Could someone edit the Wikipedia, please? To remove the faulty citations in that screenshot? I'd suggest starting the Child Prodigy section instead with:
John von Neumann was a child prodigy who by the age of 12 "already had an astonishing grasp of advanced mathematics." source: The Recollections of Eugene Wigner.
Rather than the current incorrect "8 figure" claim.
I've heard sometimes it's better to ask for an editor or someone with experience, since there are lots of rules, and first-time accounts editing a big page like Johnny's might be distrusted. But I really don't know anything about Wikipedia's details.
There was much discussion of this piece. I’m thankful to everyone who replied. As I said in a footnote, I expected a few changes given the size and length of this piece. I've added in two new pieces of supporting evidence: the Mandlebrot story of Johnny forgetting a proof (someone on social media mentioned this, but I can't find the account to credit them), and also that Goldstine's second anecdote about Johnny's perfect memory is far less impressive. Additionally, I’ve made four changes to maximize accuracy and fairness: all minor, and none affecting the thesis.
1. Due to the recent scholarly evidence that Johnny likely added variable addresses to the “von Neumann” architecture, which is at least one novel part of the full whole he’s credited with, the sentence describing him as just “mixing and matching” was expanded to imply inclusions of at least some original contributions, although I don’t get into the details.
2. Eugene Wigner’s quote about Einstein vs. Johnny is no longer described as him judging the “Smartest Man” but the “Greatest Scientific Mind.” This is because it was pointed out Wigner could be read as distinguishing the two types of intelligence at another point of his recollections and specifically saying the scientific greatness is not based on speed, but on the overall qualities closer to what he assigns to Einstein (but he never gives a description of “smart”).
3. The “He did not have a photographic memory” quote is expanded to provide more context from Macrae, since there’s ambiguity around it, as Macrae says it in the context of remembering faces, but also suggests it impacts how Johnny visualized things and did math. Its original truncated version was in a paragraph merely arguing that he "couldn't remember literally everything" (a high bar) and Macrae's statement supports that. However, it could be read as the decider of all the other debates and anecdotes, being so strong, so I included more context.
4. I no longer describe some unspecified past biographies as “pop-sci slop,” I just point out some contain errors (I’m fine with books containing mistakes, it’s inevitable, although I think some of these tall tales did need more checking).
I assume this is partly meant as a disagreement with my https://www.astralcodexten.com/p/book-review-the-man-from-the-future , but sorry, I still think I'm (mostly) right. I've read the same von Neumann biographies you have, and as far as I can tell you're citing them in a misleading, bordering on deceptive, way. I'll just give a few examples but I think the pattern they present extends to the rest of the piece also.
TALE OF TWO CITIES: You discount Goldstine's account of him reciting Tale of Two Cities from memory, because you say MacRae gives "critical context that deflates it"- that he memorized this book in particular to learn English. But here's what MacRae actually says:
>> "But [von Neumann] did intend early that he should escape from Europe to America if he could, and he hoped his facility in languages could help toward this. He had devised a cunning way of getting the syntax in foreign languages, such as English, right. He read selected books in the language he wanted to get the feel of, very quickly but with enormous concentration, so that every word in the passages he chose was implanted in his mind. Through this practice, he was able at age fifty in the early 1950s to baffle Herman Goldstine by quoting the first dozen pages of Dickens’s Tale of Two Cities word for word."
There is no claim here that Tale of Two Cities was unique or that Goldstine specifically chose Tale of Two Cities because he knew it was the one English book JvN has memorized. All that MacRae says is that JvN was reading all of these English books because he wanted to learn English to go to America. So when you say that Goldstine's choice of TOTC "couldn't have been a random book" because "it was the exact same passage Macrae reports Johnny purposefully memorized years before", this is disingenuous - MacRae mentions a general tendency to read things "with enormous concentration" (is this the same as deliberate memorization? I'm not sure), and brings up TOTC only in the context of it being the one Goldstine tested him on!
I don't really believe JvN memorized "every book he ever read", but I do think it's plausible that he memorized large sections of many books basically for fun and for language practice, then remembered them for decades. It's always possible that JvN specifically suggested TOTC to Goldstine because it was one of the few he had memorized, and that Goldstine lied about this, but there is no positive evidence for this story.
GREEK: You say that Goldstine "makes a number of much-cited wild claims about Johnny, including about his early life, things that Goldstine could not possibly have known for sure. First, he says that 'He and his father joked together in Classical Greek'” - here you are implying that he must be making this up, because how could he have known this? But Goldstine very clearly explains how he knows this - he says von Neumann told him! And MacRae also writes that von Neumann told "Princeton colleagues" about his tendency to joke in Greek at age 6 (I don't know if he only means Goldstine, or if others reported this too). He does include the line "It has to be said that the rest of the family denies this," but this statement has no cited source. I assume the source was JvN's brother Nicholas, who I think was the only family member still alive for MacRae to interview, but Nicholas was eight years younger than John and would not have been alive when John was six. So contra the way you frame this, Goldstine has a perfectly valid source but I don't know how MacRae would know this.
EIGHT-DIGIT NUMBERS: You say it must be false that von Neumann could calculate eight-digit numbers in his head at age 6, because you found the claim in Harry Henderson, and you think he is misquoting MacRae, who was talking about adulthood. But Henderson is correctly quoting Paul Halmos, who worked as von Neumann's assistant, and who said it about age six specifically (although he describes it as an "unverifiable" "story in circulation"). See https://gwern.net/doc/math/1973-halmos.pdf .
MacRae does use the "half wrong" framing device, but it's unclear in what sense he thinks the claim that JvN could multiply eight-digit numbers was half-wrong. At the point where he should explain this, all he says is:
>> "The most important use to which he had put his memory was that he had stuffed an unprecedented number of mathematical constants and equations into it. Most of us have very few mathematical constants in our mind, perhaps only the up-to-twelve-times multiplication table. Johnny had put in his mind layers and layers of algebraic verities. These were the explanation of his extraordinary powers of mental calculation. He was not actually better than many other mathematicians — or indeed than some vaudeville freaks — at multiplying one eight-digit figure by another. But he used his accumulation of mathematical constants and equations to become a startling problem-solver and extraordinary concept-expander."
By the claim that he was "no better than vaudeville freaks", MacRae seems to imply that he could multiply eight-digit numbers, but so could some other freakish people, so it doesn't matter. I'm not sure what the claim that this is based on memorized "algebraic verities" is supposed to imply, or why this takes away from his achievement. I think it's misleading to take the fact that MacRae used a framing device saying this was "half-wrong" out of context, and then accuse people who mention it of saying "half-wrong" things, unless you understand the sense in which this was wrong better than I do. In any case, I don't know what MacRae's source is, and whether he is just repeating Halmos.
WEEPING MATH TEACHER: You argue against the claim that "His first math teacher wept when he met Johnny (false)". You mock people who believe this, calling it "a gem I heard from the brain trust on social media". But you relegate the evidence to a footnote, and as far as I can tell your evidence only shows that there was another math teacher before the one who wept, and so the weeper was the second math teacher. Given how irrelevant a disagreement this is - I don't even think most versions of this claim online specify that it was the first - I think it's bizarre how hostile you are to the people who believe it; anyone who didn't go through the footnotes one by one would think you had disproven that a math teacher wept at JvN's talent at all.
NO PHOTOGRAPHIC MEMORY: You quote MacRae saying he had "no sort of photographic memory" to imply that his memory wasn't really that good, and contrasting this with the claim that he could "remember literally anything he wants". But here's the MacRae paragraph that includes this phrase:
>> "His memory and feel for words, plus unsurpassed feel for mathematical symbols, had not extended to memory for faces. All his life he was embarrassed by not knowing people who clearly knew him. He had no sort of photographic memory. This imposed some limitations on his mathematics (he was not good at envisaging shapes) but probably also added to some of his strengths. It is difficult for anybody with a photographic mind to think in terms of more than three dimensions. Johnny had no difficulty in thinking in terms of a quarter of a dimension or of some minuscule fraction of a dimension or several hundred thousand dimensions or infinite dimensions. He just moved the algebraic symbols for these across the chessboard of his mind."
MacRae is saying that, although his memory for words and mathematical symbols was amazing, it was not "photographic" in the sense of applying to faces.
I think this also gets at a more general objection I have to this section. MacRae is telling all of these amazing stories about the crazy things JvN can do, and you're quote-mining all of them for a few words you can take out of context to suggest that he wasn't really all that great. Not only would someone who went back to the source find you are citing the few words in a misleading way, but they would find them placed in the middle of countless anecdotes about how great JvN indeed was, which you pretend don't exist because you can't fit them into your debunking project.
(continued in reply because full comment is too long)
MacRae concludes - and you sort of also conclude, but then back away from at the last moment - the obvious explanation, which is that Jews were succeeding all over Europe, and Budapest had more Jews than anywhere else (cf. Berlin 3% Jewish vs. Budapest 23%), plus a better selection filter for the richest and smartest Jews. I think this matches very well with what you call "pop hereditarianism", as does the fact that von Neumann's father was also a genius, his mother came from a family of geniuses, that his brother and daughter (while not quite geniuses) were both accomplished in their fields, etc.
You mention that there were two non-Jewish Nobel Prize winners in the Budapest of the age, but neither of those are considered among the Martians. Two non-Jewish Nobel Prize winners is well within the base rate for European cities of the time - for example, Madrid during the same period also contained two current-or-future Nobel Prize in Medicine winners. Nobody is surprised that Budapest included two Nobel Prize winners - they're surprised about the Martians!
