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I recently read an article that discussed--rather briefly--the issues of infinitesimals and the criticism of them by Berkeley. The author of the article (which, of course, I cannot find, as I read it on my phone) claimed that Berkeley's criticism led to a bunch of revisions to calculus textbooks, including authors adding a bit more rigor--and that those changes and additions made calculus harder to teach, learn, and understand.

The author shows a derivation for the derivative of $y = x^2 + x$ using an approximation of non-standard analysis. We define the average slope of the curve over an interval as $m = \frac{y_1-y_0}{x_1-x_0}$. Then, we shrink the size of the interval such that $y_1-y_0 = dy$ and $x_1-x_0 = dx$, where $dx$ and $dy$ are infinitesimals--values so close to zero that they might as well be zero. Then plugging $y+dy$ and $x+dx$ into the original function, we get:

$$ \begin{align*} y+dy &= (x+dx)^2 + x+dx \\ y+dy &= x^2+2xdx+dx^2 + x + dx \\ dy &= 2xdx+dx^2+dx \\ \frac{dy}{dx} &= 2x + dx+1 = 2x+1 \end{align*} $$

and since the value of $dx$ is so close to zero that it might as well be zero, we handwave it away to get $y' = 2x+1$.


Though Newton and Leibniz had attempted to apply some rigor to the creation of infinitesimals, they mostly handwaved it away as in the above example, and Berkeley's criticism of that perhaps justified (even if the manner of critique was a bit ridiculous). But the critique was also seen as a personal affront not just to Newton (maybe) and Leibniz, but to all those working with the newly-created system of calculus. Revisions were made, eventually ending up with the Weierstrass $\delta / \varepsilon$ formulation that we have at the beginnings of modern calculus textbooks... where students become readily confused.

Apparently there have been A/B pedagogical studies showing that non-standard analysis leads to better learning outcomes than traditional calculus--but even those methods involve spending a chapter defining hyperreal/surreal numbers in order to produce a "rigorous" system.

I argue that students learning calculus for the first time don't need a rigorous treatment, whether hyperreals, or $\delta / \varepsilon$, or something else. I saw the derivation above yesterday and thought "Wow, that would have made calculus make sense a lot faster. That's just algebra with some added bits and bobs." (I took my first calculus course nearly 30 years ago; the idea of it being algebra with bits added shocked me with its simplicity.)

HS juniors/seniors/college freshman don't need the analysis to be justified logically or in a mathematically rigorous way in order to perform calculus--and the vast majority of those taking calculus are not going on to take advanced mathematics courses where such rigor becomes important. I mean... we teach arithmetic from kindergarten on, but at no point does anyone other than a math major (or a self-taught adult with odd interests) learn about justifying arithmetic with set theory. We don't teach students an axiomatic formalization of elementary algebra--we present them with axioms or identities directly and don't worry about how we know they're true.

If all we need to teach differentiation is to define infinitesimals as handwaveable quantities, then use the basic slope formula to find differentials, do we really need the formalization of $\delta$ and $\varepsilon$, and of limits?

I know I've babbled on a bit here. To summarize:

  1. Is the authors' premise that calculus would be easier if we dumped the formalism correct/reasonable?
  2. If we were to drop the formalism, would there be any downsides for non-math majors?
  3. Was it really Berkeley's critique that caused this shift, or would it have happened anyway?
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    $\begingroup$ Of the 3 questions at the end, only number 3 is historical. $\endgroup$ Jan 30, 2022 at 1:26
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    $\begingroup$ Even number 3 is not really about history. What might have happened otherwise is not history, although it is hard to imagine that Lagrange, Gauss and Cauchy were dissatisfied with lack of rigor in calculus because they read Berkeley. $\endgroup$
    – Conifold
    Jan 30, 2022 at 5:59
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    $\begingroup$ Some of your questions might be on-topic on the Mathematics Educators stack. I never post there so you would need to check first. $\endgroup$
    – mdewey
    Jan 30, 2022 at 13:33
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    $\begingroup$ I would argue that the header question and questions 2 and 3 are all within scope -- especially since there is in some degree historical evidence for the influence and effect of Berkeley's critique, and enough evidence to show that it was not a unique event -- indeed far from that, it was the third attack of its kind. There is also some historical evidence for the attitudes of past mathematicians to the practical utility of the criticised techniques. Question 1 on the other hand seems too indefinite to answer in the absence of an identified author or details of proposal. $\endgroup$
    – terry-s
    Jan 31, 2022 at 19:54
  • $\begingroup$ Just to add this -- the plan of derivation for the derivative of x^2 + x in the question looks to me hardly different (in complexity or simplicity) from a derivation in terms of limits. Only replace dx and dy by δx and δy to indicate finite rather than infinitesimal increments: Then on the last line, the derivative dy/dx is in the result just the limit of δy/δx as δx tends towards zero, and on the right-hand side the limit of δx under the same condition is of course itself zero -- as the difference from zero can be made smaller than any finite ε that may be specified. $\endgroup$
    – terry-s
    Feb 1, 2022 at 21:36

