> 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.
When I was a teen, I was introduced to a friend of my step-father's that had the same affliction — he slept only 3 or 4 hours a day. He had told me it was unusual. I don't know how unusual it is. If it is common enough then von Neumann and Edison having the same inclination is of no significance.
He would get rejected by a hiring committee. Not from the right school and not a social fit due to his preference of working with very loud German march music.
I've only really seen this mentioned in comments. All references I could find state that he often slept until late (for example, he liked to sleep until after 10AM).
I would be wary of glorifying extremely short sleep patterns, they are unhealthy for 90% of the population.
Well if you can solve problems of functional analysis or linear programming while at a party with others I must congratulate you, you're probably half way there lol
Possibly, funnily enough his daughter mentions in her book that in hindsight one of the first signs of cancer was that he started sleeping for 7 or 8 hours a day!
If sleeping 8h/day would grant him, say, ten more years, that would more than paid those extra 4h of awake time over his entire adult life, as 34x(1/6) < 10x(2/3).
It's possible a lack of sleep reduced his immune response to the cancer. The body has some natural cancer fighting mechanisms, and it's only the mutations that escape these that become malignant tumors.
It’s possible. But what’s more possible is spending the latter half of his life working on nuclear weapons and observing bomb tests took their toll.
It’s interesting how you guys are latching onto this man’s lack of sleep to speculate why he died young instead of the more obvious possibility, given his background.
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.
Or perhaps there are no causes at all, and the laws of physics are deterministic and time reversible. Perhaps causes appear only in our models of predicting the world, because our models contain agents and counterfactuals which don’t exist in the laws of physics but are useful for predicting some phenomena.
Deterministic time-reversible things can still have causality. Just today I used gdb's reversible-debugging feature on a deterministic program to locate a bug. The existence of a bug implies causality in the program's execution state.
Deterministic time-reversible things could still have causality in a universe that has causality. But that's not an effective response to the claim that there's no causality at all.
I suppose you could still talk about causes in a deterministic time-reversible system, but it doesn’t match the normal sense of the word “cause” given that future states determine past states just as much as past states determine future states.
Getting better and better at solving problems (here: in a broad range of mathematics) at an unconventional rate makes it uncanny.
It seems like magic. As everyone doing magic tricks knows there is a lot of effort going into a good one. The "innate abilities" or "genius" is the "magic" we all like to experience.
In the case of "von Neumann" he isn't isolated.
He is often lumped together with other Hungarian Jews fleeing Nazism and refered to as A marslakók ("The Martians"): Erdős, Kármán, Pólya, Szilárd, Teller, Wigner ...
Hungarians are a curious bunch, aren't they?
Arguably the strongest female chess player of all time (in the same class as the strongest men) is also Hungarian: Judit Polgár. How do they do it?
There is no magic, no genius, her Hungarian father wanted to prove exactly this to the (western) world (in the east the notion isn't prevalent).
And he simply conducted the experiment[0] consequently his youngest daughter became a so-called chess genius. Or as a Korean or Chinese might say she is doing the hard work.
We all like to tell ourselves stories and it is fine, it is a great and very effective way to convey certain things but sometimes we lose sight of what is really going on because of our infatuation with the narrative devices. Instead of learning and trying to explain things, we sometimes just want to indulge in the "magic moments". But as Feynman put it nicely: you can do both one doesn't exclude the other.
> Arguably the strongest female chess player of all time
More than arguably. Judit is beyond any doubt the strongest female of all time, by a long way. Her sisters also weren't bad, Susan becoming women's world champion. Judit was so good she never bothered playing for the women's title. Hou Yifan, the second strongest female of all time, women's world champion at age 16, is ranked about #90 in the world. Judit was #8.
Wigner himself said the only genius amongst The Martians was von Neumann so he at least thought von Neumann was special compared to the others. For your example a mathematician who would fit better would be Norbert Wiener, who was taught from a young age by his father to be a prodigy.
