> Most anecdotes about von Neumann abide by this three-act structure: a question that baffles the best minds; their sweaty, pointless deliberations; von Neumann’s swift, soaring leap to the solution. (Sometimes, as in the Rand Corporation tale, the question itself goes undescribed. Its sole attribute of importance is its impenetrability.)
This type of storytelling reinforces the tale of mental ability being an innate talent that doesn't need work or practice that American media loves to tell. In practice, it is much more interesting to (try to) work out how people were able to work out difficult questions on the spot.
Unfortunately there has been lack of academic literature on this too on how he was able to solve problems so quickly. Much of what is known can only gathered from anecdotes from those who knew him personally. Edward Teller said in an interview that von Neumann enjoyed thinking more than anything else and did it all the time. This fits in well with other anecdotes. He was known to leave parties for periods of time to go to his room and write things down. He slept for only 4 hours a day and at night worked on problems. From what's been said of him he clearly spent basically his entire time thinking and working through problems. Then of course his favored way of attacking problems was axiomatically so he had some systematic way of going about this too. The end result of someone who spends a lot of time solving problems is that they know how to solve a lot of problems when they come up again.
Neumann could divide 8-digit numbers in his head when he was 8-years old. At age 8 he already mastered basic differential and integral calculus. Neuman was genius who studied math over several decades.
I propose the following:
(1) People like him are born, not created. They have innate abilities.
That story caught my eye and I had a different take. This is just my experience and anecdote should always be taken with caution, but here we are.
I worked a little with someone, now deceased, who was maybe comparable to von Neumann in some ways as an academic or intellectual figure. Few in history are comparable to von Neumann really, but there are a lot of similarities in this person with reference to a different and narrower field. This person is cited in blog posts that show up on HN from time to time in disparate but related fields, and there's a similarity in a lot of respects at a smaller scale.
My impression with that person was that they were smart, very smart, but not actually smarter than a lot of other people I knew and respected in academics. They had this sort of persona, almost like a role they were playing, with an an accompanying reputation that sort of got amplified and mythologized into legend. I have a lot to say about this, but at some point it became like an ignition process, where fame led to more citations and attention, which fed off itself. A lot of what they wrote or thought about wasn't really that different from their colleagues, who were more cautious or prudent or less extravagant. Their reputation got kind of turned into something outsized from reality.
Stories like this with von Neumann (where "the question itself" or some important feature "go undescribed") kind of reminded me of my experiences in this regard. The details are fuzzy in these stories because they're exaggerated or obscure scrutiny, or are based on someone's subjective impression rather than the details.
Call me jaded, but as my career progressed and I've worked with more people, I've become increasingly skeptical of claims to great minds or genius. It's a tricky subject because I wouldn't say there's no such thing as cognitive ability, but as claims become more extreme, there's usually a different explanation. It's the regression to the mean phenomenon. Usually they're just in the right millieu, with the right colleagues, and/or have a certain character they project, or something.
Young man, in mathematics you don't understand things. You just get used to them.
— John von Neumann
Does not sound to me like a man who thought he had the answers. He did the best he could, and his best is among the best all time, but he certainly didn’t know everything.
FWIW over the years I've heard variations of the quote from a number of people. Though I'm not versed in advanced mathematics or physics the idea makes a lot of sense to me. I say that because developing expertise in fields requiring mastery of highly abstract concepts (and skill in applying the concepts) works pretty much the same way.
But then again, it's interesting how exposure to target subject matter improves "understanding things". An example is when I needed to learn what the confusing areas of shadow and light on a PET scan image were all about. Viewing such images repeatedly became sort of magic. After a while the images start to actually make sense, structures and processes come into focus and become identifiable. Eventually they're old friends, "yeah I know what's going there..."
Here's my speculation: beyond a certain level of complexity, learning via linear, step-at-a-time algorithms no longer works because the number of items to keep "online" simultaneously exceeds what can be held in working memory. The need to "get used to it" implies ability to "page" sets of symbolic representations into and out of working memory. This is an experiential learning process coalescing as an intuition for the discipline.
I think much more consequential than this educational quote are his takes on nuclear warfare and the extent to which he relied on game theory and idealized rational actors to go as far as to recommend a first strike.
This humility of accomplished mathematicians when it comes to math but ironically complete lack or overuse of it in other domains honestly seems quite common.
I have but a naive incomplete conjecture that von Neumann's ability to solve very hard mathematical problems are a direct result of him spending so much time initially at the axiomatic level of mathematical foundations. He must have developed a first-principles level of understanding and intuition about how to tackle mathematical problems, or why their solution would be impossible.
