Was the early calculus inconsistent?

Question

This question does NOT concern the RIGOR, or lack thereof, of the early calculus. Rather the question is of its CONSISTENCY.

George Berkeley wrote in 1734 with reference to the early calculus that such a method is "a most inconsistent way of arguing, and such as would not be allowed in Divinity". This passage is quoted by William Dunham in 2004. Dunham concludes: "Bishop Berkeley had made his point. Although the results of the calculus seemed to be valid ... none of this mattered if the foundations were rotten". See page 72 of http://books.google.co.il/books?id=QnXSqvTiEjYC&source=gbs_navlinks_s

On the other hand, Peter Vickers in 2007 challenged "The ubiquitous assertion that the early calculus of Newton and Leibniz was an inconsistent theory" at http://philsci-archive.pitt.edu/3477/ (soon to appear in book form at Oxford University Press), and concluded that this only holds in a limited sense and "can only be imputed to a small minority of the relevant community".

Was the early calculus consistent as far as most practitioners were concerned, as Vickers contended, or was it a most inconsistent way of arguing, as did Berkeley and Dunham?

Note 1. Berkeley claimed that calculus was based on an inconsistency that can be expressed in modern notation as $(dx\not=0)\wedge(dx=0)$. Thus he was using the term "inconsistent" in much the same sense it is used in modern logic.

Note 2. For a closely related thread, see https://math.stackexchange.com/questions/445166/is-mathematical-history-written-by-the-victors

Note 3. There is a related thread at the history SE: https://hsm.stackexchange.com/questions/3301

Answer 1

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.

Answer 2

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.”

Answer 3

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...".

Answer 4

I do not know whether the early calculus was consistent, but it surely can be made as consistent as modern mathematics, with practically no modifications of the basic setup. This goes under the name Synthetic differential geometry (SDG). Like Robinson's nonstandard analysis it is a calculus with infinitesimals. SDG should be closer to the 17th century ways of doing things because it works with nilpotent infinitesimals whereas nonstandard analysis does not. I believe the 17th century calculus used nilpotent infinitesimals. Can someone confirm this?

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[Edit: many thanks to Lee Mosher for transcribing a piece of Berkeley's text. Here is the same piece of text, as it would be written in SDG in the 21st century.]

We would like to compute the derivative of $f(x) = x^n$ where $n$ is a positive integer. Let $x \in R$ and let $o$ be any nilpotent infinitesimal of degree 2. Then by the Binomial theorem $$(x + o)^n = x^n + n o x^{n-1} + \frac{n^2 - n}{2} o^2 x^{n-2} + \text{etc}.$$ Because $o$ is nilpotent of degree 2, we have $o^2 = 0$ and so all terms but the first two equal zero. Thus we get $$(x + o)^n = x^n + n o x^{n-1}$$ hence $$(x + o)^n - x^n = n o x^{n-1}$$ or $$f(x + o) - f(x) = n x^{n-1} o$$ Because $o$ here is an arbitrary infinitesimal (i.e., the equation holds for all $o$ whose square iz zero), we may use the Axiom of Microaffinity to conclude that $$f'(x) o = n x^{n-1} o$$ Now we use the Cancelation Principle to cancel $o$ on both sides, which yields $f'(x) = n x^{n-1}$.

I must say Berkeley's writting was a great deal more picturesque. The Axiom of Microaffinity and the Cancelation Principle are an axiom and a theorem of SDG, respectively. They circumvent the problem that Berkeley was complaining about, namely that first we pretend that $o$ is not zero (so that we can cancel it on both sides of equation), but then we pretend it is zero so that all those higher terms disappear. Instead, we can do the following: assume that $o^2 = 0$ (which does not imply that $o = 0$ because we are not assuming classical logic) so that the higher terms disappear, but then use a sort of weak cancelation property of infinitesimals which allows us to cancel them under certain conditions, even though they are not invertible.

Axiom of Microaffinity: For every $f : R \to R$ and $x \in R$ there exists a unique number $f'(x)$, called the derivative of $f$ at $x$, such that for all infinitesimals $o$ we have $f(x + o) - f(x) = f'(x) o$.

Cancelation principle: Let $a, b \in R$. If $a \cdot o = b \cdot o$ for all $o \in \Delta$ then $a = b$.

Is this weird? Yes, it sure is if you are classically trained. It gets weirder: if we let $\Delta = \lbrace o \in R \mid o^2 = 0 \rbrace$ be the set of square-nilpotent infinitesimals then

1. Potentially there exist non-zero infinitesimals: $\lnot \forall o \in \Delta, o = 0$.
2. There are no infinitesimals which are distinct from zero: $\lnot \exists o \in \Delta, o \neq 0$.

But it is precisely what we need to explain all the confusion about infinitesimals. Remember it this way: potentially there are some non-zero ones (we cannot exclude their existence) but they are all potentially zero (they are so small we cannot distinguish them from zero). Just don't ask yourself whether an infinitesimal is zero and all will be fine.

Here $R$ is the "smooth real line", which is an ordered field. Of course, it does not satisfy the Archimedean axiom, as that would force all infinitesimals to be zero. So it is a different kind of animal than the usual $\mathbb{R}$.

