Aristotle against Infinite Regress, III

Aristotle’s third argument also occurs in Physics VIII, 5. Let’s suppose (for contradiction) that there is an infinite regress of motions, M₁ moved by M₂, M₂ by M₃, etc. We’re going to simplify somewhat Aristotle’s actual argument. Aristotle proceeds by way of a dilemma, setting up a disjunction of two alternatives: either (i) in all but a finite number of cases, the effect is a merely contingent consequence of its cause, or (ii) in infinitely many cases, the motion results from its cause with necessity. We think that in the first alternative Aristotle is considering something like a “Humean” account of causation (after the 18^th century Scottish philosopher David Hume), in which there is no real connection between causes and effects. In such a world, causes could not be counted on to have any effect at all, and so the continuity of motion cannot be ensured. In a Humean world, the continuity of motion would involve and infinite number of unexplained events. So, we will consider only the second alternative.

Aristotle next assumes that there is only a finite number of kinds or species of motion. If so, the infinite series will cycle through at least one kind infinitely many times. Let K be one of these cyclically recurring species of motion. Suppose that M_j and M_k are two motions in the series, with M_j preceding M_k, and with both M_j and M_k exemplifying species K of motion. Since we are dealing with a series of moving causes, transitivity holds. So, we can say that substance experiencing motion M_j gives the subject of motion M_k its K-motion, and it does so precisely because it is being moved K-wisely.

As Aristotle states, we can think of M_j and M_k as both being instances of being thrown from location M to location L. Or M_j and M_k could both be instances of learning some piece of geometry. Aristotle argues that such suppositions are absurd. If one teaches another person geometry, it can’t be that one teaches that student by virtue of being taught the same thing at the same time.

It’s true that such cyclical causation seems absurd in some cases, like the teaching/learning case. It would be absurd to think that a teacher could only confer the movement toward a piece of geometrical knowledge G at time t by being himself in a state of movement toward G at t. Clearly, what’s necessary is for the teacher to already have the knowledge of G at time t, not to be in a state of movement toward G.

However, in the case of locomotion, the corresponding state doesn’t seem absurd at all. Think of an infinite number of dominoes in a line, each falling as a result of being struck by its predecessor. In this case, each domino has to be moving in a certain way at the time of contact in order to confer the same kind of movement on its successor. Nothing absurd about that.

In order to reconstruct Aristotle’ s argument, we must first ask: How are motions individuated? That is, when do we have one motion rather than many? It is reasonable to suppose that motions are individuated by the thing being moved, the end or telos of the motion (the direction in which the thing is being moved) and the time at which the motion occurs. If we take these three factors to define a species of motion, then we can conclude that each motion is sui generis: that is, there is at most one motion in each species. Now, if the number of species of motion occurring at a particular instant of time is finite, it follows that the number of particular motions occurring at that time must also be finite. Consequently, a simultaneous infinite regress of motions would have to have involve literally circular causation, with each motion absurdly causing itself.

Aristotle assumes that there can be only a finite number of species of motion (at least, at a single instant of time). This is a weak point of this proof. First, it seems to depend on Aristotle’s assumption that the physical universe is finite in scale. An infinite universe would seem to have “room” enough for an infinite number of species of motion. Second, even assuming Aristotle’s finite world, it doesn’t seem to be the case that the universe can contain only finitely many species of motion. Each species is defined in part by the telos or end of the motion. Although space cannot be actually divided into infinitely many parts, it is potentially divisible without limit. This potentially divisibility would seem to be all that we need to define an infinite number of distinct species of locomotion (motion through space).

There’s a way to repair this gap in Aristotle’s argument, one that is admittedly anachronistic but is very much in the spirit of Aristotle’s argument. Where Aristotle says ‘finite’ we could understand to him mean measured by a particular cardinal number k in Cantorian set theory. Where Aristotle says ‘infinite’, we could understand him to mean greater than k. It is certainly reasonable to suppose that there is a particular cardinal number, say k, which is the number of possible species of motion at a time. If an infinite regress R has a larger number of members, say 2^k, then R will have be circular, containing multiple copies of at least one species. Consequently, Aristotle’s argument can be taken to rule out the possibility of infinite series with cardinalities greater than k.

But what about regresses that are infinite but no larger than k? Here again, we could apply the Avicenna-Aquinas aggregation strategy, arguing that any regress no longer than k can be treated as a single, composite motion, requiring a simultaneous mover. The only sort of regress that cannot be aggregated into a single motion would be a regress with absolutely infinitely many members, a proper class of simultaneous motions. Aristotle gives us compelling reason to reject the possibility of such an absolutely infinite regress.