# Cauchy's limit concept

Berkeley criticized Newton’s and others’ infinitesimals: They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities? (Wikipedia).

He was pointing to a major paradox which was resolved in the 19th century: In the early 1820’s, through his lectures at the École Polytechnique, Augustin Louis Cauchy (1789-1857) clarified the concept of a limit and was able to provide strictly arithmetical definitions of continuity, the derivative, and the definite integral, http://www.me.berkeley.edu/faculty/casey/Calculus.pdf

Thinking about the parallels in physics of quantum mechanic and relativity, you might expect that this breakthrough lead to a series of new applications in physics and other fields but it seems that the technique was already so widely used that not much came forward. Is that wrong or what do you think of this?

Should Cauchy’s limits concept be given more credit?

• Discoveries come from working on things and playing with them. The formalization of a concept, in this case limits, doesn't affect much on how you play with a tool, specially one that you sort of have an idea of how it is supposed to be played with. Instead, its role is more in telling how not to used it. That is why it may not produce any significant change in the output of good results, but in preventing the bad ones from appearing and being endlessly discussed. Formalization that improves notation or language are the ones having more impact on the volume of results. – OR. Jul 30 '13 at 20:01
• Let me say I do not know if in fact Cauchy's formalization of limits did or did not impact significantly the production of new results. I just gave a reason that may explain why, if it is the case that it didn't. – OR. Jul 30 '13 at 20:03
• Also, when you ask "Should Cauchy’s limits concept be given more credit?" I guess you mean "more credit than it is currently given by historians of mathematics"? If so, this seems to be a history question and not a math question at all. Could you perhaps rephrase your question to be more explicitly mathematical and less subjective? Otherwise I think it might not be appropriate for this site. – Pete L. Clark Jul 30 '13 at 20:06
• Dear Mikael, Am I write that you are asking whether Cauchy's breakthrough development of a rigorous approach to calculus led to important new discoveries? (And that the comments about QM, relativity, steam engines, etc., are just intended to illustrate breakthroughs in other areas of science that did lead to important new discoveries?) If so, I suggest that you rewrite your question a little to make this clear, since it seems to have been unclear to many people. Once you do this, it should be easy to have your question reopened. Regards, – Matt E Jul 30 '13 at 20:54
• ... Also, I think that Cauchy's achievement is fairly well-known are recognized by mathematicians. So I'm not sure that his work is lacking either for credit or for applications/new developments. Regards, – Matt E Jul 30 '13 at 21:50

You seem to ask why the specific contribution by Cauchy to the foundations of analysis in terms of his limit concept has not had major applications in physics and other fields (as have many of his other contributions, to complex analysis, combinatorics, elasticity theory, differential geometry, etc). There are two parts to this question:

(1) did Cauchy really make such a contribution, namely a strictly arithmetical definition of limit?

(2) Why doesn't a strictly arithmetical definition of limit have more applications?

I think the answer to (1) is clearly negative; Cauchy never gave a strictly arithmetical definition of limit. He repeatedly defined limit in kinetic or kinematic language, in terms of what appears to be a primitive notion in Cauchy namely that of a variable quantity, akin to Leibniz. What you take to be Cauchy's strictly arithmetical definition is actually Weierstrass's, in terms of epsilon-delta and alternating quantifiers.

Specifically, the notion of Cauchy sequence, while its mathematical equivalent is found in Cauchy's work, was actually defined by Cauchy in the language of infinite indices and the property of the corresponding terms in the sequence being infinitely close, or more precisely partial sums being infinitesimal. This makes Cauchy's procedures closer to their proxies in Robinson's framework than the Weierstrassian framework.

Cauchy on occasion exploited arguments that seem to go in the direction of our epsilon-delta proofs, but in none of them did he give an explicit formula for delta in terms of epsilon, a tell-tale of a modern epsilon-delta proof.

As far as the apparent lack of scientific applications of Weierstass's purely arithmetic notion of limit, it must be attributed to the fact that this tremendous accomplishment by Weierstrass had mainly applications in establishing firmer foundations for some procedures in analysis, especially study of Fourier series, where the 19th century literature was plagued by errors. In other words, this accomplishment is mainly significant for mathematicians rather than physicists and other scientists.

• Mikhail, do you know who first used the term "Cauchy sequence"? – Andrés E. Caicedo Jun 11 '17 at 16:04
• @AndrésE.Caicedo, I responded in the body of my answer. – Mikhail Katz Jun 11 '17 at 16:15