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.

19.1. The meaning of “existence”: Lessons from infinity

Based on 19.0. Can infinite quantities exist?

The topic is the concept of existence, not its fact—not why there's something rather than nothing—but the bare concept brings its own austere delights. Philosophical problems arise from our conflicting intuitions, but “existence” is a primitive element of thought because our intuitions of it are so robust and reliable. Of course, we disagree about whether certain particulars (such as Moses) have existed and even about whether some general kinds (such as the real numbers) exist, but disputes don’t concern the concept of existence itself. If Moses’s existence poses any conceptual problem, it concerns what counts as being him, not what counts as existence. Adult readers never seriously maintain that fictitious characters exist; they disagree about whether a given character is fictitious. Even the question of the existential status of numbers is a question about numbers rather than about existence. As will be seen, sometimes philosophers wrongly construe these disputes as being about existence.

When essay 19.0 asked “Can infinite quantities exist?” existence’s meaning wasn't in play—infinity’s was. Existence is well-suited for the role as a primitive concept in philosophy because it is so unproblematic, but it’s unproblematic nature can be thought of as a kind of problem, in that we want to know why this concept is uniquely unproblematic. We would at least like to be able to say something more about it than merely that it’s primitive, but in philosophy, we acquire knowledge by solving problems, and existence fails to provide any but the unhelpful problem of its being unproblematic. The problem of infinity provides, in the end, some purchase on the concept of existence, which concept I assumed in dealing with infinity.

In one argument against actual infinity, I proposed as conceptually possible that separate things might be distinguishable only concerning their being separate things. If we assume that infinite sets can exist, the implication is the contradiction that an infinite set and its successor—when still another point pops into existence—are the same set because you can’t distinguish them. (In technical terms, the only information that could distinguish the set and its successor, given that their members are brutely distinguishable, is their cardinality, which is the same—countably infinite—for each set.)

What’s interesting is the role of existence, which imposes an additional constraint on concepts besides the internal consistency imposed by the mathematics of sets. Whereas we are unable to distinguish existing points, we are able—in a manner of speaking—to distinguish points that exist from those that don’t exist. While no proper subsets are possible for existing brutely distinguishable points, the distinction within the abstract set of points between “those” that exist and “those” that don’t exist allows us to extend the successor set by moving the boundary, resulting in contradiction.

If finitude is a condition for existence, we’ve learned something new about the concept of existence. Its meaning is imbued with finitude, with definite quantity. Everything that exists does so in some definite quantity. Existence is that property of conceptual referents such that they necessarily have some definite quantity.

Existence is primitive because almost everyone knows the term and can apply it to the extent they understand what they’re applying it to. The alternative to primitive existence is primitive sensation, as when Descartes derived his existence from his “thinking.” But sensationalism is incoherent; “experiences” inherently lacking in properties (“ineffable”) are conceived as having properties (“qualia”). The heirs of extreme logical empiricism, from Rudolf Carnap to David Lewis, have challenged existence’s primitiveness. Carnap defined existence by the place of concepts in a fruitful theory. Lewis applies this positivist maxim to conclude that all possible worlds exist. Lewis isn’t impelled by an independent theory of logical existence, such as a Platonic theory that posits actually realized idealizations. Rather, the usefulness of possible worlds in logic requires their acceptance, according to Lewis, because that’s all that we mean by “exists.” Lewis is driven by this theory of existence to require infinitely many existing possible worlds, which disqualifies it on other grounds. But the grounds aren’t separate. When you don’t apply the constraints of existence because you deny their intuitive force, you lose just that constraint imposing finitude. The incoherence of sensationalism and actual infinitism argues for a metaphysics upholding the primacy of common-sense existence.

19.0. Can infinite quantities exist?

1. The actuality of infinity is a paramount metaphysical issue.

Some major issues in science and philosophy demand taking a position on whether there can be an infinite number of things or an infinite amount of something. Infinity’s most obvious scientific relevance is to cosmology, where the question of whether the universe is finite or infinite looms large. But infinities are invoked in various physical theories, and they seem often to occur in dubious theories. In quantum mechanics, an (uncountable) infinity of worlds is invoked by the “many worlds interpretation,” and anthropic explanations often invoke an actual infinity of universes, which may themselves be infinite. These applications make actual infinite sets a paramount metaphysical problem—if it indeed is metaphysical—but the orthodox view is that, being empirical, it isn’t metaphysical at all. To view infinity as a purely empirical matter is the modern view; we’ve learned not to place excessive weight on purely conceptual reasoning, but whether conceptual reasoning can definitively settle the matter differs from whether the matter is fundamentally conceptual.

