That's a very interesting article, I'm surprised it doesn't seem to be more widely known.
966 karma · joined August 19, 2020
A quick Google search of the author would also show that they have published several similar proofs before [0] which were also wrong.
https://old.reddit.com/r/math/comments/w1zyn8/favourite_math...
Of those listed there I would like to highlight this one:
https://www.youtube.com/c/Aleph0/videos
A few others that I've found recently also in math/physics:
https://www.youtube.com/c/PhysicsExplainedVideos/videos
https://www.youtube.com/c/SimulatingPhysics/videos
https://www.youtube.com/channel/UC3L8AcX1xJEhTofnCF2MAaQ/vid...
https://www.youtube.com/c/ScienceClicEN/videos
The French version of the above channel has a lot more videos but unfortunately aren't translated to English yet.
Proper lectures by professors of various math/physics topics unfortunately are scattered around YouTube and I'm surprised no one has gathered a big list of them given that during COVID a lot of stuff has been put online. I've only found the following two channels which gather various lectures given however at a graduate/research level so you're unlikely to learn much unless you have a significant background:
http://geosci.uchicago.edu/~kite/doc/von_Neumann_1955.pdf
(see page 8 of the pdf)
[0]: https://time.com/magazine/us/5264136/may-14th-2018-vol-191-n...
[1]: https://www.nytimes.com/2017/05/20/world/asia/china-cia-spie...
[2]: https://www.nytimes.com/2021/10/05/us/politics/cia-informant...
Most biographies of him strangely enough don't really go into too much detail on his work at the AEC, perhaps because there's already so much to talk about but also perhaps because a lot of this stuff was classified. It doesn't help either that the earliest biographies were only written more than 20 years after he died, and back in his lifetime in the first half of the 20th century not as much was written down as these days. I am assuming you are talking about his life in the 1940s and 1950s when you say "this portion of his life" so I'll give recommendations based off that. All the main biographies generally cover it but I'll give a few notes to help you make a decision. Not in any particular order.
- The Martian's Daughter: A Memoir
His daughter's autobiography, first part of the book talks about their relationship and given she was born in 1935 most of it covers the last parts of his life.
- John von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life and Death
This is the first biography by Heims; goes more into detail about their war work but I would point out though that Heims pretty clearly aims to denigrate von Neumann as a war hawk, none of the quotes are false or anything like that but when the book first came out several of von Neumann's colleagues criticized the book for trying to character assassinate him.
- The Computer from Pascal to von Neumann
Goldstine talks about the history of the computer but about two-thirds of the book covers work von Neumann and him both did together and there is plenty of biographical material on von Neumann. Obviously more focused on computers though.
- John von Neumann and the Origins of Modern Computing
Unlike Goldstine's book this is a biography von Neumann, however unlike Goldstine Aspray didn't know von Neumann as a close collaborator. Like Goldstine's it also focuses more on computing, not as much Goldstine however a lot of von Neumann's automata work is tied up with computers so it's a good read regardless.
- Prisoner's Dilemma: John Von Neumann, Game Theory, and the Puzzle of the Bomb
This book focuses more on game theory with less biographical material on von Neumann, however it does have a dedicated chapter to his last years and what he was doing then.
- John von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game Theory, Nuclear Deterrence, and Much More
This is probably the biggest biography, it is very even describing each stage of von Neumann's life. It is a bit more of a hero worship compared the other books which you'll notice if read the reviews of the book. Less technical compared to the others too.
- The World as a Mathematical Game: John Von Neumann and Twentieth Century Science
This one tries to evaluate von Neumann's contributions to 20th century science overall, as it was written in 2009 it does source to the previous books quite a bit. Shorter compared to the others.
- The Man from the Future: The Visionary Life of John von Neumann
This one was only released this year, it's a biography but it kind of tries to describe the world around von Neumann at his time and how he fits into it. Obviously by now almost everyone who knew him personally aside from his daughter and a handful of others are dead so it focuses more on his technical contributions rather than anecdotes on his influence.
- Turing's Cathedral: the Origins of the Digital Universe
This one is also more focused on computing, has some things that aren't written in the others as Dyson did a particularly good job going through some of the archives, but not as much focused on von Neumann compared to the others.
- A Fiery Peace in a Cold War: Bernard Schriever and the Ultimate Weapon
Isn't a biography of von Neumann but of Schriever, who worked with von Neumann to setup the first ICBM programs in the USA. Obviously the main focus here is on the military and political side of von Neumann's work in the 1940s and 50s, and not much else in terms of his life. However contains lots of information on this particular part of his life that other books lack.
If you look at the "Further Reading" section of his Wikipedia page there is a long list of books that feature him heavily in case you want more. Happy reading!
