Abstract
It is widely known that Aristotle rules out the existence of actual infinities but allows for potential infinities. However, precisely why Aristotle should deny the existence of actual infinities remains somewhat obscure and has received relatively little attention in the secondary literature. In this paper I investigate the motivations of Aristotle’s finitism and offer a careful examination of some of the arguments considered by Aristotle both in favour of and against the existence of actual infinities. I argue that Aristotle has good reason to resist the traditional arguments offered in favour of the existence of the infinite and that, while there is a lacuna in his own ‘logical’ arguments against actual infinities, his arguments against the existence of infinite magnitude and number are valid and more well grounded than commonly supposed.
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Notes
I have followed the standard translations in generally translating ‘apeiros’ as ‘infinite’ (rather than, e.g., ‘unlimited’) and ‘peperasmenos’ as ‘finite’ (rather than, e.g., ‘limited’), but have found it necessary to render ‘perainein’ as ‘to limit’ because of the lack of an appropriate verb (and it is worth keeping in mind, especially with regard to (D) and (F) [see below], that rendering ‘apeiros’ as ‘unlimited’ is often more perspicuous).
The literature is largely divided between two camps. One view, associated especially with Hintikka (1966) and Bostock (1972), focuses on Aristotle’s remarks likening the infinite’s manner of existence to that of processes (e.g. Phys. 206a18–25, a31–3, b13–14). On this view, only processes may be infinite (i.e. unending) and, in claiming that the infinite exists only potentially, Aristotle means to express either: (a) that the processes in question are composed of temporal parts which are never there all at once and whose parts do not endure (Hintikka); or (b) that the processes in question are unending and cannot be completed (Bostock). Another view, associated especially with Lear (1979), focuses on Aristotle’s remarks likening the infinite’s manner of existence to that of matter (e.g. 206b14–16; cf. 207a21–5, 207a35–b1, 207b34–208a4). On this view, something is potentially infinite iff ‘there will always be possibilities that remain unactualized’ (Lear 1979, p. 191). Thus, for instance, magnitude is potentially infinite because while its capacity (dunamis) for division may be (partially) actualized through someone dividing it, it can always be further actualized through further divisions. The same thought applies to number being potentially infinite through addition. For further discussion, see Charlton (1991), Coope (2012).
Translations of Physics 3–4 follow Hussey (1983) with occasional modifications. Translations of Aristotle’s other texts follow Barnes (1984). The Greek text of the Physics used is that of Ross (1936). Translations of Euclid, Alexander of Aphrodisias, Simplicius, Philoponus, and Aquinas rely upon Heath (1926), myself, Edwards (1994), Urmson (2002), and myself respectively.
Plato’s Timaeus is, seemingly, the notable exception (cf. Cael. 279b4–280a35; Phys. 251b17–18).
The gloss I provide concerns Aquinas’s remarks and is tentative. Avicenna offers a more detailed discussion. Note that, for Aristotle, the now is an instantaneous point in time without duration (analogous to a point on a line) which serves as a limit to divide the time before and the time after (cf. Phys. 218a3–30; 233b33–234a31; Waterlow 1984).
For further discussion, see White (1992, pp. 28–30, 193–220).
On the prospects of Aristotle’s finitist mathematics with reference to discussions of parallel lines and other phenomena, see Hussey (1983, pp. 93–96, 178–179).
‘But if someone were to say that a limit is relative (pros ti) (since a limit is a limit of something), still it is not relative to something else (ou pros allo), but relative to itself (pros auto)’ (Simpl. In Phys. 516, 18–20).
Further, we should notice that while Aristotle accepts that every body is either finite or infinite (e.g. Cael. 274a30–31), Aristotle denies (1) in its current form or else restricts the domain of the quantifier to a limited domain as it is not the case that anything whatsoever is finite or infinite. For instance, points (stigmai) and pathē are neither (Phys. 202b32–4). Since a point is the limit of a line (Top. 141b19–22), some limits will be neither finite nor infinite (cf. Alex. Aphr. In Top. 30, 27–31, 26).
For instance, in early and mid-twentieth century discussions of perception, a lot of mileage was gained from the notion that if, in perceptual experience, I am aware of something as being red, then there must be something, of which I am aware, that is red.
On the one hand, Aristotle seems to takes (at least some) mathematical claims to be (literally) true (Metaph. 1077b31–33). On the other hand, he seems to take (at least some) mathematical claims, notably those which treat the relevant objects as being ‘separated’, as being best understood along fictionalist lines (Metaph. 1078a17–31; cf. Phys. 193b31–5). (These claims are perfectly consistent.) He also seems to claim that (at least some) mathematical objects do not exist in the manner typically supposed (Metaph. 1077b15–17) and that (at least some) mathematical objects do exist in the manner typically supposed (1077b32–4). Aristotle here appeals to the fact that existence is spoken of in different ways (either in actuality [entelecheia] or in the manner of matter [hulikōs], 1078a28–31) in a manner which parallels the discussion of the infinite in the Physics (see above).
