The question is the title of a 2013 publication in the Notices of the American Mathematical Society, by twelve authors (of which I am one). The contention is that traditional history of mathematics is based on the assumption of an inevitable evolution toward the real continuum-based framework as developed by Cantor, Dedekind, Weierstrass (referred to as the "great triumvirate" by Carl Boyer here) and others. Taking some seminal remarks by Felix Klein as their starting point, the authors argue that the traditional view is lopsided and empoverishes our understanding of mathematical history. Have the historians systematically underplayed the importance of the infinitesimal strand in the development of analysis? Editors are invited to submit reasoned responses based on factual historical knowledge, and refrain from answers based on opinion alone.

To be even more explicit, we ask for additional examples from history that support either Boyer's viewpoint or the NAMS article viewpoint. That is, limit the question to facts and not opinions (based on a comment by Willie Wong at meta).

Note 1. For a closely related MO thread see this.

Note 2. A reaction to the Notices article by Craig Fraser was published here.

Note 3. Another would-be victor Gray is analyzed in this MSE thread.

Note 4. The Notices article originally contained a longish section on Euler, which was eventually split off into a separate article. The article shows, using the writings of Ferraro as a case study, how an assumption of default Weierstrassian foundations deforms a scholar's vision of Euler's mathematics. The article was recently published in 2017 in Journal for General Philosophy of Science.

Note 5. A response to Craig Fraser's reaction was published in 2017 in Mat. Stud.; see this version with hyperlinks.

Note 6. Further insight into the mentality of some math historians can be gleaned from a recent (2022-23) exchange in The Mathematical Intelligencer; see the answer https://math.stackexchange.com/a/4725050/72694 below.

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    $\begingroup$ The history of Linear Algebra is almost certainly written by the vectors. $\endgroup$
    – Emily
    Commented Jul 16, 2013 at 16:37
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    $\begingroup$ I think Claude Lobry would like this paper. I was introduced to non-standard analysis by a little book of him:"Et pourtant ils ne remplissent pas N". Excellent read, even if you are a die hard Weierstrassian. $\endgroup$ Commented Jul 16, 2013 at 16:55
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    $\begingroup$ +1 for damn good question, All the math history I ever did might be just propaganda. $\endgroup$
    – jimjim
    Commented Jul 17, 2013 at 8:12
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    $\begingroup$ I see the following question: "Have the historians systematically underplayed the importance of the infinitesimal strand in the development of analysis?" I'm no expert on the history of the subject, but considering you had to find your source material for your article somewhere, I'd say the answer is an emphatic no. What else is there to discuss? $\endgroup$ Commented Jul 17, 2013 at 14:29
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    $\begingroup$ Following on from Raskolnikov's comment, what are you trying to achieve? No one is stopping anyone from writing a book on the history you cite, and no one is stopping anyone from trying to convince educators to give infintesimals another look. There may be others who disagree with you, but that's a different issue. For the record, I have no strong opinion. $\endgroup$
    – user452
    Commented Jul 17, 2013 at 17:33

5 Answers 5


Certainly the victors write the history, generally. But when the victory is so complete that there is no further threat, the victors sometimes feel they can beneficently tolerate "docile" dissent. :)

Srsly, folks: having been on various sides of such questions, at least as an interested amateur, and having wanted new-and-wacky ideas to work, and having wanted a successful return to the intuition of some of Euler's arguments ... I'd have to say that at this moment the Schwartz-Grothendieck-Bochner-Sobolev-Hilbert-Schmidt-BeppoLevi (apologies to all those I left out...) enhancement of intuitive analysis is mostly far more cost-effective than various versions of "non-standard analysis".

In brief, the ultraproduct construction and "the rules", in A. Robinson's form, are a bit tricky (for people who have external motivation... maybe lack training in model theory or set theory or...) Fat books. Even the dubious "construction of the reals" after Dedekind or Cauchy is/are less burdensome, as Rube-Goldberg as they may seem.

Nelson's "Internal Set Theory" version, as illustrated very compellingly by Alain Robert in a little book on it, as well, achieves a remarkable simplification and increased utility, in my opinion. By now, having spent some decades learning modern analysis, I do hopefully look for advantages in non-standard ideas that are not available even in the best "standard" analysis, but I cannot vouch for any ... yet.

Of course, presumably much of the "bias" is that relatively few people have been working on analysis from a non-standard viewpoint, while many-many have from a "standard" viewpoint, so the relative skewing of demonstrated advantage is not necessarily indicative...