Some of his colleagues at the Institute of Advanced Study and in other places resented this. They said he was wasting his time, wasting his talent, on this work that could be done by other people, while his mathematical brain could be doing academic research that others could not do. Just shortly after being appointed commissioner of the US Atomic Energy Commission (AEC), the pinnacle of his non-academic career, he was diagnosed with cancer. Within 2 years he would be dead. At the time of his first diagnosis the main academic subject he was dealing with was his theory of automata, however at first he was optimistic about his cancer and continued working heavily on things to do with the AEC. After some time the doctors made it clear to him that he was going to die soon, and he should wrap up any affairs that he wanted to complete quickly before he died.
Now he panicked, after living his life and having so many incomplete things he wanted to do he was going to die and he was running out of the one thing he could not escape from - time. He tried to finish the topic he was currently working on, the theory of automata, however cancer affected him quicker and quicker and he could not. He wouldn't even finish a lecture he was asked to give - Yale's Silliman lecture, although the lecture he didn't finish would be cobbled together and published as a book, The Computer and the Brain, as would his work on automata, which was edited by Arthur Burks. He had grand aspirations for his theory of automata, it would be his greatest work, something he created entirely on his own, combining mathematical logic, information theory and biology. However, he put other things first, and he never got to finish it, indeed it seemed like he wanted to write far more, the book edited by Burks covered only 2 or 3 of the planned set of 5 lectures, and this was only the first set of five.
After he died several of his colleagues again made comments when interviewed that they felt that his talents were wasted. Considering his working life was only about 30 years they felt much of the last 10 years of his life, primarily spent consulting and working with the government, would be better spent on things that only von Neumann could do, his treatise on von Neumann algebras, his work on automata (incidentally, his theory of automata hasn't really made much progress since he died, especially in comparison to other fields), many other things that he worked on for a bit, got interested in other things, and said he would come back to later.
I am not sure what conclusion I should make of this, but I hope this little story is interesting to others too.
Funny you mention that, only recently I was reading a review paper on the state of constructive quantum field theory [0]. In the outlook section the author writes
>Why haven’t these models of greatest physical interest been constructed yet (in any mathematically rigorous sense which preserves the basic principles constantly evoked in heuristic QFT and does not satisfy itself with an uncontrolled approximation)? Certainly, one can point to the practical fact that only a few dozen people have worked in CQFT. This should be compared with the many hundreds working in string theory and the thousands who have worked in elementary particle physics. Progress is necessarily slow if only a few are working on extremely difficult problems
But they also say
> It may also be the case that a completely new approach is required
This kind of mathematical physics is generally considered a part of mathematics rather than physics, and this paper is talking about formulating a rigorous mathematical framework and elucidating conceptual ideas rather than about making new predictions, but the idea that new mathematics is required is certainly not a crazy one.
On the topic of document, quite a few times I've seen mathematicians say work on theorem provers and the sort are a waste of time and there's nothing to be learnt and blah blah blah and I think this is an excellent example of why this isn't the case. Kevin found a deficiency in his knowledge in algebraic geometry and upon spending some time on it related it to some wider topics in the foundations of mathematics (the notion of equality between set theory and homotopy type theory, for example see https://mathoverflow.net/questions/263024/does-equality-betw...). I would think there are many other things that we will discover that have been otherwise assumed or taken for granted when more mathematics is formalized by theorem provers and other tools.
For example, The Abel Prize [1] does interviews with their award winners every year and quite often they are asked about this question, and most of them agree that as a mathematician your peak "mathematical powers" so to speak and energy are at their highest in your 20s and 30s, while afterwards you have to rely more on experience to make up for it as you grow older.
https://www.youtube.com/playlist?list=PLVV0r6CmEsFzeNLngr1Jq...
You are judged before you meet someone by your social media page. You can't know everything about a person before meeting them, or really without being a good friend of theirs, so social media becomes a proxy for this, and not a particularly good one because you put what you want on it and thus teenagers try to game it in various ways (for example only posting holiday pics to show how glamorous your life is).
It becomes a race towards the bottom in so many ways and in particular for girls, because so much value is placed on physical appearance (either by others or by the girls themselves) and how 'cool' are you and thus it makes you constantly compare and judge yourself with others. It is no surprise that teenage mental health is plummeting when a lot of time is spent on social media, it constitutes a large part of the status game, and there is no reasonable alternative. This is why I think it's a sole factor because I can imagine you would have the same problem even if you took away other economic/political/environmental problems.
Not to say those also don't have an effect, because they obviously do, but unlike those other things, every teenager wants friends, everyone wants a social life, it's not something only half of them know and the other half don't know much about because they read it on the news or learnt about it from school, it's something that affects every single one of them every single day. You cannot escape from it.
Surprisingly despite being a guy I don't know many guys with mental health issues, maybe they don't want to talk about it to other guys, maybe I am not close enough with them, but most girls I know do have mental health issues, and every single one of them (n~=50), literally without exception, has self esteem and body image issues. And it worries me, a lot, because how can we have it that 50% of the population suffers like this every day for years on end and nothing changes? What will the end result be?
Plus arXiv papers are freely accessible ;)
"It would appear that we have reached the limits of what is possible to achieve with computer technology, although one should be careful with such statements, as they tend to sound pretty silly in five years."
- attributed to von Neumann, 1949.