I know you have a special interest in aristocratic tutoring, but you don't provide any evidence that aristocratic tutoring was any better in Budapest than anywhere else that there were aristocrats. You claim that Budapest "may have been one of the few places and times in all of history where you could hire a bevy of top-notch professional private tutors and governesses without being obscenely wealthy", but this isn't true - you could do the same in any college town in America today (I can't remember if I got this example from you - if so, thank you).
The "superb high schools in Budapest" are only judged superb retroactively because they produced so many geniuses. I have looked pretty hard for what teaching techniques they did differently or better from anywhere else, and I can't find them - except maybe the Hungarian math competitions, which are great but have failed to produce Martians post-1950 despite still going on. I've already discussed why I think your attempts to disprove that von Neumann was smart long before high school are tenuous, but it doesn't matter - there's equally good evidence for many of the other Martians (for example, TMWLON claims that age four Erdos could calculate in his head the number of seconds people had lived, given their age). Also, I think Nicholas says JvN's high school teachers complained he didn't do the assignments anyway.
No piece can ever be entirely without errors, but I think the overly aggressive style of this essay - the insistence that you've debunked everyone else, that they only read Wikipedia but you are a virtuous fact checker, that everyone who disagrees with you is "pop hereditarians" (are you a "pop nurturist"? why not?) places a higher burden on you to get things right.
I think a more honest version of this piece would have said that some of these stories come from stories told by JvN's Princeton friends, and others from stories told by JvN himself, and all of these could have been tall tales or grown in the telling; if you don't think this level of intelligence is possible, then you have to believe this. But I don't think your framing - that you have absolutely debunked this and that everyone except you is a credulous fool - matches the strength of your evidence.
There are lots of people who talk and write about John von Neumann. So no, this piece isn't aimed specifically at you (unless you identify as a pop-hereditarian, but if so, that’s sort of your choice).
However, a note. You use pretty strong language here. You seem to think that the point of the piece is to perfectly balance compliment of Johnny with criticism. That's the point of a full biography, not here. And Johnny’s been complimented more than enough! I also clearly praise him many times, very explicitly.
Driven by that perceived injustice, you accuse me of presenting points in “in a misleading, bordering on dishonest, way” and of overhyping the debunking. However, everything you list ends up being either (a) contradicted by further sources, or (b) a minor quibble. Only one even gets close to mattering at all, and there's other supportive evidence on my side for it.
I don’t think I’m overhyping the main thrust either. For instance, it does impact what you've written yourself. Here's how you originally set up the main motivation for your take on Johnny and The Martians that you’ve expanded on over the years:
>>> “By age ten, John von Neumann, greatest of the Hungarian supergeniuses, already spoke English, French, German, Italian, and Ancient Greek, knew integral and differential calculus, and could multiply and divide 8-digit numbers in his head. Wikipedia notes that on his first meeting with his math teacher, the math teacher “was so astounded with the boy’s mathematical talent that he was brought to tears”. This doesn’t sound like a guy whose potential was kindled by formal education. This sounds like a guy who would have become one of history’s great mathematicians even if his teachers had slept through his entire high school career.”
I think we can both agree that most of this is debunked here. So I’m not sure why you wouldn't start closer to “Okay, most of this stuff is good to know” and go from there, instead of starting with HERE ARE ALL THE PROBLEMS and the problems are things like, “I can imagine less parsimonious explanations.” As always though, I do appreciate the amount of time you put into these many responses. However, I believe that all the sources end up being on my side in the examples. I’ll probably make a small change due to your comments to reflect the one ambiguity, but that might be tomorrow. I've ordered them by ease of resolution.
EIGHT-DIGIT NUMBERS:
We simply cannot deny Macrae specifically writes “half wrong” about multiplying such numbers in Johnny’s head. You ask what I mean to take away from his achievement by quoting that? Nothing. You point to more complimentary passages from Macrae. I’m not sure the one you use, where he is “no better than vaudeville freaks” is that supportive, but okay. The important point is that Harry Henderson misremembered something ascribed “half-wrongly” (whatever exactly you feel that means) to an *adult* Johnny. You admit that’s true, but skip over it as somehow minor in order to defend Johnny from "half wrong," whereas it’s the thing itself.
WEEPING MATH TEACHER:
>>> “You argue against the claim that "His first math teacher wept when he met Johnny (false)…. But you relegate the evidence to a footnote, and as far as I can tell your evidence only shows that there was another math teacher before the one who wept, and so the weeper was the second math teacher."
He had a long succession of math teachers before the weeping story. I hoped that was clear. First, he had math teachers before he was 10 because of preschool teachers/tutors who came to his house. Then he has Ratz. THEN he has Fekete. For years. Probably some others. Then Szego. Who weeps. But the weeping story is quite late, maybe 15 years old. The age and order matter because I’m implying that the tutoring had an effect. I’m sorry it ended up in a footnote but it's nitpicking to be mad about that.
GREEK:
>>> “First, he says that 'He and his father joked together in Classical Greek'” - here you are implying that he must be making this up, because how could he have known this? But Goldstine very clearly explains how he knows this - he says von Neumann told him! …. He does include the line "It has to be said that the rest of the family denies this," but this statement has no source.”
Macrae’s denial is citing both Johnny’s brother and also his second wife, I believe. Other evidence contradicts the story though, which I didn't mention. In Johnny’s brother’s account Greek is specifically listed as something that the Lutheran Gymnasium taught, starting at age 14. Max sends them to the Lutheran school so as to get a classical education and learn well things like Latin and Greek. And Ulam specifically notes “He remembered his school Latin and Greek remarkably well.” So then, do we have any reason then to believe he was fluent as a child? I’m not even sure he was fluent as an adult! We see stories and jokes from Johnny get misinterpreted all the time (the phone book, the Byzantine professor, etc). It was probably a joke or slight exaggeration, and then Goldstine picked it up.
NO PHOTOGRAPHIC MEMORY:
I agree that he’s talking about Johnny’s memory of faces, but nothing about its usage in the [original] text is wrong. It’s right next to “could rarely remember a name” and it’s in a paragraph with the point of saying he couldn’t remember “literally anything.” I think the most you could say is that other readers will apply that to all other things, and so Macrae would be a conclusive disproof of all Johnny’s abilities, whereas I mean it as a disproof of the topic of that paragraph. I’ll probably add a small (he meant their faces) here.
[EDIT: Since this was only thing about which I thought Scott had a point, it led to a minor change. Scott was wrong to say it's just for faces, because Macrae says lacking a photographic memory impacted Johnny's mathematics in the next sentence. But without further context a reader might take the quote to imply the case is entirely closed from it alone. So I instead just made the "no photographic memory" quote longer to let the quote speak for itself, as there's so much disagreement about it. This change impacts nothing about the actual thrust or argumentation or thesis overall, since none of it relies on that (much later used) quote. And the longer quote still perfectly supports what that specific section is saying too, just as the original did]
TALE OF TWO CITIES:
This is the only one that I think bears on the main topic of the post directly. But your theory cannot be true. Macrae specifically gives us additional context that the books were being studied to learn English. Goldstine does not in his reporting of it. So he is not relying just on Goldstine's anecdote, as you say. Instead, he is connecting the dots to something he knows from elsewhere. That's why he says "in these two cases," and picks out that they were "select books" and makes the specific claim about learning English, which Goldstine doesn't.
I didn't get to mention that Goldstine's other example, btw, right below the Dickens, is that von Neumann gave a lecture and used his old letters and symbols as he'd done in the original research 20 years ago. That's clearly incredibly weak tea as an anecdote.
MARTIAN STUFF
I don’t really see anything that deals the “it was baked in from the beginning of the education pipeline so why are you surprised?” argument, especially for such a small n. At this point, we're just explaining the position of the immigrants in Budapest, and seem to be letting Budapest do the work.
TUTORING STUFF
I don't think it's easy to replicate the socio-intellectual conditions of Budapest in 21st century America. Perhaps our differences is just that I have tried to do things like look for good tutors and it is *extremely* difficult and time consuming. I almost started an entire company just to solve this problem!
How dare you question Rationalist Myths! Von Neumann was a demigod, and don't you forget it! Too bad the hysteria over on X hasn't reached a level where Elon chimes in and uses his god-like intelligence to decide whether JvN qualifies as a demigod.
On the last point---have you written on some of the factors that prevent an aristocratic tutoring company from starting? It seems we already have tutoring-centric social customs in place for musical talent---so what keeps people from adopting similar customs for mathematical and linguistic talent?
"the obvious explanation, which is that Jews were succeeding all over Europe, and Budapest had more Jews than anywhere else (cf. Berlin 3% Jewish vs. Budapest 23%)"
If you want to explain mathematical productivity of different areas primarily in terms of general European Jewish excellence rather than in terms of idiosyncracies of culture in the region (even if we focus specifically on the regional Jewish culture), shouldn't we be looking at absolute numbers rather than per capita? Page at https://encyclopedia.ushmm.org/content/en/gallery/jews-in-prewar-germany-1 lists Jewish populations in some major cities around 1933, including Berlin with around 160,000, Budapest is not included there but https://www.yadvashem.org/righteous/stories/budapest-historical-background.html says the population immediately before the war was around 200,000. So if general Jewish excellence alone is the answer wouldn't we expect the number of Jewish mathematical geniuses from Berlin to be almost as large as the number from Budapest?
