In von Neumann: The Unparalleled Genius (Part I), we describe von Neumann's early days: coming to Earth as a child prodigy; and his achievements in set theory, game theory, quantum mechanics, and operator theory after entering university and during the Göttingen period. In the next part, we continue his genius story. Von Neumann is known to have immigrated to the United States because of the Nazi regime' rise to power, and von Neumann was one of them. Here he continued to exert his extraordinary talents, especially in mathematics, and the study of traversal theory alone was "enough to ensure his mathematical immortality"; and he also contributed to the work of the Manhattan Project, computer science, etc. Finally, we can also see a life of von Neumann, the Earthman side of the "Martian".
Written by | Jørgen Veisdal (Assistant Professor, Department of Industrial Economics and Technology Management, Faculty of Economics and Management, Norwegian University of Technology)
Compile the | Nezha
In the United States
In October 1929, Johann von Neumann was invited to give a lecture on quantum theory at Princeton University, his first visit to the United States, when he was a non-faculty lecturer at the University of Hamburg. From 1930 to 1933 he was a visiting professor at Princeton University. In the same year at the end of his term, Hitler came to power in Germany for the first time, leading von Neumann to completely abandon his academic position in Europe and make a statement to the Nazi regime,
"If these boys were to do this for two more years, they would have ruined at least a generation of German science."
Of course, many facts have proved that von Neumann's prediction was correct. The following year, when the Nazi education minister asked, "Now that we have escaped Jewish influence, how has mathematics progressed in Göttingen?" Hilbert is said to have replied:
"There is no math in Göttingen anymore."
At Princeton University (1930-1933)
In the mid-1930s, the situation of von Neumann (and many other leading mathematicians and physicists) in Princeton, New Jersey, is now well known.
According to Wigner (Macrae, 1992), at the recommendation of Princeton University, von Neumann was recruited by Princeton University professor Oswald Veblen along with fellow Lutheran church high school classmate Eugene Wigner:
“...... Don't just invite one person, but at least two, and the two of them already know each other and don't suddenly feel like they're being put on an island with no personal connections. Johnny's name was already world-famous by then, so they decided to invite Johnny von Neumann. They looked at who was writing with von Neumann? They found Mr. Wigner. So I was also sent a telegram. ”
—From Norman Macrae, John von Neumann (1992)
Von Neumann first came to Princeton as a visiting professor in 1930. Regarding his work there, von Neumann himself placed particular emphasis on Ergodic theory later in life.
Traverse theory
Traversal theory is a branch of mathematics that studies the statistical properties of deterministic dynamical systems. Formally, it studies the state of a dynamical system with an invariant measure. In layman's terms, think about how the planets in the solar system moved according to Newtonian mechanics: the planets were in motion, but the rules governing their motion remained the same. In two papers published in 1932, von Neumann made fundamental contributions to the theory of such systems, including Von Neumann's mean ergodic theorem, which is considered the first rigorous mathematical basis for the statistical mechanics of liquids and gases. The two papers are titled "Proof of the Quasi-Ergodic Hypothesis" and "Physical Applications of the Ergodic Hypothesis."
Proof of quasi-traversal hypothesis paper Credit: von Neumann, J. (1932). Proof of the Quasi-ergodic Hypothesis. Proceedings of the National Academy of Sciences 18 （1） pp. 70–82.
The paper "The Physical Application of the Traversal Hypothesis" Image source: von Neumann, J. (1932). Physical Applications of The Ergodic Hypothesis. Proceedings of the National Academy of Sciences 18（3） pp. 263–266.
