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How Euler Did It, by Ed Sandifer

eulerarchive.maa.org

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Re: How Euler Did It, by Ed Sandifer

#51

Post by feoren »

norir wrote: post #44
feoren wrote: post #15

Earlier quoted context omitted.

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…

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 it…

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.

Re: How Euler Did It, by Ed Sandifer

#52

Post by vehicles2b »

johngossman wrote: post #45

Is there a book other than Bell’s “Men of Mathematics” that broadly covers the lives of a lot of mathematicians? I loved that book when I read it as an undergraduate, but I wouldn’t mind having a second opinion.

Princeton Companion to Mathematics has a section (part) on mathematicians

Re: How Euler Did It, by Ed Sandifer

#53

Post by jacquesm »

seanhunter wrote: post #4
nickcw wrote: post #2

I have read a few of these and enjoyed them greatly. Reading them you realise that Euler really did invent a huge swathe of mathematics in use today. In particular I read this one: http://eulerarchive.maa.org/hedi/HEDI-2009-02.pdf And I realised that Euler had found two formulae for Pi which can be used to calculate any hex digit of Pi. I wrote this up in a paper: "In 1779 Euler discovered two formulas for π which ca…

Part of the problem is he wrote so much it has taken a while to go through it all. I believe the "Opera Omnia" project to publish all his works has been going for over a hundred years and is just about getting to the end now. So I would expect there's a huge amount that just hasn't been fully appreciated/digested.

A while indeed. Euler and Bach are similar in that sense: to properly ingest their life's output you need more than one life.

Re: How Euler Did It, by Ed Sandifer

#54

Post by feoren »

lp4vn wrote: post #42
feoren wrote: post #15

Earlier quoted context omitted.

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…

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.

Re: How Euler Did It, by Ed Sandifer

#55

Post by m_mueller »

seanhunter wrote: post #4
nickcw wrote: post #2

I have read a few of these and enjoyed them greatly. Reading them you realise that Euler really did invent a huge swathe of mathematics in use today. In particular I read this one: http://eulerarchive.maa.org/hedi/HEDI-2009-02.pdf And I realised that Euler had found two formulae for Pi which can be used to calculate any hex digit of Pi. I wrote this up in a paper: "In 1779 Euler discovered two formulas for π which ca…

Part of the problem is he wrote so much it has taken a while to go through it all. I believe the "Opera Omnia" project to publish all his works has been going for over a hundred years and is just about getting to the end now. So I would expect there's a huge amount that just hasn't been fully appreciated/digested.

How is this humanly possible? Was Euler even an order of magnitude faster at producing new math than, say, Gauss or von Neumann?

Re: How Euler Did It, by Ed Sandifer

#56

Post by zyklu5 »

feoren wrote: post #15
norir wrote: post #7

It's interesting to me that I can't think of anyone remotely comparable to Euler in the public consciousness today. Even the work of someone like Erdos seems very esoteric by comparison and also was largely done in collaboration. Was Euler just born at the right time and picking all the low hanging fruit? Or maybe his immense creative production was a unique consequence of wealth plus limited distractions? I'm inclin…

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…

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)

Re: How Euler Did It, by Ed Sandifer

#57

Post by g9yuayon »

nickcw wrote: post #2

I have read a few of these and enjoyed them greatly. Reading them you realise that Euler really did invent a huge swathe of mathematics in use today. In particular I read this one: http://eulerarchive.maa.org/hedi/HEDI-2009-02.pdf And I realised that Euler had found two formulae for Pi which can be used to calculate any hex digit of Pi. I wrote this up in a paper: "In 1779 Euler discovered two formulas for π which ca…

I always feel amazed that Euler wrote faster than people could publish or understand his work, even after he became completely blind.

Re: How Euler Did It, by Ed Sandifer

#59

Post by testless »

norir wrote: post #7

It's interesting to me that I can't think of anyone remotely comparable to Euler in the public consciousness today. Even the work of someone like Erdos seems very esoteric by comparison and also was largely done in collaboration. Was Euler just born at the right time and picking all the low hanging fruit? Or maybe his immense creative production was a unique consequence of wealth plus limited distractions? I'm inclin…

For modern mathematics, I think Alexander Grothendieck was one of the most influencial talents in the last century. He unified large branches of mathematics in a very short period. He was more a bird than a frog [1], however.

[1] Bird vs. Frogs. https://www.ams.org/notices/200902/rtx090200212p.pdf

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