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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?

Andrews
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Mikael Jensen
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  • I don't get it. Is the question about philosophy of science/physics/quantum mechanics in analogy to the philosophical and meta-mathematical implications of Cauchy's definition of the limit using finitary means? – Asaf Karagila Jul 30 '13 at 19:49
  • It is mainly a statement about what happens when a stumbling block is removed or a new invention sees the light of day. Alternatively, think about the steam engine. – Mikael Jensen Jul 30 '13 at 19:54
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    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
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    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
  • I don't understand the question either. In particular, the logic of the sentence "Thinking about the parallels in physics of quantum mechanic and relativity, you might expect that this breakthrough lead to a series of new discoveries but it seems that the technique was already so widely used that not much came forward." is completely obscure to me. Could you confirm that you mention relativity and quantum mechanics as an analogy only (at first reading I thought otherwise)? Also, what "parallels" are you making here? – Pete L. Clark Jul 30 '13 at 20:04
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    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
  • "It is mainly a statement..." Then it isn't a question. – Thomas Andrews Jul 30 '13 at 20:16
  • Wow, this question got closed pretty fast. If he is still interested, I hope the OP will revise his question and flag for reopening. – Pete L. Clark Jul 30 '13 at 20:36
  • @PeteL.Clark: Dear Pete, My understanding was that the OP was suggesting that whereas the breakthroughs of QM and relativity led to a series of new discoveries, the breakthrough of Cauchy's discovery of a rigorous approach to analysis didn't lead to correspondingly important new discoveries. (I don't really agree with this statement, by the way; i.e. when the OP writes "Is that wrong ... ?", my reply would be "yes, that is wrong".) Cheers, – Matt E Jul 30 '13 at 20:50
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    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
  • @Matt: Thanks. If this is what the OP is asking, then (of course?) I agree that it is wrong. I guess I also detected a whiff of historiography in "Should Cauchy's limits concept be given more credit?" Unfortunately "Why was Cauchy's limits concept important for mathematics?" is to my taste too broad a question for a site like this. You usually interpret questions with great generosity and argue on the side of keeping them open: I wonder if you can help out here. – Pete L. Clark Jul 30 '13 at 20:54
  • @PeteL.Clark: Dear Pete, I don't think the question is as broad as "why was Cauchy's limits concept important for mathematics". I think a good answer could just mention that Cauchy initiated a new phase in the rigorous interpretation of mathematical intuition (thinking of Euclid as giving a rigorous interpretation of geometric intuition), which led to basically the entire modern basis for doing mathematics; one could then mention a few specific examples in modern math that are directly descended from Cauchy's ideas, such as topology, functional analysis, and analytic number theory. Cheers, – Matt E Jul 30 '13 at 20:58
  • @RBG, You are referring to administrative formalization. The formalization of soccer rules by the English schools has prevented alternatives rules “from appearing and being endlessly discussed”. I believe there is a little more than that buried in the limit concept. – Mikael Jensen Jul 30 '13 at 21:07
  • @Others The history tag is “for questions concerning history of mathematics, historical primacies of results, and evolution of terminology”. It is somewhat related to the latter but I must admit it is a philosophical-historical issue. The tag “Philosophy” is defined fairly broad: Questions involving philosophy of mathematics My question is close to the interpretation by Matt E, but I am used to encounter some resistance with all philosophical issues. My impression is that you more or less understand the question but I am not going to get much more information. Let us leave it. – Mikael Jensen Jul 30 '13 at 21:08
  • Dear Mikael, If my interpretation is more-or-less correct, then let me amplify my second comment to Pete Clark and say that Cauchy's work did lead to a lot of new discoveries in mathematics; indeed, to a pretty large extent it provided the whole framework in which mathematics was done from that time on. It also led to the extension of analytic ideas to a much wider arena of applicability --- topological spaces, topological groups, $p$-adic numbers, functional analysis, and many other areas of mathematics which depend on the rigorous foundations of analysis for their own foundations. ... – Matt E Jul 30 '13 at 21:48
  • @Mikael: If you're asking for an explanation of what important mathematics came from Cauchy's limits concept, Matt E has sketched out an answer to that question which others would surely be willing to flesh out. (That question is broad for my taste, but that doesn't make it inappropriate for the site, necessarily. It probably is appropriate, in fact.) If none of us have fully understood your question yet, why don't you edit it to make yourself fully understood? – Pete L. Clark Jul 30 '13 at 21:48
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    ... 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
  • When I look back at Matt’s comments I guess I got an answer along the road, and I should probably consider the question answered and also thank him. This view of Cauchy was never conveyed to me during my (limited) math education. – Mikael Jensen Jul 31 '13 at 09:34
  • This is an interesting and legitimate question. I suggest reopening it. – Mikhail Katz Jul 31 '13 at 11:55
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    I think there is an interesting question lurking here. The problem is that it remains hidden. I do not quite understand how we could be made even more aware than we are of the significance of Cauchy's contribution. I would appreciate some clarification of the way such an awareness is lacking. In particular, I would appreciate some evidence of this lack, in the form of data or studies or something tangible, as opposed to the OP's anecdotal evidence. – Andrés E. Caicedo Aug 02 '13 at 06:09
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    @AndrésE.Caicedo, I think the OP has in mind not the general significance of Cauchy's oeuvre (it would be absurd to doubt that) but rather the specific issue of arithmetic foundations, as detailed in my answer. I may be wrong; it is up to the OP to issue a clarification (and I hope he comes back even though he was given a cold shoulder here). – Mikhail Katz Jun 11 '17 at 16:21

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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 Katz
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Regarding my question: "Thinking about the parallels in physics of quantum mechanic and relativity, you might expect that this breakthrough lead to a series of new application....."