In other words, as a subfield of measurement theory, traversal theory focuses on the behavior of dynamical systems that allow for long-running operations. Von Neumann's traversal theorem is one of the two most important theorems in the field, and the other theorem was proposed by George David Birkhoff. According to Halmos (1958):
"The insight gained from [von Neumann's] paper is that the whole problem is inherently group-theoretic, and in particular, the solvability of measuring problems is related to the general algebraic concept of group solvability. Thus, von Neumann argues that it is the changes in groups rather than the changes in space that cause the differences; Replacing rigidly moving groups with other perfectly reasonable groups can produce unsolvable problems in R2 and solvable problems in R3. ”
—From Paul Halmos, Von Neumann on Measure and Ergodic Theory (von Neumann on the Theory of Measurement and Traversal, 1958)
"If von Neumann had never done anything else, those things would have been enough to guarantee his mathematical immortality."
—Paul Halmos (1958)
At the Institute for Advanced Study
In 1930-33, von Neumann was a visiting professor at Princeton University for three years, before he was offered the position of tenured professor at the Institute for Advanced Study (IAS). He was 30 years old at the time. Prior to this, the Institute had planned to award the position to Weil, but this was not possible (Macrae, 1992). Just three years after its founding, von Neumann became one of the top six professors at IAS, the other five being J. Alexander. W. Alexander), Einstein, Marston Morse, Oswald Wiblen, and Weil.
Institute for Advanced Study, Princeton, New Jersey
Image credit: Cliff Compton
When he joined in 1933, the institute was still located in the Mathematics Department at Fine Hall at Princeton University. Founded in 1930 by Abraham Flexner and funded by philanthropic funding from Louis Bamberger and Caroline Bamberger Fuld, the Institute for Advanced Study was and is a university unlike any other. Inspired by Flexner's experiences at the University of Heidelberg, the Collegiate of All Souls, the University of Oxford and the Collège de France, the IAS was described by biographer Sylvia Nasar as:
"A first-class research institution, with no teachers, no students, no classrooms, only researchers, protected from the changes and pressures of the outside world."
—Sylvia Nassar (1998)
In 1939, the Institute for Advanced Study moved to its own campus, Fuld Hall. In the early 1930s, the Institute for Advanced Study succeeded the University of Göttingen as the most important center of the mathematical universe. This dramatic and rapid change has been dubbed the "Great Purge" of 1933, with many top scholars fleeing Europe for fear of their own safety. In addition to von Neumann and Wigner, there were, of course, Einstein (1933), Born (1933), von Neumann's Fellow Budapest compatriots Syrard (1938) and Edward Taylor (1933), as well as Edmund Landau (1927), James Franck (1933) and Richard Curran (1933).
Part of the staff of the Institute for Advanced Study, photograph taken in the late 1940s, front row from left: Dana Munro, Whitney Oates, Einstein, Mario Laserna Pinzón, Marston Morse, Solomon Lefschetz; back row from left: von Neumann (visible in the background), Morgan Stern, Samuel Wilks.
In front of the Institute for Advanced Study Computer MANIAC, from left: Julian Bigelow, Herman Goldstine, Oppenheimer and von Neumann.
geometry
During his time at the Institute for Advanced Study, von Neumann founded the field of continuous geometry, which is an analogy to the geometry of compound insinuation, where the dimensions of subspaces are not discrete sets of 0,1,...,n, but can be any element of the unit interval [0,1].
Continuous geometry is a lattice L with the following properties:
- L is a modulus
- L is complete
-Lattice operations satisfy continuity
- Each element in L has a "complement"
- L is irreducible, i.e. only two elements, 0 and 1, have a unique "complement"
Like his theory of traversal, von Neumann published two papers on continuous geometry, one proving its existence and discussing its properties, and the other giving examples:
von Neumann (1936). Continuous geometry. Proceedings of the National Academy of Sciences 22 (2) pp. 92–100. （《连续几何学》）
von Neumann (1936). Examples of continuous geometries. Proceedings of the National Academy of Sciences 22 (2) pp. 101–108.（《连续几何学实例》）
Manhattan Project (1937-1945)
In addition to academic pursuits, from the mid-to-late 1930s, von Neumann developed expertise in explosive science, a phenomenon that is difficult to model mathematically. In particular, von Neumann became an authority on the mathematics of shaped charges, a charging technique that concentrated the energy of the explosion.