Two developments have discouraged the metaphysical exploration of actually existing infinities: the mathematical analysis of infinity and the proffer of crank arguments against infinity in the service of retrograde causes. Although some marginal schools of mathematics reject Cantor’s investigation of transfinite numbers, I will assume the concept of infinity itself is consistent. My analysis pertains not to the concept of infinity as such but to the actual realization of infinity. Actual infinity’s main detractor is a Christian fundamentalist crank named William Lane Craig, whose critique of infinity, serving theist first-cause arguments, has made infinity eliminativism intellectually disreputable. Craig’s arguments merely appeal to the strangeness of infinity’s manifestations, not to the incoherence of its realization. The standard arguments against infinity, which predate Cantor, have been well-refuted, and I leave the mathematical critique of infinity to the mathematicians, who are mostly satisfied. (See Graham Oppy, Philosophical perspectives on infinity (2006).) 

2. The principle of the identity of indistinguishables applies to physics and to actual sets, not to everything conceivable.

My novel arguments are based on a revision of a metaphysical principle called the identity of indistinguishables, which holds that two separate things can’t have exactly the same properties. Things are constituted by their properties; if two things have exactly the same properties, nothing remains to make them different from one another. Physical objects do seem to conform to the identity of indistinguishables because physical objects are individuated by their positions in space and time, which are properties, but this is a physical rather than a metaphysical principle. Conceptually, brute distinguishability, that is, differing from all other things simply in being different, is a property, although it provides us with no basis for identifying one thing and not another. There may be no way to use such a property in any physical theory, we may never learn of such a property and thus never have reason to believe it instantiated, but the property seems conceptually possible.

But the identity of indistinguishables does apply to sets of existing things (actual sets): indistinguishable actual sets are identical. Properties determine actual sets, so you can’t define a proper subset of brutely distinguishable things.

3. Arguments against actual infinite sets.

A. Argument based on brute distinguishability.

To show that the existence of an actual infinite set leads to contradiction, assume the existence of an infinite set of brutely distinguishable points. Now another point pops into existence. The former and latter sets are indistinguishable, yet they aren’t identical. The proviso that the points themselves are indistinguishable allows the sets to be different yet indistinguishable when they’re infinite, proving they can’t be infinite.

B. Argument based on probability as limiting relative frequency.

The previous argument depends on the coherence of brute distinguishability. The following probability argument depends on different intuitions. Probabilities can be treated as idealizations at infinite limits. If you toss a coin, it will land heads roughly 50% of the time, and it gets closer to exactly 50% as the number of tosses “approaches infinity.” But if there can actually be an infinite number of tosses, contradiction arises. Consider the possibility that in an infinite universe or an infinite number of universes, infinitely many coin tosses actually occur. The frequency of heads and of tails is then infinite, so the relative frequency is undefined. Furthermore, the frequency of rolling a 1 on a die also equals the frequency of rolling 2 – 6: both are (countably) infinite. But when there are infinitely many occurrences, relative frequency should equal the probability approached in a finite world. Therefore, infinite quantities don’t exist.

4. The nonexistence of actually realized infinite sets and the principle of the identity of indistinguishable sets together imply the Gold model of the cosmos.

Before applying the conclusion that actual infinite sets can’t exist, together with the principle of the identity of indistinguishables, to a fundamental problem of cosmology, caveats are in order. The argument uses only the most general and well-established physical conclusions and is oblivious to physical detail, and not being competent in physics, I must abstain even from assessing the weight the philosophical analysis that follows should carry; it may be very slight. While the cosmological model I propose isn’t original, the argument is original and as far as I can tell, novel. I am not proposing a physical theory as much as suggesting metaphysical considerations that might bear on physics, whereas it is for physicists to say how weighty these considerations are in light of actual physical data and theory.

The cosmological theory is the Gold model of the universe, once favored by Albert Einstein, according to which the universe undergoes a perpetual expansion, contraction, and re-expansion. I assume a deterministic universe, such that cycles are exactly identical: any contraction is thus indistinguishable from any other, and any expansion is indistinguishable from any other. Since there is no room in physics for brute distinguishability, they are identical because no common spatio-temporal framework allows their distinction. Thus, although the expansion and contraction process is perpetual, it is also finite; in fact, its number is unity.

The Gold universe—alone, with the possible exception of the Hawking universe—avoids the dilemma of the realization of infinite sets or origination ex nihilo.

(Edited July 25, 2013: Clarified in Section 2, last paragraph and other places, that the identity of indistinguishables applies to actual sets.)