The Maths History feed on Twitter mentioned that the 21st of August was the birthday of Augustin-Louis Cauchy, who lived from 1789 to 1857. His is one of those names you get to know very well when you’re a mathematics major, since he published 789 papers in his life, and did very well at publishing important papers, ones that established concepts people would actually use.

He’s got an intriguing biography, as he lived (mostly) in France during the time of the Revolution, the Directorate, Napoleon, the Bourbon Restoration, the July Monarchy, the Revolutions of 1848, the Second Republic, and the Second Empire, and had a career which got inextricably tangled with the political upheavals of the era. I note that, according to the MacTutor biography linked to earlier this paragraph, he followed the deposed King Charles X to Prague in order to tutor his grandson, but might not have had the right temperament for it: at least once he got annoyed at the grandson’s confusion and screamed and yelled, with the Queen, Marie Thérèse, sometimes telling him, “too loud, not so loud”. But we’ve all had students that frustrate us.

Cauchy’s name appears on many theorems and principles and definitions of interesting things — I just checked Mathworld and his name returned 124 different items — though I’ll admit I’m stumped how to describe what the Cauchy-Frobenius Lemma is without scaring readers off. So let me talk about something simpler.

Suppose you have a sequence, which is just an ordered list of numbers: the first number is, say, 1; the second number, say, 1/2, the third number, say, 1/4, and so on. A sequence might converge, which means roughly what you’d think it means from the ordinary meaning of the word, that the sequence eventually settles so close to some fixed number that you can’t see the difference after some point. Or it might not converge; for example, the sequence 1, 2, 4, 8, 16, et cetera clearly isn’t settling to anything. So too does the sequence 1, -1, 1, -1, 1, -1, and so on, not converge, even though it sticks very close to the numbers +1 and -1.

A sequence is called a Cauchy Sequence if, as you look at the M-th and the N-th terms as M and N get incredibly large, the difference between the M-th and the N-th terms gets arbitrarily close to zero. For example, consider the sequence 1, 1/2, 1/3, 1/4, and so on. The M-th term is 1/M, and the N-th term is 1/N. The difference between them is 1/M – 1/N, which is going to be incredibly tiny if M and N are very large at all. (Try M of 1,000, and N of 10,000; then try M of 1,000,000 and N of 10,000,000, if you want to convince yourself, although not your thesis committee. This tryout is just a sanity check.)

From that start — that this is a sequence where, eventually, the difference between any pair of entries gets arbitrarily close to zero — it would seem like a Cauchy sequence has to converge. And here’s a neat, counter-intuitive point: if the terms in the sequence are real numbers, then, yes, the Cauchy sequence has to converge to some real number. But if the terms in the sequence are rational numbers, then you’re not guaranteed that the sequence will converge to a rational number.

They will converge to some irrational number instead, but if you want to keep all your work in the rationals — and you might want to — then you’re left, sometimes, with a sequence that’s clearly converging to something but it’s just outside what you’re working with. If you spend a little time you can probably think of one, even if you haven’t got any mathematical training at all. Just think of the conditions: every term in the sequence has to be a rational number (if you write it out in decimals, it either ends or starts repeating itself), and eventually all the terms have to get so close together the differences between any pair are infinitesimally small, but the terms themselves get arbitrarily close to something that isn’t a rational number.

Cauchy was a very interesting man. Here are some scoops of information I have heard about him, some maybe incorrect or only partially.

-> He was incredibly arrogant – he did not read Galois’s work nor Abel’s. This (kind of) led to both of them failing to have a good financial basis at a young age. They both died at a young age. Cauchy should have given more help.

-> Supposedly he learnt the languages first then mathematics, the motivation being he would learn to communicate. All the while, his mathematical genius had shown at a young age, before he learnt the languages.

-> Was denied several important positions due to his personality, they were given to other (perhaps not so famous or as famous) mathematicians.

-> Defined the notion of a “function” in the best manner regarding his time. Euler’s idea of a function is now what we consider a continuous function. The classic function f(x) = |x| was defined as f(x)=x if x >= 0 and f(x) = -x if x<0 and Euler saw that not as a function. Cauchy defined it f(x)= sqrt(x^2), taking the ambiguity away and destroying Euler's idea. Neat thinking.

When I think of Gauss and Euler I think of a fox erasing its track with its deceptive fail as it smoothly progresses to master another problem: Gauss never told anyone how his ideas came, for brevity purposes. Euler was similar. When I think of Riemann, I think of someone who breaks down things into little blocks and looks at a geometrical vision. But Cauchy for me is the most "pure" of all – his way of thinking is to do with complete rigour. His introduction of limits and their implications for … everything (integration, differentiation and so on) took the gas, turned it into liquid and solidified it. His birthday should be a celebration of mathematical rigour. With Cauchy comes certainty.

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I’m not positive just how much is correct, particularly about matters regarding his personality, because I just haven’t read biographies dedicated specifically to Cauchy.

It’s shocking to read some of the positions that Cauchy didn’t get, and it surely would make anyone feeling depressed at her or his own job search feel better to know that a person whose name is on so many important theorems and so much impressive work was denied jobs he was surely able to fill. Of course, that comfort vanishes when you learn Cauchy was passed over in favor of someone like Legendre or Ampere instead.

My gut feeling is that Cauchy’s work in coming to the modern definition of a function is his most-overlooked-but-accessible work, probably because it’s so temptingly easy to just jump to the current definition and not follow the confusing path of the idea’s historical development. I’m tempted to learn the subject well enough to be able to write with more confident ignorance about it.

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According to Wikipedia – admittedly not exactly an authoritative resource – Cauchy was not entirely dismissive of the powerful ideas that Galois gave us. But he surely could have helped him more than he did.

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For what it’s worth MacTutor’s biography of Galois suggests that Cauchy’s treatment of Galois is at least more ambiguous than the MacTutor biography of Cauchy suggests. The Galois biography notes, for example, that Cauchy advised Galois to send a paper on solutions by radicals to Fourier for consideration for a Grand Prize which was certainly good advice. The paper was misplaced when Fourier died and that’s certainly beyond Cauchy’s control.

The biography of Abel is similarly vague on Cauchy’s treatment: Abel’s quoted as saying Cauchy “scarcely deigned to look at” a paper, but Cauchy and Legendre were named referees for it; MacTutor doesn’t say when the paper was published or under what conditions. It does mention that Abel’s Paris memoir was found by Cauchy “after much searching”, but that doesn’t say much about what Cauchy could have, or could reasonably be expected to have, done for Abel.

I admit I’m speaking out of my depths here. I don’t know what biographers generally make of Cauchy or Abel, and am only a little bit more familiar with Galois because of the great drama that he lived.

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