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The Mathematician (1947) (st-and.ac.uk)
81 points by jonnybgood 6 months ago | hide | past | web | favorite | 15 comments

John Von Neumann was a terrifyingly intelligent person. I've never read anything by him that wasn't intellectually stimulating and rewarding. Thanks for this.

Indeed. There is an interesting discussion on Quora about whether he was smarter than Einstein.


The conclusion seems to be that he had a higher raw IQ but Einstein was a 'deeper' thinker and better aesthetician. General relativity (which took Einstein 10 years of 'thought experiments') is often considered the greatest achievement of any single human mind. In Einstein's words:

"After a certain high level of technical skill is achieved, science and art tend to coalesce in aesthetics, plasticity, and form. The greatest scientists are artists as well." - Albert Einstein, 1923

From that page:

According to George Pólya:

“The only student of mine I was ever intimidated by. He was so quick. There was a seminar for advanced students in Zürich that I was teaching and von Neumann was in the class. I came to a certain theorem, and I said it is not proved and it may be difficult. Von Neumann didn't say anything but after five minutes he raised his hand. When I called on him he went to the blackboard and proceeded to write down the proof. After that I was afraid of von Neumann”.

Worth noting for those who may not know, Pólya was himself an extremely prolific mathematician with many major contributions to mathematics.

I'm not sure I entirely follow what he's saying. My understanding is that Von Neumann is saying that at its core, mathematics is about describing nature, so the mathematician should prioritize keeping those descriptions useful over rigorous. This isn't an attempt to summarize the whole article, just to see if I'm in the ballpark on what he's saying.

Yeah you've got it.

My read is he's saying for the most part mathematics arises in empiricism, in real world problems. Once the problem is translated into mathematics useful tools develop, but then over time branches will spawn that become increasingly abstract/aesthetic and are generally a waste of time.

I also liked the beginning bit where he says you generally can't understand something until you have "assimilated it in an instinctive and empirical way". You need that muscle memory, even for intellectual subjects.

Although obviously focused on his daughter, "The Martian's Daughter" [1] provides additional interesting insights into his life. What continually amazes me was his capacity to navigate large scale organizations and bureaucracy to achieve his objectives. I typically find smart people have high frustration with slow moving bureaucracy and for him, I can only imagine that it moved even slower. Perhaps this is how he could do small things like invent Game theory in the cracks between other projects.

[1] https://www.amazon.com/Martians-Daughter-Memoir-Marina-Whitm...

Does anyone happen to know more about the relationship of mathematics with philosophy and epistemology? This to me is more interesting than its relationship with the natural sciences and empiricism; although analysis of the latter relationship seems to have potential insight that is interesting as well.

The SEP is always a good place to start: https://plato.stanford.edu/entries/philosophy-mathematics/

Philosophy of mathematics is just one part of the connection between philosophy and mathematics. Many historical philosophers have been heavily influenced by mathematical thinking.

To give just a few examples from European history of philosophy: Spinoza took the geometric method (basically the axiomatic method) from Euclid. Kant is heavily drawing on geometry and the discovery of non-Euclidean geometry has been suggested as disproving him. If you look into Hegel's Encyclopedia of the Sciences, you will find a lengthy discussion of calculus.

In current analytic philosophy there are many other areas of overlap and cross-polination, for example in decision theory and metaphysics.

You might like Russel's Introduction to Mathematical Philosophy https://people.umass.edu/klement/imp/imp-ebk.pdf

From the Introduction:

"And as we need two sorts of instruments, the telescope and the microscope, for the enlargement of our visual powers, so we need two sorts of instruments for the enlargement of our logical powers, one to take us forward to the higher mathematics, the other to take us backward to the logical foundations of the things that we are inclined to take for granted in mathematics."

The literature on this subject is both vast and populated by the work of the most eminent philosophers -- both continental and analytic -- of the last two hundred years.

Here are some books that I liked. However, there are three caveats. First, being a great philosopher does not mean that one is a great writer. Second, I don't agree with all of these philosophers (indeed, most of them don't agree with _themselves_, since they changed their minds over time). Finally, I myself am not a philosopher, so this list is necessarily somewhat idiosyncratic:

- Edmund Husserl's Philosophy of Arithmetic and Logical Investigations. In the first, he studies what we mean by numbers, and comes to (in my view) a very modern cognitive-scientific/computer-scientific view (ie, that we have direct access only to very small numbers and that our understanding of bigger numbers is symbolic). In the second, he turns around and rejects the idea that reductionism can turn logic into psychology, inventing phenomenology in the process and authoring the root source of modern continental philosophy.

- Alain Badiou's _Number and Numbers_. If you need a Maoist continental philosopher who would disdain the entire Internet Bayesian rationalist community for being a bunch of squishy relativists who don't genuinely believe in capital-T Truth or capital-R Reality, then have I got the man for you! (After all, one should believe in true things because they are true, not because it would let you win a bloody bet. You should believe in true things even if that would make you lose a bet.)

Badiou is a hard-core set-theoretic Platonist realist who passionately hates everything in logic that I love ("[...] if it is true that mathematics, the highest expression of pure thought, in the final analysis consists of nothing but syntactical apparatuses, grammars of signs, then a fortiori all thought falls under the constitutive rule of language"), and I love him for it.

- Ludwig Wittgenstein's _Tractatus Logico-Philosophicus_and Philosophical Investigations_. I find PI much more convincing than the Tractatus, but it's worth reading because it makes a strongly-argued case for logical realism. The big danger with Wittgenstein is that his epigrammatic style is infectious (he's the best writer on this list by a country mile).

- David Lewis's books _Convention_ and _Counterfactuals_. I read _Convention_ as formalizing Wittgenstein's idea of a language-game in terms of game theoretic ideas of equilibria (as well as less formal ideas like Schelling points). _Counterfactuals_ is good (despite Lewis's bonkers modal realism) because counterfactuals and causation are an important part of actual real-world-applicable logic (as opposed to formal logic).

- Michael Dummett's On the Logical Basis of Metaphysics. This is probably the book that has had the biggest impact on how I think about everything. His analysis of realism anti-realism is beautiful and subtle, and makes one of the best arguments for intuitionistic mathematics I have ever read.

I can also recommend Proofs and Refutations by Imre Lakatos. It's an unusually easy read for a philosophy book (fewer than 200 pages and written as a sort of Socratic dialogue) and engages with some very deep problems related to the nature of mathematical proofs.

Some kind of kerning or OCR error lead to the appearance of "modem mathmatics" several times in the article.

Von Neumann predicted the Internet and arxiv and the polymath project ;-)


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