Infinite Regress Arguments as per impossibile Arguments in Aristotle: De Caelo 300^a30–^b1, Posterior Analytics 72^b5–10, Physics V.2 225^b33–226^a10

Introduction

Infinite regress arguments are a powerful tool in Aristotle. Aristotle uses such arguments to attack and defend positions in: natural philosophy (De Caelo 274^b12, 311^b31, Physics 204^a21–30), epistemology (Posterior Analytics 72^b8, 82^a39), argument theory (De Interpretatione 20^b40), metaphysics (Metaphysics 1000^b28, 1006b12–29, 1031^b15–1032^a10), politics and ethics (Politics 1257^b34–40, Nicomachean Ethics 1094^a20). However, this style of argument has received relatively little attention, especially compared to other common styles of argument in Aristotle, such as the indifference arguments discussed by Makin (1993). Moreover, improving our understanding of infinite regress arguments has become more pressing since recent scholars have pointed out that it is not clear whether Aristotle’s infinite regress arguments are, in general, effective or indeed what the logical structure of these arguments is. A better understanding of infinite regress arguments in Aristotle is significant not only for those interested in Aristotle’s logic, but also for those interested in his natural philosophy and his ideas about infinity. Even if you disagree with the conclusions of this paper, I hope it provokes further reflection on how infinite regress arguments work in Aristotle.

In the literature on the use of infinite regress arguments in philosophy, two points are usually addressed. First, an infinite regress argument needs to do more than simply generate an infinite sequence. Since there are lots of permissible infinite sequences, there must be some reason to think that a regress argument generates a problematic or vicious sequence.^[1] Broadly there are two views about why an infinite series might be vicious. Passmore (1961), chapter 2 argues that infinite regress arguments show that some explanation is inadequate, roughly because that explanation itself posits something that needs a further account and such accounts cannot be deferred indefinitely. In contrast, Black (1996), p. 111 argues that infinite regress arguments show that some proposition is false, roughly because the proposition leads to an infinite sequence, which contradicts a tacit or explicit assumption that there is no such sequence.

Secondly, there is the question of the logical structure of infinite regress arguments. The dominant view here is that infinite regress arguments are reductio arguments of a certain sort.^[2] A reductio supposes some proposition, derives an unacceptable consequence from that proposition and so goes back to reject the initial proposition. A per impossibilie argument is similar in that it supposes some proposition, derives an unacceptable consequence then reject the initial proposition. But a per impossibile argument differs in that the consequence drawn is unacceptable because it is impossible. Those who think that infinite regress arguments are structured like reductiones hold that infinite regress arguments suppose some proposition, derives an infinite sequence, declares that the sequence is unacceptable, and so rejects the supposition.

It seems that Aristotle would have a ready answer both to the question of what the structure of an infinite regress argument is and to the question of what is objectionable about an infinite sequence: Aristotle’s infinite regress arguments are just a special case of Aristotle’s per impossibile arguments.^[3] Aristotle’s per impossibile arguments derive an impossibility from a set of assumptions, and, in a separate step, derive a conclusion. An infinite regress is one sort of impossibility, according to Aristotle’s finitism.^[4] So, an infinite regress argument begins with a set of assumptions, derives an infinite series, that is, an impossibility and, in a separate step, derives a conclusion. Although most arguments in which Aristotle says that some sequence goes on ‘to infinity’ (eis apeiron) do not explicitly link the infinite series to an impossibility, several arguments do.^[5] Those arguments are also those which argue indirectly, positing an assumption and then arguing that it leads to an impossible infinite sequence. So one might be tempted to conclude that infinite regress arguments are per impossibile arguments.

In this paper, I reject this reading. First, Aristotle’s finitism is qualified. There are at least three sorts of infinity that Aristotle recognises: number, time and divisibility of spatial magnitudes.^[6] Aristotle also allows that some things would grow ad infinitum, such as fire at De Anima 416^a15, although he allows this only on the assumption that there is infinite fuel, a condition that he thinks can never, in fact, be met. But in any case, just because there is an infinite sequence, it does not follow that there is an impossibility. Second, even within those cases where Aristotle explicitly links an infinite sequence to an impossibility, we will see that there is not one single argumentative strategy or reason that the infinite sequence is impossible. If Aristotle had a general commitment to finitism, it would follow from his ‘conceptual’ (logikos) argument against infinity. Instead, I argue that infinite regresses generate domain-specific impossibilities. Even in the cases where Aristotle explicitly describes an infinite sequence as ‘impossible’ (adunaton), it is not Aristotle’s general commitment to finitism that explains why the infinite regresses are vicious. Rather it is a domain-specific reason that there can only be an infinite number of entities in a given domain. Those are the cases I will examine in this paper.