There was a 1986 article by C. Henson and J. Keisler "on the strength of non-standard analysis", in J. Symbolic Logic, 1986, maybe cited by A. Robert?... which follows up on the idea that a well-packaged (as in Nelson) version of the set-theoretic subtley of existence of an ultraproduct is (maybe not so-) subtly stronger than the usual set-theoretic riffs we use in "doing analysis", even with AxCh as usually invoked, ... which is mostly not very serious for any specific case. I have not personally investigated this situation... but...

Again, "winning" is certainly not a reliable sign of absolute virtue. Could be a PR triumph, luck, etc. In certain arenas "winning" would be a stigma...

And certainly the excesses of the "analysis is measure theory" juggernaut are unfortunate... For that matter, a more radical opinion would be that Cantor would have found no need to invent set theory and discover problems if he'd not had a "construction of the reals".

Bottom line for me, just as one vote, one anecdotal data point: I am entirely open to non-standard methods, if they can prove themselves more effective than "standard". Yes, I've invested considerable effort to learn "standard", which, indeed, are very often badly represented in the literature, as monuments-in-the-desert to long-dead kings rather than useful viewpoints, but, nevertheless, afford some reincarnation of Euler's ideas ... albeit in different language.

That is, as a willing-to-be-an-iconoclast student of many threads, I think that (noting the bias of number-of-people working to promote and prove the utility of various viewpoints!!!) a suitably modernized (= BeppoLevi, Sobolev, Friedrichs, Schwartz, Grothendieck, et al) epsilon-delta (=classical) viewpoint can accommodate Euler's intuition adequately. So far, although Nelson's IST is much better than alternatives, I've not (yet?) seen that viewpoint produce something that was not comparably visible from the "standard" "modern" viewpoint.

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    $\begingroup$ This is the key, really: "I've not (yet?) seen that viewpoint produce something that was not comparably visible from the "standard" "modern" viewpoint." $\endgroup$ Commented Jul 18, 2013 at 0:56
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    $\begingroup$ I think that Cantor invented set theory after stumbling on the idea of ordinals when he was working on trigonometric series, or something similar. I think that his construction via Cauchy sequences came later. $\endgroup$
    – Asaf Karagila
    Commented Jul 18, 2013 at 7:50
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    $\begingroup$ @user72694 Thanks for the pointer! Goldbring's work is a very interesting example! $\endgroup$ Commented Jul 18, 2013 at 12:53
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    $\begingroup$ @paulgarrett It is a delicate matter, when a trigonometric series (a series of sines and cosines) is a Fourier series ($\displaystyle \frac{a_0}2+\sum_{n=1}^\infty(a_n\cos(nx)+b_n\sin(nx))$ is the Fourier series of $f(x)\in L^1(0,2\pi)$ iff $\displaystyle a_n=\frac 1{2\pi}\int_0^{2\pi} f(t)\cos(nt)\,dt$ and $\displaystyle b_n=\frac1{2\pi}\int_0^{2\pi} f(t)\sin(nt)\,dt$ for all $n$). The classic counterexample is $\displaystyle \sum_{n=2}^\infty\frac1{\log n}\sin(nx)$. This is a trigonometric series that converges everywhere, but it is not the Fourier series of any function. $\endgroup$ Commented Jul 18, 2013 at 15:31
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    $\begingroup$ @it's, your remark is astute. Leibniz already was aware of the difference between assignable and inassignable numbers (here "ordinary" numbers like $\pi$ are assignable, whereas for example infinitesimals are inassignable). Such a distinction is formalized in modern infinitesimal analysis in terms of the standard/nonstandard distinction. So the idea that there are different levels of "reality" for numbers has been there since the genesis of infinitesimal calculus. See this article in British Journal for History of Math for details $\endgroup$ Commented Oct 9, 2023 at 11:38

To give an example of the kind of answer requested here, note that one of the first examples in the NAMS text is from David Mumford, who wrote about overcoming his own prejudice (stemming from what he was taught concerning infinitesimals) in the following terms: "In my own education, I had assumed that Enriques [and the Italians] were irrevocably stuck.… As I see it now, Enriques must be credited with a nearly complete geometric proof using, as did Grothendieck, higher order infinitesimal deformations.… Let’s be careful: he certainly had the correct ideas about infinitesimal geometry, though he had no idea at all how to make precise definitions."

