The truth about infinitesimals
In what follows I am not pushing any conclusions; I’m offering an extended example with which to think about questions of truth and historical interpretation.
Russell knew how things stood with infinitesimals. Weierstrass and Cantor have solved the problem of placing the calculus on a rigorous basis. The infinitesimals of an earlier age are banished; the contradictions entailed by their use can now safely be ignored because it has been shown we can do without them. (NB: My guide in this discussion is John Bell’s The continuous and the infinitesimal, esp. ch. 4.)
Leibniz in particular is censured for having given a “wrong direction to speculation as to the Calculus”.
His belief in the actual infinitesimal hindered him from discovering that the calculus rests on the doctrine of limits, and made him regard his dx and dy as neither zero, nor finite, nor mathematical fictions, but as really representing the units to which, in his philosophy, infinite division was supposed to lead (Russell, The Principles of mathematics 325).
Leibniz’s—and his successors’—attempts to save infinitesimals were bound to come to naught, because “infinitesimals as explaining continuity must be regarded as unnecessary, erroneous, and self-contradictory” (Principles 345: his verdict on Hermann Cohen’s Neo-Kantian interpretation of the calculus).
Celebrating the accomplishments of Weierstrass, Russell makes his work a fulfillment of Zeno’s:
After two thousand years of continual refutation, these sophisms [Zeno’s paradoxes] were reinstated, and made the foundation of a mathematical renaissance, by a German professor, who probably never dreams of any connection between himself and Zeno. Weierstrass, by strictly banishing all infinitesimals, has at least shown that we live in an unchanging world, and that the arrow, at every moment of its flight, is truly at rest (347).
What Russell knows is that the key notion of the calculus—that of limit—has been defined by nineteenth-century mathematicians in terms that make no reference to other than “normal” finite quantities. In the classical ε, δ definition of convergence, those variables range over ordinary numbers like .01 or 1/π. The dx and dy of Leibniz’s calculus are thus eliminated; every proposition containing such symbols can be replaced, salve veritate and also salve deductive relations, by a proposition not containing them. Elimination by definition became a familiar strategy in analytic philosophy, carried forward in extremis by “nominalists” whose aim was to eliminate by similar means all abstracta, including the sets or classes to which, in the late nineteenth century, all other mathematical entities then conceived had been reduced. The model and origin of all such reductions was, directly or indirectly, that of Weierstrass.
The importance of that model would be difficult to exaggerate. It was seen to be the successful solution, ending all dispute, to a longstanding philosophical and mathematical problem, the problem of infinitesimals. It was Progress with a capital P.
Imagine now that a philosopher with the typical British propensity to hand out marks approaches the work of Leibniz. For all that one admires him, one sees him to be entangled in irresoluble issues to which one has now has the answer. Or rather, because one now has the answer, and because it is not Leibniz’s under any reasonable interpretation, one knows that he must be mistaken. The only question is how and why. The how we have seen: it is that Leibniz thinks that dx and dy denote a special sort of number, an infinitesimal. The why is not so clear: it is perhaps that Leibniz denied that the continuum is composed of (actual) points, or because he thinks that intensive quantity (degree) is not reducible to extensive quantity (length, area, volume). So one must in the end give him a low mark for having taken a wrong direction.
There are many things to say about that approach to history. It has some virtues: for example, it offers a clear criterion of importance. Those who promote Progress are to be commended and rewarded with a place in the Pantheon, those who don’t can still find a place, but only in the rooms devoted to Error, or rather to Philosophically Interesting Error, since mere error is relegated to the merely erudite. Leibniz for his mathematical accomplishments wins the laurel, but for his speculations is justly reproved. Hermann Cohen, from whose philosophical defense of infinitesimals nothing can be salvaged, has been dropped from anglophone histories of the philosophy of mathematics after Weierstrass (Bell, I should note, tries to do justice to the “dissenting voices” of the late nineteenth century).
It also gives the historian a readymade frame: history is the Path of Progress. Like an Olympic athlete, the spirit of philosophy may have to overcome obstacles, but they enter the story only as perturbations of the striving to truth; they have no interest in themselves. If the Path of Progress is toward a secular philosophy, then Leibniz’s theological speculations will at best have been the occasion for other, more productive, investigations. If it is toward Weierstrass, then the entire discourse of infinitesimals goes to the basement—or the curiosity cabinet.
The stinger here is that Russell was wrong about infinitesimals. There are not one but two distinct ways to model the notion in mathematical analysis.
The first, and better known, is the “Nonstandard Analysis” discovered by Abraham Robinson. For the present discussion the significant property of the “nonstandard real numbers” is that among them are numbers ε such that for all numbers n = 1, 2, 3, …, nε is smaller than 1. You cannot fill out the interval between 0 and 1 with segments of length ε, no matter how many you set end to end. The nonstandard number ε is thus an infinitesimal.
The construction of the nonstandard reals guarantees that a subset of them will satisfy all the first-order sentences satisfied by the standard reals. On the other hand, there will be numbers not in that subset that satisfy sentences not satisfied by any standard number, like the sentence above satisfied by x.
Robinson showed that you can interpret much of what Leibniz and others thought was true of their dx and dy if you take terms referring to infinitesimal differences to denote nonstandard numbers like x. The naïve derivations of formulas in the calculus are valid under this interpretation. For example, since dx2 is infinitesimal relative to dx itself, one can write
(x + dx)2 – x2 =
x2 + 2xdx + dx2 – x2 =
2xdx + dx2 =
(x + dx)2 – x2 =
x2 + 2xdx + dx2 – x2 =
2xdx + dx2 =
just as an eighteenth-century mathematician would have.