The next section will flesh out in more detail Aristotle’s ideas about indirect per impossibile arguments and some of his general conceptual reasons that infinity is impossible. Sections 3, 4 and 5 will show that his general conceptual reasons that infinity is impossible do not obtain in three arguments where Aristotle argues using per impossible reasoning and infinity. The infinities in these arguments are impossible for domain-specific reasons, either physical, cognitive or metaphysical.

Aristotle on per impossibile argument and impossible infinity

To see precisely how we can understand Aristotle’s infinite regress arguments, this section will outline Aristotle’s idea of an indirect, per impossibile argument and Aristotle’s finitism. I will argue that Aristotle does have some general conceptual reasons that infinities are impossible. But we will see in subsequent sections that these reasons are not in play in the particular arguments which employ per impossibile reasoning with an infinite sequence.

Aristotle tells us what a per impossibile argument is in Prior Analytics I.23, which aims to prove a metatheorem about Aristotle’s logic, namely that ‘every syllogism comes about in one of the figures’ (40^b20–22). It turns out that Aristotle means that every valid deductive argument either is a syllogism in the figures or has such a syllogism as a proper part. However, some important valid arguments do not seem to obey this theorem, that is arguments that begin with a supposition, derive an impossibility (e. g. a contradiction or a necessary falsehood) and conclude by asserting the contradiction of the supposition (41^a30–32). Scholars sometimes represent these arguments schematically in this way:

(Per Impossibile) Γ, suppose not-q; Γ, not-q |-- r; but it is not possible that r; therefore q.^[7]

This scheme represents per impossibile arguments as described by Aristotle. The part ‘Γ, suppose not-q; Γ, not-q |-- r’ represents the direct inference. The ‘but it is not possible that r; therefore q’ reflects an inference rule of reductio. Aristotle is committed to the idea that any finite direct inference can be restated as a series of syllogistic inferences. Per impossibile arguments involve some premises, Γ, and a supposition, not-q. The premises and the supposition jointly entail an impossibility, r. Because the supposition and the premises jointly entail an impossibility, the premises and supposition together must be false. Since we are holding the premises fixed, we conclude that the supposition is false and hence the contradictory of the supposition is true. Aristotle separates the deduction of the impossibility from proving the conclusion (i. e., the contradictory of the supposition). Deducing the impossibility happens by a formal syllogism; proving the conclusion takes an extra, non-syllogistic, step.

Aristotle gives the example of proving the incommensurability of the diagonal (Prior Analytics 41^a25–32). In that example, the proof begins by supposing that the diagonal is commensurable and takes as premises the axioms of geometry. With these, the mathematician deduces that an odd number is equal to an even number, which is impossible. Since the supposition that the diagonal is commensurable entails an impossibility, the supposition must itself be false. So the contradictory, that the diagonal is incommensurable, must be true.

We could view infinite regress arguments as per impossible arguments by giving an inference rule for infinite regress arguments:

(Infinite Regress Rule) From a finite set of premises A, the supposition B, and a deduction of an impossible infinite sequence from A and B, one may infer the contradictory of B.

This inference rule works like the rule for per impossibilie or reductio. From a supposition, and some set of premises, some unacceptable result is derived from which we infer the contradiction of the supposition. The only difference is that what is derived is an impossible infinity of some kind.

There are obvious advantages to viewing infinite regress arguments as per impossibile arguments. First, it explains how an infinite regress argument can derive a conclusion. In general, per impossibile arguments prove a conclusion by supposing something, deriving an impossibility from that, and then rejecting the supposition. Infinite regress arguments would work in a similar way: by supposing something, deriving an (impossible) infinite sequence, and so rejecting the supposition. According to Aristotelian finitism, (certain) infinite series are impossible. In fact, Aristotle defines infinities as impossibilities (Physics 204^a2–7). Since an infinite regress argument makes some suppositions and derives an infinite series, where that series is impossible, an infinite regress argument is a per impossibile argument. Thus, Aristotle could have reason, from his theory of argument and from his wider metaphysical commitments, to think that infinite regress arguments are simply a kind of per impossibile argument.