In my own capacity, I've seen solutions to very hard problems as "obvious" or the impossibility of such solutions as "obvious" compared to other engineers who lack a formal training in Computer Science, for example. To use a naive example, what types of problems could be done with regular expressions, and what not, because they are regular languages, and knowing the theory of regular languages. He must have gone so much further down the rabbit hole that things were seemingly obvious, or if a solution was possible at all, it would have to follow from certain first-principles, and could reason his way back to the highest level of abstraction, with immense ease.
Freeman Dyson also commented on this "Johnny’s unique gift as a mathematician was to transform problems in all areas of mathematics into problems of logic. He was able to see intuitively the logical essence of problems and then to use the simple rules of logic to solve the problems" - https://www.ams.org/notices/201302/rnoti-p154.pdf
At its core, the game theory of von Neumann is the saddlepoint result.
There are two players, Red and Blue, for some positive integer n n moves, and an n x n matrix G = [g_ij] of real numbers of payoffs. So each of Red, Blue pick a move, that is, i, j = 1, ..., n, i for Red and j for Blue, and then at the same time they make, show, their moves. Then Red gains g_ij and Blue loses g_ij.
The game theory saddlepoint result says that each of Red and Blue picks their moves probabilistically independently and randomly with some probability distribution over moves 1, 2, ..., n. Each of Red and Blue gets to pick their own distribution. So, the question for Red and Blue is, what distribution to pick?
The saddlepoint result says that there is a distribution for Red and a distribution for Blue that is a saddlepoint, that is, any change by Red will result nothing better for Red and any change by Blue will result in nothing better for Blue.
For the game of paper, scissors, rock, Red and Blue have the same distribution, uniform over the three moves.
Von Neumann had a proof, but there is an easy proof early in linear programming theory. When I was teaching, I covered the result in a few minutes, about the same as needed just to read this post.
Broad, profound consequences for civilization? Naw.
To me the singular fascinating thing about Johnny was that for all his freakish intelligence and astonishing powers of recall, he apparently took Pascal's Wager on his deathbed, that is, he converted to (IIRC) Catholicism just before he died, in case God does exist (and is Catholic.)
Not to put too fine a point on it: he was a metaphysical dullard!
I don't think he converted as a logical response, but rather because he was scared shitless of not existing(I don't blame him). He also died fairly young, one would have expected him to live another 20 years.
> Game theory, faithful to its name, treats every human context as a game—a self-contained situation in which your sly rival must lose for you to win, and in which the nature of these losses and wins can be always summed up in precise numbers.
That’s a complete mischaracterization. Game theory literally has zero to say about the human condition. It is a mathematical theory.
It reads to me as if the reviewer had a ‘hook’ for the review and was going to use it, whether he could find support for it or not.
>Game theory literally has zero to say about the human condition. It is a mathematical theory.
Correct. Economics is the study of the result of rational agents engaging in optimizing (in the mathematical sense) behavior. Game theory is an extension of that when there are strategic interactions between those agents (traditional micro assumes simple price-taking behavior). Source: undergraduate was in Economics.
My professor said that a good example of the Nash equilibrium is the double yellow line on the road. You have no incentive to go past it, just like the people driving at you in the other lane have no incentive to go into your lane. Game theory can explain a lot of human actions, alone or in groups.
In the context of a person who (for zero real research) sounds as though he intellectualized life perhaps to a fault, the game theory anecdote seems to fit.
If it had nothing to say about the human condition it would be called something like "multi-party valuation theory" and not specifically designed to analyze "roulette to chess, baccarat to bridge".
> Von Neumann, who saw the underlying mathematics better than almost anyone, showed how wave and matrix mechanics were essentially the same, and how one could be expressed in the other’s language.
Maybe he did that, but Dirac was certainly the first to do so.
> By the time von
Neumann started his investigations on the formal framework of
quantum mechanics this theory was known in two different mathe-
matical formulations: the "matrix mechanics" of Heisenberg, Born
and Jordan, and the "wave mechanics" of Schrödinger. The mathe-
matical equivalence of these formulations had been established by
Schrödinger, and they had both been embedded as special cases in a
general formalism, often called "transformation theory," developed
by Dirac and Jordan. This formalism, however, was rather clumsy
and it was hampered by its reliance upon ill-defined mathematical
objects, the famous delta-functions of Dirac and their derivatives.
