrage der axiomatischen Mengenlehre, J. Reine Angew. Math. vol. 160 (1929) pp. 227-241.

12. Gödel said: "The interesting thing about this axiom is that it is a maximization principle, somewhat like Hilbert's completeness axiom in geometry. Roughly speaking, it says that any set that does not lead to contradictions in a well-defined way exists. As a maximization principle, it also explains the fact that this axiom implies the axiom of choice. I think that fundamental problems in abstract set theory, such as Cantor's continuum problem, can only be solved satisfactorily with the help of stronger axioms of this kind. Such axioms are in some sense the opposite of or complementary to the constructivist interpretation of mathematics."

13. [12]Zur Hilbertschen Beweistheorie, Math. Zeit. vol. 26 (1927) pp. 1-46.

14. [14]Zerlegung des Intervalles in abzâhlbar viele kongruente Teilmengen, Fund. Math. vol. 11 (1928) pp. 230-238.

15. [28]Zur allgemeinen Theorie des Masses, Fund. Math. vol. 13 (1929) pp. 73-116.

16. Recently pushed to its most extreme minimal form by RM Robinson.

17. [35]Algebraische Reprasentanten der Funktionen “bis auf eine Menge vom Uaasse Null,” J. Reine Angew. Math. vol. 161 (1931) pp. 109-115.

18. [45] Einige Sâtze uber messbare Abbildungen, Ann. of Math. vol. 33 (1932) pp. 574-586.

19. [51]Zum Haarschen Maass in topologischen Gruppen, Compositio Math. vol. 1 (1934) pp. 106-114.

20. [54]On complete topological spaces, Trans. Amer. Math. Soc. vol. 37 (1935) pp. 1-20.

21. [59]On inner products in linear, metric spaces. With P. Jordan. Ann. of Math, vol. 36 (1935) pp. 719-723.

22. [60]The determination of representative elements in the residual classes of a Boolean algebra. With MH Stone. Fund. Math. vol. 25 (1935) pp.353-378.

23. [64]The uniqueness of Haar's measure, Rec. Math (Mat. Sbornik) NS vol.1 (1936) pp. 721-734.

24. [69]On some analytic sets defined by transfinite induction. With C. Kuratowski. Ann. of Math. vol. 38 (1937) pp. 521-525.

25. Journal de Mathématiques, 1905, Chapter VIII.

26. [75]On infinite direct products, Compositio Math. vol. 6 (1938) pp. 1-77.

27. [24] Über die analytischen Eigenschaften von Gruppen linearer Transformationen und ihrer Darstellungen, Math. Zeit. vol. 30 (1929) pp. 3-42.

28. [48]Die Einfiihrung analytischer Parameter in topologischen Gruppen, Ann. of Math. vol. 34 (1933) pp. 170-190.

29. [50]On an algebraic generalization of the quantum mechanical formalism. With P. Jordan and E. Wigner. Ann. of Math. vol. 35 (1934) pp. 29-64.

30. [14]Zerlegung des Intervalles in abzâhlbar viele kongruente Teilmengen, Fund. Math. vol. 11 (1928) pp. 230-238.

31. [106] Zur Algebra der Funktional operatoren und Theorie der normalen Operatoren, Math. Ann. vol. 102 (1929) pp. 370-427.

32. Professor Wallace Givens is preparing a lecture that will be published shortly by Princeton Press. Another paper on continuous geometry, written in 1935, was published in the Annals of Mathematics.

This article is based on the Creative Commons License (CC BY-NC 4.0), translated from S. Ulam, John von Neumann 1903-1957, Bull. Amer. Math. Soc. 64 (1958), 1-49, original link:

https://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10189-5/S0002-9904-1958-10189-5.pdf

Produced by: Science Popularization China

Special Tips

1. Go to the "Featured Column" at the bottom of the menu of the "Fanpu" WeChat public account to read a series of popular science articles on different topics.

2. Fanpu provides a function to search articles by month. Follow the official account and reply with the four-digit year + month, such as "1903", to get the article index for March 2019, and so on.

Copyright statement: Personal forwarding is welcome. Any form of media or organization is not allowed to reprint or excerpt without authorization. For reprint authorization, please contact the backstage of the "Fanpu" WeChat public account.