The inference to (7) presupposes that not being bounded entails not being bounded by a surface. Despite not being a logical truth, this seems true, necessary, analytic, and knowable a priori. If necessary, it can be supplied as a premise (e.g. \(\langle \)if a is not bounded, then a is not bounded by a surface\(\rangle \)). Simplicius may be making an observation in this vicinity (In Phys. 477, 17–19).
Unlike some later philosophers (consider, for instance, the discussions of Al-Kindī and Avicenna mentioned below), Aristotle does not here seem to consider in any detail the possibility of bodies possessing certain boundaries but which are infinitely extended in only one or two dimensions (such as an infinitely tall skyscraper). Instead, he considers only bodies which are infinitely extended in all dimensions (cf. Phys. 204b20–1).
Al-Kindī and Avicenna gave substantial attention to arguing against the possibility of infinite magnitudes or bodies and offer alternative arguments. However, they depart from Aristotle on a number of scores (for instance, Avicenna is more tolerant of actual infinities than most working in the Aristotelian tradition) and, unlike the ‘physical’ and ‘logical’ arguments offered by Aristotle, the arguments they offer against infinitude proceed ‘geometrically’, i.e. they rely upon mathematical demonstrations and premises which Aristotle does not mention in these contexts. See Al-Kindī On the Explanation of the Finitude of the Universe; Avicenna Pointers and Reminders 2.1.11; Salvation 2.2; cf. Physics of the Healing 3.8.1; Rescher and Khatchadourian (1965), McGinnis (2010).
We can thus see why Aristotle characterises the infinite as that of which something [i.e. some part] is always outside’ (Phys. 207a1–2, 7–8) and contrasts it with what is ‘complete’ (teleios) or ‘whole’ (holos) (i.e. ‘that of which nothing [i.e. no part] is outside’, Phys. 207a8–9, 11–15; cf. Metaph. 1021b12–14). As regards spatiality, recall that Aristotle takes commonsense to be (at least largely) correct in supposing that everything there is must be somewhere [e.g. Phys. 205a10; 208a29, 208b32–3 (though he may make an exception for the unmoved mover; Cael. 279a11–18; cf. Phys. 267b6–9)].
Hussey offers the following reconstruction (1983, p. 80), but little in the way of further discussion:
(1) Any actual numbered totality is countable (taken as self-evident).
(2) For any countable totality it is possible for someone to have counted that totality (by definition of ‘countable’).
(3) To have counted an infinite is to have traversed an infinite.
(4) What is infinite cannot have been , 204a5–6).
\(\therefore \) (5) No infinite totality can have been counted (by (3) (4)).
\(\therefore \) (6) No infinite totality is countable (by (2) (5)).
\(\therefore \) (7) No actual numbered totality is infinite (by (1) (6)).
For reasons that will immediately become clear I don’t think that this interpretation allows for a clear or straightforward diagnosis of the argument.
For moderns, number terms (e.g. ‘three’, ‘eight’, etc.) are usually viewed as singular terms which refer to abstract, unique objects (e.g. Frege Grundlagen Introduction, §38, §61). In contrast, for the Greeks, there were many twos, many threes (and so forth) because there were many sets of two units, of three units (and so forth). Felix, Tibbles, and Tigger constitute a number and each unit in this number is a cat (for sets or collections of seemingly heterogeneous units, cf. Metaph. 1088a8–14). It seems that, on Aristotle’s view, when we have a set (collection, etc.) of three cats and a set of three dogs, the number of cats is equal to the number of dogs but the numbers are not—contra Frege (e.g. Grundlagen §62–3)—(numerically) identical because the numbers (i.e. the sets) in question are composed of different units (Phys. 220b8–12, 224a2–10; cf. Mayberry 2000, pp. 52–59). It has been thought prudent not to use the notion of a set in approaching such issues (e.g. Hussey 1980, p. xx), but it seems that, on the ancient conception described here, numbers (arithmoi) are in fact akin to collections or sets as conceived by moderns in several important respects (for discussion, see Burnyeat 1987, pp. 220–221, 234–238; Mayberry 2000, pp. 17–63).
There is a close connection in Greek language and thought between counting (arithmein), being countable (arithmētos), and number (arithmos). The translation of ‘arithmos’ as ‘number’, although ubiquitous, slightly obscures this fact.
This must, I think, be the sense of ‘countable’ Philoponus has in mind when he complains, against Aristotle, that advocates of the existence of the infinite would not grant that all number (arithmos) is countable (arithmētos) (In Phys. 417, 19–21).
Iamblichus seems to attribute this conception of number (as a finite plurality) to Eudoxus (Iambl. In Nicom. 10, 17–20). Pritchard (1995, pp. 23–30) argues that the conception of number as necessarily finite had wider currency among the Greeks but the evidence is not straightforward.
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Nawar, T. Aristotelian finitism. Synthese 192, 2345–2360 (2015). https://doi.org/10.1007/s11229-015-0827-9
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DOI: https://doi.org/10.1007/s11229-015-0827-9