John Bell explained all this in his excellent booklet on Syntehtic Differential Analysis.

Answer 5

Coming back to the B.Berkeley critics, there is a common denominator of all known getarounds, both the two mainstream ones (Wstrass and NSA) and exotic ones like the SDG interpretation.

That is, one considers an extension - call it $R^+$ - of the true reals R and a map $R^+ \to R\cup \{\infty\}$ - call it the valuation map. For instance,

1) $R^+$ consists of all convergent infinite real sequences and the valuation map is the `` taking the limit'' map

2) $R^+$ is a nonstandard extension of $R$ and the valuation map is the ``standard part'' map ($\infty$ for infinitely large objects)

3) Nilpotent or any other applicable exotics.

It occurs that the evaluation map cannot be a homomorphism, it always lacks something. For instance the value of a non-0 infinitesimal is 0, the value of its inverse is $\infty$, but $0\cdot \infty=1$ makes little sense in $R$.

This is I believe the only sound way to view the medieval controversies around infinitesimals. That is, accept that a non-0 infinitesimal is not equal to the real number 0, it just has the value 0. Maybe, a devoted scholar of Leibnizz, Euler, etc. (although there is no much of etc. after Euler!) can find a support of this point of view.

Obviously, a modern mathematician would ask for either a concrete mathematically defined model of both $R^+$ and the valuation map - and the two mainstream such models are listed above, with perhaps more yet to come under category 3 - or at least to set it up in the form of calculus of propositions, with rigorous rules of inference albeit w/o a fixed interpretation of objects.

Vladimir Kanovei

Answer 6

I would agree with Alexandre Eremenko's answer. The early calculus in fact was not inconsistent, as elaborated below.

Joël's answer is based on a premise that "the question is not precise enough to get a definite answer", and "does not make real sense, because 'arguments' are not results". This premise is historically incorrect. In fact, in the historical literature the claim of inconsistency of the early calculus is very specific and precise. It is routinely based on Berkeley's analysis of the typical calculations such as that of the derivative of a power, or the derivative of the product of two functions. The alleged inconsistency is presented as follows. Berkeley claims that (1) $dx$ is nonzero at the start of the calculation; (2) $dx$ is assumed to be zero at its conclusion; (3) in any consistent reasoning, $dx$ cannot be simultaneously zero and nonzero; (4) therefore the procedures of the calculus were inconsistent, Q.E.D. In modern notation, this amounts to a claimed inconsistency of the type $(dx\not=0)\wedge(dx=0)$.

Berkeley, however, did not read Leibniz carefully enough. Leibniz explicitly and repeatedly clarifies that he is working with a generalized notion of equality, where expressions equal up to a negligible term are also held to be equal. In modern terminology, this means that Leibniz is working with a binary relation which is not equality on the nose, but rather approximate equality in a suitable sense. It is in this sense that Leibniz writes formulas like $2x+dx=2x$ (note that he did not use our "=" symbol). Leibniz might not have been "rigorous" by modern standards, but he was not inconsistent, either. In fact, Leibniz's procedures were more soundly based than Berkeley's criticisms thereof. Philosopher David Sherry and I presented our analysis last year in the Notices of the AMS at http://www.ams.org/notices/201211/

Felix Klein wrote in 1908 that there were in fact not one but two separate tracks for the development of analysis: (A) the Weierstrassian one, in the context of an Archimedean continuum; and (B) the track exploiting indivisibles and/or infinitesimals. The B-track was eventually popularized by Abraham Robinson.

Everybody is familiar with the great accomplishment of Weierstrass in developing rigorous foundations for analysis, which is beyond dispute. However, historian Carl Boyer (and many others), in describing Cantor, Dedekind, and Weierstrass as "the great triumvirate", adds an anti-infinitesimal spin to their accomplishment. Namely, the traditional historical literature seeks to couple their "rigorous" accomplishment to the elimination of "inconsistent" infinitesimals, as if pursuing the A-track depended on the elimination of the B-track (Dunham's "rotten foundations"). It is the coupling of Weierstrass's accomplishment to an ill-informed critique of infinitesimals (both classical and modern) that constitutes the historical misconception pointed out by Vickers and others.

Answer 7

As it was noted by Ryan Budney and Lee Mosher and Ben Braun, the word "consistent" cannot be used here in the sense of modern mathematical logic. So one cannot investigate this question rigorously. But one can apply the word "consistent" with its everyday (fuzzy) sense, meaning "free of contradictions". Then the answer is to some extent a question of opinion.

Myself I side with the opinion of Peter Vickers: early calculus was consistent. It was not worse than arguments in most other hard sciences (physics, chemistry). But perhaps not on the level of rigor of Mathematics.

On the other hand, what Berkeley says "a most inconsistent way of arguing, and such as would not be allowed in Divinity" sounds ridiculous to me. "Divinity" is a pseudo-science which deals with the things that do not exist; thus what is "allowed" in Divinity or not allowed is completely a matter of opinion. Divinity cannot be compared with other hard sciences, while mathematics of Newton (or Leibnitz or Euler) can.

