19.2. Infinitesimals: Another argument against actual infinite sets

Argument

My argument from the incoherence of actually existing infinitesimals has the following structure:

1. Infinitesimal quantities can’t exist;

2. If actual infinities can exist, actual infinitesimals must exist;

3. Therefore, actual infinities can’t exist.

Although Cantor, who invented the mathematics of transfinite numbers, rejected infinitesimals, mathematicians have continued to develop analyses based on them, as mathematically legitimate as are transfinite numbers, but few philosophers try to justify actual infinitesimals, which have some of the characteristics of zero and some characteristics of positive numbers. When you add an infinitesimal to a real number, it’s like adding zero. But when you multiply an infinitesimal by infinity, you sometimes get a finite quantity: the points on a line are of infinitesimal dimension, in that they occupy no space (as if they were zero duration), yet compose lines finite in extent.

Few advocate actual infinitesimals because an actually existing infinitesimal is indistinguishable from zero. For however small a quantity you choose, it’s obvious that you can make it yet smaller. The role of zero as a boundary accounts for why it’s obvious you can always reduce a quantity. If I deny you can, you reply that since you can reduce it to zero and the function is continuous, you necessarily can reduce any given quantity—precluding actual infinitesimals. When I raise the same argument about an infinite set, you can’t reply that you can always make the set bigger; if I say add an element, you reply that the sets are still the same size (cardinality). The boundary imposed by zero is counterpoint for infinitesimals to the openness of infinity, but actual-infinitesimals’ incoherence suggests that infinity is similarly infirm.

Can more be said to establish that the conclusion about actual infinitesimal quantities also applies to actual infinite quantities? Consider again the points on a 3-inch line segment. If there are infinitely many, then each must be infinitesimal. Since there are no actual infinitesimals, there are no actual infinities of points.

But this conclusion depends on the actual infinity being embedded in a finite quantity—although, as will be seen, rejecting bounded infinities alone travels metaphysical mileage. For boundless infinities, consider the number of quarks in a supposed universe of infinitely many. Form the ratio between the number of quarks in our galaxy and the infinite number of quarks in the universe. The ratio isn’t zero because infinitely many galaxies would still form a null proportion to the universal total; it’s not any real number because many of them would then add up to more than the total universe. This ratio must be infinitesimal. Since infinitesimals don’t exist, neither do unbounded infinities (hence, infinite quantities in general, their being either bounded or unbounded).

Infinitesimals and Zeno’s paradox

Rejecting actually existing infinities is what really resolves Zeno’s paradox, and it resolves it by way of finding that infinitesimals don’t exist. Zeno’s paradox, perhaps the most intriguing logical puzzle in philosophy, purports to show that motion is impossible. In the version I’ll use, the paradox analyzes my walk from the middle of the room to the wall as decomposable into an infinite series of walks, each reducing the remaining distance by one-half. The paradox posits that completing an infinite series is self-contradictory: infinite means uncompletable. I can never reach the wall, but the same logic applies to any distance; hence, motion is proven impossible.

The standard view holds that the invention of the integral calculus completely resolved the paradox by refuting the premise that an infinite series can’t be completed. Mathematically, the infinite series of times actually does sum to a finite value, which equals the time required to walk the distance; Zeno’s deficiency is pronounced to be that the mathematics of infinite series was yet to be invented. But the answer only shows that (apparent) motion is mathematically tractable; it doesn’t show how it can occur. Mathematical tractability is at the expense of logical rigor because it is achieved by ignoring the distinction between exclusive and inclusive limits. When I stroll to the wall, the wall represents an inclusive limit—I actually reach the wall. When I integrate the series created by adding half the remaining distance, I only approach the limit equated with the wall. Calculus can be developed in terms of infinitesimals, and in those terms, the series comes infinitesimally close to the limit, and in this context, we treat the infinitesimal as if it were zero. As we’ve seen, actual infinity and infinitesimals are inseparable, certainly where, as here, the actual infinity is bounded. The calculus solves the paradox only if actual infinitesimals exist—but they don’t.

Zeno’s misdirection can now be reconceived as—while correctly denying the existence of actual infinity—falsely affirming the existence of its counterpart, the infinitesimal. The paradox assumes that while I’m uninterruptedly walking to the wall, I occupy a series of infinitesimally small points in space and time, such that I am at a point at a specific time the same as if I had stopped.

Although the objection to analyzing motion in Zeno’s manner was apparently raised as early as Aristotle, the calculus seems to have obscured the metaphysical project more than illuminating it. Logician Graham Priest (Beyond the Limits of Thought (2003)) argues that Zeno’s paradox shows that actual infinities can exist, by the following thought experiment. Priest asks that you imagine that rather than walking continuously to the wall, I stop for two seconds at each halfway point. Priest claims the series would then complete, but his argument shows that he doesn’t understand that the paradox depends on the points occupied being infinitesimal. Despite the early recognition that (what we now call) infinitesimals are at the root of the paradox, philosophers today don’t always grasp the correct metaphysical analysis.

Distinguishing actual and potential infinities

Recognizing that infinitesimals are mathematical fictions solidifies the distinction between actual and potential infinity. The reason that mathematical infinities are not just consistent but are useful is that potential infinities can exist. Zeno’s paradox conceives motion as an actual infinity of sub-trips, but, in reality, all that can be shown is that the sub-trips are potentially infinite. There’s no limit to how many times you can subdivide the path, but traversing it doesn’t automatically subdivide it infinitely, which result would require that there be infinitesimal quantities. This understanding reinforces the point about dubious physical theories that posit an infinity of worlds. It’s been argued that some versions of the many-worlds interpretation of quantum mechanics that invoke an uncountable infinity of worlds don't require actual infinity any more than does the existence of a line segment, which can be decomposed into uncountably many segments, but an infinite plurality of worlds does not avoid actual infinity. We exist in one of those worlds. Many worlds, unlike infinitesimals and the conceptual line segments employing them, must be conceived as actually existing.

[Edit September 15, 2013.] Corrected claim that many-worlds theories of quantum mechanics posit an infinity of worlds. Some many-worlds theories do, and some don't. This argument applies only to those versions positing infinite worlds.