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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.

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Nice question. But what period is Robinson exactly talking about? How far does "around" in "aronud 1900" extends? Does it contain the works of Weierstrass ? or even earlier, of Cauchy ? Or is it restricted to one or two decades after and before 1900 ? In the latter case, it is of course much more difficult to see what lapses he is talking about, but this interpretation seems difficult to conciliate with the phrase "early proponent of the doctrine of limits". Perhaps providing a wider context for the passage of Robinson you quote would be helpful. –  Joël Apr 15 '13 at 16:45
Another comment, which is not especially addressed to the OP, but to the whole MO community, and perhaps should be on meta-MO (but wondering about this should be on meta-meta-MO): what's the meaning of this tag, "history-overview". Shouldn't we create a tag history-of-mathematics or even perhaps history-of-calculus-and-analysis and retag most of the history-overview tag? –  Joël Apr 15 '13 at 16:48
The "History-overview" category is probably borrowed from the ArXiv. –  katz Apr 15 '13 at 16:55
@Joël: As katz said the ho.history-overview tag corresponds to a math arXic category; more generally, the tags with a two letter prefix correspond precisely to the math arXiv categories. (There can be temporary deviations, when somebody creates something, but these are considered as "errors" and treated as such.) These arXiv-tags serve as top-level tags on MO; each question, if reasonably possible, should have at least one such tag. (Compare the explanatory text on the page where one asks a questions just below teh field to enter a tag.) Of course, this should not prevent the... –  quid Apr 16 '13 at 11:42
...creation of more specialised tags 'below' the different top-level tags, if there is a need. However, the/a top-level tag should still be used in addition to more specialised tags. So, for the specifc case one could consider tagging it say history-of-calculus in addition to, yet not instead of, ho. So, a re-tagging definietly should not be done. One could however consider tagging with something in addition. (For bigger activities of this kind it would be however good to check before on meta,; and, not do too many tag-edits at the same time, not to clutter the frontpage) –  quid Apr 16 '13 at 11:53

3 Answers 3

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.

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Excellent answer. You might want to elaborate a bit on Abel summation. –  katz Apr 16 '13 at 11:35
So you would send (1 - 1) + (1 - 1) + (1 - 1) +- ... = 0 to 1/2? I would not join you in that ABC, not even with Abel, Borel, and Cesaro. –  Rhett Butler Apr 16 '13 at 18:19
@Rhett, I never invited you to join me! But if you won't go with me then where will you go? Frankly my dear.... –  Aaron Meyerowitz Apr 16 '13 at 20:49
@Rhett Butler: Of course not! But if you have an issue with this (ie there being a difference here), it seems you would have to have issues with todays convergent series, too. –  quid Apr 16 '13 at 21:09
@quid: The question of the OP concerned Robinson and his critique of the "lapses of the early proponents of the doctrine of limits". These proponents did not consider Abel- or Cesario-summation. They simply applied parameters in sum formulas out of their acceptable range (see even $q$ = 2). And there is no issue in excusing that practice. –  Rhett Butler Apr 17 '13 at 5:54

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


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.

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This interesting work does not seem to date from the 1900s. –  katz Apr 4 '13 at 12:57
Sorry, it seems that I misunderstood you. I only wanted to show why Robinson was entitled to speak of the leniency accorded to the lapses of Leibniz and the early proponents of the doctrine of limits. –  Rhett Butler Apr 4 '13 at 14:45
Sorry, but, I do not understand this at all. From the question quoting Robinson (my emphasis): "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 " –  quid Apr 4 '13 at 16:34
Thank you for the reply. I now understand what you had in mind. (Wheter or not you misunderstood the question, I guess OP is best placed to clarify.) –  quid Apr 4 '13 at 17:08
I don't understand what is wrong with Leibniz's subtraction. The three bounds $\infty$ are the same infinite number and the algebra works as usual, giving the right answer, as it should. The problematic subtraction $\infty-\infty$ only occurs when you believe that all infinite quantities are the same. –  François G. Dorais Apr 14 '13 at 21:25

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

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@katz: With all respect, but today we have "interpretations" in mathematics (as well as in justice) which makes nearly anything correct or excusable - from a suitable point of view. In my opinion Euler has been one of the greatest, if not the greatest at all. Nevertheless, in the mathematics of real numbers as we know and teach it, his calculations are wrong and good for nothing but confusing newbies. –  Rhett Butler Apr 15 '13 at 7:47
My point was precisely that Euler was <em>not</em> working with our real number system, as argued by Detlef Laugwitz in Archive for History of Exact Sciences link.springer.com/article/10.1007%2FBF00329867, Patrick Reeder (see mathoverflow.net/questions/126986/…) as well as Bair et al. –  katz Apr 15 '13 at 7:56
@katz: I have not looked into Euler's original paper. Did he explicitly say that his calculation is invalid in the real numbers? At least many of those who are reporting his results use the customary notation and do not mention any deviations from the real numbers. I find this misleading and incorrect. –  Rhett Butler Apr 15 '13 at 10:08
That's a good point. Euler's RESULTS are usually formulated in terms familiar to us, such as the series expansion of $e^x$. However, in proving such results, Euler exploited TECHNIQUES that routinely rely, for example, on infinite (and therefore not REAL) numbers. Thus, he obtains the expansion of $e^x$ by applying the binomial formula to an INFINITE exponent, and simplifying the coefficients. Moreover, Euler explicitly asserts that it is impossible to do calculus without infinitesimals and infinite numbers. –  katz Apr 15 '13 at 11:43
@katz: Your comment reminds me of how Euler calculated log2. I could add it to my answer but am too lazy. Certainly you know it anyway. $\log2 = \log2\infty - \log\infty$ . And again we have two harmonic series subtracted. –  Rhett Butler Apr 15 '13 at 13:11

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