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Interpreting historical mathematicians involves a recognition of the fact that most of them viewed the continuum as not being made out of points. Rather they viewed points as marking locations on a continuum taken more or less as a primitive notion. Modern foundational theories starting around 1870 are based on a continuum made of points and therefore cannot serve as a basis for interpreting the thinking of the earlier mathematicians as far as the foundations are concerned.

What one can however seek to interpret are the techniques and procedures (rather than foundations) of the earlier authors, using techniques and procedures available in modern frameworks. In the case of analysis, the modern frameworks available are those developed by Weierstrass and his followers around 1870 and based on an Archimedean continuum, as well as more recently those developed by Robinson and his followers, and based on a continuum containing infinitesimals, and other frameworks such as the one developed by Lawvere, Kock, and others.

I was therefore a bit puzzled by the following comment by a historian expressed here:

Recently there have been attempts to argue that Leibniz, Euler, and even Cauchy could have been thinking in some informal version of rigorous modern non-standard analysis, in which infinite and infinitesimal quantities do exist. However, a historical interpretation such as the one sketched above that aims to understand Leibniz on his own terms, and that confers upon him both insight and consistency, has a lot to recommend it over an interpretation that has only been possible to defend in the last few decades. It is parsimonious and requires no expert defence for which modern concepts seem essential and therefore create more problems than they solve (e.g. with infinite series). The same can be said of non-standard readings of Euler; etc.

Question 1. Is this historian choosing one foundational framework over another in interpreting the techniques and procedures of the historical authors?

Question 2. What exactly is a Parsimonious History?

Question 3. Gray and Bottazzini reportedly make a rather poetic proposal in the following terms: "The best policy is to read on in a spirit of dialogue with the earlier authors." The proposal of such a conversation with, say, Euler sounds intriguing. I am only wondering about Gray's comment here that "Euler’s attempts at explaining the foundations of calculus in terms of differentials, which are and are not zero, are dreadfully weak." Isn't such an opening line in a conversation likely to be a conversation-stopper?

Question 4. In connection with the work of Laugwitz mentioned by one of the responders, one could ask why Gray does not cite explicitly the work of any of the authors that he wishes explicitly to criticize for using modern infinitesimals? Laugwitz's article (Laugwitz, Detlef. Definite values of infinite sums: aspects of the foundations of infinitesimal analysis around 1820. Arch. Hist. Exact Sci. 39 (1989), no. 3, 195–245) appeared in Archive for History of Exact Sciences, clearly a reputable journal since Jeremy Gray happens to be its Editor-in-Chief. Similarly, the article McKinzie, Mark; Tuckey, Curtis. Hidden lemmas in Euler's summation of the reciprocals of the squares. Arch. Hist. Exact Sci. 51 (1997), no. 1, 29–57 appeared in the same journal and exploited Robinson's framework to clarify some of Euler's procedures; it, too, is being stonewalled by the Editor-in-Chief.

