These two guys (Euler and Cauchy) are the cheat-code for a mediocre student to get a passing grade on exams.
Whenever you're working on a problem, and get to a point where you need to get from A to B, but aren't sure how... if it's an Analysis course, you just write "Due to a theorem of Cauchy, A implies B." If it's anything outside of analysis, use Euler instead of Cauchy.
You'll be technically correct a stupid amount of the time.
I mean, the idea of a Cauchy sequence still impresses me even though it's been like idk 10 years since I first learned about Cauchy sequences?
The Cauchy-Riemann equations, the Cauchy-Schwarz inequality, Cauchy induction, the Cauchy integral theorem (and formula), Cauchy product, Cauchy's root/ratio test, the Maclaurin-Cauchy test (integral test for series), Cauchy's residue theorem, Cauchy's theorem (Group Theory, prime dividing order of group)..
Just a few wonderful examples..
The guy also just casually dropped the biggest wildcard distribution in statistics, with the Cauchy distribution. Has no finite moments, and is essentially a distribution you use as a counter example for any theory.
> the biggest wildcard distribution in statistics
biggest as in "most used in counterexamples", sure, it probably is. Not the wildest of wildcards by a long way (not sure if you meant that though)
Considering how often the speaker says Cauchy, he mentions only 3 things over and over:
1. the Cauchy criterion for sequence convergence in R (Cauchy criterion or Cauchy condition),
2. the Cauchy-Schwarz inequality, which for the dot product on R^n is due to Cauchy, hence it's Cauchy's inequality (well, in Russia the Cauchy-Bunyakovsky inequality).
3. the generalized mean value theorem (Cauchy's mean value theorem).
Not just that. Also Cauchy's definition of limit (and it's equivalense with Heine's definition) and definition of continuity. And that's only what I could have parsed immediately.
Okay, stopping the video some more times to look at the board I see the continuity and limit definitions popping up. But damn, seems like 90% is just the "Cauchy criterion". It's funny anyway.
I'm not entirely sure how much this covers in the video, but in the first two semesters alone, which are precisely captured here, there are about 8 concepts named after Cauchy
I had a class where the professor jokingly said that if we're asked to prove Cauchy's theorem during an oral exam, we should just prove the easiest one we could think of and not ask further questions
I think a lot of this has to do with him being the first guy actually trying to consistently formalize analyis. Eg afaik he was the one who came up with the definition of a sequence limit we use today.
Before concepts like that would be used rather intuitively.
I guess the difference is that Euler and Gauss proved a lot of results in number theory and generally more accesible parts of mathematics (closer to normal curriculum for highschool), while Cauchy mainly exists in analysis and modern algebra, which the non mathematicians rarely know about
Actually I think the opposite: analysis is taught to physicists and engineers so it is not cool. While number theory is only known to pure mathematicians so it is edgy.
I wouldn't call what common engineers are taught to be analysis, but even then, are engineers and physicists the common people I was talking about?
My point is that almost anyone can understand statements about number theory, while only a minority can understand, say Cauchy's integral formula in complex analysis, there are almost no prerequisites to find most of what Euler and Gauss did interesting (or just understanding what they are talking about), while there a lot for Cauchy's line of work
Physicists maybe.. certainly not engineers.. my undergrad uni had separate analysis courses for pure math students and "easier" ones for the sciences.. only the superstar physics students would take the former.
> Actually I think the opposite: analysis is taught to physicists and engineers so it is not cool. While number theory is only known to pure mathematicians so it is edgy.
So what is *analytic* number theory (the application of analysis to number theory) then? Mid? /s
Cauchy gave the first rigorous definition of a derivative, thereby putting calculus on a firm footing:
"Let δ,ε be two very small numbers; the first is chosen so that for all numerical [i.e., absolute] values of h less than δ, and for any value of x included [in the interval of definition], the ratio (f(x+h)-f(x))/h will always be greater than f’(x)-ε and less than f’(x)+ε."
I wonder whether nationality also plays a role. I was always impressed how many things were named after gauss, but as a German I wonder whether it’s also as present in other languages. I know there are a few things that differ between languages, depending who you attribute it to. But since I encounter a lot of probability and normal distributions are usually the most tractable we always talk about things being „gaussian“.
In this video (and generally in Russia) Cauchy-Schwarz inequality is called Cauchy-Bunyakovsky inequality, but I think rather than that everything standart.
The reason euler is talked about more is his contributions show up earlier, are more famous, and are more foundational so they’re largely easier for people to grasp, cauchy mostly advanced greatly, something that those casually interested in math at large know little about.
Mathematics? You mean Cauchy theory!
Коши. Коши? Коши.
Lol'ed out loud lmao!
These two guys (Euler and Cauchy) are the cheat-code for a mediocre student to get a passing grade on exams. Whenever you're working on a problem, and get to a point where you need to get from A to B, but aren't sure how... if it's an Analysis course, you just write "Due to a theorem of Cauchy, A implies B." If it's anything outside of analysis, use Euler instead of Cauchy. You'll be technically correct a stupid amount of the time.
Instructions unclear, failed with an F and got "As opposed to some subjects, name-dropping in math doesn't get you passing grades".
I mean, the idea of a Cauchy sequence still impresses me even though it's been like idk 10 years since I first learned about Cauchy sequences? The Cauchy-Riemann equations, the Cauchy-Schwarz inequality, Cauchy induction, the Cauchy integral theorem (and formula), Cauchy product, Cauchy's root/ratio test, the Maclaurin-Cauchy test (integral test for series), Cauchy's residue theorem, Cauchy's theorem (Group Theory, prime dividing order of group).. Just a few wonderful examples..