According to Macrae, by 1937 von Neumann had anticipated the coming of war. Although he was clearly suited for advanced strategic and operational work, he humbly applied to become a Lieutenant in the U.S. Army Ordnance Reserve. Being part of the Officers' Reserve meant he had unhindered access to a wide variety of explosion statistics, which he found more attractive (Macrae, 1992).
Id photo of von Neumann at Los Alamos
Von Neumann talks to Feynman and Ulam in Los Alamos.
There is no doubt that von Neumann's main contribution to the atomic bomb was not as a lieutenant in the Reserve of the Ordnance Department, but in the concept and design of the explosion lens. The plutonium core used by the "fat man" that was later dropped in Nagasaki used a related design.
In 1944, as a member of the Manhattan Project, von Neumann demonstrated that the blast wave reflected from the solids increased the pressure than previously thought, depending on its angle of incidence. This discovery prompted them to decide to detonate the atomic bomb a few kilometers above the target instead of at the site of impact (Macrae, 1992). On July 16, 1945, in the nevada desert, the "Trinity" atomic bomb was successfully detonated, and von Neumann was also present. This was the first successful atomic bomb test.
Philosophical work
Von Neumann lectured at the American Philosophical Society in 1957
Image credit: Alfred Eisenstaedt
Macrae notes that von Neumann was not only one of the most important mathematicians of his life, but in many ways he should also be considered one of the most important philosophers of his time. John Dorling, professor of philosophy at the University of Amsterdam, highlighted von Neumann's contributions to the philosophy of mathematics (including set theory, number theory, and Hilbert space), physics (especially quantum theory), economics (game theory), biology (cellular automata), computers, and artificial intelligence.
His research on computers and artificial intelligence (AI) first appeared in the mid-1930s. He was in Princeton for the first time with 24-year-old Alan Turing, who spent a year at the IAS in 1936-37. Turing's career began in the same field as von Neumann—set theory, logic, and Hilbert's decision problems. In 1938, when he completed his Ph.D. in Princeton, Turing extended the work of von Neumann and Gödel, introducing the concepts of ordinal logic and relative computation, augmenting his previously designed Turing machine with the so-called Oracle machine, allowing the study of problems beyond the capabilities of Turing machines. Although von Neumann asked Turing if he would like to work as a postdoctoral research assistant after he received his PhD, Turing refused and returned to the UK (Macrae, 1992).
Work on computing
After a month here, I talked to von Neumann about all kinds of inductive processes and evolutionary processes, and he said, "Of course, that's what Turing said." I asked, "Who is Turing?" He said, "Look up the 1937 Journal of the London Mathematical Society." ”
In fact, there is a general-purpose machine to imitate all other machines ... Only von Neumann and a few people understood. When he understands this, he knows what we can do. —Julian Bigelow
—From George Dyson, Turing's Cathedral (Turing's Cathedral, 2012)
Despite Turing's departure, von Neumann was still thinking about computer problems in the late 30s and during the war. Based on his experience working on the Manhattan Project, in the summer of 1944 he was first attracted to the ENIAC program of the Moore School of Engineering at the University of Pennsylvania. After observing the large number of calculations required to predict the radius of the explosion, plan the trajectory path, and crack the encryption scheme, von Neumann quickly realized the need to greatly increase computing power.
In 1945, von Neumann proposed a computer architecture, now known as the von Neumann architecture, which included the basic elements of modern electronic digital computers:
Processing unit containing arithmetic logic units and processor registers;
Control unit containing an instruction register and a program counter;
Storage units that can store data and instructions;
External memory;
Input and output mechanisms;
Von Neumann and the IAS machine, sometimes referred to as the "von Neumann machine", were placed in the basement of the "Fulley Building" from 1942 to 1951 (photo: Alan Richards)
That same year, in the field of software engineering, von Neumann invented the so-called Merge Sort Alogrithm algorithm, which splits an array in half, then recursively sorts, and then merges. Von Neumann personally wrote the first 23 pages of sorting programs for the EDVAC computer in ink.