There is also a formal reason to think that infinite regress arguments are per impossiblie arguments. If we think of the above example as typical, infinite regress arguments tend to rely on a nested series of suppositions and inferences. Examples below will make the point clearer, but often an infinite regress argument will make a supposition, derive some further conclusion and then use this supposition for a subsequent iteration of the same sort of argument. This process will then go on indefinitely. According to many scholars, Aristotle’s per impossible arguments allow such a structure too, including nesting one argument in another.^[8] The difference might be that, unlike a typical per impossibile argument, infinite regress arguments do not discharge by some reductio inference rule, but seemingly keep on iterating ad infinitum. This, in turn, suggests that infinite regress arguments cannot be proofs in Aristotle’s sense, since they are not finitely long. But, again, in this respect, they resemble reductio arguments, which are not proofs either.

Some textual evidence supports the idea that infinite regress arguments are per impossibile arguments. In several prominent cases, Aristotle connects the infinite sequence to an impossibility. Specifically, Aristotle posits an assumption and then, in order to reject it, derives an infinite sequence he claims to be impossible (De Caelo 300a30–b1, Posterior Analytics 72b5–10, Physics 201a10–11, possibly Metaphysics 1030b35). There are many more cases of arguments involving infinite regresses in which Aristotle does not explicitly say that the infinite series is impossible.^[9] But the fact is that, in some important cases, Aristotle does connect infinite regresses to per impossibile indirect arguments.

However, there are some a priori reasons against this interpretation. The first of these is that Aristotle holds that not all infinities are impossible. Indeed, some infinities exist in the physical world. In particular, time (Physics 206^a10–11), division of magnitudes (Physics 203^b16–18) and numbers (Physics 206^a11–12) are widely acknowledged to be infinite on Aristotle’s account.^[10] Indeed, some have argued that Aristotle does not reject infinite pluralities (Rosen 2020). So, it is not enough to say that infinite regress arguments are per impossibile arguments, since some infinities are not impossible.^[11]

Which are the possible infinities according to Aristotle? Recent scholarship has emphasised the way in which infinity is indefinitely extendible finitude:^[12] ‘Generally, the infinite exists this way, by something else always being taken and what is taken is always finite, but always different’ (Physics 206^a27–29). Roughly, the idea here is that being infinite is an attribute that something has if one can keep on taking finite, non-identical portions of it, for as long as one likes. For example, time, number and magnitude are such that no matter how much time or number you take, or how many divisions of a magnitude, you have taken, you can always take another finite but non-identical amount. This is what it means for something to be potentially infinite: no matter how much you take, there will still be more to take.

There is some debate as to precisely why these infinities are possible. One reading, associated with Hintikka (1966), is to think that these potential infinities are a matter of it being theoretically possible, that is, the potentially infinite only exists in thinking (Metaphysics 1048^b14–17) and there are no actual infinities in the sense that there are no mind-independent infinities. The objectionable, impossible infinities would simply be mind-independent infinities. On this view, if a supposition leads to positing a mind-independent infinity, that supposition should be rejected by reductio. The alternative interpretation, associated with Lear, is that division of magnitude, time or number are potentially infinite because of the structure of those entities.^[13] On this reading, if a supposition leads to positing an infinity that is not structurally possible, that supposition should be rejected by reductio.

These are the infinities that Aristotle allows, but for our purposes, it is helpful to focus on the reasons that Aristotle rejects some infinities. Here there is even less discussion in the literature.^[14] The core of Aristotle’s conceptual argument that there can be no actual infinities is given at Physics 204b4–10.^[15] These arguments are contrasted with the ‘physical’ (phusikos) arguments against infinity. The conceptual arguments are supposed to explain in general why infinities are not possible, as opposed to the physical arguments, which show that there are no physical infinities. Aristotle offers two conceptual arguments. The first is this:

1. If the definition of ‘body’ is ‘that which is bounded by a surface’, then there cannot be an infinite body, either as an object of thought or of sense-perception (Physics 204^b4–6. Translation Hussey).

The argument is roughly:

1. Every body is bounded by a surface;

2. No infinitely extended thing is bounded by a surface;

3. So, no body is infinitely extended.^[16]

We must supply premise (2), but once we do, we see that we have a valid syllogistic argument. If this is correct, Aristotle rejects infinitely extended bodies. Commentators have wondered how (1) can be justified, since ‘bounded by a surface’ is not usually part of Aristotle’s definition of ‘body’.^[17] This makes premise (1) seem ad hoc. Nawar (2015), p.11 offers what I take to be a plausible reply: unbounded bodies would lack both determinate spatial features but also lack identity conditions. Without a boundary, there is no point at which the body stops and the rest of the world starts. So not only is there nothing outside the body, but there is also nothing inside the body. Hence the infinite body would lack determinate location and identity.

