How Euler Did It, by Ed Sandifer
eulerarchive.maa.org
eulerarchive.maa.org
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 can be used to calculate any binary digit of π without calculating the previous digits. Up until now it was believed that the first formula with the correct properties (known as a BBP-type formula) for this calculation was published by Bailey, Borwein and Plouffe in 1997."
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to a faster-converging version which Sandifer discusses in the middle of that paper was first described by Newton in an unpublished monograph from 1684.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.
One other cool thing about Euler and BBP-type pi series: Euler seems to have derived his results in a manner similar to how the famous BBP formula
{\displaystyle \pi =\sum _{k=0}^{\infty }\left[{\frac {1}{16^{k}}}\left({\frac {4}{8k+1}}-{\frac {2}{8k+4}}-{\frac {1}{8k+5}}-{\frac {1}{8k+6}}\right)\right]}
is actually proven. A friend of mine gave the proof of the famous series result as an exercise in his honors calc 2 class one year. They had some fun with it.
https://en.wikipedia.org/wiki/Bailey%E2%80%93Borwein%E2%80%9...
If humans ever started living much longer, we'd probably see a different attitude towards work.
Today that concept is watered down. A "well rounded" education is just taking a few classes that people hate and will blow off because to graduate they need to check some boxes so they can focus on doing one thing moderately well and finding their place as a cog in a machine that will abuse their ignorance. It's all mass produced conveyor belt education that manufactures young adults with little conventional wisdom. The more you lean into behaving like a part, the more you will be treated that way.
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.
(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)
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.
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.
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.
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.
Bullets with rifling came about circa 1820 (ish) but were not widespread by 1832 and traditional dueling pistols remained the norm for quite some time, the element of chance likely factored in as part of the hand of god influencing outcomes.
https://en.wikipedia.org/wiki/Duelling_pistol
For some in the eighteenth century, duelling with less-accurate, smooth-bore weapons was preferred as they viewed it as allowing the judgement of God to take a role in deciding the outcome of the encounter.
[1] https://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/WK...
I have no data to back it up.
Not to be rude, but saying this about a Fields medallist is somehow very funny.
[1] Bird vs. Frogs. https://www.ams.org/notices/200902/rtx090200212p.pdf
I once casually flipped through the CS journal papers archived in my university library, and those two decades were really fascinating.
1) All of the papers were simple, clear, and easy to follow. No greek symbols, no fancy maths, just straighforward pseudocode.
2) The algorithms being presented were new at the time, but were rather trivial, and could have been invented by any of us. They just hadn't been invented, formalised, and written down yet. It was basically a race starting from zero, and nobody had taken very many steps yet.
3) Every new algorithm was such a huge leap that it unlocked many new avenues of advancement for other algorithms. There was a period where it was a race of publishing these follow-up developments nearly as fast as people could type.
For a modern example of this, look at the pace of progress of generative AI such as LLMs and Diffusion Models. The first papers were almost trivial, but that "one clever trick" unlocked many more ideas and the rate of publication over the last twelve months has been insane. A lot of low-hanging fruit, a lot of road-blocks suddenly removed, etc...
I own a copy of his "Elements of Algebra" and it's interesting to read because he actually talks and uses the notion of infinitesimals in this basic algebra book. And it makes sense! He essentially just says "think of the biggest number, make it even bigger!!! Now, put it under 1, and just like that we 'get almost zero'"
You would never see something like that now, or even then really, and yet the idea is so simple a kid understands. His writing just has such an optimistic and playful sense to it.
I've been wondering if an llm might be tuned to recognize insightful explanations that are accessible and powerful, to then help train one for creating such. Trying for an AI tutor that's less like drilling textbook bogosity and superficiality, and more like tutoring by that rare someone known for their outlier-deep understanding of a field. Even if an llm only manages to serve as a delivery mechanism for human curated insights, having a deployment story of very widespread impact might help motivate a novel-y broad contribution and curation effort.
A quarter of all the output in so many fields. For a whole century. When you think about everyone else who was contributing to science at that time. It's just completely staggering.
In these times, we need every whizzkid we can get.
Whizzkids will educate themselves, what's needed is giving people idle time in order to pursue things. Most influential thinkers found themselves with this in some fashion.