In addition, in a seminal paper published in 1953 titled "Probabilistic Logics and the Synthesis of Reliable Organisms from Unrealiable Components," von Neumann first introduced stochastic computing, but the idea was so groundbreaking. It was so much so that it could not be achieved for the next decade (Petrovik & Siljak, 1962). Related to this, von Neumann pioneered the field of cellular automata through a rigorous mathematical treatment of self-replicating structures, which predates the discovery of DNA structures several years earlier.
Although von Neumann was influential throughout his life, he was convinced that the core concept of modern computers actually came from Turing's 1936 paper "On Computable Numbers, with an Application to the Entscheidungsproblem" – Fraenkel (1972)
"Von Neumann firmly told me, and I'm sure he stressed to others, that this basic concept is to Turing's credit – something that Babbage, Lovelace and no one else expected."
—— Stanley Fraenkel（1972）
consultant
"The only part we want for a systematic tender is what you think when you shave: we want you to pass on to us any ideas you put into things."
—Excerpt from a letter from the head of the RAND Corporation to von Neumann (Poundstone, 1992)
During his career in the United States, von Neumann has consulted for a variety of private or public affairs and defense contractors, including the Defense Research Council (NDRC), the Weapons Systems Assessment Group (WSEG), the Central Intelligence Agency (CIA), the Lawrence Livermore National Laboratory (LLNL), and the RAND Corporation. In addition, he was an advisor to the U.S. Armed Forces Special Weapons Program (AFSWP), a member of the Atomic Energy Commission's General Advisory Committee, a member of the U.S. Air Force Scientific Advisory Group, and in 1955 he became a member of the Atomic Energy Commission (AEC).
Private life
Although von Neumann held many positions, took on many responsibilities, and had a large number of research results, as a mathematician, he had an unusual way of life. As Vonnauman (von Neumann's younger brother) and Halmos describe:
"Partying and nightlife have a special appeal to von Neumann. While teaching in Germany, von Neumann was a regular in Berlin's nightlife scene during the Cabaret era. ”——Vonnauman（1987）
Von Neumann's family often held parties, was famous, and had a long time. - Halmos (1958)
von Neumann and his wife, Klára Dán, with pet 丨Author: Alan Richards
His second wife, Klára, said he could count anything but calories.
von Neumann also liked Yiddish and yellow jokes, especially oil poems (Halmos, 1958). He does not smoke but has received complaints from the International Smoking Association because he often plays German marches very loudly on the phonograph in his office, distracting people in the next office, including Einstein. In fact, von Neumann claims that some of his best works are done in noisy, chaotic environments, such as in his living room, where the sound of television sets is always loud. Despite his poor driving skills, he enjoyed driving and often read while driving, leading to multiple arrests and accidents.
von Neumann in the Everglades, Florida in 1938
Image credit: Marina von Neumann Whitma
As a thinker
Von Neumann's close friend Ulam described von Neumann's mastery of mathematics this way:
"Most mathematicians are familiar with a certain mathematical process (method) or have a certain mathematical ability. For example, Norbert Wiener was proficient in the Fourier transform. Some mathematicians who are able to master both methods may impress those who know only one of them. And John von Neumann has mastered three methods:
1) The ability to perform symbolic operations on linear operators;
2) have an intuitive sense of the logical structure of any new mathematical theory;
3) Have an intuitive sense of the superstructure of the combinatorial aspects of the new theory. ”
Biographer Sylvia Nassar describes von Neumann's own "thinking machine", and here are anecdotes about the so-called "two train riddle":
The two cyclists began to distance themselves by 20 miles, heading north and south, maintaining a speed of 10 miles per hour. Meanwhile, a fly set off from the front wheel of the south bike at 15 miles per hour and flew north; it encountered the front wheel of the north bike, then turned around and flew again to the front wheel of the south bike, and continued like this until it was flattened by the two front wheels. Q: What is the total distance for flies to fly?
