Was the early calculus inconsistent?

The question is not precise enough to get a definite answer, but not for the reason most people say in commentaries. The problem does not lie in the ambiguous meaning of "consistent" (which just means "free of contradictions", which was as clear then as now), but in the meaning of "way of arguing". What we do have is a corpus of results from the founders of calculus (say Pascal, Descartes, Fermat, Newton, Leibniz), and a corpus of arguments they used to justify them. The corpus of results is certainly a corpus of true results, so is consistent, and was certainly recognized as such even by Berkeley (to my knowledge, the first serious contradiction involving results of calculus came 150 years after the founding period with Cauchy's theorem that a limit of continuous functions is continuous, combined with counter-examples from Fourier's theory, so is completely out of our scope).

Now is the corpus of arguments used by our fathers "consistent"? This question does not make real sense, because "arguments" are not results, and are not "true or false", either individually or in groups. They are, then and now, incomplete developments aimed at convincing one's that some results are true. The thing one can say is that, however shaky the arguments seem to us, they were used by these founders to prove only true results. In this very weak sense, their arguments were consistent.

Now, is the "way of arguing" of our fathers consistent? Again, the meaning of this question is problematic, because there is no unique way to deduce from a finite set of examples was what the "way or arguing" of our founding fathers. What is sure is that a naive reader of their arguments, trying to guess, "by induction" in the sense of natural sciences, what was the way of arguing of these people, and trying to apply this way of arguing to get new results, would easily come across contradictions (even not so naive readers, such as Cauchy, eventually did so). Actually, it took almost 200 years for mathematicians to find a "consistent way of arguing" in which the arguments of the founder can be reformulated without too much distortion: it is the $\epsilon,\delta$ approach of Weierstrass and others. It took almost one more century to construct a second consistent approach, which perhaps has the slight advantage on the classical one to reformulate with even less distortion the arguing of the founders of calculus. Yet priority has a great weight in science, and this is the most obvious reason for which the non-standard analysis has not supplanted the traditional one.

I want to finish by a side, wittgensteinian, remark: we are not in a qualitatively different situation than our founding fathers were: there is no way to be sure that our current "way of arguing" is consistent, because there is no way to be sure what our current "way of arguing" exactly is. By this I am not thinking at all at the problem that since Gödel we doubt that ZF or any other system is consistent, but to the much most basic problem that even with a "certainly consistent set of axioms" (say the axioms of the theory of groups, to fix ideas), we are not really sure what our way of arguing (that is the logico-formal rules which allow us transform statements into other statements, from axioms to theorems) is. To be sure, we mathematicians now take great care to begin a treatise by explaining carefully those "formal rules" or reasoning. Yet this formal rules use notions that are not completely clear (such as the notions of "intuitive integer") and skill that we can not be sure to posses (for example the capacity to recognize, in a finite expression, all occurrence of a given free variable). What we do is we see other people working using those rules, we try to do the same by imitation, getting punished if we do it wrong, and after some times do not make mistake anymore -- so we deduce that we understand the rules as the others do. But there is no way to be really sure of that.

Contrary to Andrej Bauer’s contention, seventeenth-century calculus looks very little like SDG. Unlike in SDG, the integrals were construed as infinite sums, the intermediate value theorem was assumed to hold for continuous curves and, more to the point, for the most part the infinitesimals that were employed were invertible rather than nilpotent. For a while, the Dutch mathematician, Bernard Nieuwentijt, in his debate with Leibniz, argued in favor of the use of nilpotent infinitesimals, but eventually came to believe that his attack on Leibniz was ill-founded and returned to the then standard use of invertible infinitesimals. Of course, I’m not suggesting that nilpotent infinitesimals were not used—they were from time to time—but only that their use was not the main view. After all, following Leibniz, most mathematicians wanted their infinitesimals to behave like real numbers.

Nilpotent infinitesimals along with invertible infinitesimals were employed by a number of differential geometers in the nineteenth century and entered mainstream mathematics around the turn of the twentieth-century (in systems of dual numbers), when geometers such as Hjelmslev and Segre became interested in geometries in which two points need not determine a unique straight line, and Grothendieck (and others) later employed them in algebraic geometry.