The jews of Hungary may have had a very unique blend of genes(Central asian, central european and jewish*) that made them uniquely talented in sciences and particularly mathematics.
But we know from DNA this is not likely true, German Jews were frequently descended from more eastern Jews who moved there recently, and all Ashkenazi Jews descend from people who once lived in Germany around the Middle Ages. Modern analysis has German and Hungarian Jews, quite close, only difference is German Jews are slightly more Levantine, they are 95% identical. Although surnames frequently changed in Jewish communities, the fact is most Hungarian Jews had Germanic surnames( i.e. Von Neumann, Wigner), with some Magyarizing them from 1848 on to assimilate more.
Can you provide a citation for the DNA claims? One would assume that they had some hungarian dna since they were there for at least several centuries. It's also the thing that the hungarias were jewdified to the extend that reportedly around 7% of them have at least 25% of Jewish dna.
There is a long series of studies on Ashkenazi DNA, many helpful, some that are not useful at all, like Eran Elhaik's. This is an area where I believe AIs that can pull up info from lots of different sources to create a decent unified assessment. This is what I got from Chat GPT:
Hungarian Jew vs German Jew genetically:
95%+ identical core Ashkenazi ancestry
Small regional differences in European admixture (East-Central vs West-Central Europe)
+1–3% more East-Central European ancestry, on average
Both fall within the same tight Ashkenazi genetic cluster
Also, many Jews did intermarry with Hungarians, but if they did so, their descendants would overwhelmingly not be a part of the Jewish community, and would be assimilated into the Hungarian population.
As someone without a dog in this fight, this seems confusingly nitpicky. Like, the Tale of Two Cities thing you cite seems totally consistent with someone who's memorized some passages as a party trick?
>Grandfather Kann had gone straight from commercial high school into founding his business, but he proved to be demonic in his capacity for arithmetic manipulation. He could add in his head monstrous columns of numbers or multiply mentally two numbers in the thousands or even millions. The six-year-old Johnny would laboriously perform the computation with pencil and paper, and announce with glee that Grandfather had been absolutely on the mark.
So MacRae *explicitly* says 6yo JvN could not do multiplication in his head!
Hi Scott, do you happen to know if there is any evidence that JvN was tutored in math before the age of 10?
(Also, do you, in all honesty, believe that JvN could multiply 8 digit numbers in his head at age 6? Halmos says he could *divide* 8 digit numbers (would the answer be a 1 digit number?); and Halmos also suggests some of the stories he relays are fictional)
I don't know anything about pop-hereditarianism, but what I do know about hereditarians, including (actually especially) the most extreme ones, is that what they advocate is exactly the system that produced von Neumann, namely that you take the most talented pupils and pump all your resources into giving them the best education instead of throwing it down the drain on the general population. Not only that, but they are the *only* people who advocate such a system. So, while this may demonstrate my ignorance of social-media subcultures, it seems to me that this is all an argument in favour of the people you are attacking.
The obvious model here is that competence equals the product of genes times environment. That leaves plenty of room for environment to play a role (we're not being pure hereditarians here), and also suggests that it is more valuable to focus on the education of geniuses, since each unit increase in "education" will buy you a greater increase in "competence" if your "genes" multiplier is larger.
For the public to benefit from this arrangement, we should also focus on instilling a sense of noblesse oblige on the beneficiaries of the tutoring. There's little point in pouring all our resources into educating a few geniuses if they choose to go into high-frequency trading, or start a new cryptocurrency, or code up a misaligned AI, or write esoteric math proofs which never find application, or build an even-more-addictive social media app. In fact, it could actually be counterproductive! The impact of moral education seems very under-studied. Why is it that so many geniuses are currently wasting their talent?
Furthermore, any genetic engineering for IQ increase should be accompanied by genetic engineering for virtue. Otherwise we risk creating a race of supervillains. Just look at how humans treat animals in factory farms for a preview of how we might be treated. https://www.palladiummag.com/2025/11/01/factory-farming-is-a-blight/
Both the extreme nurture and extreme nature position appear to absurd as to be caricatures. Who would say that had Michael Jordan never touched a basketball until age 18 that he would have become a six-time MVP? But who would say that a person 5 foot 3 person with poor hand to eye coordination could become an NBA champion with the right education?
The difference is that I have never met anyone who actually advocated the extreme nature position, only people trying to refute it. But the extreme nurture position is the default orthodoxy of polite society.
It's worth noting that Wigner's memoir was published when he was 90 years old, coauthored by a collaborative memoirist. This line, thus, doesn't necessarily reflect any depth or nuance in Wigner's thoughts on the topic, and it's obviously a leap to present this single line written 35 years after the death of von Neumann as reflective of the mindset of the Martians, collectively.
Interestingly, Feynman - a colleague of von Neumann and Wigner - was apparently averse to even acknowledging *individual* differences in ability - let alone, group differences (see here: https://x.com/carl_feynman/status/1758846050268303857).
Conceivably, that could shed light on Wigner's apparent comment. That is, perhaps there was a taboo in such circles in discussing even glaring truths like the fact that they were much more intellectually gifted than average. Thus, we should infer even less about what the Martians collectively actually thought about group differences, and the reason for their incredible talents from an apparent protestation to the very concept on one of their parts.
"Let me insert here a word or two about the Jewish family structure which is not irrelevant to the Jewish tradition of learning. At all times, the young learned man, and especially the rabbi... was always a match for the daughter of the rich merchant. Biologically this led to a situation in sharp contrast to that of the Christians of earlier times. The Western Christian learned man was... less fertile than the community around him. On the other hand, the Jewish scholar was very often in a position to have a large family. Thus the biological habits of the Christians tended to breed out of the race whatever hereditary qualities make for learning, whereas the biological habits of the Jew tended to breed these qualities in. To what extent this genetic difference supplemented the cultural trend for learning among the Jews is difficult to say. But there is no reason to believe that the genetic factor was negligible."
Incidentally, Feynman and his colleagues at the Manhattan Project largely shared their Ashkenazi ethnicity, although they varied in country of origin and background.
Einstein, whom this article insists was smarter than von Neumann, precipitated the project, and wasn't Hungarian. Per his Wiki, it sounds like he had only limited tutoring, mentioning just a single tutor who quickly felt that young Einstein was out of his league.
Feynman was born in the US to a homemaker and sales manager, attended a public high school, apparently received no tutoring, and largely taught himself college level mathematics. The same goes for several others, there.
Often followed up by "there's nothing to see there" and the insinuation that less successful people are just morally inferior in some way. Like a beautiful girl who insists it all just comes from within.
This is gratifying to read. A few months back, someone I thought seemed interesting and thoughtful DM’d me on Substack and asked if I wanted to join a Discord of Substackers who are interested in open dialogue and ideas etc. I thought it sounded cool so I joined, only for it to be chock full of classic anti-woke, “heterodox” edge lord type stuff, but the most frequent posters were diehard hereditarians. They went so far as saying they’d try to “convert” me ha.
They said it was dispassionate and idea-driven, but that wasn’t my experience. Many of them had read a few genetics papers and would go on and on about alleles and use other borrowed vocabulary etc. but upon close inspection this well-oiled machine of intellectual scaffolding was really like a Rube Goldberg propping up pretty conventional biases. Just consistently reiterating that their evidence wasn’t dispositive was enough to really raise their hackles, so much that one of the moderators and Discord owner got angry at me and DM’d me an apology a few days later. I thought it would be too dramatic to announce my departure so I still get a ping every few weeks but I still find it funny.
Thats not to say I don’t think there is any truth in hereditarianism. I just think that people who think it’s the keystone to a deep understanding of the world’s paradoxes rely too much on the conspiracy that it’s being suppressed. I find it helpful to remember that most conspiratorial thinking is usually the result of fear—that life is complex and problems are complex and it’s much easier to say “there are insurmountable genetic gaps in ability” than “the world is unjust and inefficient, and we perpetuate this in some way just by existing in it.”
This isn't your thrust here, but it seems worth pointing out to me: your doing somewhere between a disservice and misrepresentation of ol' johnny's mathematical contributions. In particular his contributions to areas of pure mathematics that are more abstract and so less relatable to a non-mathematician. Eg game theory gets a mention but operator algebras doesn't, though his contributions to the latter has massively impacted/defined many fields of pure math. Granted that for the same reason you don't discuss them they don't contribute much to his mythos, that's still badfeels in the context of a comprehensive report like this one
A couple other pure math specific comments
- in this field it doesn't matter much what people say about one's results/work right when it's completed, eg a proof being kludgy, developments being annoying technical elaborations, the work being not that original. The arc of academic time does a near perfect signal filtering in pure math thx to the deduction underpinning - the ideas/frames/concepts/etc that endure are the signal. Their endurance is the signal. Trivial/superficial contributions are never remembered
- ugly proofs that moved the needle somewhere genuinely new are commonly how new ideas emerge - they have the nucleus of the thing that later gets clarified. Johnny did a lot of that
- formalizing ideas correctly is also a massive and central part of pure math. They let the deductive machinery proceed forward, others grasp the thing. Crystallizing out the right formalism, ie discovering the appropriate frame, is how fields flourish and underpins how metaphors between fields (the main activity of value in pure math research) can be found
These feel kinda like nitpicks to me too, but also kinda not. This dude's legacy is hugely defined in terms of his mathematical work and it seems a bit like you don't get it. Said with love
No, I actually totally agree with you, but it's more a limitation of the format thing than a purposeful choice to downplay anything. If someone paid me to write a biography of John von Neumann, I would include a lot more about his work and its positive influence. I'm just coming from a place where so many other biographies have basically done this (although even in them, interestingly enough, operator algebras don't crop up a bunch, likely because it's more abstract than his other stuff). And, frankly, this was almost 8,000 words and a ton of work. So I have to make explicit choices about what to include and what not to include, and what I think readers will actually get through.