Although von Neumann himself attempted at first, in collaboration
with Hilbert and Nordheim [l], to edify the quantum-mechanical formalism along similar lines, he soon realized that a much more
natural framework was provided by the abstract, axiomatic theory of
Hilbert spaces and their linear operators [2], This mathematical
formulation of quantum mechanics, whereby states of the physical
system are described by Hilbert space vectors and measurable quan-
tities by hermitian operators acting upon them, has been very suc-
cessful indeed. Unchanged in its essentials it has survived the two
great extensions which quantum theory was to undergo soon: the
relativistic quantum mechanics of particles (Dirac equation) and the
quantum theory of fields.
Leon van Hove, "Von Neumann's Contributions to Quantum Mechanics", 1958
In terms of deep contributions to a variety of mathematical science type fields, probably no one. People might say Terence Tao, and of course no offense to him he is extraordinary smart and widely knowledged, but in my opinion he hasn't made the same level of deep contributions to mathematics let alone other fields. From what I've heard and read historically to be considered one of the "greats" so to speak you had to have full mastery of a field, either by starting it or revolutionizing it (funnily enough von Neumann was criticized for this by other mathematicians in his time). An easy example is Alexander Grothendieck, who is considered one of the top mathematicians of the 20th century. He revolutionized algebraic geometry and homological algebra. For von Neumann the most obvious field would be operator algebras. Of course the reason for mathematicians these days not being as "creative" or "revolutionary" may just be that all the low hanging fruit so to speak has already been picked, which in my view is the most obvious reason, so it's significantly harder to make deep contributions compared to times past, but it is what it is.
Ask me in 30 years i.e. I don't think you can really spot genius as it's happening. von Neumann was just a smart guy people liked to work with even in his prime, to an extent at least.
[+] [-] ttyprintk|4 years ago|reply
[+] [-] rustybolt|4 years ago|reply
This type of storytelling reinforces the tale of mental ability being an innate talent that doesn't need work or practice that American media loves to tell. In practice, it is much more interesting to (try to) work out how people were able to work out difficult questions on the spot.
Famous examples: [1] https://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te#Adria... (I imagine trigonometric identities were kind of Viète's thing) [2] https://en.wikipedia.org/wiki/Taxicab_number (Ramanujan had worked on 'taxicab numbers' before)
[+] [-] mjreacher|4 years ago|reply
[+] [-] nabla9|4 years ago|reply
Neumann could divide 8-digit numbers in his head when he was 8-years old. At age 8 he already mastered basic differential and integral calculus. Neuman was genius who studied math over several decades.
I propose the following:
(1) People like him are born, not created. They have innate abilities.
(2) People like him still need to study.
[+] [-] WEUJD883jjdAJ|4 years ago|reply
I worked a little with someone, now deceased, who was maybe comparable to von Neumann in some ways as an academic or intellectual figure. Few in history are comparable to von Neumann really, but there are a lot of similarities in this person with reference to a different and narrower field. This person is cited in blog posts that show up on HN from time to time in disparate but related fields, and there's a similarity in a lot of respects at a smaller scale.
My impression with that person was that they were smart, very smart, but not actually smarter than a lot of other people I knew and respected in academics. They had this sort of persona, almost like a role they were playing, with an an accompanying reputation that sort of got amplified and mythologized into legend. I have a lot to say about this, but at some point it became like an ignition process, where fame led to more citations and attention, which fed off itself. A lot of what they wrote or thought about wasn't really that different from their colleagues, who were more cautious or prudent or less extravagant. Their reputation got kind of turned into something outsized from reality.
Stories like this with von Neumann (where "the question itself" or some important feature "go undescribed") kind of reminded me of my experiences in this regard. The details are fuzzy in these stories because they're exaggerated or obscure scrutiny, or are based on someone's subjective impression rather than the details.
Call me jaded, but as my career progressed and I've worked with more people, I've become increasingly skeptical of claims to great minds or genius. It's a tricky subject because I wouldn't say there's no such thing as cognitive ability, but as claims become more extreme, there's usually a different explanation. It's the regression to the mean phenomenon. Usually they're just in the right millieu, with the right colleagues, and/or have a certain character they project, or something.
[+] [-] chongli|4 years ago|reply
— John von Neumann
Does not sound to me like a man who thought he had the answers. He did the best he could, and his best is among the best all time, but he certainly didn’t know everything.
[+] [-] jrapdx3|4 years ago|reply
But then again, it's interesting how exposure to target subject matter improves "understanding things". An example is when I needed to learn what the confusing areas of shadow and light on a PET scan image were all about. Viewing such images repeatedly became sort of magic. After a while the images start to actually make sense, structures and processes come into focus and become identifiable. Eventually they're old friends, "yeah I know what's going there..."