This first argument, however, still seems quite specific. It might give us some idea of which infinities are impossible, but it certainly does not offer a reason to reject infinity that exports from the case of body. The first argument does not even show there could not be an infinite number of bounded bodies. Aristotle’s second conceptual argument offers a more general reason why infinities might be impossible, arguing that there are no separate infinite numbers:^[18]

1. Neither can there be a separated infinite number: for number, or what has number, is countable (arithmētos), and so, if it is possible to count (arithmein) what is countable, it would be possible to traverse (diexerchomai) the infinite (Physics 204^b7–10. Translation Hussey)

Spelled out, the argument goes like this:

1. Every number and everything that has number is countable;

2. Everything that is countable can be counted;

3. Everything that can be counted can be traversed;

4. Nothing infinite can be traversed;

5. So, nothing that has number is infinite.

What is important here is that premise (2) expresses that when we are counting something ‘countable’ the counting can be completed. The aorist infinitive arithmesai with the endechetai indicates this. Anything about which the counting can be completed can be traversed, but infinite things cannot be traversed. Aristotle draws attention to a non-ending traverse (Physics 204^a4). What is traversed is the right sort of thing, but the traverse cannot be completed or is unending.^[19] It cannot be traversed because the traversing process cannot get finished. This idea that a traverse cannot get finished could itself be spelled out, since there may be two reasons a traverse could not be completed. Some processes cannot be completed because you cannot accomplish the final item in the sequence since there is no final item. Counting the natural numbers up from zero is an example of such a process. The other sense in which a traverse might be unending is that there is a definite endpoint that, however, cannot be reached. Each step in the traverse just throws up more steps to take, in a manner reminiscent of Zeno’s dichotomy paradox.

Premise (3) seems true if we take ‘traversable’ in the ‘can be completed’ sense. After all, for Aristotle, an infinite number of things just is a number of things such that no matter how many I take, there is always another finite number to take. When I have counted to an arbitrarily large number, I have completed the traverse up to and including that number. What seems doubtful in this argument is whether premise (2) is true. Everything that is countable can be counted in the sense that the count can be completed. There is a perfectly ordinary sense of counting where something is counted if it is brought into one-to-one correlation with the natural numbers. But it seems that, on this account, there are countable things where the count cannot be completed. I can count the even numbers, for example. But clearly, this count cannot be completed.

Summing up Aristotle’s reasons for thinking that infinities are impossible is not straightforward, but if there is a general reason Aristotle is suspicious of infinities, it will be found in Aristotle’s conceptual argument that nothing infinite is traversable. The impossible infinities are the ones which do not have a traverse that can be completed. But even in that case, we would have to look at each infinite regress argument on a case-by-case basis to decide whether it involves an impossible sort of infinity. In the following sections, I will show how infinity is impossible for different reasons in different domains.

The natural rest argument, De Caelo 300^a30–^b1

In De Caelo III.2, Aristotle is arguing that a body naturally rests in the centre of the cosmos:

1. So it is clear that there is something in the middle. If, then, this rest is natural to it, clearly, motion to this place is natural to it. If, on the other hand, it rests by force, what stops it from moving? If it is something at rest, we shall simply repeat the same argument; and either we shall come to something which naturally rests where it is, or necessarily we shall go on to infinity (εἰς ἄπειρον), which is impossible (ἀδύνατον). (De Caelo 300^a30–^b1. Translation Stocks, modified).

Aristotle has distinguished two kinds of rest: rest by force, in which something is at rest because some external force constrains it, and natural rest, in which something is at rest despite there being no external force constraining it. Aristotle then asks whether the body in the centre of the cosmos rests by force or by nature. If it is by force, then we can ask of the external, constraining thing whether it, in turn, is at rest by force or by nature, and so on ad infinitum.^[20] And this is impossible.

But what precisely continues ad infinitum? One option, suggested by Stocks’ translation, is that it is the argument that can be repeated ad infinitum. We ask, ‘Why does something rest in the centre?’; if the reply is ‘it rests naturally’, then Aristotle gets the conclusion he wants. But if the reply is ‘something forces it to rest’, we can repeat the same argument against this explanation. Unless the interlocutor mentions something that naturally rests, the reiteration of the same argument can go on and on. Thus, every explanation stands in need of a further explanation. What is impossible here is indefinitely continuing the argument because explanations must come to an end somewhere.

The worry with this broadly explanationist reading is that it seems to slide from the properties of an argument to the properties of what the argument is about. It is true that the argument threatens to be reiterated infinitely, and infinite iteration of an argument – at least in a dialectical context – is impossible. But it does not follow from the fact that an infinitely long argument is impossible that an infinite number of things, each holding another in place, is impossible.