Ya, does make sense re cutting/curating for length + audience
But if your thesis is that johnny's work was not so genius - and you did weave in a bunch of deets to that effect - then leaving out this field is pretty disingenuous
Like, we don't characterize Grothendeick's life's work as ineffectual activism
"if your thesis is that johnny's work was not so genius"
That's not my thesis. I think I'm clear he's one of the great geniuses of the 20th century, as I specifically say that (or other variants) a few times.
I would recommend the one by Norman Macrae. It has an error in the tutoring progression, and he plays down Johnny's role in the EDVAC fiasco to his subject's benefit, and he has some ambiguous language around Johnny's mental powers. But it's a good biography overall.
Did you read Turing’s Cathedral by George Dyson? It is an early history of computing that covers the personalities involved. I honestly found it quite dull but it corroborated a lot of what you said about the intensity of his tutoring in Hungary, and how the “Martians” operated a peer model partly out of ease but also because they wanted to get their friends out of Europe as prospects began to darken
Nice balanced discussion. One other point is that the strong hereditarians (those who believe genius like Von Neumann almost entirely genetic rather than mix of innate and environmental advantages) tend to assume mathematical genius is primarily just a matter of high level of "general intelligence" applying itself to mathematical problems, but when mathematicians talk about the thought processes behind highly developed mathematical talent they often seem to disagree with this model. Instead they seem to think that high-achieving mathematicians have achieved their results in large part due to becoming highly developed in the use of specific techniques or mental frameworks which can be applied to problems in innovative ways. For example from https://www.ams.org/notices/199701/comm-rota.pdf
'A long time ago an older and well-known number theorist made some disparaging remarks about Paul Erdös’s work. You admire Erdös’s contributions to mathematics as much as I do, and I felt annoyed when the older mathematician flatly and definitively stated that all of Erdös’s work could be “reduced” to a few tricks which Erdös repeatedly relied on in his proofs. What the number theorist did not realize is that other mathematicians, even the very best, also rely on a few tricks which they use over and over. Take Hilbert. The second volume of Hilbert’s collected papers contains Hilbert’s papers in invariant theory. I have made a point of reading some of these papers with care. It is sad to note that some of Hilbert’s beautiful results have been completely forgotten. But on reading the proofs of Hilbert’s striking and deep theorems in invariant theory, it was surprising to verify that Hilbert’s proofs relied on the same few tricks. Even Hilbert had only a few tricks!'
Terence Tao also talks about particular mental frameworks and visualization techniques he uses at https://mathoverflow.net/a/38882 and David Bessis talks in the thread at https://x.com/davidbessis/status/1816459945824395648 about how 'how I progressed from undergrad to grad student to career mathematician, the epiphanies along the way, how I broke through my perceived glass ceiling. In the end, it no longer felt innate, it felt more like there was an untaught "method"', and at https://x.com/davidbessis/status/1816461991130607813 he says 'No-one can copy what you do in your head, your secret mental tricks. You just get better and better. No-one understands why and they just think you’re gifted.'
Stanislaw Ulam's book Adventures of Mathematician analyzes von Neumann's talent (as well as Norbert Wiener's) in similar terms on p. 95-97:
'Wiener had a sense of what is worth thinking about, and he understood the possibilities of using mathematics for seemingly more important and more visible applications in theoretical physics. He had a marvelous technique for using Fourier transforms, and it is amazing how much the power of algorithms or symbolism could accomplish. I am always amazed how much a certain facility with a special and apparently narrow technique can accomplish. Wiener was a master at this. I have seen other mathematicians who could do the same in a more modest way. For instance, Steinhaus obtained quite penetrating insights into other fields, and his student, Mark Kac, now at Rockefeller University, surpassed him. Antoni Zygmund in Chicago, another Pole, is a master of the great field of trigonometric series. Several of his students have obtained epoch-making results in other fields-for example, Paul Cohen, who did this in set theory, the most general and abstract part of mathematics.
I don't think Wiener was particularly fond of combinatorial thinking or of working on foundations of mathematico-logical or set theoretical problems. At the beginning of his career, he may have gone in this direction, but later he applied himself to other fields and to number theory.
Von Neumann was different. He also had several quite independent techniques at his fingertips. (It is rare to have more than two or three.) These included a facility for symbolic manipulation of linear operators. He also had an undefinable "common sense" feeling for logical structure and for both the skeleton and the combinatorial superstructure in new mathematical theories. This stood him in good stead much later, when he became interested in the notion of a possible theory of automata, and when he undertook both the conception and the construction of electronic computing machines. He attempted to define and to pursue some of the formal analogies between the workings of the nervous system in general and of the human brain itself, and the operation of the newly developed electronic computers.'
Excellent writing. I have read many Neumann biographies (his Hungarian name with Hungarian name order is Neumann János, 'von' was just added abroad). There is an understandable mythos-creating mechanism in his story. He wasn't a child prodigy in the sense that e.g. Mozart was, but he was much much more talented than his age group. His fantastic teacher at Fasori Gimnázium, László Rácz recognized that in time.
> So we have four fluent languages, his ability to add “el” to words, and at least some unclear amount of Latin and Greek from school as an adult—deeply impressive, except for the Spanish!
I wanted to add that this looks similar to von Neumann's joke about knowing numbers. It also is a regular American joke to do that. It's probably his humor.
You're spot on. There's actually a ton of the stories like that. Just nowadays no one understands wit, since that requires reading comprehension.
E.g., there's the much-repeated anecdote (which goes something like this, I'm just paraphrasing here) about the professor of history who says to Johnny's wife: "I will come to your next party, but only if we don't discuss Byzantine history. I'm supposed to be the world expert on it, and if people hear Johnny talk, they won't think it's true anymore!"
And then pop biographers use it to show how smart he his, like "Wow, that was a real ask, he definitely was stipulating to not do that because Johnny knew so much more. That Johnny! That brain!" While actually it was either a joke or Johnny was just... annoying.
It's wit and probably manners and basic social interactions. The internet probably isn't any bit helpful given there are few forced social interactions one has to maintain.
It also makes you wonder whether the story about him joking in Greek as a child was just that… a joke.
Like he was a kid, I’m sure he just had a couple jokes or puns. I have made a few (dumb) jokes in languages I am nowhere near fluent in.
Have you seen the documentary from 1966? It contains a lot of firsthand accounts from his contemporaries and although I watched it some years ago, I recall the anecdotes to feel much more real and revealing of his personality.
Great piece! Brings to mind tales of supposed savants who just studied mnemonics and methods of calculating pi, etc. In addition to his tutors, young vN was raised in a general intellectual environment, with lots of smart people regularly coming over for dinner and discussing their specialties. Here's hoping we can all afford the modern equivalent of governesses and tutors.
I'd love for your next book to be on the topic of parenting and/or tutoring - and particularly how to bring these methods to the modern era. I've enjoyed all of your pieces that touch on tutoring - or related topics like teaching early reading.
Of course he knew the numbers in the Manhattan phone book; it's easy. They're assigned by the NANPA NPA-NXX rules like all phone numbers in North America, so the first three are 212 for the area code, followed by three digits to identify blocks of 10,000 four digit numbers, 0000-9999, skotch some special use numbers. There you have it!
Which ones out of the possible numbering space are assigned and in-service, and to who, well, now you're getting clever:-)
But in an important sense, once you know the rules - the algorithm - you do in fact know which numbers *can* be in it and which *cannot*. Nothing that starts with 202 or 415 will be in it, for example. No number longer than 10 digits will be in it. No float will be in it, only integers, and positive ones at that. And it's a reasonable guess that the numbering plan will be pretty full as it's Manhattan after all, so nearly all the possible numbers will be in there.
And you know, we've learned something here! It's possible to compress this large mass of data into a short generative problem; just the compression is lossy. Really lossy:0
Could someone edit the Wikipedia, please? To remove the faulty citations in that screenshot? I'd suggest starting the Child Prodigy section instead with:
John von Neumann was a child prodigy who by the age of 12 "already had an astonishing grasp of advanced mathematics." source: The Recollections of Eugene Wigner.
Rather than the current incorrect "8 figure" claim.
I've heard sometimes it's better to ask for an editor or someone with experience, since there are lots of rules, and first-time accounts editing a big page like Johnny's might be distrusted. But I really don't know anything about Wikipedia's details.
Should be simple since you provided clear sourcing.
In fact it was already done yesterday!
Errata:
There was much discussion of this piece. I’m thankful to everyone who replied. As I said in a footnote, I expected a few changes given the size and length of this piece. I've added in two new pieces of supporting evidence: the Mandlebrot story of Johnny forgetting a proof (someone on social media mentioned this, but I can't find the account to credit them), and also that Goldstine's second anecdote about Johnny's perfect memory is far less impressive. Additionally, I’ve made four changes to maximize accuracy and fairness: all minor, and none affecting the thesis.