Here's my speculation: beyond a certain level of complexity, learning via linear, step-at-a-time algorithms no longer works because the number of items to keep "online" simultaneously exceeds what can be held in working memory. The need to "get used to it" implies ability to "page" sets of symbolic representations into and out of working memory. This is an experiential learning process coalescing as an intuition for the discipline.
[+] [-] Barrin92|4 years ago|reply
This humility of accomplished mathematicians when it comes to math but ironically complete lack or overuse of it in other domains honestly seems quite common.
[+] [-] laichzeit0|4 years ago|reply
In my own capacity, I've seen solutions to very hard problems as "obvious" or the impossibility of such solutions as "obvious" compared to other engineers who lack a formal training in Computer Science, for example. To use a naive example, what types of problems could be done with regular expressions, and what not, because they are regular languages, and knowing the theory of regular languages. He must have gone so much further down the rabbit hole that things were seemingly obvious, or if a solution was possible at all, it would have to follow from certain first-principles, and could reason his way back to the highest level of abstraction, with immense ease.
[+] [-] mjreacher|4 years ago|reply
https://link.springer.com/chapter/10.1007/1-4020-4040-7_11
https://direct.mit.edu/posc/article-abstract/18/4/480/15281/...
https://journals.openedition.org/etudes-benthamiennes/7901?l...
Freeman Dyson also commented on this "Johnny’s unique gift as a mathematician was to transform problems in all areas of mathematics into problems of logic. He was able to see intuitively the logical essence of problems and then to use the simple rules of logic to solve the problems" - https://www.ams.org/notices/201302/rnoti-p154.pdf
[+] [-] ElephantsMyAnus|4 years ago|reply
The thinking of geniuses seem to be mastery of simplification, rather than the contrary.
The best intelligence test might be
1. Solve programming problems with the shortest code possible.
2. Where is Waldo.
[+] [-] graycat|4 years ago|reply
There are two players, Red and Blue, for some positive integer n n moves, and an n x n matrix G = [g_ij] of real numbers of payoffs. So each of Red, Blue pick a move, that is, i, j = 1, ..., n, i for Red and j for Blue, and then at the same time they make, show, their moves. Then Red gains g_ij and Blue loses g_ij.
The game theory saddlepoint result says that each of Red and Blue picks their moves probabilistically independently and randomly with some probability distribution over moves 1, 2, ..., n. Each of Red and Blue gets to pick their own distribution. So, the question for Red and Blue is, what distribution to pick?
The saddlepoint result says that there is a distribution for Red and a distribution for Blue that is a saddlepoint, that is, any change by Red will result nothing better for Red and any change by Blue will result in nothing better for Blue.
For the game of paper, scissors, rock, Red and Blue have the same distribution, uniform over the three moves.
Von Neumann had a proof, but there is an easy proof early in linear programming theory. When I was teaching, I covered the result in a few minutes, about the same as needed just to read this post.
Broad, profound consequences for civilization? Naw.
[+] [-] carapace|4 years ago|reply
Not to put too fine a point on it: he was a metaphysical dullard!
[+] [-] oh_sigh|4 years ago|reply
[+] [-] smegger001|4 years ago|reply
or just hedging his bets.
[+] [-] JackFr|4 years ago|reply
That’s a complete mischaracterization. Game theory literally has zero to say about the human condition. It is a mathematical theory.
It reads to me as if the reviewer had a ‘hook’ for the review and was going to use it, whether he could find support for it or not.
[+] [-] DebtDeflation|4 years ago|reply
Correct. Economics is the study of the result of rational agents engaging in optimizing (in the mathematical sense) behavior. Game theory is an extension of that when there are strategic interactions between those agents (traditional micro assumes simple price-taking behavior). Source: undergraduate was in Economics.
[+] [-] servytor|4 years ago|reply
[+] [-] smitty1e|4 years ago|reply
[+] [-] morelisp|4 years ago|reply
[+] [-] fgh|4 years ago|reply
Maybe he did that, but Dirac was certainly the first to do so.
[+] [-] n4r9|4 years ago|reply
Leon van Hove, "Von Neumann's Contributions to Quantum Mechanics", 1958
https://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958...
[+] [-] elchief|4 years ago|reply
[+] [-] mjreacher|4 years ago|reply
[+] [-] Dracophoenix|4 years ago|reply
As far as others are concerned, they're probably members of the Prometheus Society.
https://en.m.wikipedia.org/wiki/Prometheus_Society
[+] [-] random3|4 years ago|reply
[+] [-] mhh__|4 years ago|reply