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John von Neumann thought he had the answers
- fgh 5y ago> 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.
- n4r9 5y ago> 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 https://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958-10206-2/S0002-9904-1958-10206-2.pdf https://www.ams.org/journals/bull/1958-64-03/S0002-9904-1958...
- fgh 5y agoThanks, that clarifies his actual contribution.
- n4r9 5y agoI 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.
- chongli 5y agoYoung 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.
- Barrin92 5y agoI 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.
- mjreacher 5y agoHis 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.
- ttyprintk 5y agoOr when the actors are no longer nations modeled on an individual decision-maker but, eg trucker convoys purchasing their way into the club.
- Qem 5y agoI'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.
- rustybolt 5y ago> 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. Famous examples: [1] https://en.wikipedia.org/wiki/Fran%C3%A7ois_Vi%C3%A8te#Adriaan_van_Roomen's_problem 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 https://en.wikipedia.org/wiki/Taxicab_number (Ramanujan had worked on 'taxicab numbers' before)
- nabla9 5y agoReality is multicausal. 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.
- tshaddox 5y ago> Reality is multicausal. 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.
- MichaelBurge 5y agoDeterministic 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.
- laichzeit0 5y agoI 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.
- ElephantsMyAnus 5y ago>it would have to follow from certain first-principles, and could reason his way back to the highest level of abstraction, with immense ease. 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.
- mjreacher 5y agoThere's plenty of academic literature on the topic, available through everyone's favorite scholarly website if not available to you otherwise. https://link.springer.com/chapter/10.1007/1-4020-4040-7_11 https://link.springer.com/chapter/10.1007/1-4020-4040-7_11 https://direct.mit.edu/posc/article-abstract/18/4/480/15281/Von-Neumann-s-Methodology-of-Science-From https://direct.mit.edu/posc/article-abstract/18/4/480/15281/... https://journals.openedition.org/etudes-benthamiennes/7901?lang=en 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 https://www.ams.org/notices/201302/rnoti-p154.pdf
- elchief 5y agoso, who's today's von Neumann?
- mhh__ 5y agoAsk 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.
- Dracophoenix 5y agoWhen it comes to pure mathematics, Terrance Tao seems to be von Neumann's equivalent. As far as others are concerned, they're probably members of the Prometheus Society. https://en.m.wikipedia.org/wiki/Prometheus_Society https://en.m.wikipedia.org/wiki/Prometheus_Society
- jbotz 5y agoProbably 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.
- outoftheabyss 5y ago
- JackFr 5y ago> 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.
- morelisp 5y agoIf 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".
- smitty1e 5y agoIn 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.
- servytor 5y agoMy 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.
- DebtDeflation 5y ago>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.
- lisper 5y ago
- carapace 5y agoTo 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!
- smegger001 5y ago>Not to put too fine a point on it: he was a metaphysical dullard! or just hedging his bets.
- oh_sigh 5y agoI 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.
- graycat 5y agoAt 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.
- ttyprintk 5y agohttps://archive.ph/Q6awE https://archive.ph/Q6awE