After a few years of discussion here, I believe in retrospect that a relevant answer is: the breakthrough I was looking for is found in the definition of real numbers where the Cauchy limit constitutes an important component.

Mikael Jensen
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  • Mikael, as already discussed here, Cauchy did not give an epsilon-delta definition of limit. So the frequently used expression "Cauchy limit" is an ahistorical fabrication by historians of Boyer's ilk, who perpetuate the myth of the historical evolution of mathematical analysis as a triumphant march toward the yawning heights of the "great triumvirate" and the Weierstrassian epsilontic. – Mikhail Katz Jun 18 '23 at 13:01
  • @Mikhail In retrospect I would rephrase my question: Why has the modern limit concept not led to other types of breakthrough (in mathematics or otherwise), and my own answer is that it has, in that it is used in the definition of real numbers, an important modern concept. – Mikael Jensen Jun 19 '23 at 18:15
  • Well, as you know the real numbers can be defined via unending decimal expansions, and unending decimal expansions were already known to Simon Stevin in the 16th century. Notably, Cauchy used them freely; he didn't need an abstract definition via Cauchy sequences (of course the idea of a Cauchy sequence is due to Cauchy). – Mikhail Katz Jun 20 '23 at 08:54
  • Mikhail I have two comments: i) a formal comment that any unending expansion is problematic in that "Mathematics does not have "an actual sum of an infinity of numbers in store" according to . – Christian Blatter (Nov 23, 2013 at 20:46 in a now deleted question of mine) who's point, I believe, is that addition cannot be defined for an infinity of numbers. ii) the other is the question whether a - "legal" - infinitive series might help take away Newton's problems with infitesimals. – Mikael Jensen Jun 21 '23 at 14:09
  • Mikael, this amounts to changing the subject. Who was talking about infinite sums? It is well known that real numbers can be defined via infinite decimal strings. Daniel Barlet (the famous complex analyst) once tried to convince me that this is in fact the preferred definition of a real number. I don't know if I was convinced but this is certainly a possible, and well known, definition. Here each string ending with an infinite tail of 9s needs to be identified with the corresponding terminating decimal. Not that Stevin was aware of this, but Euler already was, and certainly Cauchy :-) – Mikhail Katz Jun 21 '23 at 14:13
  • Mikael, can you provide a link to this deleted question? I certainly don't find Blatter's comment on this page. – Mikhail Katz Jun 21 '23 at 14:15
  • Incidentally, you may enjoy the discussion of Cauchy here: https://math.stackexchange.com/a/4722832/72694 – Mikhail Katz Jun 21 '23 at 14:17
  • The question was: Does a sum of infinite numbers exist? [closed] It was asked Nov 23, 2013 at 20:22. It was closed by a "bot" which I assume is an policing algorith. – Mikael Jensen Jun 21 '23 at 14:21
  • Do you have a link? – Mikhail Katz Jun 21 '23 at 14:22
  • https://math.stackexchange.com/questions/578491/does-a-sum-of-infinite-numbers-exist – Mikael Jensen Jun 21 '23 at 14:23
  • That question seems pretty confused, so I did not try to undelete it. You could make an effort to clarify it, though; perhaps it can become a better question. Meanwhile, I took the opportunity of answering another question of yours about the real numbers: https://math.stackexchange.com/a/4722935/72694 Hope this helps. – Mikhail Katz Jun 21 '23 at 14:30
  • Mikhail Goimg with the original question - rephrased - would infinite series describing "unending decimal expansions" have silenced bishop Berkeley in his critique of infitesimals "fluxions"? I believe we would need epsilon-delta reasoning which seems to me a few steps ahead of infinite series. – Mikael Jensen Jun 21 '23 at 18:25
  • as an amateur I take your word for it that the ono-to-one relation with infinite series and real number is enoungh to define them, but I wonder why we are reading so much about Cantor and Dedekind. Didn't they add something important? Se e.g. Erich Reck on "Dedekind’s Contributions to the Foundations of Mathematics",
    • – Mikael Jensen Jun 22 '23 at 07:49
  • Very good question. Since you mentioned Reck's article, let me quote from it directly: "As is well known, the inventors of the calculus relied on appeals to 'infinitesimal' quantities, typically backed up by geometric or even mechanical considerations, although this was seen as questionable from early on. The nineteenth-century 'arithmetizers' found a way to avoid infinitesimals (in terms of the epsilon-delta characterization of limits familiar from current introductions to the calculus). Yet this again, or even more, led to the need for a systematic characterization of various... – Mikhail Katz Jun 22 '23 at 09:01
  • ... quantities conceived of as numerical entities, now in the form of a unified treatment of rational and irrational numbers. Dedekind faced this need directly, etc." According to Reck, part of Dedekind's accomplishment was "to avoid infinitesimals". At the time, the analysts working in the foundations were unable to formalize the notion of infinitesimal which until then was prevalent in the practice of analysis; therefore they sought to eliminate them. The result was a rigorous foundation (which is certainly important), but also an impovished one ... – Mikhail Katz Jun 22 '23 at 09:02
  • -- something that did not become clear until Abraham Robinson's work. – Mikhail Katz Jun 22 '23 at 09:02