There are two ways to answer this question. One way is to calculate the distance covered by the fly for each leg of the two bikes, and finally add them up to get an infinite number of progressions. An easier approach is to observe the bikes rendezvous an hour after departure, so that the flies have only an hour to travel; so the answer is definitely 15 miles. When the question was presented to von Neumann, he immediately solved it, thus disappointing the questioner: "Oh, you must have heard of this trick before!"
"What trick?" von Neumann asked, "All I did was sum the infinite series. ”
—From Nasar, A Beautiful Mind (A Beautiful Mind, 1998)
As a mentor
In a 1934 paper, "Szeged In 1934," Edgar R. Lorch described his work experience as von Neumann's assistant in the 1930s, and his responsibilities included:
Attend von Neumann's lectures on operator theory, take notes, complete unfinished proofs, and distribute them to the libraries of all universities in the United States;
Assist von Neumann in his work as editor of the Annals of Mathematics, reading every manuscript accepted for publication; drawing a red line under the Greek letter and a green line under the German letter, italicizing, writing notes to the printing house in margins; going to the printing house once a week to instruct them in typesetting;
Translated von Neumann's extensive 100-page papers into English;
"His fluid train of thought is difficult for those who lack talent to understand. He would write equations in a hurry on a small part of the blackboard and erase them before the students had time to copy them, which made him infamous. ”
--Excerpt from N.A. Vonneuman, John von Neumann: As Seen by his Brother (von Neumann in the Eyes of Brothers, 1987)
old age
In 1956, President Eisenhower, left, awarded President John von Neumann the Medal of Freedom
In 1955, von Neumann was diagnosed with cancer, possibly bone, pancreatic or prostate cancer (there are different theories about which diagnosis was made first), at the age of 51. After two years of suffering, he was eventually confined to a wheelchair. von Neumann died on 8 February 1957 at the age of 53. It is said that on his deathbed, he recited the first few lines of each page of Goethe's Faust verbatim as a way to appease his brother (Blair, 1957).
Von Neumann was buried in Princeton Cemetery in New Jersey, where his lifelong friends Eugene Wigner and Gödel were buried years later. Gödel wrote him a letter a year before his death, and now the letter has been made public. Computer scientist Hartmanis discussed the letter in detail in The Structural Complexity Column (primarily on the P=NP problem). An excerpt follows:
Letter dated 20 March 1956 from Gödel to von Neumann:
Dear Mr. von Neumann,
I am very sad to hear that you are sick. The news came unexpectedly. Morgan Stern told me last summer that you had a show of weakness, but at the time he didn't think it was a big deal. As far as I know, you have received thorough treatment over the past few months and I am glad that this treatment has been as successful as expected and that you are getting better now. I hope and wish that your condition will improve as soon as possible and, if possible, the latest medical discoveries will make you fully recover.
……
I am glad to hear from you personally. If there is anything you need my help with, please let me know. Best wishes and best wishes to you and your wife.
You sincerely
Kurt Gödel
P.S. I sincerely congratulate the U.S. government on the honor it has given you
TV interviews
Notably, in the early 1950s, the NBC program America's Youth Wants to Know included a video interview with von Neumann (please go to "Back to Park"):
For any reader interested in learning more about the life and work of John von Neumann, I especially recommend his friend Ulam's 1958 article "John von Neumann 1903-1957," published in the Bulletin of the Bulletin of the American Mathematical Society, and McRae's biography John von Neumann (The Pioneer of Genius – von Neumann). Neumann Biography).
(End of full text)
This article is translated from Jørgen Veisdal, The Unparalleled Genius of John von Neumann
https://www.cantorsparadise.com/the-unparalleled-genius-of-john-von-neumann-791bb9f42a2d