1. Due to the recent scholarly evidence that Johnny likely added variable addresses to the “von Neumann” architecture, which is at least one novel part of the full whole he’s credited with, the sentence describing him as just “mixing and matching” was expanded to imply inclusions of at least some original contributions, although I don’t get into the details.
2. Eugene Wigner’s quote about Einstein vs. Johnny is no longer described as him judging the “Smartest Man” but the “Greatest Scientific Mind.” This is because it was pointed out Wigner could be read as distinguishing the two types of intelligence at another point of his recollections and specifically saying the scientific greatness is not based on speed, but on the overall qualities closer to what he assigns to Einstein (but he never gives a description of “smart”).
3. The “He did not have a photographic memory” quote is expanded to provide more context from Macrae, since there’s ambiguity around it, as Macrae says it in the context of remembering faces, but also suggests it impacts how Johnny visualized things and did math. Its original truncated version was in a paragraph merely arguing that he "couldn't remember literally everything" (a high bar) and Macrae's statement supports that. However, it could be read as the decider of all the other debates and anecdotes, being so strong, so I included more context.
4. I no longer describe some unspecified past biographies as “pop-sci slop,” I just point out some contain errors (I’m fine with books containing mistakes, it’s inevitable, although I think some of these tall tales did need more checking).
I assume this is partly meant as a disagreement with my https://www.astralcodexten.com/p/book-review-the-man-from-the-future , but sorry, I still think I'm (mostly) right. I've read the same von Neumann biographies you have, and as far as I can tell you're citing them in a misleading, bordering on deceptive, way. I'll just give a few examples but I think the pattern they present extends to the rest of the piece also.
TALE OF TWO CITIES: You discount Goldstine's account of him reciting Tale of Two Cities from memory, because you say MacRae gives "critical context that deflates it"- that he memorized this book in particular to learn English. But here's what MacRae actually says:
>> "But [von Neumann] did intend early that he should escape from Europe to America if he could, and he hoped his facility in languages could help toward this. He had devised a cunning way of getting the syntax in foreign languages, such as English, right. He read selected books in the language he wanted to get the feel of, very quickly but with enormous concentration, so that every word in the passages he chose was implanted in his mind. Through this practice, he was able at age fifty in the early 1950s to baffle Herman Goldstine by quoting the first dozen pages of Dickens’s Tale of Two Cities word for word."
There is no claim here that Tale of Two Cities was unique or that Goldstine specifically chose Tale of Two Cities because he knew it was the one English book JvN has memorized. All that MacRae says is that JvN was reading all of these English books because he wanted to learn English to go to America. So when you say that Goldstine's choice of TOTC "couldn't have been a random book" because "it was the exact same passage Macrae reports Johnny purposefully memorized years before", this is disingenuous - MacRae mentions a general tendency to read things "with enormous concentration" (is this the same as deliberate memorization? I'm not sure), and brings up TOTC only in the context of it being the one Goldstine tested him on!
I don't really believe JvN memorized "every book he ever read", but I do think it's plausible that he memorized large sections of many books basically for fun and for language practice, then remembered them for decades. It's always possible that JvN specifically suggested TOTC to Goldstine because it was one of the few he had memorized, and that Goldstine lied about this, but there is no positive evidence for this story.
GREEK: You say that Goldstine "makes a number of much-cited wild claims about Johnny, including about his early life, things that Goldstine could not possibly have known for sure. First, he says that 'He and his father joked together in Classical Greek'” - here you are implying that he must be making this up, because how could he have known this? But Goldstine very clearly explains how he knows this - he says von Neumann told him! And MacRae also writes that von Neumann told "Princeton colleagues" about his tendency to joke in Greek at age 6 (I don't know if he only means Goldstine, or if others reported this too). He does include the line "It has to be said that the rest of the family denies this," but this statement has no cited source. I assume the source was JvN's brother Nicholas, who I think was the only family member still alive for MacRae to interview, but Nicholas was eight years younger than John and would not have been alive when John was six. So contra the way you frame this, Goldstine has a perfectly valid source but I don't know how MacRae would know this.
EIGHT-DIGIT NUMBERS: You say it must be false that von Neumann could calculate eight-digit numbers in his head at age 6, because you found the claim in Harry Henderson, and you think he is misquoting MacRae, who was talking about adulthood. But Henderson is correctly quoting Paul Halmos, who worked as von Neumann's assistant, and who said it about age six specifically (although he describes it as an "unverifiable" "story in circulation"). See https://gwern.net/doc/math/1973-halmos.pdf .
MacRae does use the "half wrong" framing device, but it's unclear in what sense he thinks the claim that JvN could multiply eight-digit numbers was half-wrong. At the point where he should explain this, all he says is:
>> "The most important use to which he had put his memory was that he had stuffed an unprecedented number of mathematical constants and equations into it. Most of us have very few mathematical constants in our mind, perhaps only the up-to-twelve-times multiplication table. Johnny had put in his mind layers and layers of algebraic verities. These were the explanation of his extraordinary powers of mental calculation. He was not actually better than many other mathematicians — or indeed than some vaudeville freaks — at multiplying one eight-digit figure by another. But he used his accumulation of mathematical constants and equations to become a startling problem-solver and extraordinary concept-expander."
By the claim that he was "no better than vaudeville freaks", MacRae seems to imply that he could multiply eight-digit numbers, but so could some other freakish people, so it doesn't matter. I'm not sure what the claim that this is based on memorized "algebraic verities" is supposed to imply, or why this takes away from his achievement. I think it's misleading to take the fact that MacRae used a framing device saying this was "half-wrong" out of context, and then accuse people who mention it of saying "half-wrong" things, unless you understand the sense in which this was wrong better than I do. In any case, I don't know what MacRae's source is, and whether he is just repeating Halmos.
WEEPING MATH TEACHER: You argue against the claim that "His first math teacher wept when he met Johnny (false)". You mock people who believe this, calling it "a gem I heard from the brain trust on social media". But you relegate the evidence to a footnote, and as far as I can tell your evidence only shows that there was another math teacher before the one who wept, and so the weeper was the second math teacher. Given how irrelevant a disagreement this is - I don't even think most versions of this claim online specify that it was the first - I think it's bizarre how hostile you are to the people who believe it; anyone who didn't go through the footnotes one by one would think you had disproven that a math teacher wept at JvN's talent at all.
NO PHOTOGRAPHIC MEMORY: You quote MacRae saying he had "no sort of photographic memory" to imply that his memory wasn't really that good, and contrasting this with the claim that he could "remember literally anything he wants". But here's the MacRae paragraph that includes this phrase:
>> "His memory and feel for words, plus unsurpassed feel for mathematical symbols, had not extended to memory for faces. All his life he was embarrassed by not knowing people who clearly knew him. He had no sort of photographic memory. This imposed some limitations on his mathematics (he was not good at envisaging shapes) but probably also added to some of his strengths. It is difficult for anybody with a photographic mind to think in terms of more than three dimensions. Johnny had no difficulty in thinking in terms of a quarter of a dimension or of some minuscule fraction of a dimension or several hundred thousand dimensions or infinite dimensions. He just moved the algebraic symbols for these across the chessboard of his mind."
MacRae is saying that, although his memory for words and mathematical symbols was amazing, it was not "photographic" in the sense of applying to faces.
I think this also gets at a more general objection I have to this section. MacRae is telling all of these amazing stories about the crazy things JvN can do, and you're quote-mining all of them for a few words you can take out of context to suggest that he wasn't really all that great. Not only would someone who went back to the source find you are citing the few words in a misleading way, but they would find them placed in the middle of countless anecdotes about how great JvN indeed was, which you pretend don't exist because you can't fit them into your debunking project.
(continued in reply because full comment is too long)
SOURCE OF MARTIANS: I tried to respond to arguments like yours here in my posts on von Neumann, including https://www.astralcodexten.com/p/book-review-the-man-from-the-future . I don't think they're very plausible.
MacRae concludes - and you sort of also conclude, but then back away from at the last moment - the obvious explanation, which is that Jews were succeeding all over Europe, and Budapest had more Jews than anywhere else (cf. Berlin 3% Jewish vs. Budapest 23%), plus a better selection filter for the richest and smartest Jews. I think this matches very well with what you call "pop hereditarianism", as does the fact that von Neumann's father was also a genius, his mother came from a family of geniuses, that his brother and daughter (while not quite geniuses) were both accomplished in their fields, etc.
You mention that there were two non-Jewish Nobel Prize winners in the Budapest of the age, but neither of those are considered among the Martians. Two non-Jewish Nobel Prize winners is well within the base rate for European cities of the time - for example, Madrid during the same period also contained two current-or-future Nobel Prize in Medicine winners. Nobody is surprised that Budapest included two Nobel Prize winners - they're surprised about the Martians!
I know you have a special interest in aristocratic tutoring, but you don't provide any evidence that aristocratic tutoring was any better in Budapest than anywhere else that there were aristocrats. You claim that Budapest "may have been one of the few places and times in all of history where you could hire a bevy of top-notch professional private tutors and governesses without being obscenely wealthy", but this isn't true - you could do the same in any college town in America today (I can't remember if I got this example from you - if so, thank you).
