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Comment by feoren | Hacker News Reader

Part of it is low-hanging fruit, certainly. Euler lived at a time when it was still possible to "know all math". That breadth of knowledge is simply not possible for a single human anymore; the discipline of mathematics is orders of magnitude larger. Comparable mathematicians today like Erdős and Terence Tao collaborate because they really can't learn the intricate details of every corner of math, so they collaborate with people who work in those corners instead. I think these are all people that have a deep understanding of the interconnected structures within math, and that makes them incredibly productive. I'm not going to try to compare the "level of genius" between Euler, Erdős, and Tao (although I think the latter two would readily claim Euler wins), but once you have a gift like that, there's a big difference between having all of mathematics in your head and not.

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I find the notion of 'low-hanging fruit' in such contexts profoundly ahistorical. If graph theory was low-hanging why did it take thousands of years since Sumer or ancient Egypt? Or consider something from number theory: every other batch of students in a math camp I'm familiar with has someone who has 'proved' quadratic reciprocity for themselves -- and how could they not ? -- since childhood they have been immersed in a culture which points at it; while it took Euler roughly 40 years to even formulate the idea and then Gauss to prove it. It's not that - to use an anachronistic term - class field theoretic phenomena was not known to other cultures thousands of years ago.

(Btw there are many contemporary mathematicians at least at the level of Terence Tao but for some reason haven't been blessed by lay popularity -- mostly because Tao's math looks more school math like / familiar to non-math people than say Peter Scholze's)

It's low-hanging fruit because it depended on a bunch of mathematics and technology that Euler benefitted from: algebra, the printing press, and mass-produced paper. Sure, the Ancient Egyptians had papyrus but that is nothing compared to the volume of paper Euler had available to him.

Euler lived nearly three centuries after the invention of the printing press. He had vast numbers of books available to him and essentially unlimited paper to write on. He also had been tutored in algebra which remains the most important development in the history of mathematics. The abstract manipulation of symbols made possible by algebra is such an enormous leap over the geometric methods of the ancient mathematicians. It allows one to solve countless problems trivially in seconds which would take days to solve geometrically.

Your first sentence can be used for anything:

Etale cohomology was a low-hanging fruit because it depended on a bunch of mathematics and technology that Grothendieck benefitted from: algebra, the printing press, and modern transportation. Sure, Euler had horse drawn carriages but that is nothing compared to the speed of modern transportation that Grothendieck had available to him.

> If graph theory was low-hanging why did it take thousands of years since Sumer or ancient Egypt?

Because it wasn't low hanging thousands of years ago. It was only low hanging after an enormous body of foundational work was laid down over those thousands of years. And Euler knew all of it. It's no longer possible to know all of mathematics.

> every other batch of students in a math camp I'm familiar with has someone who has 'proved' quadratic reciprocity for themselves

This is exactly my point though: things get easier to understand over time as the more foundational mathematics gets laid out to prepare for them. Nowadays some of this stuff is considered basic. It's very "low hanging fruit" now, it's just that those summer-camp kids aren't making the discovery for the very first time. What point exactly are you defending here?

> Btw there are many contemporary mathematicians at least at the level of Terence Tao but for some reason haven't been blessed by lay popularity

I'm not sure why this needs to devolve into a contest. Terence Tao, Peter Scholze, whoever: they can't know all math anymore, like Euler did. That is ultimately why there are no more Eulers.

The difference is, to put in Rumsfeldian fashion, between known unknowns (kids proving QR now) and unknown unknowns (euler sensing QR, gauss proving it).

That Euler knew most of the math of his time is irrelevant. If one had any serious learning at any time in most of human history one would know all the math of their time.

To be clear, I am not saying that Erdos and Tao are less talented than Euler. That seems impossible to say. But the magic of Euler is that his fingerprints are all over so many of the fundamental things that can be understood by a bright high school student but were mostly unknown before him. The work of figures like Erdos and Tao seems far, far less accessible in its present form at least and thus more limited in its overall impact.

Part of that, too, is the "founding father" effect. Most humans alive today are related to Genghis Khan and/or Charlemagne. It just takes time for new ideas to dissipate and cross-breed.

The low hanging fruit argument only takes you so far. How many other mathematicians in his epoch or before were able to pick as many low hanging fruits as him?

By all means he was a crazy outlier generational genius. A few others in history, like Archimedes and Newton, have been accused of "not leaving anything for anyone else to discover" as well. The question was: why do we not seem to see these crazy outliers anymore? The answer is certainly not that truly exceptional people simply stopped being born after the year 1800. The nature of what it could mean to "know everything" about a field has completely changed.

Also they lived in a time with far less education and communication.

Newton and Leibniz discovered calculus simultaneously. If calculus were a hot new idea now, dozens or hundreds of people would be discovering it simultaneously.

Look at NN/LLM AI for an example.

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