I don’t think it is far fetched to claim that such a complex system as the human brain has a genetic basis regarding intelligence as well, especially towards outliers of the IQ bell curve.
For example, Ashkenazi Jews have very specific neurological diseases, but those same genes could also give much increased intelligence when they are paired with their normal counterparts. Here is a very interesting blog post on an investigation behind “The Martians” for example: https://slatestarcodex.com/2017/05/26/the-atomic-bomb-consid... which basically finds this same reason for their “superhuman” intelligence.
My rhetoric got a bit over my head, but I really think that every (healthy) newborn human has the potential to become what is called a "genius". This is the norm.
The limiting factor isn't some predetermined thing like genetics.
It may come into play when we look into a very narrow set of like-minded and like-socialized where one could argue that "von Neuman" was the most "gifted" of the Martians bunch. But even there one could also argue that he just spent more time on problems. Does this maybe count as intelligence, too? The perseverance? From the Polgár-Sisters the youngest - Judit - wasn't necessarily the smartest but no one in her family would argue that she wasn' the most strong-willed by a big margin.
I definitely think the genetic factor is way overblown (like by one order of magnitude) in Western societies. From personal experience from East-Asian or Middle-East (like Iran) countries, they have a more pragmatic outlook (not denying some genetic factor but they really don't care so much, in our parts it is seems like some fetish in comparison).
> My rhetoric got a bit over my head, but I really think that every (healthy) newborn human has the potential to become what is called a "genius". This is the norm.
I really have to disagree with this take. Sure, nurture has a huge effect, but similarly to how I could train all my life and do improve day-by-day, I would never be an olympic gold medal winner, intelligence ought to be the same way. You do need a good basis on which you can build. There are plenty of people having extraordinary discipline, they can achieve impressive results, but they won’t be geniuses without having talent on a fundamental level. But of course, talent is not enough either without practice.
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.
His quote was only relevant to preventing other nations from obtaining nuclear weapons. While for now nuclear peace still exists it still remains to be seen in the long term, especially as more and more nations think of developing nuclear weapons, whether the idea of allowing nuclear proliferation was a good one in the end.
I'm pessimistic here. The countries that once pursued/had nuclear weapons, but then signed treaties and gave up on them were screwed one after another: Iraq, Lybia, Iran and now Ukraine. Contrast with North Korea, that pursued them not matter what. Nobody ever screwed them. The failure of the great powers in keep their word with countries that actually took measures to stop or limit nuclear proliferation will seriously backfire.
Yes they were screwed and lots of people did die unnecessarily. However, Saudi Arabia wants nukes, Shinzo Abe (ex Japan PM) said Japan should get nukes, the future of the long term program of Iran is still unsettled. Apartheid South Africa had nukes and only gave them up because they didn't want the native Africans to have them. The problem becomes as more and more countries develop nuclear weapons, the risk of conflict using them grows higher. Already we have a long term conflict between Pakistan and India where both nations are nuclear armed and I very much doubt Pakistan's nuclear security as anywhere as good as say America's or China's. What would happen if 20 or 30 or 40 countries had nukes? They would threaten to use them over every small thing and eventually I guess someone would use them, and who knows what happens then.
Can we claim that if the Soviets develop the bomb first, they wouldn’t have used it on the US as soon as possible? With that in mind, his take is strong, but not irrational.
He also claimed/“proved” there were no hidden variables possible in QM even though de Brogile had one already and Bohm would soon show his version (which was very similar).
He also claimed human (and possibly) animal consciousness were the only thing able to collapse wave functions.
For your first claim I think you'll find what von Neumann claimed is a lot more limited than you think, and even then what exactly was interpreted is still up for debate, see from last year:
Likewise, for your second claim I think you'll find von Neumann said that the collapse of the wavefunction can be placed anywhere in a measurement chain, from the first measuring device to the observer's conscious perception of the measurement result. The stronger claims about human consciousness specifically causing the collapse of wave functions come from London and Bauer's book "The Theory of Observation in Quantum Mechanics" and Wigner's "Remarks on the Mind-Body Question". Again, this viewpoint is reflected in the literature too:
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.