Another way to take the point would be that the impossibility attaches not to the potentially infinitely long argument but to the infinite series of things. This would give the argument a structure like this:

1. Every resting thing, x, rests either by force or by nature

2. If some x rests by force, there is some other thing, y, which forces x to rest

3. There is a resting thing in the centre of the cosmos, a

4. a rests either by force or by nature

5. | If a rests by force, then there is some other thing, b, which forces a to rest

6. |b is either moving or at rest

7. ||Suppose b is at rest

8. ||If b is at rest, then b either rests by force or by nature

9. ||If b rests by force, there is some other thing, c, which forces b to rest

10. ||c either rests by force or by nature

11. || And so on ad infinitum

12. So, there is no resting a which is forced to rest by something else (Discharge IR)

This argument leaves open the possibility that what holds a in place is a moving thing, and Aristotle attempts to deal with this possibility in a separate argument (De Caelo III.2 300^b1–8).^[21] This argument tries to show that there cannot be a b which forces a to rest and is itself at rest and forced to rest by something. So, either b rests naturally or there is nothing that forces a to rest, that is, a rests naturally. But in either case, something rests naturally. Incidentally, the argument is not valid. It could be the case that a forces b to rest and b forces a to rest. So neither a nor b rests naturally, but there is not an infinite series of objects forcing the other to rest.

But for our purposes, what is important about T2 is that Aristotle explicitly says that the series of steps that goes on indefinitely is impossible. Aristotle also connects infinite iteration to the impossible in other places.^[22] This is one reason to think that Aristotle’s infinite regress arguments are per impossibile arguments. It is also important to note that the argument does not derive a contradiction. There is nothing contradictory about supposing that the thing in the centre of the cosmos is held in place by something external, which is itself held in place by something external, etc. So, the series is not impossible because of a straightforward contradiction. Rather, it is impossible because of the infinite series.

But it seems that the infinite sequence here should be impossible for some reason or other; otherwise, Aristotle could not possibly think that rejecting the supposition that something rests in the centre by force is a valid move. So in what sense is the infinite sequence generated here impossible? One answer would appeal to Aristotle’s argument that there is no indefinitely extended body.^[23] But the passage at T3 does not argue that there is a body (soma) at rest in the centre of the universe. T3 asks whether there is a thing (ti) which is at rest at the centre, while Aristotle’s argument in Physics 204^a34–206^a7 is primarily aimed at showing that there cannot be infinite perceptible bodies.

There are two sorts of considerations that might explain why Aristotle thinks the infinite sequence of things holding other things in the centre by force, discussed in T3, is impossible. At Physics 205^b35, Aristotle says:

1. Overall, if place cannot be infinite, and if every body occupies place, then there cannot be an infinite body. Now, anything that is somewhere is in a place, and anything that is in a place is somewhere. But just as the infinite cannot be a quantity (because then it would have to be a particular quantity, such as 2 or 3 feet long – that is what ‘quantity’ means), so also anything that is in a place is so because it is in some particular place, and that is to say that it is above or below or in one of the other six directions. But each of these directions has a limit (Physics 205^b35–206^a7. Translation Waterfield, modified).^[24]

Overall T4 argues that if place cannot be infinite, then body cannot be infinite. The thought seems to be this. Every body occupies a place. Place cannot be infinite. So, bodies cannot be infinite. Why can place not be infinite? Well, just as a quantity cannot be infinite because something that has a quantity has a definite quantity, such as being two feet long, so, too, place cannot be infinite, as something that is in a place is in some definite place: one of the six directions. But each of these directions has a limit. So place has a limit. But, since each body must occupy a place, no body can be infinite.

Something like this could help with the natural centre argument. If there is an infinite number of things needed to keep the thing in the centre there by force, that infinite number of things must be located at one of the six directions (above, below, left, right, front and back) relative to the thing in the centre or at some combination of those. But, if those directions are finite, there simply cannot be enough locations for that infinite number of bodies. So, it is impossible that an infinite number of things exist hold something in the centre.

Could Aristotle answer my other objection, that a and b might be two parts of a thing that rests in the centre, each of which forces the other to rest? Again, considerations from the Physics might help, but this time considerations about place. Think about the composite object, ab, such that a forces b to rest in the centre and b forces a to rest in the centre and ab is a whole composed of parts a and b. Can ab occupy the centre? Not if the centre is a simple place. Because, as Aristotle says,

1. There has to be an exact correspondence between place and body, since the total place cannot be larger than the body capable of filling it, and the body cannot be larger than the place either (Physics 205^a29–34. Translation Waterfield.).