Mikhail Katz
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  • The link is for an undergraduate mathematics series book! – Narasimham May 30 '16 at 17:43
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    It’s impossible to say for sure without seeing the historical interpretation that he offered. It sounds, however, as if he’s arguing against the idea that Leibniz et al. were using an informal version of infinitesimals on the grounds that a defensible, formal version didn’t exist until relatively recently. This is obviously a specious argument. – Brian M. Scott May 30 '16 at 18:10
  • @BrianM.Scott, thanks for your comment. Why do you feel that this is a specious argument? – Mikhail Katz May 31 '16 at 08:44
  • What's wrong with an undergraduate mathematics series textbook, @Narasimham ? – Mikhail Katz May 31 '16 at 09:47
  • Nothing wrong at all. I wondered if the advanced concepts could be appreciated or assimilated, when understanding the masters could be such a daunting task for learning undergraduates. – Narasimham May 31 '16 at 10:02
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    @BrianM.Scott I read it more narrowly, that he says arguing that Leibniz et al. were using some informal version of rigorous modern non-standard analysis - such as Robinson's for example - is inadequate. Rather, they had their own concepts of infinitesimals, which, I guess, would have significant differences to any given modern theory (and of course also significant similarities). And one should aim to understand their own concepts when writing their history. Which sounds quite sensible to me. – Daniel Fischer May 31 '16 at 19:11
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    @Daniel: I question whether there's a real difference between an informal version of rigorous modern non-standard analysis and any notion of infinitesimals that they may plausibly have had. I don't see that rigorous has any real effect: surely no one would suggest that their notion(s) were close enough to ours for the details implicit in rigorous to be meaningful or relevant. I certainly don't see that there is any conflict between trying to understand their concepts and suggesting that those concepts were to some considerable extent informal versions of concepts that we have formalized. – Brian M. Scott May 31 '16 at 20:38
  • @Brian As I understand it, the "rigorous" just serves to assert that the modern theories of non-standard analysis stand on an equal footing (with respect to rigour) as modern standard analysis. (In contrast to everything pre 18xx, where rigorous foundations had not yet been laid.) My understanding of the quote [which may be wrong, I haven't read anything by Gray, could be that that would make my interpretation untenable] is that Gray argues against interpreting them in terms of a specific modern theory. – Daniel Fischer May 31 '16 at 21:10
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    That their "concepts were to some considerable extent informal versions of concepts that we have formalized" is more than plausible. But one should try to understand their concepts (as far as one can reconstruct them) independently of the modern theories, and then see whether they correspond to the modern theory's concepts or deviate from them, rather than shoehorning their concepts into the modern theory. – Daniel Fischer May 31 '16 at 21:10
  • Re the Edit, note that a definition of the concept of "Parsimonious History" (not mentioned by the author of the quote) is not needed to found the conclusion that one historical interpretation is more parsimonious than another. – Did Jun 01 '16 at 07:20
  • @Did, would "parsimonious historical interpretation" be more accurate than "parsimonious history"? – Mikhail Katz Jun 01 '16 at 07:35
  • @DanielFischer, thanks for your comments. I think you have reformulated my question in a very illuminating way. Namely, if one sets aside issues of foundations and focuses on procedures, how does one justify studying modern proxies of historical procedures in terms of one modern theory rather than another? – Mikhail Katz Jun 01 '16 at 08:00
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    Can you please stop adding new questions here? After there are already answers to the original question, it's unfair to the answerers to add further questions. Create a new question for the further questions that arose. – Daniel Fischer Jun 01 '16 at 14:29
  • OK, I won't add any additional questions here. At any rate the original question is listed first. Since the other questions concern directly the same passage from Gray, I am not sure if it would be appropriate to ask separate questions dealing with the same passage. – Mikhail Katz Jun 01 '16 at 14:35
  • @Did, which interpretation is more parsimonious in the case of the infinitesimals of Leibniz, Euler, and Cauchy: one that involves alternating quantifiers or one that doesn't? – Mikhail Katz Jun 05 '16 at 09:18
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    @EricWofsey, It is hard to see what your goal was with this kind of attempted closing. The opinion-based objection does not apply to this kind of historical question. – Mikhail Katz Jun 05 '16 at 14:35

4 Answers4

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Note: [2016-06-01] Please note that OP has changed the title which was:

How can a modern historian interpret historical mathematicians?

and he has added question 2. This answer addresses the original question which is at the time question 1.

To me it seems Jeremy Gray, the author of the referenced book, put some words of warning that interpreting the work of the old masters in terms of one or two modern concepts only is too often superficial and not appropriate to fully grasp what was really going on.

If we look at the last sentence of OPs cited paragraph we can see that the historical development is more than that.

From section 1.5: But it is to say that the foundations of the calculus were for at least two centuries the subject of shifting, partial, and largely coherent speculations that form the opening chapters of the history of analysis.

To better see the author's position I like to cite from Hidden Harmony-Geometric Fantasies: The Rise of Complex Function Theory written by Jeremy Gray and Umberto Bottazzini.

From the Introductory Section: In any history of ideas, the historian seeks to show how things once thought about in one way became thought about in another. As complex function theory developed many ideas were first introduced naively and only slowly refined.

Definitions were lacking, and when provided were sometimes inadquate by later standards. Moreover precision, when it became available, could be misleading: mathematicians on occasion offer a clear definition with very few ideas about its deepest implications - as the example of continuity in real analysis shows.

Sometimes these problems can be confronted directly, as with the very definition of an analytic function, but more often one has to ride out a long period of some vagueness.

Let us note some specific issues: Cauchy, for example, often used the phrases continuous and finite and continuous very loosely to mean something like complex analytic. Similar problems occcur with counting roots according to their multiplicities, with $\lim$ versus $\limsup$, and points of infinity and poles. ...