The second way of restoring infinitesimals, Smooth Infinitesimal Analysis, was proposed in the 1960s by William Lawvere.
One supposes first of all that there are “smooth spaces”—it helps here simply to think of continuous curves or surfaces—and “smooth maps” of such objects into or onto one another (imagine, for example, wrapping a straight line around a cylinder like the stripe on a candy cane).
One supposes that in the one-dimensional continuum—which is essentially a straight line—there is an “infinitesimal segment” around 0. This segment, call it Δ, can be stretched or displaced by any “smooth map” of it into a smooth space, but it will always remain unbroken and straight. (In technical terms, every smooth map f from Δ to the one-dimensional continuum R is affine, i.e. of the form f(x) = a + bx for unique a, b in R.) If you map it into a circle, it looks like an “infinitesimal side” of the infinite-sided polygon which, like Leibniz, you imagine the circle to be.
It turns out that for any element ε of Δ, ε2 = 0. So in particular the derivation given above for the “differential” of the function x2 works in Smooth Infinitesimal Analysis too if we take dx to be an element of the infinitesimal line segment Δ. (According to Bell, Leibniz’s contemporary Nieuwentijdt was even closer in spirit to Smooth Infinitesimal Analysis than Leibniz.)
Infinitesimal line segment mapped to an infinitesimal tangent on the curve f(x).
The world or category of smooth spaces and smooth maps has many nice features that make it a hospitable world in which to do calculus or analysis. The “pathologies” that beset late nineteenth-century analysis, and that led to a “crisis of intuition” to which Russell’s later philosophy of mathematics is in part a response, cannot occur in the “smooth” world.
The key point is that all of this is consistent. It is also quite beautiful, I think, but for present purposes we need only truth. Infinitesimals—that is to say, the elements of the infinitesimal line segment Δ—have just the properties they need to have to make the propositions of elementary calculus, which takes every (differentiable) function to be “linear in the small”, not only true but straightforwardly derivable. A classical analyst will see, along with the benefits, certain costs—e.g. the restriction of maps to smooth maps, and, more significantly, the failure of the Axiom of Choice in the smooth world. But at the same time paradox and pathology are avoided, and the language of the calculus, with its dx and dy, receives a direct interpretation in which those are denoting terms.
Back to Russell. On the one hand, he didn’t know enough—he lived just long enough that he could have witnessed the beginnings of Nonstandard and Smooth Infinitesimal Analysis, but of course at the time he was writing the Principles and the Introduction to mathematical philosophy neither of them existed except perhaps in Platonic heaven. On the other hand, he knew too much—or rather “knew”—: for him it was true that the continuum consists of points, that among the real numbers there are no infinitesimals, and that calculus involves finite quantities only. And so it was true that Leibniz had got it wrong, with all that that implied, in Russell’s opinion, for the interpretation of his philosophy.
Would it be a mistake for the historian to draw on Smooth Infinitesimal Analyis by way, not of vindicating infinitesimals and thereby doing justice of a sort to Leibniz, but of understanding what Leibniz himself intended, what his dx was supposed to mean? We might distinguish (as Rorty sometimes does) between an ideal Leibniz to whom we teach the mathematics necessary for him to comprehend Smooth Infinitesimal Analysis and the actual Leibniz. The actual Leibniz knew nothing of categories, rings, morphisms; he knew of some instances to which those concepts apply, but that is not the same. He couldn’t have entertained the thought of a nilpotent ring of infinitesimals whose every morphism into the real numbers is affine.
But understanding Leibniz is not only a matter of grasping his intentions; it is, for us, a matter also of bringing what he said into our discourse; and here the “teaching” may go in either direction.
Agamemnon is King (for so we translate basileus) by virtue of his similarity with the Kings of England (the Anglo-Saxon kings in fact called themselves “basileus”). But clearly when Homer refers to him as King he doesn’t have in mind the titular head of a constitutional monarchy. To describe the actual rank and function of Agamemnon would require explanation; one could translate basileus as “priest/judge/warrior having supreme authority in all things” (following Aristotle Politics iii.15) but that would be cumbersome and still not exact.Like cicerones of every sort, the historian needs to be flexible. Who is the historian writing for, after all? Not Leibniz or his contemporaries, but for “us”, one’s egocentrically defined audience of fellow specialists and (one hopes) fellow philosophers. For them analogies with current concepts may be helpful; the key here is to remember that they are analogies. If one likens Leibniz’s dx to the elements of the ring of infinitesimals Δ described above, that may help the modern reader—not to grasp what Leibniz actually had in mind but—to emulate Leibniz’s thought by thinking of analogous lines of thought in Smooth Infinitesimal Analysis.
There was nothing inevitable about the discovery of either Nonstandard or Smooth Infinitesimal Analysis. The model theory upon which Nonstandard Analysis depends, and in particular the work of Skolem that Robinson drew on, arose in the study of formal logical systems, a study that though one might find some element of its origins in Leibniz, needed to be developed much further before the constructions of Nonstandard Analysis could be conceived. Similarly Smooth Infinitesimal Analysis, though inspired by eighteenth-century theories of infinitesimals, requires for its development a general concept of algebraic structure that arose much later. To suppose that either was present, even “in germ”, in Leibniz or Newton would be to deny the creativity required to develop the two theories that, after the fact, can be taken to show that the use of infinitesimals need not founder in incoherence.