So, since ab has parts, it cannot be wholly in the centre. Each of a and b is a body and so must occupy its own place. Each place cannot be larger than a or b. But, since there is a unique central place, and a and b must be at different places, at most one of a and b can be located at the centre. So ab cannot be located at the centre. So it cannot be the case that there is an item with two parts, each holding the other in place, in the centre.

If this is right, then the natural centre argument does rely on some commitments that Aristotle’s opponents need not share. But it does present a way to understand Aristotle’s claim that this infinite sequence is impossible. Note, however, that it is domain-specific in two ways. First, it depends on Aristotle’s own physical views. Second, the reason that the infinite sequence is impossible is distinctive to physics. The argument rests on arguing physically (physikôs, Physics 204^b10) rather than conceptually (logikôs, Physics 204^b4–10). Indeed, Aristotle’s worry is that there will not be enough room for the bodies that are generated. So, we would not expect the same reason that infinite sequences are impossible to be exportable to other domains of inquiry. We will see how this fails to export to the cognitive domain in the next section.

Understanding in Posterior Analytics 72^b5–10

In a famous discussion of the starting points of a science, Aristotle is arguing that a demonstration must be finitely long, non-circular and begin with some primitive starting points.^[25] Again, this is one of the few places where he explicitly links an infinite regress argument with an impossibility. He argues for the first claim in the following text:

1. For, one party, supposing that one cannot understand in another way^[26], claims that we are led back ad infinitum on the ground that we would not understand what is posterior because of what is prior if there are no primitives; and they are right – for it is impossible (adunaton) to go through infinitely many items. (Posterior Analytics 72^b5–10. Translation Barnes, slightly modified).

Aristotle agrees with the party who claims that there are primitives. Part of the reasoning for this conclusion is that it is impossible to go through infinitely many things. Again, you might think Aristotle’s point is that what is impossible is ever explaining how one can understand a proposition Pn, since the explanation appeals to understanding a Pn+1. So the explanation fails to explain. But the text is clear that what is impossible is ‘going through’ infinitely many things. If these things were steps in the regress argument, the argument would confuse a feature of the argument with a feature of what the argument is about. There must be something about understanding which makes an infinite series impossible.

So how would we understand this part of the argument to be a per impossibile argument? Here is my reconstruction of the reasoning:

1. For all P_n, if a understands P_n, then there is a P_n+1 such that a understands P_n+1 and a infers P_n from P_n+1 (Premise)

2. There is a P₁ such that a understands P₁ (Supposition for reductio)

3. | If a understands P₁, then there is a P₂ such that a understands P₂ and a infers P₁ from P₂ (Instantiation 1,2)

4. | There is a P₂ such that a understands P₂ and from which a infers P₁ from P₂ (1,3)

5. ||If a understands P₂, then there is a P₃ such that a understands P₃ and a infers P₂ from P₃ (Instantiation 2,4)

6. || There is a P₃ such that a understands P₃ and a infers P₂ from P₃ (MP, 4, 5)

7. || …

8. ||| So if a understands P₁ then a understands infinitely many Ps

9. ||| a understands infinitely many Ps (MP 2,8)

10. ||| Impossibility

11. So it is not the case that there is a P₁ such that a understands P₁ (Discharge IR)

I’ve formulated this argument as best I can, but it is not valid as it stands. (7) is not a formally valid move and is at best an informal step meaning something like ‘etc’. So, it is not even obvious that this argument validly derives an impossibility. Premise (8) is supposed to capture what (7) suggests, namely that the steps (2)–(6) can be iterated infinitely. We need some premise whose content mentions infinity; otherwise, the argument again risks confusing the properties of the argument with the properties of what the argument is about.

Barnes has argued that this argument can be construed as valid (Barnes 1993, pp. 104f.). Barnes alludes to the principle of mathematical induction, the principle that if an arbitrary number n has a feature F, and the successor of n has F, then every number has F. Applying mathematical induction to this case you get the following. If a understands P₁ by inferring P₁ from P₂ and understanding P₂, and a understands P₂ on the basis of inferring P₂ from P₃ and understanding P₃, then, for every n a understands P_n on the basis of inferring P_n from P_n+1 and understanding P_n+1. This is worrying because it implies that to understand P₁, a needs to perform an infinite number of inferences.