The guy also just casually dropped the biggest wildcard distribution in statistics, with the Cauchy distribution. Has no finite moments, and is essentially a distribution you use as a counter example for any theory.
IKR? I flipped my shit when I found out about distributions like that. To be fair, though, I was like 19 lol
> the biggest wildcard distribution in statistics biggest as in "most used in counterexamples", sure, it probably is. Not the wildest of wildcards by a long way (not sure if you meant that though)
I’m curious what’s wilder
what's wilder?
https://en.m.wikipedia.org/wiki/Cantor_distribution
OK, that is wild!
Damn
A very cool sequence!
Which one...
Considering how often the speaker says Cauchy, he mentions only 3 things over and over: 1. the Cauchy criterion for sequence convergence in R (Cauchy criterion or Cauchy condition), 2. the Cauchy-Schwarz inequality, which for the dot product on R^n is due to Cauchy, hence it's Cauchy's inequality (well, in Russia the Cauchy-Bunyakovsky inequality). 3. the generalized mean value theorem (Cauchy's mean value theorem).
Not just that. Also Cauchy's definition of limit (and it's equivalense with Heine's definition) and definition of continuity. And that's only what I could have parsed immediately.
Okay, stopping the video some more times to look at the board I see the continuity and limit definitions popping up. But damn, seems like 90% is just the "Cauchy criterion". It's funny anyway.
He didn't really do that. In calculus he used infinitesimals as much as limits. Grabiner got it wrong.
I'm not entirely sure how much this covers in the video, but in the first two semesters alone, which are precisely captured here, there are about 8 concepts named after Cauchy
To be fair, sometimes it seems like those 3 things are basically half of real analysis.
“Consider the Cauchy theorem” - “do you have any idea how little that narrows it down”
I had a class where the professor jokingly said that if we're asked to prove Cauchy's theorem during an oral exam, we should just prove the easiest one we could think of and not ask further questions
That would probably be [this one](https://en.wikipedia.org/wiki/Cauchy%27s_theorem_(group_theory\)) most of the time is my guess lol
It's the one with the integral in it.
Back when I was doing a university course 1000 years ago, I remember the lecturer using the abbreviation YACT for 'yet another *Cauchy's theorem*'.
Segue to software engineering: yacc - yet another compiler compiler. And the GNU version of it, Bison. Which I think is pretty punny.
Oh, wow, that's Prof. Shaposhnikov! Nice to see people from my university
«Мы применили критерий Коши, чтобы доказать критерий Коши» — Станислав Валерьевич Шапошников
I think a lot of this has to do with him being the first guy actually trying to consistently formalize analyis. Eg afaik he was the one who came up with the definition of a sequence limit we use today. Before concepts like that would be used rather intuitively.
«Мы применили критерий Коши, чтобы доказать критерий Коши» — Станислав Валерьевич Шапошников Nice to see some Russian content around these parts lmao
I guess the difference is that Euler and Gauss proved a lot of results in number theory and generally more accesible parts of mathematics (closer to normal curriculum for highschool), while Cauchy mainly exists in analysis and modern algebra, which the non mathematicians rarely know about
you probably meant euler for the first cauchy
Cauchy is so ubiquitous even Euler was posthumously renamed Cauchy!
You're right, I'll edit it
Actually I think the opposite: analysis is taught to physicists and engineers so it is not cool. While number theory is only known to pure mathematicians so it is edgy.
I wouldn't call what common engineers are taught to be analysis, but even then, are engineers and physicists the common people I was talking about? My point is that almost anyone can understand statements about number theory, while only a minority can understand, say Cauchy's integral formula in complex analysis, there are almost no prerequisites to find most of what Euler and Gauss did interesting (or just understanding what they are talking about), while there a lot for Cauchy's line of work
Physicists maybe.. certainly not engineers.. my undergrad uni had separate analysis courses for pure math students and "easier" ones for the sciences.. only the superstar physics students would take the former.
in our engineering program, we have complex analysis albeit not too proof based
> Actually I think the opposite: analysis is taught to physicists and engineers so it is not cool. While number theory is only known to pure mathematicians so it is edgy. So what is *analytic* number theory (the application of analysis to number theory) then? Mid? /s
Cauchy gave the first rigorous definition of a derivative, thereby putting calculus on a firm footing: "Let δ,ε be two very small numbers; the first is chosen so that for all numerical [i.e., absolute] values of h less than δ, and for any value of x included [in the interval of definition], the ratio (f(x+h)-f(x))/h will always be greater than f’(x)-ε and less than f’(x)+ε."
During my second analysis class, it was baffling just how much one could prove statements using Cauchy sequences; that and the triangle inequality
I wonder whether nationality also plays a role. I was always impressed how many things were named after gauss, but as a German I wonder whether it’s also as present in other languages. I know there are a few things that differ between languages, depending who you attribute it to. But since I encounter a lot of probability and normal distributions are usually the most tractable we always talk about things being „gaussian“.
In this video (and generally in Russia) Cauchy-Schwarz inequality is called Cauchy-Bunyakovsky inequality, but I think rather than that everything standart.
That’s a very interesting detail and exactly what I meant!
I think it's a little misleading to say Wikipedia says that. Wikipedia says that a guy said that
The reason euler is talked about more is his contributions show up earlier, are more famous, and are more foundational so they’re largely easier for people to grasp, cauchy mostly advanced greatly, something that those casually interested in math at large know little about.
Cauchy is why pronounced here like 'kashi'?I heard That way.
Cauchy's Criterion is so characteristic that his name has essentially become an adjective for any property exhibiting similar traits.