Answer 8

I think that the question is sufficiently precise if we think at a realistic meaning of the word “inconsistent”. Also nowadays, for non logicians the adjective “inconsistent” doesn't really mean “free of contradictions” (this is only the obvious meaning given by modern Mathematical Logic), but rather it means not acceptable by a large or important part of the scientific community.

Also nowadays, some of our works in some parts of modern Mathematics are not accepted as sufficiently rigorous by other parts. These works are hence perceived only as not sufficiently precise “ways of arguing”. Therefore, these “foreign argumentations” are perceived as potentially inconsistent, and need a different reformulation to be accepted. I know of relationships of this type between some parts of Geometry and Analysis, to mention only an example. It is the same problem occurring in the relationships between (some parts of) Physics and Mathematics because these two disciplines are really completely different “games”: in Physics the most important achievement is the existence of a dialectic between formulas and a part of nature, even if the related Mathematics lacks in formal clarity and is hence not accepted by several mathematicians.

Analogously, early calculus was consistent until the community accepted these “ways of arguing” and discovered statements which could be verified as true by a dialogue with other part of knowledge: Physics and geometrical intuition in primis.

Since in the early calculus the formal intuition (in the modern sense of manipulation of symbols, without a reference to intuition) was surely weak, the dialectic between proofs and intuition was surely stronger (I mean statistically, in the distribution of 17th century mathematicians). In my opinion, this is the reason of the discovering of true statements, even if the related proofs are perceived as “weak” nowadays. Once the great triumvirate Cantor, Dedekind, and Weierstrass decided that it was time to make a step further, the notion of “inconsistent” changed for this important part of the community and hence, sooner or later, for all the others.

Also from the point of view of rules of inference, the consistency of early calculus has to be meant in the sense of dialectic between different parts of knowledge and acceptance by the related scientific community.

Therefore, in this sense, in my opinion early calculus is as consistent as our (and the future) calculus.

I agree with Joel that “we are not in a qualitatively different situation”: probably in the near future all proofs will be computer assisted, in the sense that all the missing steps will be checked by a computer (whose software will be verified, once again, by a large part of the community) and we will only need to provide the main steps. Necessarily, articles will change in nature and, I hope, they will be more focused on those ideas and intuitions thanks to which we were able to create the results we are presenting. Therefore, young students in the future will probably read disgusted at our papers saying: “how were they able to understand how all these results were created? These papers seems like phone books: def, lem, thm, cor, def, lem, thm, cor... without any explanation of discovery rules and several missing formal steps!”.

Finally, I think that only formally, but not conceptually, this early calculus may look similar to NSA or SDG. In my opinion, one of the main reason of the lack of diffusion of NSA is that its techniques are perceived as “voodoo” by all modern mathematicians (the majority) that rely their work on the dialogue between formal mathematics and informal intuition. Too much frequently the lack of intuition is too strong in both theories. For example, for a person like Cauchy, what is the intuitive meaning of the standard part of the sine of an infinite number (NSA)? For people like Bernoulli, what is the intuitive meaning of properties like $x\le0$ and $x\ge0$ for every infinitesimal and $\neg\neg\exists h$ such that $h$ is infinitesimal (but not necessarily there exists an infinitesimal; SDG)? Moreover, as soon as discontinuous functions appeared in the calculus, the natural reactions of almost every working mathematicians (of 17th century and nowadays) looking at the microaffinity axiom is not to change Logic switching to the intuitionistic one, but to change this axiom inserting a restriction on the quantifier “for every $f:R\longrightarrow R$”.

The apparently inconsistent argumentation of setting $h\ne0$ and finally $h=0$, can be faithfully formalized using classical calculus rather than using these theories of infinitesimals. We can say that $f:R\longrightarrow R$ (here $R$ is the usual Archimedean real field) is differentiable at $x$ if there exists a function $r:R\times R\longrightarrow R$ such that $f(x+h)=f(x)+h\cdot r(x,h)$ and such that $r$ is continuous at $h=0$. It is easy to prove that this function $r$ is unique. Therefore, we can assume $h\ne0$, we can make freely calculations to discover what is the unique form of the function $r(x,h)$ for $h\ne0$ and, in the final formula, to set $h=0$ because $r$ is clearly continuous for all the examples of functions of the early calculus. This is called the Fermat-Reyes methods, and it can be proved also for generalized functions like Schwartz distributions (and hence for an isomorphic copy of the space of all the continuous functions). Moreover, in my opinion, both Cauchy and Bernoulli would had perfectly understood this method and the related intuition. On the contrary, they would not be able to understand all the intuitive inconsistencies they can easily find both in NSA and SDG.

Answer 9

The completeness of the real number implies that there are no infinitesimals. If $\epsilon$ is infinitesimal, then $n\epsilon<1$ for all $n\in \mathbb N$. This bounded increasing sequence has no least upper bound, although it should by completeness.

In the form of Archimedes' axiom, completeness has been a part of mathematics since ancient times. Archimedes himself used it to solve some problems of calculus. I always thought that Berkeley spotted this inconsistency and rightfully complained about it.