Later on the authors continue with their preferred approach to the history of mathematics and mathematicians:

There is therefore no truly satisfactory way to represent the originial ideas of mathematicians when they are like this. To say nothing is to produce confusion. To silently bring them into line with modern standards not only introduces anachronisms but also brings in historical falsehoods and nullifies the purpose of history.

To correct them in more than the most egregious cases is to encumbeer genuine blunders and thereby diminish the work of major mathematicians.

The best policy is to read on in a spirit of dialogue with the earlier authors, aware, as one might be, of the limitations and false implications of their papers and books, and waiting to see when, if at all in the period, better light was shone on the subject. In this way one can grapple with more of the complexity, and the drama, of the past.

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Markus Scheuer
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    You (or perhaps the authors of the book from where you quote) have captured the spirit of mathematical development over time. +1 for the nice answer. – Paramanand Singh May 31 '16 at 09:54
  • @ParamanandSingh: Perhaps me, but for sure the authors from where I quote. :-) Many thanks for your nice comment. – Markus Scheuer May 31 '16 at 10:09
  • What do you think they mean when they claim that "Definitions were lacking"? They go on to say that sometimes definitions were provided, but what does the original comment mean? – Mikhail Katz May 31 '16 at 11:24
  • Well the issue of foundations is kind of precisely the problem. What they seem to be saying is that set-theoretic definitions were lacking, though they don't mention this explicitly. I don't see them introducing the distinction between definitions and foundations clearly enough. The example they give, namely that of continuity, is particularly ill-chosen because Cauchy's definition of continuity was actually quite precise. Do they give any evidence in the book that he used "continuous" in the sense of "complex analytic"? I find this surprising. – Mikhail Katz May 31 '16 at 12:09
  • Since you are the one who brought it up in your answer can you give a sample of their insights? – Mikhail Katz May 31 '16 at 15:44
  • Bottazzini's comment that "The best policy is to read on in a spirit of dialogue with the earlier authors" is at best a dodge, since it does not clarify what the content of such a dialogue will be. For example, if Gray is interested in confronting Euler on allegedly "dreadfully weak" foundations, the dialogue is not like to be productive. So even after Bottazzini commits himself to resolving issues through dialogue, the question still remains: what is on the agenda? Is it foundations (as Gray's comment seems to suggest) or procedures? Bottazzini and Gray leave this question unanswered. – Mikhail Katz Jun 02 '16 at 06:51
  • I have read this interesting book but it mostly deals with complex analysis rather than real analysis. In fact on page 111 they criticize historians for focusing on real analysis at the expense of complex analysis. I don't see them saying that much relevant to my question. Incidentally, they do cite Laugwitz profusely, but only his work on Riemann. This at any rate acknowledges that he was a first-rate historian, and only underscores the question why Gray stonewalls Laugwitz's research on Cauchy published in Gray's own journal! At any rate Bottazzini-Gray do not shed any light at all... – Mikhail Katz Jun 02 '16 at 08:33
  • ...on the foundations vs procedures dichotomy, nor on the question why and how would a historian choose to prefer one modern foundational framework over another in interpreting the work of historical authors. – Mikhail Katz Jun 02 '16 at 08:34
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Comment

What about Detlef Laugwitz's comments in his: Infinitely small quantities in Cauchy's textbooks, Hist.Math. 14(1987), no.3, 258–274 [that you cited elsewhere in SE]:

As a historian of mathematics one cannot but take an author’s own intentions and reasons seriously: Infinitely small quantities are fundamental in Cauchy’s analysis, they are compatible with rigor, and they produce simplicity.

[Cauchy's theorems on continuity and convergence] are incorrect when interpreted in the by now common conceptual framework of analysis (which obviously cannot have been Cauchy’s framework). Both theorems become correct as soon as one adds assumptions on uniformity (which, at least in the form by now common, were never used by Cauchy). The theorems are correct in any of the modern theories of infinitesimals (which, apart from being unknown to Cauchy, lack the “simplicity of infinitesimals,” at least in the version of Robinson).

The three attitudes mentioned (Cauchy erred; Cauchy forgot about essential assumptions; Cauchy was correct, but only when put against a modern background) are unsatisfactory from the point of view of a historian. [...] The only satisfactory attitude should be: Try and understand Cauchy’s theorems and their proofs from his own concepts.

Note: according to my understanding of J.Gray's point of view, a "parsimonious history" is an approach aimed at understanding past theories and concepts "in their original environment", avoiding to overload them with recent developments.

