Did Euler prove theorems by example?

Question

In his 2014 book, Giovanni Ferraro writes at beginning of chapter 1, section 1 on page 7:

Capitolo I

Esempi e metodi dimostrativi

1. Introduzione

In The Calculus as Algebraic Analysis, Craig Fraser, riferendosi all'opera di Eulero e Lagrange, osserva:

A theorem is often regarded as demonstrated if verified for several examples, the assumption being that the reasoning in question could be adapted to any other example one chose to consider (Fraser [1989, p. 328]).

Le parole di Fraser colgono un aspetto poco indagato della matematica dell'illuminismo.

I am not fluent in Italian but the last sentence seems to indicate that Ferraro endorses Fraser's position as expressed in the passage cited in the original English without Italian translation.

I was rubbing my eyes as I was reading this so I decided to check in Fraser's original, thinking that perhaps the comment is taken out of context. I found the following longer passage on Fraser's page 328 quoted by Ferraro:

The calculus of EULER and LAGRANGE differs from later analysis in its assumptions about mathematical existence. The relation of this calculus to geometry or arithmetic is one of correspondence rather than representation. Its objects are formulas constructed from variables and constants using elementary and transcendental operations and the composition of functions. When EULER and LAGRANGE use the term "continuous" function they are referring to a function given by a single analytical expression; "continuity" means continuity of algebraic form. A theorem is often regarded as demonstrated if verified for several examples, the assumption being that the reasoning in question could be adapted to any other example one chose to consider.

Let us examine Fraser's hypothesis that in Euler and Lagrange, allegedly "a theorem is often regarded as demonstrated if verified for several examples."

I don't see Fraser presenting any evidence for this. Now Wallis sometimes used a principle of "induction" in an informal sense that a formula verified for several values of $n$ should be true for all $n$, but for this he was already criticized by his contemporaries, a century before Euler and Lagrange.

Several articles were recently published examining Euler's proof of the infinite product formula for the sine function. The proof may rely on hidden lemmas, but it is a sophisticated argument that is a far cry from anything that could be described as "verification for several examples."

It seems to me that this passage from Fraser is symptomatic of an attitude of general disdain for the great masters of the past. Such an attitude unfortunately is found among a number of received historians. For example, we find the following comment:

Euler's attempts at explaining the foundations of calculus in terms of differentials, which are and are not zero, are dreadfully weak.

(p. 6 in Gray, J. ``A short life of Euler.'' BSHM Bull. 23 (2008), no.1, 1--12).

In a similar vein, in a footnote on 18th century notation, Ferraro presents a novel claim that

for 18th-century mathematicians, there was no difference between finite and infinite sums.

(footnote 8, p. 294 in Ferraro, G. ``Some aspects of Euler's theory of series: inexplicable functions and the Euler-Maclaurin summation formula.'' Historia Mathematica 25, no. 3, 290--317.)

Far from being a side comment, the claim is emphasized a decade later in the Preface to his 2008 book:

a distinction between finite and infinite sums was lacking, and this gave rise to formal procedures consisting of the infinite extension of finite procedures

(p. viii in Ferraro, G. The rise and development of the theory of series up to the early 1820s. Sources and Studies in the History of Mathematics and Physical Sciences. Springer, New York.)

Grabiner doesn't hesitate to speak about

shaky eighteenth-century arguments

(p. 358 in Grabiner, J. ``Is mathematical truth time-dependent?'' Amer. Math. Monthly 81 (1974), 354--365); it is difficult to evaluate her claim since she does not specify the arguments in question.

Instead of viewing Fraser's passage as problematic, Ferraro opens his book with it, which is surely a sign of endorsement. The attitude of disdain toward the masters seems to have permeated the field to such an extent that it has acquired the status of a sine qua non of a true specialist.

In my study of Euler I have seen sophisticated arguments rather than proofs by example, except for isolated instances such as de Moivre's formula. On the other hand Euler's oeuvre is vast.