However, it is not obvious that mathematical induction is a valid principle unless F is a mathematical feature. In this case, F is an epistemic feature. But mathematical induction is not generally valid for epistemic features: if I can calculate the prime factors of numbers up to n, and the prime factors of numbers up to n+1, it does not follow that I can calculate the prime factors for every number. Now maybe you think that ‘understanding’ is just a mathematical feature, since for Aristotle understanding P just is to have inferred it from some P_n+1. But then the argument just cashes out to ‘to have inferred P, I must have made an infinite number of inferences’, which begs the question. I doubt there will be a satisfactory formulation of this argument using mathematical induction, at least not in a straightforward way. So I suggest we return to look at the argument as formulated with conditional premises.

On that reading, although (8) reflects the impossibility we would expect if this were a per impossibile argument, it is unclear how (8) is supposed to be derived. The instantiation and modus ponens steps can be repeated indefinitely, but there is no obvious way to discharge this feature of the argument into a premise. A certain sub-set of the steps of the argument can be repeated again and again. But this does not entail a premise whose content involves an infinity. What, I think, the reiteration of these premises does allow is a step of the form:

(8’) There is a P_n such that a understands P_n and a infers P_n-1 from P_n

for an arbitrarily large n. That is, we just stop the chain of modus ponens moves at an arbitrary point. Though (8’) isn’t itself impossible, it does not entail (8). In fact, all (8’) entails is that to understand P₁, an agent must understand an arbitrarily large number of things. But, to put the point in Aristotelian terms, ‘traversing’ an arbitrarily large, but finite, number of things is not, in general, impossible. So, the infinity cannot be impossible for the conceptual reason that I would have to traverse infinitely many things; there must instead be some domain-specific reason. For example, this traverse might be impossible for beings like us who cannot understand an arbitrarily large number of things.

One further point to make here concerns the move from (8) to (9): Why is it impossible to understand infinitely many propositions? This move will depend on some particular view in cognitive psychology. It seems independently plausible that an agent cannot understand an infinite number of propositions, at least, cannot explicitly understand them. Indeed, one might hesitate to say that an agent understands a mathematical proof that is infinitely long, or hesitate to call such an infinitely long argument a proof at all. However, Aristotle may have made a rod for this own back with his notion of potential infinities. While I might agree that it is impossible to explicitly understand an infinitely large number of propositions, it is not so clearly impossible to potentially understand an infinite number of propositions.

At this stage in the discussion, however, Aristotle focuses on what he calls understanding simpliciter:

1. We think we understand a thing simpliciter […] when we think we know of the explanation because of which the object holds that it is its explanation, and also that it is not possible for it to be otherwise (Posterior Analytics 71^b10–13. Translation Barnes.)

In the context of a similar argument, Aristotle is perfectly clear that:

1. You can define every substance of this type, but you cannot go through infinitely many items in thought (Posterior Analytics 83^b6. Translation Barnes, slightly modified. Cf. Posterior Analytics 84^a3)

Presumably, Aristotle is confident of these cognitive facts because of his view that a finite thing cannot traverse an infinity in a finite time (De Caelo 272^a3, ^a29, Physics 238^a33 cited by Barnes 1993, p. 105). This suggests that there is a simple argument that understanding P₁ is impossible.

1. Understanding P₁ requires understanding infinitely many Ps;

2. Understanding infinitely many Ps requires traversing infinitely many Ps in thought;

3. It is not possible to traverse infinitely many Ps in thought;

4. So, understanding P₁ is not possible.

The argument seems valid, and it seems clear that, in the case of an infinitely long chain of derivations, the point is not that the derivations cannot get started: I can understand P₁ based on an inference of P₁ from P₂. Rather, Aristotle’s point is that this infinite sequence cannot get finished, because there is no end point, either because there simply isn’t one or because there is a loop of derivations. We cannot go through such an infinite sequence to attain understanding, Aristotle would say, because understanding is a matter of being still, coming to the end of a process, stopping, rather than a continuous process (Metaphysics 994b23–24, De Anima 407^a32–33, Metaphysics 1067^b10–11, 1075^a7–9, Physics 247^b1–13, Posterior Analytics 72^a37–39). Clearly, if understanding is the endpoint of a process, it is not possible to get to understanding if that process has no endpoint.

All this is to say that this infinite regress argument has the form of a per impossibile argument, but what is impossible is domain-specific. Aristotle’s point is not a simple finitist one that infinite sequences are impossible. Nor is it the conceptual idea that we end up with an impossible infinity because there is no finite point at which we can arbitrarily stop. Nor is it the same point that we saw in the resting centre argument, that there is not enough room, in some sense. Aristotle’s reasons that infinite chains of understanding are not possible are based squarely in his cognitive psychology: understanding is the end of a process, but since we cannot go through infinitely many things in thought, the infinite process has no end. So, there can be no understanding. The impossibility is not simply an infinity; rather, the impossibility is domain-specific infinity.