I suspect that the misconception that seventeenth-century calculus looks like SDG can be traced in part to John Bell’s wonderful expository writings on SDG. Bell was taken to task for this by the historian-mathematician Detlef Laugwitz in his otherwise very positive review (for Mathematical Reviews MR1646123 (99h:00002)) of the first edition of Bell's A Primer of Infinitesimal Analysis (1998). Moreover, I am not aware of any of the many serious writings on the history of the calculus that supports the view suggested by (my friend) John.

Response to Mikhail Katz:

Mikhail: Fermat’s work was one I had in mind when I said nilpotent infinitesimals were used from time to time. However, his work, which was largely concerned with tangent constructions and lacked generality, predates the work of Newton and Leibniz, never caught on, and is not characteristic of the mainstream approaches to the calculus of the 17th century, which is what I said I was talking about. Moreover, Fermat’s work is notoriously unclear and, by my lights, the similarities with SDG are vague at best.

Many thanks, however, for the reference to Cifoletti’s work, which I will take a look at. I hasten to add, however, that the following passage from the Mathematical Reviews review of the work, which you yourself cite, does not inspire confidence.

“In the second part of the book, the author embarks on an investigation of the link between modern synthetic differential geometry, originally proposed by F. W. Lawvere in 1967 and afterwards largely developed by Lawvere and other mathematicians, and Fermat's mathematics.

In many situations, for the most part informal ones, Lawvere himself and other mathematicians working in this research field expressed their feelings that there had to be some kind of affinity between synthetic differential geometry and seventeenth-century mathematical practice. The author has tried to make explicit these general feelings, but this part of the book is mathematically weak and somewhat naive.

The best example is footnote 29, page 208, where the author claims to have established a direct connection between Fermat and synthetic differential geometry, on the basis of having been able to convince G. Rejes, during a talk she had with him about Fermat's work, to name a particular axiom of one possible formulation of the theory after Fermat.”

I found a copy of the relevant passage from Berkeley's works at this web site. I have cut and pasted from that site, and I have reformatted the mathematics; apologies to the good Bishop for any alterations in meaning.

XIV. To make this Point plainer, I shall unfold the reasoning, and propose it in a fuller light to your View. It amounts therefore to this, or may in other Words be thus expressed. I suppose that the Quantity $x$ flows, and by flowing is increased, and its Increment I call $o$, so that by flowing it becomes $x + o$. And as $x$ increaseth, it follows that every Power of $x$ is likewise increased in a due Proportion. Therefore as $x$ becomes $x + o$, $x^n$ will become $(x + o)^n$: that is, according to the Method of infinite Series, $$x^n + nox^{n-1} + \frac{n^2-n}{2} o^2 x^{n-2} + \text{ etc.}$$ And if from the two augmented Quantities we subduct the Root and the Power respectively, we shall have remaining the two Increments, to wit, $$o \text{ and } nox^{n-1} + \frac{n^2-n}{2} o^2 x^{n-2} + \text{ etc.}$$ which Increments, being both divided by the common Divisor o, yield the Quotients $$1 \text{ and } nx^{n-1} + \frac{n^2-n}{2} ox^{n-2} + \text{ etc.}$$ which are therefore Exponents of the Ratio of the Increments. Hitherto I have supposed that $x$ flows, that $x$ hath a real Increment, that $o$ is something. And I have proceeded all along on that Supposition, without which I should not have been able to have made so much as one single Step. From that Supposition it is that I get at the Increment of $x^n$, that I am able to compare it with the Increment of $x$, and that I find the Proportion between the two Increments. I now beg leave to make a new Supposition contrary to the first, i. e. I will suppose that there is no Increment of $x$, or that $o$ is nothing; which second Supposition destroys my first, and is inconsistent with it, and therefore with every thing that supposeth it. I do nevertheless beg leave to retain $nx^{n - 1}$, which is an Expression obtained in virtue of my first Supposition, which necessarily presupposeth such Supposition, and which could not be obtained without it: All which seems a most inconsistent way of arguing, and such as would not be allowed of in Divinity.

It looks to me that Berkeley's argument amounts to an argument raised by every discerning student in a nonrigorous first semester calculus course: ``Is the increment zero? or not zero? How can it be both? That's inconsistent!'' In which case I would invite the good Bishop to come to my office hours where I would introduce him to $\epsilon$, $\delta$ proofs.

I bet I could even convince the Bishop that Divinity would allow it: "Suppose the Devil gives you any $\epsilon > 0$. This $\epsilon$, although positive, might be very, very, very small, as small as the Devil likes...".