Modern scientific endeavors are a stultified and cultish mess of bureaucracy pushing against ever greater natural barriers of complexity. Even assuming someone like vN could survive 16 years of extraordinarily deficient modern American schooling to begin an approach to academia, it would be for naught.
Assuming his talent isn't passed over for belonging to a white male, he then joins a research group where he barely has access to his PI while investigating an extremely specific and relatively unimportant question for which funding was secured during the prior year. Use of equipment in deviation from this program is grounds for a grant audit. Regardless of results, the group must publish at least one paper per year to maintain funding. Data may be falsified.
Graduate coursework is even more strictly defined than undergraduate majors, and interdisciplinary study is strongly discouraged, as his credit-hours would be supported through the department by an R/TA. There is simply no time for him to waste studying such queer novelties as geophysics or molecular biology.
Midway through his PhD, he is informed in no uncertain terms that he must participate in a potentially hazardous medical trial or be suspended from the program. He declines, says his goodbyes to his group members (one of whom barely speaks English but seems like a nice guy), and departs for a happier life raising chickens, which he can afford because he was an early adopter of Bitcoin.
Another reason why Dyson may have called von Neumann a prototypical frog is that the dominant path of von Neumann's work (at least the mathematical parts) was aimed towards solving specific problems. His early work on set theory and Hilbert's proof theory was aimed at providing a rigorous foundation for mathematics. Likewise, his mathematical foundations for quantum mechanics was aimed at providing a solid framework for quantum mechanics. After realising this framework would fail for quantum field theory[0][1], he invented more mathematics to try resolve this. From these attempts came his work on quantum logic, rings of operators (von Neumann algebras), continuous geometries, Jordan algebras, etc. If you look at his collected papers, the dominant theme in his work in pure mathematics and mathematical physics is solving problems, either of others (for example his work on the Banach-Tarski paradox or Hilbert's fifth problem), or his own, as I mentioned above. That is not to say he didn't do others things too, he did plenty, but his work where he used mathematics to solve problems was probably the areas of his work Dyson himself was most familiar with.
This also explains von Neumann's lack of publishing in number theory and topology, because none of that was necessary to know for him to solve the problems he was trying to solve. In comparison, David Hilbert and Hermann Weyl, both of whom you could probably describe as slightly more broader than JvN in pure mathematics, were not so focused on solving big specific problems, but had more philosophical views on why they did mathematics.
As a side note, I think it speaks volumes that there has been no one like von Neumann since he died. As you mention, fields are much more detailed and have been studied much more than they had been in von Neumann's time, but access to knowledge is a lot easier with the internet, not purely just for learning things, but for communicating too. If you look at some of von Neumann's selected letters published by Redei, you'll notice the time gap between some of them when he was exchanging ideas with some foreign scientist could be months. Obviously now you can communicate basically instantly if you so wish. There was also many more parents who push their children to be child protégées and to make them learn lots and lots of things and succeed to a far greater level than some of their other peers at school, making them learn university level mathematics while still in middle school for example, yet there don't seem to be people who achieve as much as JvN did in terms of breadth. I would think that his prodigious memory could have played a very big role here which is something that very few people have. I would also note his love for thinking, he seemed to be almost obsessed with it based on the way Ed Teller described it in JvN's documentary. We must consider whether the culture we have today in education is really based on enjoying learning and thinking as goal itself, or whether it is more based on the idea that we enjoy those things as a pathway to some other goal, whether it be money or fame or something else.
Excellent article. Thank you for republishing it, i could have missed it if you didn't.
Modern scientific endeavors are a stultified and cultish mess of bureaucracy pushing against ever greater natural barriers of complexity. Even assuming someone like vN could survive 16 years of extraordinarily deficient modern American schooling to begin an approach to academia, it would be for naught.
Assuming his talent isn't passed over for belonging to a white male, he then joins a research group where he barely has access to his PI while investigating an extremely specific and relatively unimportant question for which funding was secured during the prior year. Use of equipment in deviation from this program is grounds for a grant audit. Regardless of results, the group must publish at least one paper per year to maintain funding. Data may be falsified.
Graduate coursework is even more strictly defined than undergraduate majors, and interdisciplinary study is strongly discouraged, as his credit-hours would be supported through the department by an R/TA. There is simply no time for him to waste studying such queer novelties as geophysics or molecular biology.
Midway through his PhD, he is informed in no uncertain terms that he must participate in a potentially hazardous medical trial or be suspended from the program. He declines, says his goodbyes to his group members (one of whom barely speaks English but seems like a nice guy), and departs for a happier life raising chickens, which he can afford because he was an early adopter of Bitcoin.
As someone who does jiu jitsu and amateur economics, your BJJ/Wrestling comparison was top notch.
Nice piece, thanks for writing. I enjoyed the recent Von Neumann biography as well.
BTW, I think you meant to write 'overstated' in this sentence: "The importance of this building block to modern economics cannot be understated".
Another reason why Dyson may have called von Neumann a prototypical frog is that the dominant path of von Neumann's work (at least the mathematical parts) was aimed towards solving specific problems. His early work on set theory and Hilbert's proof theory was aimed at providing a rigorous foundation for mathematics. Likewise, his mathematical foundations for quantum mechanics was aimed at providing a solid framework for quantum mechanics. After realising this framework would fail for quantum field theory[0][1], he invented more mathematics to try resolve this. From these attempts came his work on quantum logic, rings of operators (von Neumann algebras), continuous geometries, Jordan algebras, etc. If you look at his collected papers, the dominant theme in his work in pure mathematics and mathematical physics is solving problems, either of others (for example his work on the Banach-Tarski paradox or Hilbert's fifth problem), or his own, as I mentioned above. That is not to say he didn't do others things too, he did plenty, but his work where he used mathematics to solve problems was probably the areas of his work Dyson himself was most familiar with.
This also explains von Neumann's lack of publishing in number theory and topology, because none of that was necessary to know for him to solve the problems he was trying to solve. In comparison, David Hilbert and Hermann Weyl, both of whom you could probably describe as slightly more broader than JvN in pure mathematics, were not so focused on solving big specific problems, but had more philosophical views on why they did mathematics.
As a side note, I think it speaks volumes that there has been no one like von Neumann since he died. As you mention, fields are much more detailed and have been studied much more than they had been in von Neumann's time, but access to knowledge is a lot easier with the internet, not purely just for learning things, but for communicating too. If you look at some of von Neumann's selected letters published by Redei, you'll notice the time gap between some of them when he was exchanging ideas with some foreign scientist could be months. Obviously now you can communicate basically instantly if you so wish. There was also many more parents who push their children to be child protégées and to make them learn lots and lots of things and succeed to a far greater level than some of their other peers at school, making them learn university level mathematics while still in middle school for example, yet there don't seem to be people who achieve as much as JvN did in terms of breadth. I would think that his prodigious memory could have played a very big role here which is something that very few people have. I would also note his love for thinking, he seemed to be almost obsessed with it based on the way Ed Teller described it in JvN's documentary. We must consider whether the culture we have today in education is really based on enjoying learning and thinking as goal itself, or whether it is more based on the idea that we enjoy those things as a pathway to some other goal, whether it be money or fame or something else.
[0]: https://www.sciencedirect.com/science/article/abs/pii/S1355219896000172
[1]: https://link.springer.com/chapter/10.1007/978-94-017-2012-0_18