2 Answers 2

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There are four questions here (including the one in the heading) that appear to involve aspects of the history of calculus methods. For clarity in case of future amendment, the questions as they are addressed here are:

(Heading) Did Berkeley's criticism of infinitesimals hobble calculus pedagogy?

(This heading-question was also amplified/explained in the text: [Did] Berkeley's criticism [lead] to ... revisions to calculus textbooks, including ... more rigor -- and [did] those changes and additions [make] calculus harder to teach, learn, and understand.)

(1) Is the authors' premise that calculus would be easier if we dumped the formalism correct/reasonable?

(2) If we were to drop the formalism, would there be any downsides for non-math majors?

(3) Was it really Berkeley's critique that caused this shift, or would it have happened anyway?

First, about questions (Heading) and (3):-

Berkeley's (1685–1753) criticism (in "The Analyst; or a Discourse Addressed to an Infidel Mathematician" (1734)), which targeted Leibnitz's infinitesimal calculus in L'Hopital's version of 1696, was the third of three widely-known critiques of calculus-like methods that appeared from about the mid-17th to the mid-18th century. All three critiques had a similar central point: to condemn the ambiguity or contradiction said to arise from inconsistent treatments of infinitesimals (or earlier, indivisibles). For the critics, infinitesimals seemed to be treated as non-zero for one purpose and then as zero for another purpose in the same demonstration. This was offensive to those who held to an early mathematical ideal, which was to deliver certain knowledge through proofs that were also certain and indubitable. Ambiguities and contradictions were repugnant to this ideal.

On the first occasion of such critiques, in the mid-17th century, the target was the 'method of indivisibles' proposed by Bonaventura Cavalieri (c.1598–1647) in 'Geometria indivisibilibus continuorum ...' (1635), and then, with a reponse to criticisms, in 'Exercitationes geometricae sex' (1647). Critics here were Paul Guldin (1577-1643), in vol.4 of his "Centrobaryca seu de centro gravitatis trium specierum quantitatis continuae" (1635-1641), and André Tacquet (1612-1660). Cavalieri's method remained under a cloud of doubt and disrepute even though it was acknowledged (e.g. by Torricelli and Newton) to yield many useful results.

On the second occasion, at the end of the 17th century and at the turn of the 18th, the main critic was Michel Rolle (1652-1719), who vigorously attacked, especially, the Leibnizian infinitesimal methods published by L'Hopital in 1696 in 'Analyse des infiniment petits'. His attacks have already been discussed in this forum, in Did Michel Rolle say that the calculus is "a collection of ingenious fallacies"?. On that second occasion, the attacks were conducted in public in Paris and were defended among others by Pierre Varignon, whose arguments were notable for their appeal to Newton's method of limits (the 'method of first and last ratios' given in Book 1 section 1 of the 'Principia', see below.)

Berkeley's 1734 version of the critique did have effects on the writing of at least one textbook: 'Treatise of Fluxions' (1742) by Colin Maclaurin (1698-1746). This book was regarded by Jean-Étienne Montucla (1725-1799) in his Histoire des Mathématiques (2nd ed., vol.3), as an effective rebuttal of Berkeley's criticism and as a good defence of the calculus methods of both Newton and Leibnitz. Montucla's 'Histoire' gave this special appreciation of Maclaurin's defence of the calculus (in passages from pp.116 and 118, here in my translation):--

"Nobody is unaware these days that the infinitesimal calculus is in its fundamentals absolutely the same as what Newton has called the calculus of fluxions. Now, this latter has nothing that is not in conformity with the most rigorous principles of Geometry, as has been shown at full length. Accordingly, both the one and the other must enjoy the same degree of certainty."