How much talent is wasted making people jump through hoops in academia/finance/ad-tech?
A lot of pre-industrial thinkers were associated with the clergy because they received tax money from peasants.
There's a robust multi-level system of "Olympiads", starting from the neighborhood level, and going all the way up to the national level. Every student knows about them, and more importantly, "magnet schools" scoop up students who do well in competitions.
This works really well for math, and so pretty much every Fields medal award ceremony has awardees from the xUSSR countries.
I'm really surprised that this kind of system is not more widespread, especially in the US. After all, sports and competition is kinda a thing here?
Only those you see becoming one.
You never hear of all the "Einsteins" who never leave the patents office because they never got inspired for some passion or various other stupid reasons.
I wouldn't be so pessimistic.
I don't think it's that complex. My personal premises are
* if the kid doesn't like to get up and go to school or is too tired over longer periods, the school is doing something wrong, not the kid. * if there are A LOT of kids like this, the school authority is doing something wrong (and so on, up to government) * an older kid needs support on every topic at any time when interest arises. It's the kids' choice not the schools'. * learning at school must be fun, at least 90% of the time.
That way whizzkids are easily detected, will be less frustrated and thus perform better.
I got some good example anecdotes but better won't "textwall" here :-)
Just imagine a world, where Einstein would get optimum support from young age so he would start professional physics years earlier.
Or one where he didn't go to the patent office but one where he worked in some factory or other place, not finding the spare time he needed to do physics.
Way too much luck involved if you ask me.
One example: someone elsewhere in the thread mentioned John Carmack. Carmack's not a thief, or at least it would be a grotesque oversimplification to label him as one. But he stole an Apple II when his parents wouldn't buy him one.
What's critical is that once talent is identified, it's nurtured to the greatest extent possible.
You just can't know, how much more Einsteins there would be. People that would never steal an Apple II or would never even get that opportunity.
Heck, if Einstein became a younger father, there's a good chance he never looked deeper into physics despite talent and interest without someone (like a teacher) pushing him.
I claim that it's pure coincidence. Carmack didn't make it because of public education but despite of it. We can't afford this anymore.
And the world is full of parents who are neglectful with no excuse at all. I don't think early fatherhood would stop an Einstein.
Instead, the risk is that talented people will die of starvation, disease, or warfare before they have the chance to become who they are. Or that their career will be cut short by similar circumstances. See Ramanujan, or the even more-tragic but lesser-known https://en.wikipedia.org/wiki/Oleg_Losev .
That's where the luck factor really comes into play... the luck to be born someplace peaceful. The luck to be born male, if you have to be born into a culture driven by social or religious biases. The luck to receive the nutrition (never mind the Apples) you need as a growing child. The luck that just plain keeps other people the fuck out of your way.
And that's what we have to work on as a civilization. It's a bigger problem than simply arguing over how public education should work, or whether scientist X or inventor Y would have benefited from policy Z.
15% failure rate is optimal for learning
Of course, only as an addition to the current mix. For math problems, this will be easier than for other contexts.
I like text books for pop science or really hard things - like university level education. But I am surprised to see them as an option for basic education.
The process of learning when to trust it and when not to is learning.
The anecdote goes that with his eyesight already severely deteriorated, he would dictate papers with a grandkid or two in his lap and a cat on his shoulder.
An example of this is when, in solving the then notorious Basel problem, he factors trig functions into infinite products of (x +- k*pi) terms just by analogy with root factorization in finite polynomials.
I could imagine that in 300 years time people think that Elon Musk single-handedly invented the Turing machine, at the age of 16, during a weekend, while reading Hacker News.
How would one go about disproving such an hypothesis? I've had similar doubts about Leonardo da Vinci, but I'm afraid Euler was actually just brilliant.
I get why someone not familiar with him may think this, but when I think of the word "genius" only this man, von Neumann, Ramunajan and Grothendieck come to my mind. They simply saw the world differently.
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?
The certificate is signed using a SHA-1 signature, which is considered insecure by all major browsers afaik.
Although, inspecting the certificate also gives me validity:
Not After Fri, 05 Dec 2014 12:00:00 GMT
Even the CA cert it shows me has
Not After Sun, 03 Apr 2022 00:00:00 GMT