Mauro ALLEGRANZA
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  • Thanks for this. Laugwitz clearly disagrees with Gray's thesis here. Would you include Laugwitz's paper in Archive for History of Exact Sciences here as well? – Mikhail Katz May 31 '16 at 16:03
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    @MikhailKatz "Try and understand Cauchy’s theorems and their proofs from his own concepts." is exactly how I understand the quote of Gray in your question. – Daniel Fischer May 31 '16 at 19:14
  • @DanielFischer, the question is, why should one modern theory as opposed to another be excluded apriori from providing proxies for historical procedures. – Mikhail Katz Jun 01 '16 at 08:01
  • Mauro, what I am trying to understand is the precise meaning of "recent developments". Does this refer to foundational issues of justification of the kind of entities the historical authors used (this might be called an ontological issue), or rather the procedures the historical authors used and their proxies in modern theories? – Mikhail Katz Jun 01 '16 at 08:23
  • Leaving aside the question of whether Agnesi's source was Newton and Leibniz, what is pertinent in this thread is the question of procedures. Namely, which modern framework accounts better for Agnesi's procedures? Is it a modern Weierstrassian one, as one assumes Gray would say, or a modern infinitesimal one? – Mikhail Katz Jun 01 '16 at 08:57
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    There seem to be unbalanced parentheses in your answer. – Mikhail Katz Jun 03 '16 at 07:33
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Gray's "parsimonious" argument could be termed the Gray sword analogously with the Occam razor, and if applied in the context of a proper focus on procedures would in fact yield the opposite result of the one Gray seeks.

Consider for example Cauchy's definition of continuity, namely an infinitesimal change $\alpha$ in the variable $x$ always produces an infinitesimal change $f(x+\alpha)-f(x)$ in the function. In a modern infinitesimal framework one copies this over almost verbatim to get a precise definition of continuity.

If one wishes to work in a traditional Weierstrassian framework, one needs to interpret Cauchy's definition as "really" saying that, for example, for every epsilon there is a delta such that for every $x$, etc.

Such logical complexity involving alternating quantifiers will surely fall by the (Gray) sword. Alternatively, one could seek to interpret Cauchy by means of sequences, which is not much better because Cauchy explicitly says in defining an infinitesimal that a sequence becomes an infinitesimal (rather than an infinitesimal being a sequence). So apparently Gray should be saying the following:

Since Boyer (at least) there have been attempts to argue that Leibniz, Euler, and even Cauchy could have been thinking in some informal version of rigorous modern Weierstrassian analysis. However, a historical interpretation such as the one sketched above that aims to understand Leibniz on his own terms, and that confers upon him both insight and consistency, has a lot to recommend it over an interpretation that has only been possible to defend since Weierstrass came along. It is parsimonious and requires no expert defence for which modern alternating quantifiers seem essential and therefore create more problems than they solve.

Mikhail Katz
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This is in response to a claim by Gray and Bottazzini that allegedly "Definitions were lacking". So what? It is a basic (meta)mathematical problem:

  • you are given a proof that contains no definitions (or, perhaps, refers to excessively narrow definitions)
  • recover the definitions.

This is what Bourbaki was doing when he was recovering real definitions from the boring and repetitive proofs of the early functional analysis.

Why this approach cannot be used as a tool for analysis of the history of mathematics? Mathematicians of the past were using unstated definitions and unstated assumptions. Why are research mathematicians of nowadays are forbidden from doing some kind of reverse engineering and recovering their ancient colleagues' definitions and assumptions?

www.borovik.net/selecta

Mikhail Katz
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Alexandre Borovik
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    "Why are research mathematicians of nowadays are forbidden from doing some kind of reverse engineering and recovering their ancient colleagues' definitions and assumptions?" 1. Who forbids whom of doing that, exactly? 2. What you call reverse engineering is actually often what mathematicians turned historians practice and support, at the expense of what historians view as being history. 3. Are you suggesting that Gray is not a mathematician? – Did Jun 01 '16 at 19:14
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    This does not appear to be an answer to the question, which is asking for an explanation of Gray's views, not a rebuttal of them. – Eric Wofsey Jun 01 '16 at 19:54
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    @EricWofsey, it can be debated whether this answer is conclusive, but it seems to me that the intention is to provide an analysis of misconceptions behind Gray's views which would explain Gray's comment. – Mikhail Katz Jun 02 '16 at 07:07