The War Against the Infinitesimals

Sep 30, 2024

In a previous post, Joseph Ehrenfried Hofmann Demonstrates that Gottfried Wilhelm Leibniz Did Not Plagiarize Isaac Newton, I focused on the accusation of plagiarism of the infinitesimal calculus that was launched in 1711 by the Royal Society, in defense of Newton (1643-1727) against Leibniz (1646-1716). Therein, Hofmann (1900-1973) explains that during Leibniz’s second visit to London, John Collins (1625-1683) gives Leibniz access to some of the notes of James Gregory (1638-1675) and Newton. Hofmann notes that the excerpts that Leibniz copied have nothing to do with the infinitesimals1:

Leibniz’ excerpts are confined exclusively to series-expansions and the general remarks accompanying them: the sections relating to infinitesimals in the De Analysi remain completely disregarded—for the obvious reason that they offered nothing new to Leibniz. [Hofmann, p.278-279, my emphasis]

What is important to underline is that Newton’s and Leibniz’s philosophical principles were completely different. This is made clear in an article by Mikhail Katz and David Sherry2, who claim that Newton inherits more directly from Archimedes than does Leibniz, for the indivisibles used by Archimedes differ significantly from the infinitesimals used by Leibniz:

Archimedes’ kinematic method is arguably the forerunner of Newton’s fluxional calculus, but his infinitesimal methods are less arguably the forerunner of Leibniz’s differential calculus. Archimedes’ infinitesimal method employs indivisibles. For example, in his heuristic proof that the area of a parabolic segment is 4/3 the area of the inscribed triangle with the same base and vertex, he imagines both figures to consist of perpendiculars of various heights erected on the base. The perpendiculars are indivisibles in the sense that they are limits of division and so one dimension less than the area. Qua areas, they are not divisible, even if, qua lines they are divisible. In the same sense, the indivisibles of which a line consists are points, and the indivisibles of which a solid consists are planes. We will discuss the term “consist of” shortly.
Leibniz’s infinitesimals are not indivisibles, for they have the same dimension as the figures that consist of them. Thus, he treats curves as composed of infinitesimal lines rather than indivisible points. Likewise, the infinitesimal parts of a plane figure are parallelograms. The strategy of treating infinitesimals as dimensionally homogeneous with the objects they compose seems to have originated with Roberval or Torricelli, Cavalieri’s student, and to have been explicitly arithmetized by Wallis. [Katz and Sherry, p.5, my emphasis]

I have previously written several posts about Archimedes and his use of indivisibles. See, for example, Archimedes Is Reticent to Publish Proofs Using Indivisibles.

During the eighteenth century, Newton’s approach to physics, with the assumption of action-at-distance, became the dominant paradigm, as opposed to the assumption of local action supported by René Descartes (1596-1650), Cristiaan Huygens (1629-1695) and Leibniz. In mathematics, on the other hand, it was Leibniz’s approach to the calculus that became dominant, especially once the idea of function was formally defined by Leonhard Euler (1707-1783).

This did not mean that every aspect of Leibniz’s calculus was accepted. In particular, the infinitesimals were often called ill-defined, or worse, to the point that in today’s mathematics classes in university, it is not uncommon to first see the infinitesimals in a logic class, of all places, because it was Abraham Robinson (1918-1974) who formalized in 1966 the infinitesimals using model theory, a branch of logic, in what is today called non-standard analysis3.

The first to attack the infinitesimals was George Berkeley (1685-1753) in his essay The Analyst4, first published in 1734. Therein, Berkeley spends most of his energy attacking Newton’s fluxions, but still has some time to focus on Leibniz and his successors:

§5. The foreign Mathematicians are supposed by some, even of our own, to proceed in a manner, less accurate perhaps and geometrical, yet more intelligible. Instead of flowing Quantities and their Fluxions, they consider the variable finite Quantities as increasing or diminishing by the continual Addition or Subduction of infinitely small Quantities. Instead of the Velocities wherewith Increments are generated, they consider the Increments or Decrements themselves, which they call Differences, and which are supposed to be infinitely small. The Difference of a Line is an infinitely little Line; of a Plain an infinitely little Plain. They suppose finite Quantities to consist of Parts infinitely little, and Curves to be Polygones, whereof the Sides are infinitely little, which by the Angles they make one with another determine the Curvity of the Line. Now to conceive a Quantity infinitely small, that is, infinitely less than any sensible or imaginable Quantity, or than any the least finite Magnitude, is, I confess, above my Capacity. But to conceive a Part of such infinitely small Quantity, that shall be still infinitely less than it, and consequently though multiply’d infinitely shall never equal the minutest finite Quantity, is, I suspect, an infinite Difficulty to any Man whatsoever; and will be allowed such by those who candidly say what they think; provided they really think and reflect, and do not take things upon trust. [Berkeley, pp.166-167, my emphasis]

So Berkeley, like the Pythagoreans, assumes that material objects and mathematical objects are of the same nature. Unfortunately, Parmenides and Zeno weren’t around to mock Berkeley. See my posts:

In other words, Berkeley does not understand that Leibniz’s infinitesimals are fictions, i.e., creations of the mind. So it is not surprising that he is even less able to get his head around higher-order differentials, which can be dropped during calculations if they will not affect the final finite result:

§6. And yet in the calculus differentialis, which Method serves to all the same Intents and Ends with that of Fluxions, our modern Analysts are not content to consider only the Differences of finite Quantities: they also consider the Differences of those Differences, and the Differences of the Differences of the first Differences. And so on ad infinitum. That is, they consider Quantities infinitely less than the least discernible Quantity; and others infinitely less than those infinitely small ones; and still others infinitely less than the preceeding Infinitesimals, and so on without end or limit. Insomuch that we are to admit an infinite succession of Infinitesimals, each infinitely less than the foregoing, and infinitely greater than the following. As there are first, second, third, fourth, fifth, &c. Fluxions, so there are Differences, first, second, third, fourth, &c., in an infinite Progression towards nothing, which you still approach and never arrive at. And (which is most strange) although you should take a Million of Millions of these Infinitesimals, each whereof is supposed infinitely greater than some other real Magnitude, and add them to the least given Quantity, it shall be never the bigger. For this is one of the modest postulata of our modern Mathematicians, and is a Corner-stone or Ground-work of their Speculations. [Berkeley, pp.167-168, my emphasis]

Suspicion of the infinitesimals continued. For example, in 1797 Joseph-Louis Lagrange (1736-1813) published a text for the École Polytechnique, entitled Théorie des fonctions analytiques, contenant les principes du calcul différentiel, dégagés de toute considération d’infiniment petits ou d’évanouissans, de limites ou de fluxions, et réduits à l’analyse des quantités finies5 [Theory of analytical functions containing the principles of the differential calculus, with no consideration of infinitesimal or vanishing values, of limits or fluxions, and reduced to the analysis of finite quantities].

In the end of the nineteenth century, it appeared that the final coup de grâce had been given to the infinitesimals with the work of the “triumvirate”, the name given by Carl Boyer (1906-1976) to the trio of Karl Weierstrass (1815-1897), Richard Dedekind (1831-1916) and Georg Cantor (1845-1918). The latter is considered to be one of the founders of set theory and is responsible for the introduction of transfinite numbers.

Notwithstanding this “final blow”, a number of mathematicians of the same time, including Paul du Bois-Reymond (1831-1889), Otto Stolz (1842-1905), Giuseppe Veronese (1854-1917) and Tullio Levi-Civita (1873-1941), began the work on non-Archimedean systems, thereby making infinitesimals actual. Veronese wrote a very interesting essay entitled “On Non-Archimedean Geometry”6, which I will write about in another post. The reaction from Cantor was vicious, as can be seen in these two paragraphs of an article by Karin Usadi Katz and Mikhail Katz7:

Thus, Cantor published a “proof-sketch” of a claim to the effect that the notion of an infinitesimal is inconsistent. By this time, several detailed constructions of non-Archimedean systems had appeared, notably by Stolz and du Bois-Reymond.
When Stolz published a defense of his work, arguing that technically speaking Cantor’s criticism does not apply to his system, Cantor responded by artful innuendo aimed at undermining the credibility of his opponents. At no point did Cantor vouchsafe to address their publications themselves. In his 1890 letter to Veronese, Cantor specifically referred to the work of Stolz and du Bois-Reymond. Cantor refers to their work on non-Archimedean systems as not merely an “abomination”, but a “self contradictory and completely useless” one. [Usadi Katz and Katz, p.18, my emphasis]

And so, with the triumvirate, the infinitesimals were banished, at least in mainstream mathematics. It would not be until Abraham Robinson, as mentioned above, demonstrated that infinitesimals can be added to the reals, to create the non-standard reals, that infinitesimals would be considered to be legitimate. Nevertheless, these still remain marginal in most mathematical circles.