The "superb high schools in Budapest" are only judged superb retroactively because they produced so many geniuses. I have looked pretty hard for what teaching techniques they did differently or better from anywhere else, and I can't find them - except maybe the Hungarian math competitions, which are great but have failed to produce Martians post-1950 despite still going on. I've already discussed why I think your attempts to disprove that von Neumann was smart long before high school are tenuous, but it doesn't matter - there's equally good evidence for many of the other Martians (for example, TMWLON claims that age four Erdos could calculate in his head the number of seconds people had lived, given their age). Also, I think Nicholas says JvN's high school teachers complained he didn't do the assignments anyway.
No piece can ever be entirely without errors, but I think the overly aggressive style of this essay - the insistence that you've debunked everyone else, that they only read Wikipedia but you are a virtuous fact checker, that everyone who disagrees with you is "pop hereditarians" (are you a "pop nurturist"? why not?) places a higher burden on you to get things right.
I think a more honest version of this piece would have said that some of these stories come from stories told by JvN's Princeton friends, and others from stories told by JvN himself, and all of these could have been tall tales or grown in the telling; if you don't think this level of intelligence is possible, then you have to believe this. But I don't think your framing - that you have absolutely debunked this and that everyone except you is a credulous fool - matches the strength of your evidence.
I stand by most of the claims I made in https://www.astralcodexten.com/p/book-review-the-man-from-the-future , which I think were appropriately caveated, and encourage anyone who remains uncertain to read MacRae's biography and see which of us treated it more faithfully.
There are lots of people who talk and write about John von Neumann. So no, this piece isn't aimed specifically at you (unless you identify as a pop-hereditarian, but if so, that’s sort of your choice).
However, a note. You use pretty strong language here. You seem to think that the point of the piece is to perfectly balance compliment of Johnny with criticism. That's the point of a full biography, not here. And Johnny’s been complimented more than enough! I also clearly praise him many times, very explicitly.
Driven by that perceived injustice, you accuse me of presenting points in “in a misleading, bordering on dishonest, way” and of overhyping the debunking. However, everything you list ends up being either (a) contradicted by further sources, or (b) a minor quibble. Only one even gets close to mattering at all, and there's other supportive evidence on my side for it.
I don’t think I’m overhyping the main thrust either. For instance, it does impact what you've written yourself. Here's how you originally set up the main motivation for your take on Johnny and The Martians that you’ve expanded on over the years:
>>> “By age ten, John von Neumann, greatest of the Hungarian supergeniuses, already spoke English, French, German, Italian, and Ancient Greek, knew integral and differential calculus, and could multiply and divide 8-digit numbers in his head. Wikipedia notes that on his first meeting with his math teacher, the math teacher “was so astounded with the boy’s mathematical talent that he was brought to tears”. This doesn’t sound like a guy whose potential was kindled by formal education. This sounds like a guy who would have become one of history’s great mathematicians even if his teachers had slept through his entire high school career.”
I think we can both agree that most of this is debunked here. So I’m not sure why you wouldn't start closer to “Okay, most of this stuff is good to know” and go from there, instead of starting with HERE ARE ALL THE PROBLEMS and the problems are things like, “I can imagine less parsimonious explanations.” As always though, I do appreciate the amount of time you put into these many responses. However, I believe that all the sources end up being on my side in the examples. I’ll probably make a small change due to your comments to reflect the one ambiguity, but that might be tomorrow. I've ordered them by ease of resolution.
EIGHT-DIGIT NUMBERS:
We simply cannot deny Macrae specifically writes “half wrong” about multiplying such numbers in Johnny’s head. You ask what I mean to take away from his achievement by quoting that? Nothing. You point to more complimentary passages from Macrae. I’m not sure the one you use, where he is “no better than vaudeville freaks” is that supportive, but okay. The important point is that Harry Henderson misremembered something ascribed “half-wrongly” (whatever exactly you feel that means) to an *adult* Johnny. You admit that’s true, but skip over it as somehow minor in order to defend Johnny from "half wrong," whereas it’s the thing itself.
WEEPING MATH TEACHER:
>>> “You argue against the claim that "His first math teacher wept when he met Johnny (false)…. But you relegate the evidence to a footnote, and as far as I can tell your evidence only shows that there was another math teacher before the one who wept, and so the weeper was the second math teacher."
He had a long succession of math teachers before the weeping story. I hoped that was clear. First, he had math teachers before he was 10 because of preschool teachers/tutors who came to his house. Then he has Ratz. THEN he has Fekete. For years. Probably some others. Then Szego. Who weeps. But the weeping story is quite late, maybe 15 years old. The age and order matter because I’m implying that the tutoring had an effect. I’m sorry it ended up in a footnote but it's nitpicking to be mad about that.
GREEK:
>>> “First, he says that 'He and his father joked together in Classical Greek'” - here you are implying that he must be making this up, because how could he have known this? But Goldstine very clearly explains how he knows this - he says von Neumann told him! …. He does include the line "It has to be said that the rest of the family denies this," but this statement has no source.”
Macrae’s denial is citing both Johnny’s brother and also his second wife, I believe. Other evidence contradicts the story though, which I didn't mention. In Johnny’s brother’s account Greek is specifically listed as something that the Lutheran Gymnasium taught, starting at age 14. Max sends them to the Lutheran school so as to get a classical education and learn well things like Latin and Greek. And Ulam specifically notes “He remembered his school Latin and Greek remarkably well.” So then, do we have any reason then to believe he was fluent as a child? I’m not even sure he was fluent as an adult! We see stories and jokes from Johnny get misinterpreted all the time (the phone book, the Byzantine professor, etc). It was probably a joke or slight exaggeration, and then Goldstine picked it up.
NO PHOTOGRAPHIC MEMORY:
I agree that he’s talking about Johnny’s memory of faces, but nothing about its usage in the [original] text is wrong. It’s right next to “could rarely remember a name” and it’s in a paragraph with the point of saying he couldn’t remember “literally anything.” I think the most you could say is that other readers will apply that to all other things, and so Macrae would be a conclusive disproof of all Johnny’s abilities, whereas I mean it as a disproof of the topic of that paragraph. I’ll probably add a small (he meant their faces) here.
[EDIT: Since this was only thing about which I thought Scott had a point, it led to a minor change. Scott was wrong to say it's just for faces, because Macrae says lacking a photographic memory impacted Johnny's mathematics in the next sentence. But without further context a reader might take the quote to imply the case is entirely closed from it alone. So I instead just made the "no photographic memory" quote longer to let the quote speak for itself, as there's so much disagreement about it. This change impacts nothing about the actual thrust or argumentation or thesis overall, since none of it relies on that (much later used) quote. And the longer quote still perfectly supports what that specific section is saying too, just as the original did]
TALE OF TWO CITIES:
This is the only one that I think bears on the main topic of the post directly. But your theory cannot be true. Macrae specifically gives us additional context that the books were being studied to learn English. Goldstine does not in his reporting of it. So he is not relying just on Goldstine's anecdote, as you say. Instead, he is connecting the dots to something he knows from elsewhere. That's why he says "in these two cases," and picks out that they were "select books" and makes the specific claim about learning English, which Goldstine doesn't.
I didn't get to mention that Goldstine's other example, btw, right below the Dickens, is that von Neumann gave a lecture and used his old letters and symbols as he'd done in the original research 20 years ago. That's clearly incredibly weak tea as an anecdote.
MARTIAN STUFF
I don’t really see anything that deals the “it was baked in from the beginning of the education pipeline so why are you surprised?” argument, especially for such a small n. At this point, we're just explaining the position of the immigrants in Budapest, and seem to be letting Budapest do the work.
TUTORING STUFF
I don't think it's easy to replicate the socio-intellectual conditions of Budapest in 21st century America. Perhaps our differences is just that I have tried to do things like look for good tutors and it is *extremely* difficult and time consuming. I almost started an entire company just to solve this problem!
How dare you question Rationalist Myths! Von Neumann was a demigod, and don't you forget it! Too bad the hysteria over on X hasn't reached a level where Elon chimes in and uses his god-like intelligence to decide whether JvN qualifies as a demigod.
On the last point---have you written on some of the factors that prevent an aristocratic tutoring company from starting? It seems we already have tutoring-centric social customs in place for musical talent---so what keeps people from adopting similar customs for mathematical and linguistic talent?
"the obvious explanation, which is that Jews were succeeding all over Europe, and Budapest had more Jews than anywhere else (cf. Berlin 3% Jewish vs. Budapest 23%)"
If you want to explain mathematical productivity of different areas primarily in terms of general European Jewish excellence rather than in terms of idiosyncracies of culture in the region (even if we focus specifically on the regional Jewish culture), shouldn't we be looking at absolute numbers rather than per capita? Page at https://encyclopedia.ushmm.org/content/en/gallery/jews-in-prewar-germany-1 lists Jewish populations in some major cities around 1933, including Berlin with around 160,000, Budapest is not included there but https://www.yadvashem.org/righteous/stories/budapest-historical-background.html says the population immediately before the war was around 200,000. So if general Jewish excellence alone is the answer wouldn't we expect the number of Jewish mathematical geniuses from Berlin to be almost as large as the number from Budapest?
The jews of Hungary may have had a very unique blend of genes(Central asian, central european and jewish*) that made them uniquely talented in sciences and particularly mathematics.