An infinite chain of change, Physics V.2 225^b33–226^a10

My final example of an infinite regress argument that derives an impossibility only to reject an initial supposition shows a third sense in which the infinity derived is impossible. In the Metaphysics and a parallel passage in the Physics, Aristotle argues that there is change only in the categories of quality, quantity and location. He offers an argument by elimination. In particular, Aristotle wants to argue that there is no change in the category of action and passion. At best, there is coincidental change in the category of action and passion. Broadly, Aristotle’s argument is that change in the category of action and passion entails change of change. But there is no change of change, or so Aristotle argues using an infinite regress argument:

1. Again, it will go on to infinity, if there is to be transformation of transformation and becoming of becoming. If the latter is to be, necessarily the former is also to be. For example, if a simple becoming became at some point, that which becomes also became; so that that which becomes simply was not yet, but something was already that is becoming [a] becoming [thing]. And again that became at some point, so that at the time it was not yet becoming a becoming thing. But since of infinities there is no first thing, the first [of these] will not be, so that neither will the consecutive one. So, nothing can come to be, change or undergo transformation. (Physics 225^b33–226^a10. Cf. Metaphysics 1068^a34–^b6).

This is the second of four arguments to show that there is no change of change, but this argument is of special interest because Aristotle formulates it as an infinite regress argument. If a process of becoming is the source of another process of becoming, then the latter process of becoming is not grounded. A becoming is a process for an end product. A becoming of becoming is both the end product of an earlier becoming and the process for a later product. But as long as a process is going on, the end product does not yet exist. If a process now ongoing has as its end product a further process, then that further process does not yet exist. So at any given point in time, neither the process nor the product will exist. In more detail, we could reconstruct the argument this way:

1. Every becoming is a becoming becoming

2. A becoming becoming is not yet a becoming

3. Every becoming becoming needs a becoming whose product every such becoming becoming is

4. These further becomings are also becoming becomings [by (1)]

5. So, these further becomings also need a becoming whose product they are [by (3)]

6. The presence of any becoming (no matter how many further processes/becomings are embedded in it) requires that there be some becoming that doesn’t need a previous becoming responsible for its presence. [This is the rejection of going to infinity]

7. There can be no first becoming whose product some becoming in this sequence could be, because being first it would not be a becoming becoming.

First, one worry with the argument is that the conclusion seems too strong. Aristotle’s text ends with the claim that there are no changes, transformations or becomings. But such a conclusion cannot be justified by the argument I have been able to reconstruct, which sets out to show that there is no becoming of becoming and moves from one instance of becoming of becoming to an infinite regress. The infinite regress should lead us to reject becoming of becoming but not change of any kind. So, we need to note that Aristotle is entitled only to a weaker conclusion than he advertises, namely, that there is no first becoming in this sequence.

This sort of argument seems a strong candidate for an explanationist reading. There can be no change of change because if a is a product of process b, and b is becoming, a has not been explained. Positing a further process, c, to explain b, just defers the explanation. I think the difficulty with this reading is that Aristotle phrases his argument in terms of a chain of becomings that has no beginning, not a chain that lacks explanation. Aristotle’s regress is a metaphysical one, not an epistemic one. Aristotle is assuming that there must be a first process in a sequence of changes. Change of change leads to an infinite series that needs no first process. So, change of change is impossible because it contradicts a specific assumption of Aristotle’s metaphysics. But we can now see that Aristotle’s argument here does not rely on general finitist considerations. Aristotle’s point depends on specific views about the nature of becoming, in particular, that chains of becomings must start somewhere.

Conclusion

In this paper, I argued that for three of those cases where Aristotle argues that something is impossible because of an infinite sequence, the infinite sequence is impossible for different reasons in each case. In the case of the natural centre argument, the impossibility depends on features of Aristotle’s physics, especially the finitude of space. In the understanding argument, features of Aristotle’s cognitive psychology come into play to rule out an infinite chain of understanding coming to an end. Finally, the chain of change argument showed that an infinite sequence of changes which has a change as their source cannot have a first change, which is, again, a different reason for an infinity to be impossible. So, are Aristotle’s infinite regress arguments per impossibile arguments? The simple story, that Aristotle’s infinite regress arguments generate an infinite sequence that Aristotle rejects because of his strict finitism, is not correct. Even if we restrict our attention to cases where Aristotle explicitly says that an infinite sequence generated by a supposition is impossible, the infinite sequences are impossible for different reasons in each case.