It would take no less than Kurt Gödel (1906-1978), the great logician of the twentieth century, to give short shrift to that kind of argument. Below is from the Preface to the Second Edition of Robinson’s book:

A more definite opinion has been expressed in a statement which was made by Kurt Gödel after a talk that I gave in March 1973 at the Institute for Advanced Study, Princeton. The statement is reproduced here with Professor Gödel’s kind permission.
‘I would like to point out a fact that was not explicitly mentioned by Professor Robinson, but seems quite important to me; namely that nonstandard analysis frequently simplifies substantially the proofs, not only of elementary theorems, but also of deep results. This is true, e.g., also for the proof of the existence of invariant subspaces for compact operators, disregarding the improvement of the result; and it is true in an even higher degree in other cases. This state of affairs should prevent a rather common misinterpretation of non-standard analysis, namely the idea that it is some kind of extravagance or fad of mathematical logicians. Nothing could be farther from the truth. Rather there are good reasons to believe that non-standard analysis, in some version or other, will be the analysis of the future.
One reason is the just mentioned simplification of proofs, since simplification facilitates discovery. Another, even more convincing reason, is the following: Arithmetic starts with the integers and proceeds by successively enlarging the number system by rational and negative numbers, irrational numbers, etc. But the next quite natural step after the reals, namely the introduction of infinitesimals, has simply been omitted. I think, in coming centuries it will be considered a great oddity in the history of mathematics that the first exact theory of infinitesimals was developed 300 years after the invention of the differential calculus. I am inclined to believe that this oddity has something to do with another oddity relating to the same span of time, namely the fact that such problems as Fermat’s, which can be written down in ten symbols of elementary arithmetic, are still unsolved 300 years after they have been posed. Perhaps the omission mentioned is largely responsible for the fact that, compared to the enormous development of abstract mathematics, the solution of concrete numerical problems was left far behind.’ [Robinson, pp.xv-xvi, my emphasis]

This post was just an introduction to the topic of the infinitesimals, and the resistance thereto. This is such an important topic that I will repeatedly return to it.

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Joseph E. Hofmann. Leibniz in Paris 1672-1676: His Growth to Mathematical Maturity. Cambridge University Press, 1974. Revised translation of Die Entwicklungsgeschichte der Leibnizschen Mathematik während des Aufenthalts in Paris (1672-1676). Munich: R. Oldenbourg Verlag, 1949.

Mikhail G. Katz and David Sherry. Leibniz’s Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond. 2012. https://arxiv.org/abs/1205.0174v1

Abraham Robinson. Non-Standard Analysis, Princeton University Press, 1974.

George Berkeley. De Motu and The Analyst. A Modern Edition, with Introductions and Commentary. Edited and translated by Douglas M. Jesseph. Springer, 1992.

J.L. Lagrange. Théorie des fonctions analytiques, contenant les principes du calcul différentiel, dégagés de toute considération d’infiniment petits ou d’évanouissans, de limites ou de fluxions, et réduits à l’analyse des quantités finies. Paris: Imprimerie de la République, Prairial an V (1797).

Giuseppe Veronese. On Non-Archimedean Geometry. Invited Address to the 4th International Congress of Mathematicians, Rome, April 1908. Translated by Mathieu Marion (with editorial notes by Philip Ehrlich). In Philip Erlich, editor, Real Numbers, Generalizations of the Reals, and Theories of Continua. Kluwer, 1994.

Karin Usadi Katz and Mikhail G. Katz. A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography. 2012. https://arxiv.org/abs/1104.0375v4

Fiat Lux

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