I enjoyed paul garrett's answer though it is steered in a slightly different direction, namely the effectiveness of NSA in cutting-edge research, whereas my question is mostly concerned with historical interpretation and getting an accurate picture of the mathematical past.

To give another example, Fermat's procedure of adequality involves a step where Fermat drops the remaining "E" terms; he carefully chooses his terminology and does not set them equal to zero. Similar remarks apply to Leibniz. Yet historians often assume that there is a logical contradiction involved at the basis of their methods, which can be summarized in the notation of modern logic as $(dx\not=0)\wedge(dx=0)$. Such remarks often go hand-in-hand with claims that the alleged logical contradiction was finally resolved around 1870. Without detracting from the greatness of the accomplishment around 1870, such criticism of the early pioneers of the calculus may not be on target.

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    $\begingroup$ I'm a little unsure of why this quote by Mumford is relevant to your question, which is about whether historians of mathematics have given short shrift to infinitesimals in the development of analysis. The "infinitesimals" Mumford describes are not infinitesimal numbers; they are infinitesimal deformations. Although there are some analogies between them (and Mumford makes an analogy between Enriques and Leibniz), they are really not the same. $\endgroup$ Commented Jul 18, 2013 at 7:57
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    $\begingroup$ In particular there is nothing "nonstandard" (i.e., ultraproducts, internal sets, transfer principles...) in the formalism of infinitesimal deformations. Rather it requires the enlargement of the class of rings which are viewed as rings of algebraic functions to include rings with nilpotent elements. So for instance this involves replacing $\mathbb{C}$ by $\mathbb{C}[t]/(t^n)$. Could you clarify whether you are saying that you view this as part of "the infinitesimal strand in the development of analysis", and if so, why? $\endgroup$ Commented Jul 18, 2013 at 8:00
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    $\begingroup$ Of course they are not the same, but there does tend to exist an attitude of distrust toward all historical forms of infinitesimal reasoning, as illustrated by Mumford's comment. This is usually accompanied by a belief that historical infinitesimals were allegedly "logically inconsistent". Whether or not this is the case was discussed at length in this thread. $\endgroup$ Commented Jul 18, 2013 at 8:06
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    $\begingroup$ "I did not say a word about NSA in my question (though I did add the NSA tag later, since it is relevant)." What a strange thing to say. You included it as a tag in the question, which means that you regard it as one of the keywords. The question asks for a response to your article, which has NSA all over it. And your question is all about infinitesimals in the history of analysis, a subject in which NSA is one of the most important, if not the single most important, development. $\endgroup$ Commented Jul 18, 2013 at 9:54
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    $\begingroup$ Honestly your comments are convincing me that I am not so interested in what mathematical historians have to say about this issue. (Should I be? Why?) Let me also reiterate that I have never accused Leibniz of inconsistent procedures. As a research mathematician, I well know that any method that you use on a large variety of problems and get correct answers must have a high degree of procedural consistency. So that's not such a fascinating issue to me either... $\endgroup$ Commented Jul 18, 2013 at 11:29

(This is meant as a response to a comment by Pete L. Clark on whether the history of analysis was a "linear progression". Due to its length I decided to post it as a separate answer) I agree that focusing on the term "linear" is not the issue. What does seem to be a meaningful issue is the following closely related question.

Is it accurate to view the formalisation of analysis around 1870, an extremely important development by all accounts, as having established a "true" foundation of analysis in the context of the Archimedean continuum and by eliminating infinitesimals?

An alternative view is that the success of the Archimedean formalisation in fact incorporated an aspect of failure, as well, namely a failure to formalize a ubiquitous aspect of analysis as it had been practiced since 1670: the infinitesimal.

According to the alternative view, there is not one strand but two parallel strands for the development of analysis, one in the context of an Archimedean continuum, as formalized around 1870, and one in the context of what could be called a Bernoullian continuum (Johann Bernoulli having been the first to base analysis systematically and exclusively on a system incorporating infinitesimals). This strand was not formalized until the work of Edwin Hewitt in the 1940s, Jerzy Los in the 1950s, and especially Abraham Robinson in the 1960s, but its sources are already in the work of the great pioneers of the 17th century.

To give an example, In his recent article (Gray, J.: A short life of Euler. BSHM Bulletin. Journal of the British Society for the History of Mathematics 23 (2008), no. 1, 1--12), Gray makes the following comment:

"At some point it should be admitted that Euler's attempts at explaining the foundations of calculus in terms of differentials, which are and are not zero, are dreadfully weak" (p. 6). He provides no evidence for this claim.