But we know from DNA this is not likely true, German Jews were frequently descended from more eastern Jews who moved there recently, and all Ashkenazi Jews descend from people who once lived in Germany around the Middle Ages. Modern analysis has German and Hungarian Jews, quite close, only difference is German Jews are slightly more Levantine, they are 95% identical. Although surnames frequently changed in Jewish communities, the fact is most Hungarian Jews had Germanic surnames( i.e. Von Neumann, Wigner), with some Magyarizing them from 1848 on to assimilate more.
Can you provide a citation for the DNA claims? One would assume that they had some hungarian dna since they were there for at least several centuries. It's also the thing that the hungarias were jewdified to the extend that reportedly around 7% of them have at least 25% of Jewish dna.
There is a long series of studies on Ashkenazi DNA, many helpful, some that are not useful at all, like Eran Elhaik's. This is an area where I believe AIs that can pull up info from lots of different sources to create a decent unified assessment. This is what I got from Chat GPT:
Hungarian Jew vs German Jew genetically:
95%+ identical core Ashkenazi ancestry
Small regional differences in European admixture (East-Central vs West-Central Europe)
+1–3% more East-Central European ancestry, on average
Both fall within the same tight Ashkenazi genetic cluster
Also, many Jews did intermarry with Hungarians, but if they did so, their descendants would overwhelmingly not be a part of the Jewish community, and would be assimilated into the Hungarian population.
As someone without a dog in this fight, this seems confusingly nitpicky. Like, the Tale of Two Cities thing you cite seems totally consistent with someone who's memorized some passages as a party trick?
Actors do stuff like that all the time.
Hey guys look what I found in MacRae!
>Grandfather Kann had gone straight from commercial high school into founding his business, but he proved to be demonic in his capacity for arithmetic manipulation. He could add in his head monstrous columns of numbers or multiply mentally two numbers in the thousands or even millions. The six-year-old Johnny would laboriously perform the computation with pencil and paper, and announce with glee that Grandfather had been absolutely on the mark.
So MacRae *explicitly* says 6yo JvN could not do multiplication in his head!
He also doesn't say how big the calculations are, nor if Johnny was adding or multiplying.
Hi Scott, do you happen to know if there is any evidence that JvN was tutored in math before the age of 10?
(Also, do you, in all honesty, believe that JvN could multiply 8 digit numbers in his head at age 6? Halmos says he could *divide* 8 digit numbers (would the answer be a 1 digit number?); and Halmos also suggests some of the stories he relays are fictional)
I don't know anything about pop-hereditarianism, but what I do know about hereditarians, including (actually especially) the most extreme ones, is that what they advocate is exactly the system that produced von Neumann, namely that you take the most talented pupils and pump all your resources into giving them the best education instead of throwing it down the drain on the general population. Not only that, but they are the *only* people who advocate such a system. So, while this may demonstrate my ignorance of social-media subcultures, it seems to me that this is all an argument in favour of the people you are attacking.
The obvious model here is that competence equals the product of genes times environment. That leaves plenty of room for environment to play a role (we're not being pure hereditarians here), and also suggests that it is more valuable to focus on the education of geniuses, since each unit increase in "education" will buy you a greater increase in "competence" if your "genes" multiplier is larger.
For the public to benefit from this arrangement, we should also focus on instilling a sense of noblesse oblige on the beneficiaries of the tutoring. There's little point in pouring all our resources into educating a few geniuses if they choose to go into high-frequency trading, or start a new cryptocurrency, or code up a misaligned AI, or write esoteric math proofs which never find application, or build an even-more-addictive social media app. In fact, it could actually be counterproductive! The impact of moral education seems very under-studied. Why is it that so many geniuses are currently wasting their talent?
Furthermore, any genetic engineering for IQ increase should be accompanied by genetic engineering for virtue. Otherwise we risk creating a race of supervillains. Just look at how humans treat animals in factory farms for a preview of how we might be treated. https://www.palladiummag.com/2025/11/01/factory-farming-is-a-blight/
Both the extreme nurture and extreme nature position appear to absurd as to be caricatures. Who would say that had Michael Jordan never touched a basketball until age 18 that he would have become a six-time MVP? But who would say that a person 5 foot 3 person with poor hand to eye coordination could become an NBA champion with the right education?
The difference is that I have never met anyone who actually advocated the extreme nature position, only people trying to refute it. But the extreme nurture position is the default orthodoxy of polite society.
Agree very much with all the rest.
"Let me begin by making it clear it was not a matter of genetic superiority. Let us leave such ideas to Adolf Hitler."
Ugh. Anyone who talks like this, it's immediately discrediting on the topic. They're too emotional for dispassionate analysis.
It's worth noting that Wigner's memoir was published when he was 90 years old, coauthored by a collaborative memoirist. This line, thus, doesn't necessarily reflect any depth or nuance in Wigner's thoughts on the topic, and it's obviously a leap to present this single line written 35 years after the death of von Neumann as reflective of the mindset of the Martians, collectively.
Interestingly, Feynman - a colleague of von Neumann and Wigner - was apparently averse to even acknowledging *individual* differences in ability - let alone, group differences (see here: https://x.com/carl_feynman/status/1758846050268303857).
Conceivably, that could shed light on Wigner's apparent comment. That is, perhaps there was a taboo in such circles in discussing even glaring truths like the fact that they were much more intellectually gifted than average. Thus, we should infer even less about what the Martians collectively actually thought about group differences, and the reason for their incredible talents from an apparent protestation to the very concept on one of their parts.
Notably, von Neumann's friend Norbert Wiener did recognize the impact of heredity on Ashkenazi intelligence. He writes in his autobiography (https://direct.mit.edu/books/oa-monograph/3147/chapter-standard/86086/A-Russian-Irishman-in-Kansas-City):
"Let me insert here a word or two about the Jewish family structure which is not irrelevant to the Jewish tradition of learning. At all times, the young learned man, and especially the rabbi... was always a match for the daughter of the rich merchant. Biologically this led to a situation in sharp contrast to that of the Christians of earlier times. The Western Christian learned man was... less fertile than the community around him. On the other hand, the Jewish scholar was very often in a position to have a large family. Thus the biological habits of the Christians tended to breed out of the race whatever hereditary qualities make for learning, whereas the biological habits of the Jew tended to breed these qualities in. To what extent this genetic difference supplemented the cultural trend for learning among the Jews is difficult to say. But there is no reason to believe that the genetic factor was negligible."
Incidentally, Feynman and his colleagues at the Manhattan Project largely shared their Ashkenazi ethnicity, although they varied in country of origin and background.
Einstein, whom this article insists was smarter than von Neumann, precipitated the project, and wasn't Hungarian. Per his Wiki, it sounds like he had only limited tutoring, mentioning just a single tutor who quickly felt that young Einstein was out of his league.
Feynman was born in the US to a homemaker and sales manager, attended a public high school, apparently received no tutoring, and largely taught himself college level mathematics. The same goes for several others, there.
Often followed up by "there's nothing to see there" and the insinuation that less successful people are just morally inferior in some way. Like a beautiful girl who insists it all just comes from within.
This is gratifying to read. A few months back, someone I thought seemed interesting and thoughtful DM’d me on Substack and asked if I wanted to join a Discord of Substackers who are interested in open dialogue and ideas etc. I thought it sounded cool so I joined, only for it to be chock full of classic anti-woke, “heterodox” edge lord type stuff, but the most frequent posters were diehard hereditarians. They went so far as saying they’d try to “convert” me ha.
They said it was dispassionate and idea-driven, but that wasn’t my experience. Many of them had read a few genetics papers and would go on and on about alleles and use other borrowed vocabulary etc. but upon close inspection this well-oiled machine of intellectual scaffolding was really like a Rube Goldberg propping up pretty conventional biases. Just consistently reiterating that their evidence wasn’t dispositive was enough to really raise their hackles, so much that one of the moderators and Discord owner got angry at me and DM’d me an apology a few days later. I thought it would be too dramatic to announce my departure so I still get a ping every few weeks but I still find it funny.
Thats not to say I don’t think there is any truth in hereditarianism. I just think that people who think it’s the keystone to a deep understanding of the world’s paradoxes rely too much on the conspiracy that it’s being suppressed. I find it helpful to remember that most conspiratorial thinking is usually the result of fear—that life is complex and problems are complex and it’s much easier to say “there are insurmountable genetic gaps in ability” than “the world is unjust and inefficient, and we perpetuate this in some way just by existing in it.”
Nice exposition + solid research, ty for that
This isn't your thrust here, but it seems worth pointing out to me: your doing somewhere between a disservice and misrepresentation of ol' johnny's mathematical contributions. In particular his contributions to areas of pure mathematics that are more abstract and so less relatable to a non-mathematician. Eg game theory gets a mention but operator algebras doesn't, though his contributions to the latter has massively impacted/defined many fields of pure math. Granted that for the same reason you don't discuss them they don't contribute much to his mythos, that's still badfeels in the context of a comprehensive report like this one
A couple other pure math specific comments
- in this field it doesn't matter much what people say about one's results/work right when it's completed, eg a proof being kludgy, developments being annoying technical elaborations, the work being not that original. The arc of academic time does a near perfect signal filtering in pure math thx to the deduction underpinning - the ideas/frames/concepts/etc that endure are the signal. Their endurance is the signal. Trivial/superficial contributions are never remembered
- ugly proofs that moved the needle somewhere genuinely new are commonly how new ideas emerge - they have the nucleus of the thing that later gets clarified. Johnny did a lot of that
- formalizing ideas correctly is also a massive and central part of pure math. They let the deductive machinery proceed forward, others grasp the thing. Crystallizing out the right formalism, ie discovering the appropriate frame, is how fields flourish and underpins how metaphors between fields (the main activity of value in pure math research) can be found
These feel kinda like nitpicks to me too, but also kinda not. This dude's legacy is hugely defined in terms of his mathematical work and it seems a bit like you don't get it. Said with love
No, I actually totally agree with you, but it's more a limitation of the format thing than a purposeful choice to downplay anything. If someone paid me to write a biography of John von Neumann, I would include a lot more about his work and its positive influence. I'm just coming from a place where so many other biographies have basically done this (although even in them, interestingly enough, operator algebras don't crop up a bunch, likely because it's more abstract than his other stuff). And, frankly, this was almost 8,000 words and a ton of work. So I have to make explicit choices about what to include and what not to include, and what I think readers will actually get through.
