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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

#61

Post by lordnacho »

testless wrote: post #25
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…

Is Euler really well known in public? Einstein is way more well known than Euler, I'd guess. Even if you will ask most famous mathematician, it is unlikely that he is named. Probably Pythagoras. I have no data to back it up.

This is purely an indictment of modern mathematics education. Anyone doing ordinary high school math should have had their ears full of Euler's contributions for at least a couple of years towards the end. Math teachers who don't mention Euler need to be put onto a mock Königsberg and forced to search for a solution.

Re: How Euler Did It, by Ed Sandifer

#62

Post by chongli »

zyklu5 wrote: post #56
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…

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…

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.

Re: How Euler Did It, by Ed Sandifer

#63

Post by Cyph0n »

vacuity wrote: post #43
testless wrote: post #19

Earlier quoted context omitted.

Ramanujan died early. He is another one which comes to mind.

Riemann died young, too. I only really know him for the Riemann sum formulation of integrals (I have yet to learn complex analysis), but I wouldn't be surprised if he has some popular reach through the Riemann hypothesis.

Galois is yet another.

Re: How Euler Did It, by Ed Sandifer

#64

Post by feoren »

zyklu5 wrote: post #56
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…

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…

> 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.

Re: How Euler Did It, by Ed Sandifer

#65

Post by jacobolus »

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…

> In particular I read this one: http://eulerarchive.maa.org/hedi/HEDI-2009-02.pdf [...] wrote this up in a paper https://scholarlycommons.pacific.edu/euleriana/vol3/iss1/3/

Neat! It's not clear that Euler ever realized anything about calculating an arbitrary binary digit, but it wouldn't have been too far a leap to get there.

For what it's worth, the formula (13) your paper credits to Hutton was also known to Machin in 1706. As was the formula about which Sandifer says "Without citing any particular formula, Euler proclaims that ...". The famous "Machin formula" just happened to be the one that Jones published along with an accurate π approximation in Synopsis Palmariorum Matheseos, but Machin had worked out several others.

See Tweddle, Ian (1991). "John Machin and Robert Simson on Inverse-tangent Series for π". Archive for History of Exact Sciences. 42 (1): 1–14. doi:10.1007/BF00384331. JSTOR 41133896.

The transformation of the series for arctan to a faster-converging version which Sandifer discusses in the middle of that paper was first described by Newton in an unpublished monograph from 1684. See:

Roy, Ranjan (2021) [1st ed. 2011]. Series and Products in the Development of Mathematics. Vol. 1 (2 ed.). Cambridge University Press. pp. 215–216, 219–220.

Newton, Isaac (1971). Whiteside, Derek Thomas (ed.). The Mathematical Papers of Isaac Newton. Vol. 4, 1674–1684. Cambridge University Press. pp. 526–653.

Re: How Euler Did It, by Ed Sandifer

#67

Post by vkazanov »

testless wrote: post #60
guimplen wrote: post #58

Yeah, the greatest Russian mathematician.

I think Swiss. He was born in Basel. [1] https://en.wikipedia.org/wiki/Leonhard_Euler

I think what parent meant was that Euler spent a few decades in Russia because of the funding provided by the empire. He spoke fluent Russian, even though there was a large German-speaking community there.

But it was typical for scientists to travel far for money. Some of the Bernoullis, a family famous for mathematicians, also worked in Russian for quite a while.

Does it really matter who payed 'em and what languages they spoke?

Re: How Euler Did It, by Ed Sandifer

#68

Post by karmakurtisaani »

m_mueller wrote: post #55
seanhunter wrote: post #4

Earlier quoted context omitted.

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?

If I recall correctly, Euler has the most pages of published math. Erdős has the most papers (some of them not more than a handful of sentences).

Re: How Euler Did It, by Ed Sandifer

#69

Post by karmakurtisaani »

OldGuyInTheClub wrote: post #27
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…

Euler is in a class of one. Judging by the volume of work and the pleasure he takes in collaborations, Terence Tao seems to be having a positive impact. I am not a mathematician but I hear good things about the quality of his work.

> I am not a mathematician but I hear good things about the quality of his work.

Not to be rude, but saying this about a Fields medallist is somehow very funny.

Re: How Euler Did It, by Ed Sandifer

#70

Post by TuringTourist »

karmakurtisaani wrote: post #68
m_mueller wrote: post #55

Earlier quoted context omitted.

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

If I recall correctly, Euler has the most pages of published math. Erdős has the most papers (some of them not more than a handful of sentences).

Erdős has a huge amount of credits in the papers of others (hence Erdős number) because he would just travel all over the country helping people get unstuck on their work.
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