It seems to me that Gray's sweeping claim is coming from a "linear progression" school of thinking where Weierstrass is credited with eliminating logically faulty infinitesimals, so of course Euler who used infinitesimals galore would necessarily be "dreadfully weak" without any further explanation needed.

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    $\begingroup$ To my eye your question is primarily one lying at the border of history, philosophy and personal taste. The quotation marks around true point to this, as the truth you are talking about is not mathematical truth: it is more like a political truth. If you construe the questions purely mathematically the answers are easy: was Weierstrassian analysis correct and sound? Yes, of course. Moreover it has been very useful in all the years since. Did it fail to formalize some of the earlier forms of analytic reasoning including infinitesimals? Yes, of course. $\endgroup$ Commented Jul 19, 2013 at 9:19
  • $\begingroup$ Is there something "missing" in the lack of treatment of infinitesimals in Weierstrassian analysis? That's equivalent to asking whether it is possible to add to W. analysis a rigorous and useful theory of infinitesimals, to which work of Robinson,...,Gromov,Green-Tao,Goldbring answers: yes, of course. $\endgroup$ Commented Jul 19, 2013 at 9:23
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    $\begingroup$ (i) Most mathematicians are extremely uninterested in the Xology of mathematics. Mathematics is one of the most ahistorical academic fields: as a profession we do not regard history or historical scholarship as important. It might be an interesting thing to talk about over drinks, but we have much less of a commitment to primary source material, accurate scholarship, and so forth. (Most of us are openly contemptuous of philosophy and education...) $\endgroup$ Commented Jul 19, 2013 at 10:01
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    $\begingroup$ (ii) Mathematicians who do get involved in other academic aspects of mathematics usually do so in a highly critical, individualistic way. If we somehow get interested in Euler's mathematics, we will read Euler. And we will think we understand Euler ever so much better than the historians/philosophers/educators did and thus not place much truck in what they say about him. $\endgroup$ Commented Jul 19, 2013 at 10:05
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    $\begingroup$ Final comment: the professional painter, be he journeyman or star, will find the work of the old masters to be masterful, more so than the critics. When I read an old paper by a master, I'm always struck by how knowledgeable and smart they are despite not having sophisticated modern tools. I've watched Gauss prove theorem after theorem about class groups of quadratic forms. This should be impossible: the notion of a group did not exist back then! So he works harder than we would have to in some places. But he does it, better than almost any of us could do in his place. $\endgroup$ Commented Jul 19, 2013 at 10:23

A recent discussion at https://math.stackexchange.com/questions/455871/cauchys-limit-concept is a good illustration of the influence of feedback-style ahistory (to borrow Grattan-Guinness's term), when Weierstrass's ideas are read into an earlier author whether or not they belong there. To be sure, there is a considerable amount of historical controversy concerning Cauchy. J. Grabiner emphasizes the importance of the germs of epsilon, delta procedures that can be found in certain arguments in Cauchy's oeuvre. However, the actual epsilon, delta definition of limit (as opposed to procedures found in certain arguments) was not introduced by Cauchy but rather by later authors (usually Weiestrass is credited even though an earlier occurrence is found in Dirichlet).

In any case, the formalisation of the epsilontic limit concept certainly does not lie with Cauchy. What Cauchy did write about limits is that a variable quantity has limit $L$ if its values indefinitely approach $L$. With Cauchy, the primitive notion is that of a variable quantity, and limits are defined in terms of the latter in a fashion almost identical to what Newton wrote a few centuries earlier. Recently young scholars like Bråting and Barany have challenged received views on Cauchy.

Meanwhile, the discussion at https://math.stackexchange.com/questions/455871/cauchys-limit-concept proceeds under the explicitly stated assumption concerning alleged "Cauchy's formalization of limits", which is contrary to fact. The assumption was not challenged by any of the participants. This indicates that the community is often not aware of the true nature of Cauchy's work in analysis, including his definition of continuity expressed in terms of infinitesimals rather than epsilon, delta.