Ya, does make sense re cutting/curating for length + audience
But if your thesis is that johnny's work was not so genius - and you did weave in a bunch of deets to that effect - then leaving out this field is pretty disingenuous
Like, we don't characterize Grothendeick's life's work as ineffectual activism
"if your thesis is that johnny's work was not so genius"
That's not my thesis. I think I'm clear he's one of the great geniuses of the 20th century, as I specifically say that (or other variants) a few times.
Ya true. I think you know what I'm pointing at
Regardless, much appreciation for this work
Amazing piece. Is there biography you do recommend?
I would recommend the one by Norman Macrae. It has an error in the tutoring progression, and he plays down Johnny's role in the EDVAC fiasco to his subject's benefit, and he has some ambiguous language around Johnny's mental powers. But it's a good biography overall.
https://www.amazon.com/John-Von-Neumann-Scientific-Deterrence/dp/082182676X
The other biographies I investigated all contained more factual errors and I wouldn't recommend them.
Did you read Turing’s Cathedral by George Dyson? It is an early history of computing that covers the personalities involved. I honestly found it quite dull but it corroborated a lot of what you said about the intensity of his tutoring in Hungary, and how the “Martians” operated a peer model partly out of ease but also because they wanted to get their friends out of Europe as prospects began to darken
Nice balanced discussion. One other point is that the strong hereditarians (those who believe genius like Von Neumann almost entirely genetic rather than mix of innate and environmental advantages) tend to assume mathematical genius is primarily just a matter of high level of "general intelligence" applying itself to mathematical problems, but when mathematicians talk about the thought processes behind highly developed mathematical talent they often seem to disagree with this model. Instead they seem to think that high-achieving mathematicians have achieved their results in large part due to becoming highly developed in the use of specific techniques or mental frameworks which can be applied to problems in innovative ways. For example from https://www.ams.org/notices/199701/comm-rota.pdf
'A long time ago an older and well-known number theorist made some disparaging remarks about Paul Erdös’s work. You admire Erdös’s contributions to mathematics as much as I do, and I felt annoyed when the older mathematician flatly and definitively stated that all of Erdös’s work could be “reduced” to a few tricks which Erdös repeatedly relied on in his proofs. What the number theorist did not realize is that other mathematicians, even the very best, also rely on a few tricks which they use over and over. Take Hilbert. The second volume of Hilbert’s collected papers contains Hilbert’s papers in invariant theory. I have made a point of reading some of these papers with care. It is sad to note that some of Hilbert’s beautiful results have been completely forgotten. But on reading the proofs of Hilbert’s striking and deep theorems in invariant theory, it was surprising to verify that Hilbert’s proofs relied on the same few tricks. Even Hilbert had only a few tricks!'
Terence Tao also talks about particular mental frameworks and visualization techniques he uses at https://mathoverflow.net/a/38882 and David Bessis talks in the thread at https://x.com/davidbessis/status/1816459945824395648 about how 'how I progressed from undergrad to grad student to career mathematician, the epiphanies along the way, how I broke through my perceived glass ceiling. In the end, it no longer felt innate, it felt more like there was an untaught "method"', and at https://x.com/davidbessis/status/1816461991130607813 he says 'No-one can copy what you do in your head, your secret mental tricks. You just get better and better. No-one understands why and they just think you’re gifted.'
Stanislaw Ulam's book Adventures of Mathematician analyzes von Neumann's talent (as well as Norbert Wiener's) in similar terms on p. 95-97:
'Wiener had a sense of what is worth thinking about, and he understood the possibilities of using mathematics for seemingly more important and more visible applications in theoretical physics. He had a marvelous technique for using Fourier transforms, and it is amazing how much the power of algorithms or symbolism could accomplish. I am always amazed how much a certain facility with a special and apparently narrow technique can accomplish. Wiener was a master at this. I have seen other mathematicians who could do the same in a more modest way. For instance, Steinhaus obtained quite penetrating insights into other fields, and his student, Mark Kac, now at Rockefeller University, surpassed him. Antoni Zygmund in Chicago, another Pole, is a master of the great field of trigonometric series. Several of his students have obtained epoch-making results in other fields-for example, Paul Cohen, who did this in set theory, the most general and abstract part of mathematics.
I don't think Wiener was particularly fond of combinatorial thinking or of working on foundations of mathematico-logical or set theoretical problems. At the beginning of his career, he may have gone in this direction, but later he applied himself to other fields and to number theory.
Von Neumann was different. He also had several quite independent techniques at his fingertips. (It is rare to have more than two or three.) These included a facility for symbolic manipulation of linear operators. He also had an undefinable "common sense" feeling for logical structure and for both the skeleton and the combinatorial superstructure in new mathematical theories. This stood him in good stead much later, when he became interested in the notion of a possible theory of automata, and when he undertook both the conception and the construction of electronic computing machines. He attempted to define and to pursue some of the formal analogies between the workings of the nervous system in general and of the human brain itself, and the operation of the newly developed electronic computers.'
Excellent writing. I have read many Neumann biographies (his Hungarian name with Hungarian name order is Neumann János, 'von' was just added abroad). There is an understandable mythos-creating mechanism in his story. He wasn't a child prodigy in the sense that e.g. Mozart was, but he was much much more talented than his age group. His fantastic teacher at Fasori Gimnázium, László Rácz recognized that in time.
> So we have four fluent languages, his ability to add “el” to words, and at least some unclear amount of Latin and Greek from school as an adult—deeply impressive, except for the Spanish!
I wanted to add that this looks similar to von Neumann's joke about knowing numbers. It also is a regular American joke to do that. It's probably his humor.
You're spot on. There's actually a ton of the stories like that. Just nowadays no one understands wit, since that requires reading comprehension.
E.g., there's the much-repeated anecdote (which goes something like this, I'm just paraphrasing here) about the professor of history who says to Johnny's wife: "I will come to your next party, but only if we don't discuss Byzantine history. I'm supposed to be the world expert on it, and if people hear Johnny talk, they won't think it's true anymore!"
And then pop biographers use it to show how smart he his, like "Wow, that was a real ask, he definitely was stipulating to not do that because Johnny knew so much more. That Johnny! That brain!" While actually it was either a joke or Johnny was just... annoying.
It's wit and probably manners and basic social interactions. The internet probably isn't any bit helpful given there are few forced social interactions one has to maintain.
It also makes you wonder whether the story about him joking in Greek as a child was just that… a joke.
Like he was a kid, I’m sure he just had a couple jokes or puns. I have made a few (dumb) jokes in languages I am nowhere near fluent in.
Have you seen the documentary from 1966? It contains a lot of firsthand accounts from his contemporaries and although I watched it some years ago, I recall the anecdotes to feel much more real and revealing of his personality.
Great piece! Brings to mind tales of supposed savants who just studied mnemonics and methods of calculating pi, etc. In addition to his tutors, young vN was raised in a general intellectual environment, with lots of smart people regularly coming over for dinner and discussing their specialties. Here's hoping we can all afford the modern equivalent of governesses and tutors.
I'd love for your next book to be on the topic of parenting and/or tutoring - and particularly how to bring these methods to the modern era. I've enjoyed all of your pieces that touch on tutoring - or related topics like teaching early reading.
I not equipped to evaluate any of the claims by any of the parties in this dispute, but I learned one thing for sure:
Don't mess with a nerdcel saint.
Just don't.
Great post.
Of course he knew the numbers in the Manhattan phone book; it's easy. They're assigned by the NANPA NPA-NXX rules like all phone numbers in North America, so the first three are 212 for the area code, followed by three digits to identify blocks of 10,000 four digit numbers, 0000-9999, skotch some special use numbers. There you have it!
Which ones out of the possible numbering space are assigned and in-service, and to who, well, now you're getting clever:-)
But in an important sense, once you know the rules - the algorithm - you do in fact know which numbers *can* be in it and which *cannot*. Nothing that starts with 202 or 415 will be in it, for example. No number longer than 10 digits will be in it. No float will be in it, only integers, and positive ones at that. And it's a reasonable guess that the numbering plan will be pretty full as it's Manhattan after all, so nearly all the possible numbers will be in there.
And you know, we've learned something here! It's possible to compress this large mass of data into a short generative problem; just the compression is lossy. Really lossy:0
In reality, and especially at the time, the plan is significantly more restrictive and consequently more predictable than that: https://en.wikipedia.org/wiki/North_American_Numbering_Plan
Glad you mentioned Pitts. The son of a boilermaker with no tutoring. Hopefully, he is what everyone should think of when imagining an autodidact.