No, not correct. Not even a little bit correct. Game theory explains a lot of human (and animal) behavior. It explains, for example, why I do not rob my neighbor's house when I know they are on vacation even though I could almost certainly get away with it and thereby achieve a short-term profit for myself. It explains why rich people who wring their hands over climate change continue to buy first-class airline tickets posh vacation destinations. It explains why Putin invaded Ukraine, and why the soldiers who are actually doing the dirty work continue to do that dirty work despite the fact that very few of them actually would have chosen to invade Ukraine.
It explains why discussions on the internet often degrade into flame-fests, and why this happens more rapidly if people are allowed to post anonymously or pseudonymously.
> It explains, for example, why I do not rob my neighbor's house when I know they are on vacation even though I could almost certainly get away with it and thereby achieve a short-term profit for myself.
Does game theory explain why other people do steal? Should we assume that thieves' circumstances are different enough that stealing is rational?
Game theory is sometimes a useful way of thinking about human behavior but it's a stretch to say it "explains a lot of it". The various ways people come to value things and make decisions are opaque. It's begging the question to insist that these processes are explicable by game theory, especially when so much human behavior seems inexplicable.
You said "a lot of human behavior is explained by game theory". Why can't we say that all human behavior is explained by game theory? How do we differentiate between an instance of behavior that is explained by game theory and one that isn't?
It's useful and interesting to say "here's a theory that predicts ways in which people actually behave". It's begging the question to claim the theory actually explains human behavior, which remains opaque.
> Why can't we say that all human behavior is explained by game theory?
We can. But game theory turns on the parameters of the payoff matrix, and those are "hidden variables" that are hard to measure. But they are constrained by ESS requirements, which is the thing that makes game-theory non-vacuous. It's analogous to saying that weather is explained by fluid dynamics.
Not really. It explains only the optimal play of intelligent, clear headed, self-interested actors. It does not explain why someone would jump into a river to save a drowning child that they do not know even at the cost of their own life. There are many other examples like this. Game theory does not encapsulate values or morality. It's fine for someone with sufficient psychopathy or cowardice, but to say it explains even a small fraction of human behaviour goes too far. Bakers and butchers are not seeking to maximize their utility. Some people just want to grow flowers, go to Church, and be good neighbours.
Those are all perfectly rational actions if you use a sufficiently nuanced utility function. It shouldn't surprise that modeling humans as "maximizes dollars" is overly simple.
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
I remember taking quantum mechanics as a second year mathematics undergrad in the mid 2000s and - as a mostly pure mathematician - balking at the Dirac delta function. Wasn't until the third year course on principles of QM did things start feeling steady.
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.
I think things are too different today to make comparisons like that.
In the last 50 years, our collective knowledge has grown at a pace that is incomparable to other periods of time. Something that falls out of that is that the open problems in a particular field are much more difficult to solve.
It is also true that the knowledge is much more widespread. For instance, many CS undergrads are expected to learn game theory. What was once an advanced topic is now just part of the curriculum.
Basically the common denominator has risen in a way that is much more difficult for a single mind to seem comparatively impressive.
I agree, all the easy problems are solved so only hard ones are left, but the issue of widespread knowledge doesn't seem to be as important to me. The difficulty in a lot of fields is not for example picking up undergraduate knowledge in game theory or whatever, calculus was once an advanced topic but we can teach it to 12 year olds now, but making deep contributions to fields because as you say the easy problems have been picked so either you have to introduce a new paradigm or you have to have an ever growing amount of prerequisite material to know beforehand to contribute something new of significance.
Probably not, actually. According to that Wikipedia page you linked the Prometheus Society has only 120 members. Since they accept people who are in the top 1/30,000th in terms of IQ, therefor there are 8 billion / 30,000 = 266,666 such people in the world, and so only 0.048% of those super-geniuses are members of the Prometheus Society.
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.