  • $\begingroup$ That question is so muddled that it is a terrible example to use to illustrate your point. Currently it has two votes to delete. No doubt the remarks you mention would have been rightly addressed if a reasonable version of the question had been asked instead. $\endgroup$ Commented Aug 6, 2013 at 19:35
  • $\begingroup$ @Andres Caicedo: are you referring to the question on "Cauchy's limit concept"? It could have been stated more clearly, but the thrust of the question is clear. The OP is asking what came out of what is reputed to be Cauchy's foundational contribution to formalizing analysis, in terms of recognizable breakthroughs. The OP compares this with breakthroughs in physics that resulted from contributions of Einstein, Dirac, and others. I think other editors have also interpreted his question this way; one of them went on to answer that important developments in analysis stemmed from Cauchy's rigor. $\endgroup$ Commented Aug 6, 2013 at 19:43

A recent exchange in The Mathematical Intelligencer sheds light on the mentality of some historians of mathematical analysis. The relevant articles are:

(1) Katz, M.; Kuhlemann, K.; Sherry, D.; Ugaglia, M.; van Atten, M. "Two-track depictions of Leibniz's fictions." The Mathematical Intelligencer 44 (2022), no. 3, 261-266. https://doi.org/10.1007/s00283-021-10140-3, https://arxiv.org/abs/2111.00922

(2) Archibald, Tom; Arthur, Richard T. W.; Ferraro, Giovanni; Gray, Jeremy; Jesseph, Douglas; Lützen, Jesper; Panza, Marco; Rabouin, David; Schubring, Gert. A question of fundamental methodology: reply to Mikhail Katz and his coauthors. Math. Intelligencer 44 (2022), no. 4, 360-363.

(3) Bair, J.; Borovik, A.; Kanovei, V.; Katz, M.; Kutateladze, S.; Sanders, S.; Sherry, D.; Ugaglia, M.; van Atten, M. "Is pluralism in the history of mathematics possible?" The Mathematical Intelligencer 45 (2023), no. 1, 8. http://doi.org/10.1007/s00283-022-10248-0, https://arxiv.org/abs/2212.12422

(4) Bair, J.; Borovik, A.; Kanovei, V.; Katz, M.; Kutateladze, S.; Sanders, S.; Sherry, D.; Ugaglia, M. "Historical infinitesimalists and modern historiography of infinitesimals." Antiquitates Mathematicae 16 (2022), 189-257. https://doi.org/10.14708/am.v16i1.7169, http://arxiv.org/abs/2210.14504

Most of our work involves analyzing primary sources of such pioneers as Fermat, Gregory, Leibniz, Euler, Cauchy and others (rather than disagreeing with historians). Since Archibald's "Viewpoint" text commented on our work, we did respond to their criticisms. It emerges that this recent "Viewpoint" is a good illustration of the idea that some traditional historians (though by no means all) cling to the Weierstrassian paradigm and tend to reject non-Archimedean interpretations of the work of the historical pioneers. Thus, the "Viewpoint" endorses the position of Arthur and Rabouin who claim that Leibniz never used genuine infinitesimals; rather, occurrences of the term "infinitesimal" do not refer to a mathematical entity, but rather are a (stenographic) shorthand for a purely Archimedean "exhaustion" argument. The "Viewpoint" also endorses the position of Schubring that Cauchy never used genuine infinitesimals.

We have analyzed the primary texts, with the following conclusions: Leibniz wrote in a 14/24 june 1695 letter to l'Hospital:

I use the term incomparable magnitudes to refer to [magnitudes] of which one multiplied by any finite number whatsoever, will be unable to exceed the other, in the same way [adopted by] Euclid in the fifth definition of the fifth book [of The Elements].

In modern editions of The Elements, the definition of comparability appears in Book V, Definition 4. Leibniz makes similar remarks in his article

Leibniz, G.W. Responsio ad nonnullas difficultates a Dn. Bernardo Niewentiit circa methodum differentialem seu infinitesimalem motas. Acta Erudit. Lips. (1695). In Gerhardt (GM), vol. V, pp. 320--328.

This clearly indicates that Leibniz used genuine (non-Archimedean) infinitesimals, contrary to the received view of some historians (notably, other leading historians disagree).

As far as Cauchy is concerned, our analysis of the primary documents in the text (4) cited above indicates that, while on occasion Cauchy presented arguments that can be read as prototype of Weierstrassian epsilon-delta techniques, he also clearly used genuine (non-Archimedean) infinitesimals in a number of texts, including his book on Differential Geometry from 1826, where he even used the term angle of contingence (i.e., hornangle), a classical non-Archimedean phenomenon.

These items tend to corroborate that some traditional historians tend to reject a priori those interpretations of classical pioneers that do not fit the Weierstrassian paradigm, and I thank Andy Putman for mentioning Archibald's "Viewpoint" at Math Overflow.


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