Question. Can Euler be said to have proved theorems by example in other than a handful of exceptional cases, in any meaningful sense?

Note 1. Some editors requested examples of what I described above as a disdainful attitude toward the masters of the past on the part of some historians. I provided a couple of additional ones. Editors are invited to provide examples they have encountered; I believe they are ubiquitous.

Note 2. We tried to set the record straight on Euler in this recent article and also here.

Note 3 (added in 2024): A follow-up paper appeared in Antiquitates Math:

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, https://arxiv.org/abs/2210.14504, https://mathscinet.ams.org/mathscinet-getitem?mr=4570174

Answer 1

There's some evidence that precisely the opposite can be said: that Euler is aware of the fallacies of proving theorems by example (of course, this does not necessarily mean he has never used it). One memorable instance is his Exemplum Memorabile Inductionis Fallacis, where he described how he was almost led to conjecture a recursive formula for a particular numerical sequence until he found that they disagreed on the 10th term. (There are other reasons for that formula to have been plausible; that and other topics are discussed in this article.)

(Incidentally the "right" formula is now quite well-known.)

Answer 2

"Proof by example" is a technique used by Euclid, who often proved results that hold e.g. for n integers in a typical case, say for 3 integers, as well as by Diophantus, who had to choose values for his parameters due to his lack of algebraic notation. I regard both versions as complete proofs.

This is apparently not what Fraser is referring to; Euler did generalize from examples to theorems in his Algebra, where he transferred correct results from "rings of integers" ${\mathbb Z}[i]$ to general quadratic rings without proof; but Euler wrote his algebra when he was old and completely blind, and perhaps it is fair to say that Euler was collecting evidence for his "method" rather than regarding these examples as proofs. I am not aware of a single example where Euler explicitly said that he regarded the verification of examples as a proof, but I only have read his number theoretical work in detail. The idea that Lagrange proved results by examples is ridiculous.

Answer 3

Not sure if this answer adds anything to the ones already given. I write it because It is an example where Euler explicitly writes about the necessity of giving a proof, and more importantly, calls a proof given by himself "Attempt at a proof". The following is his remarks before his "attempt at a proof" of the sum of two squares in 1758, "On numbers which are the sum of two squares". Two years later, he has another paper with the title " Proof of Fermat’s Theorem That Every Prime Number of the Form 4n + 1 is the Sum of Two Squares". In a way, even the titles of these two papers suggest an answer to your question.

All prime numbers which are sums of two squares, except 2, form this series: 5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, etc. Not only are these contained in the form 4n + 1, but also, however far the series is continued, we find that every prime number of the form 4n+1 occurs. From this, we can conclude by induction 6 that it is likely enough that there is no prime number of the form 4n+ 1 which is not also a sum of two squares. Nevertheless, induction, however extensive, cannot fulfill the role of proof. Even if no one doubts the truth of the statement that all prime numbers of the form 4n+ 1 are sums of two squares, until now mathematics could not add this to its established truths. Even Fermat declared that he had found a proof, but because he did not publish it anywhere, we properly extend confidence toward the assertion of this most profound man, and we believe that property of the numbers, but this recognition of ours rests on pure faith without knowledge. Although I labored much in vain on a proof to be discarded, nevertheless I have discovered another argument to be given for this truth, which, even it if it is not fully rigorous, still appears to be equivalent to induction connected with nearly rigorous proof.

The following is from the introduction of the second paper where Euler summarizes the first paper.

I next put forth an attempt of the proof from which the validity of this theorem is revealed much more clearly, even if it should be set aside by the standards of rigorous proof.

Answer 4

Yes, Euler demonstrated, as G. Pólya illustratively illuminates in his:

" Induction and Analogy in Mathematics; Vol. 1 of Mathematics and Plausible Reasoning".

Pólya gives an English translation of Euler's writing in Ch. 6.

It can be read with very little prior knowledge.