Lapses of "the early proponents of the doctrine of limits"

Question

I have a question that I have been wondering about for a long time without finding any answer. Concerning the period around 1900, Robinson commented in his 1966 book that "there is in the writings of this period an noticable contrast between the severity with which the ideas of Leibniz and his successors are treated and the leniency accorded to the lapses of the early proponents of the doctrine of limits. We do not propose here to subject any of these works to a detailed criticism."

I have always found the last sentence a bit disappointing. Robinson could have indeed subjected the "lapses" of these early proponents to a detailed criticism. The further we are removed from them in time, the harder it becomes to carry out such a criticism. Is anyone aware of such "lapses" and/or discussion thereof in the literature?

Note 1. As per request by Joël, here is a larger quote, borrowed from Stroyan's site http://www.math.uiowa.edu/~stroyan/InfsmlCalculus/Lecture1/HTMLLinks/Lect1_7.html "The history of a subject is usually written in the light of later developments. For over half a century now, accounts of the history of the Differential and Integral Calculus have been based on the belief that even though the idea of a number system containing infinitely small and infinitely large elements might be consistent, it is useless for the development of Mathematical Analysis. In consequence, there is in the writings of this period an noticable contrast between the severity with which the ideas of Leibniz and his successors are treated and the leniency accorded to the lapses of the early proponents of the doctrine of limits. We do not propose here to subject any of these works to a detailed criticism. However, it will serve as a starting point for our discussion to try to give a fair summary of the contemporary impression of the history of the Calculus..." I don't have Robinson's book in front of me right now, so can't provide a lengthier quote.

Answer 1

I am no expert in the history of Mathematics but I feel I must say the following since no one else has. Of course Euler ( who just had his $17\cdot 18$) did not think that $1 - 1 + 1 - 1 + 1 - 1 +\cdots=\frac12$ in that way.

Let's agree that we know what a complex series is (stick to $\mathbb{R}$ if you wish): essentially it is any "vector " in $\mathbb{C}^{\mathbb{N}}$ with a $\Sigma$ in front (or $+$ signs) to tell us that we are not looking at it as a sequence. The series form a complex vector space. There is a subspace which we would call the "usual" convergent series and we allow ourselves to say things like $\sum 2^{-n}=1$ although we might claim that really we mean that there is a linear transformation $S$ whose domain is the convergent series with range $\mathbb{C}.$ We sometimes symbolically enlarge the range in a familiar way to include $\pm \infty$ to discuss some kinds of "divergence." And of course we also care about the assertion $\sum z^{-n}=\frac{1}{1-z}$ which is unproblematic in the closed unit disk with the exception of two interesting points which deserve further thought. Integrating term by term (a bold move?) suggests that $-\sum \frac{z^{n+1}}{n+1}=\ln{(1-z)}.$ Whatever our qualms about the first series at $z=-1$, the second seems true and it looks like magic. The convergence is slow although there are ways to accelerate it. These methods applied to $\sum z^n$ at $z=-1$ lead to a not unexpected result.

There is a venerable subject of divergent series and "summation" methods for them. Such a method is a linear transformation $T$ which agrees with $S$ on the "convergent" series and perhaps satisfy a few other axioms enjoyed by $S$ (adding a few extra terms changes the sum in the obvious way.) Any method satisfying those requirements and sending $1-1+1-1+1...$ to a real $r$ would have to send it to one with $r=1-(1-1+1-1+\cdots)=1-r$ and hence to $\frac{1}{2}.$

Maybe most mathematicians have no need for the distinction between a convergent series and the number it converges to. I would certainly advise my students to avoid summing divergent series until they had a firm grasp on the usual practices. The perspective of "summing divergent sequences" feels a bit quaint these days but it is a valid subject. The amazing and brilliant things Euler and others did is the reason we now may choose to safely explore have well tamed domains such as analytic continuation of functions of one or more complex variables or regularization as practiced by Quantum Physicists making the mathematical formalism conform to measurements.

I'm not sure how all this connects to the actual question above but I'm not convinced that it is unrelated.

Answer 2

It seems that Robinson addressed some gross mistakes that happend in the early times of calculus and that were passed with silence by the posterity, the inventors of more rigor in analysis, in particular Weierstraß and his school. To give few examples:

Leibniz, Jakob Bernoulli, and Euler accepted

$\frac{1}{1-(-1)}= 1 - 1 + 1 - 1 +-... = \frac {1}{2} $

Leibniz subtracted two harmonic series, with the correct result though,

$\sum_{k \geq 2}^{\infty} \frac{1}{k^2-1} = \frac{1}{2} \sum_{k \geq 2}^{\infty} \frac{1}{k-1} - \frac{1}{2} \sum_{k \geq 2}^{\infty} \frac{1}{k+1} = \frac{3}{4} $

but in a way that today certainly would not be tolerated.

Wallis and Euler accepted

$\frac{1}{1-2}= 1 + 2 + 4 + ... = \frac {1}{-1} > \frac {1}{0} > \frac {1}{1} $

Euler calculated

$1 + 2 + 3 + ... = \frac{-1}{12}$

which, irrespective of the analytic continuation of the $\zeta$-function and Ramanujan's rediscovering it, is as wrong as the sum of the geometric series above.

With respect to such mistakes, those of d'Alembert, touched in a comment by the author of this question, seem to be negligible. In my opinion they are not the target of Robinson's remark. Some gaps in d'Alembert's proof of the fundamental theorem of algebra have been remedied by Gauss. However, even Gauss left gaps in his first proof. And nobody knows what future generations will have to criticize in proofs that presently are accepted as complete. Further d'Alembert held the opinion that irrationalities are not numbers. But that has to be understood out of his time where "number" denoted a string of digits that can be read from a sheet of paper.

Answer 3

This is mainly a comment on the discussion following the original post. Leibniz did not use the $\Sigma^\infty$ notation. If you replace $\infty$ by an infinite hyperinteger, and interpret the sum as a hyperfinite sum, the calculation becomes correct. Not only would one "dare" send such a proof to a modern referee, but in fact the literature is full of such routine calculations. Similarly, the other sums, notably in Euler, have interpretations in terms of averaging theory that make them correct, as detailed by Laugwitz, Kanovei, and others. To summarize, Leibniz's calculation only seems "wrong" from a post-Weierstrassian viewpoint where "infinite numbers" are a no-no, and illustrates well some of the misconceptions that were dealt with in our articles "Ten misconceptions from the history of analysis" (http://arxiv.org/abs/1202.4153), "Is mathematical history written by the victors?" (http://u.cs.biu.ac.il/~katzmik/bairetal.html), and a number of other recent publications, see http://u.cs.biu.ac.il/~katzmik/infinitesimals.html