[...] "It was seemingly in answer to the attacks of Berkeley that Mr Maclaurin undertook his 'Treatise of Fluxions' that appeared in 1742. There, the method of Newton is fully demonstrated without any assumption of infinitesimals or anything else capable of lending itself to controversy ... Mr Maclaurin's demonstrations are of prodigious length ... he could have limited himself to some examples ... . [But] however that may be, one can say that if any doubts could remain about the solidity of Newton's method, they are entirely dissipated by this work of Maclaurin ... ."

So one answer to questions (Header) and (3) can be this: that Berkeley's criticism, while of immediate influence, may not have been determinative of the course of calculus criticism and exposition, since other criticisms of similar kind had already been published, and were 'in the air' and known widely, even if not well, before him. On the other hand, Maclaurin's response to Berkeley was very lengthy and not easy to understand.

In response to question (2):

The controversies about calculus-related methods have mainly concerned their justificatory foundations, with much less impact on the careful practice even of some methods impugned by the critiques.

For example, in relation to the method of indivisibles, Cavalieri's successor Evangelista Torricelli (1608-1647), "worked with indivisibles, but he cautioned that their uncritical use could lead to paradoxes" (see "A History of Mathematics" (3rd Edition, 2008) V J Katz). Thus, it was probably in cautious recognition of shaky foundations for the method of indivisibles, that Torricelli took precaution after obtaining by their use -- and doubting -- a surprising result, namely, that an infinitely long solid of revolution could nevertheless have a finite volume. Thus, he found through indivisibles that the infinitely-long solid formed by rotating a hyperbola xy = k^2 around the y axis from y = a to y =∞ nevertheless has finite volume. Torricelli then quieted doubts and confirmed the result by presenting a second derivation, relying on the indubitable classical method of exhaustion.

Likewise, Newton in his 'Principia' declined to rely on the dubious justification of the method of indivisibles for his calculus methods, even though he stated specifically that 'hereby the same thing is perform'd as by the method of indivisibles' (Newton 'Principia', 1729 translation, vol.1, Book 1, sec.1). He gave reasons for setting out his method of limits ('first and last ratios') :-

"These lemmas are premised, to avoid the tediousness of deducing perplexed demonstrations ad absurdum, according to the method of the ancient geometers. For demonstrations are more contracted by the method of indivisibles : But because the hypothesis of indivisibles ... is reckoned less geometrical; I chose rather to reduce the demonstrations of the following propositions to the first and last sums and ratio's of nascent and evanescent quantities, that is, to the limits of those sums and ratio's; and so to premise, as short as I could, the demonstrations of those limits. For hereby the same thing is perform'd as by the method of indivisibles; and now those principles being demonstrated, we may use them with more safety."

So a possible answer to question (2) would seem to be this, that anything that only expresses later work on justificatory foundations for calculus-like methods might be disregarded in practical use (leaving the user in good historical company) -- provided that any shortcuts are applied with care inside the limits of well-established practice for their use.

(In view of the above, this answer does not detail 19th-century work on foundations of calculus, especially by Cauchy and Weierstrass: it seems enough for present purposes to limit the discussion to the 17th- and 18th-c. history which shows the main lines on which controversies developed. Again, the 20th-century non-standard analysis appears to concern justificatory foundations rather than everyday practice.)

(As for question (1), the questioner could not identify the author or the details of the author's suggestion: nothing can be added about that here.)

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Viewing the development of mathematics since the (abstractly justifiable) criticism of the "infinitesimal calculus": no, those criticisms did not seem to inhibit people from continuing to use ideas of calculus in many ways, but/and did stimulate some people to worry more about how to make all those got-the-right-answer methods meet a better... or at least more explicit... standard of reasoning.

Comparable issues arose with Green-Heaviside-Dirac-et-al use of "generalized functions", to obtain demonstrably correct answers to physics-y questions, for 100+ years before L. Schwartz (and then A. Grothendieck) made it all completely rigorous. Did not seem to inhibit physicists, nor physics students today, but did "trouble" mathematicians for quite a while. :)

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