Infinity, Potentiality and Actuality

Change is something with which we are all presented in some way and, as a perennial concern of human inquiry (sometimes ironically professed as the only constant), it is at the core of Aristotelian examination and understanding of the infinite, which is an idea that rests as a stark contrast to the world as it is presented in our experience, as ever-changing but always in the realm of the finite, even as it is never seen or experienced in its entirety neither by any individual nor, arguably, any collective with the capacity to perceive. The continuity of change traverses through and in relation to its infinite potentialities as realized actualities, which present themselves as finite in their existent state. It is this mediating idea of potentiality and actuality that Aristotle utilizes to address and attempt to resolve the problems presented by either the presence of infinity (e.g., an infinite body) or its absence (e.g., circumscription of divisibility and number): “‘To be’, then, may mean ‘to be potentially’ or ‘to be actually’; and the infinite is either in addition or in division” (Ackrill, 1987, 118; Physics 206^a, 13-15). Thus, the infinite must be understood in terms of possibilities and never as an actuality of what is, for it neither can nor cannot exist as neither both probability and actuality nor as neither of them.

Aristotle arrives at the examination of infinity as a necessary step to understanding change: “Change is thought to be something continuous, and the infinite is the first thing that presents itself to view in the continuous (which is why those who try to define the continuous often find themselves making use of the definition of the infinite as an auxiliary, the supposition being that what is divisible ad infinitum is continuous)” (Ackrill, 1987; Physics 111, 200^b, 16-21). Therefore, according to Aristotle, infinity is directly related to change. He maintains that continuity has been defined in terms of infinite divisibility. From this, one feature (and solely with respect to infinity as the defining component of continuity, the feature) delineating infinity is that of unbounded division. While never directly referring to it, through Aristotle’s considerations it is apparent that infinity is understood in a more general sense as a quality or feature that exhibits unboundedness[1]. This more general understanding is then once again constrained, but this time it has two directions: (1) the aforementioned direction of unbounded divisibility or taking increasingly small increments or taking away increasingly small increments ad infinitum and (2) unbounded addition, aggregation, propagation or increasing magnitudes[2].

Firstly, in the case of infinity in addition, there are various problems Aristotle exposes by considering the presence of an infinite body: “if there were a lump of the infinite body, where will that be in motion or where will it be at rest, since the place of the body of the same kind is infinite? Will it then occupy the whole of the place? How could it?...” (Ackrill, 1987; Physics, 205^a, 14-17; neither motion nor rest is possible to determine; impossibility and indeterminacy of location); “...these components will in form either be finite, of infinite in number... if they are infinite in number and simple, the places will be infinite in number too and the elements will be infinite in number... and [if] the places are finite, the whole is too” (Ackrill; Physics, 205^a, 22-28; addition of finite things to equal an infinite number or magnitude is impossible; the presence of an infinite number of things is impossible; the presence of a finite number of things and an infinite body is impossible). Consequently, it is the case that an infinite body or an infinite set of finite bodies cannot manifestly or tangibly exist: “It is manifest, then, from these considerations that there is in actual operation no infinite body” (Ackrill; Physics, 206^a, 7-8). 

Similar problems present themselves for infinity in terms of division. If addition of finite bodies cannot result in an infinite body, as has been considered, it necessarily follows that division of an infinite body cannot result in a definite and finite number for a principle of combination that results in increased magnitude, such as multiplication, is undone by division. Therefore, it must be the case that just as multiplication or addition cannot ever arrive at infinity (though it can asymptotically approach it), so too division of infinity, though even at the outset a seemingly inane notion, cannot ever arrive at a finite result, no matter the multiplicity (for, furthermore, any multiplicity of finites must be definite in number and/or magnitude and infinity is by definition unbounded and therefore indefinite in this sense). This notion is reflected in conceptual mathematics, for dividing any number by infinity results in zero and if infinity is divided by any number it remains infinity. The argument for division by infinity follows the following basic form: if (P₁) infinity is numerically unbounded it exceeds any number, (P₂) division of a number  by any number greater than one (which is the identity) always results in a lesser quotient, (P₃) the divisor has an inverse relationship to the quotient, (P₄) zero signifies the absence of quantity, (C₁) the division of any number by infinity will always result in an absence of quantity[3].

 This itself is a metaphorical notion and cannot be demonstrated in actuality because quantity cannot be destroyed and equations that include the infinite as a definite number rather than a limit cannot be undone[4] (e.g., if a number x is divided by infinity and results in zero, multiplying zero by infinity will not result in x but in zero, for any number multiplied by zero results in zero). Thus, infinity cannot exist both in addition and division.

However, the absence of the infinite likewise results in contradiction and impossibility (such as finitude of number and indivisibility of magnitude), as Aristotle points out: “But if there is, unqualifiedly, no infinite, it is clear that many impossible things result” (Ackrill, 1987; Physics, 206^a, 9-10). Hence, because it is impossible for infinity to both exist and not exist, a further problem presents itself, that of the irreconcilable nature of being and not being (a contradiction derived from the preceding contradictions of both the existence of infinity and its absence). This calls for a mediating distinction that makes infinity possible in the ways it is necessary without resulting in contradiction. According to Aristotle, this distinction suggests that infinity must somehow exist and also not exist, but not in ways that are equivalent (i.e., in different regards): “...an arbitrator is needed and it is clear that in a sense the infinite is and in a sense it is not” (Ackrill; Physics, 206^a, 13-14). It is Aristotle’s distinction between potentiality and actuality as categories of that-which-is: “‘To be’, then, may mean ‘to be potentially’ or ‘to be actually’...” (Ackrill, 118; Physics 206^a, 13-15). The role of this arbitration is to resolve the difficulties presented by infinity and their implications. 

Potential exists as a possibility of the actual. That is to say, what in actuality is at any given moment has the capacity to become something else (change) in its parts or features and/or as a whole, and its subsequent form or state at any given moment is one of its potential existences prior to the realization of this form or state, or in relation to its previous form. This potential existence of something is the only sense in which infinity can exist according to Aristotle. Because increments are divisible ad infinitum, the potentiality of a shade of red that a rose might take on can, in principle, always be different from any shade of any red rose in existence. Yet once a rose exists in a moment, tangibly and in actuality, its shade is a given shade. It can be argued that degrees of freedom play into and thus circumscribe the potentiality of a given thing, such as when biology limits the potentiality of a whale so that change will never see it fly in any given moment in its life, when the cognitive capacity of an ant will never have it reason conceptually[5], or when pigment does not change when differences become too minute to affect it. This, if it is the case for all things, would eliminate the possibility for infinity to exist even as potentiality. Could it be that the numerical capacities of division and addition which lead us to conceptualize in terms of infinity are merely abstract and metaphysical, having no real role in change and potentiality whatever?

While it is plausible that all things are finite and have a finite and definite but unknown (and likely undeterminable due to the limitations of human capacities) amount of potentialities, it is equally beyond our capacities to exhaust most if not all conceivable relations, qualities, characteristics and so on, such that no potentialities exist outside what has been inferred as possible based on various examples of its actuality. Hence, Aristotle maintains that “...[infinity] is not that of which no part is outside, but that of which some part is always outside” (Ackrill, 119; Physics 206^b, 30-34). Although infinity is inherently conceptual, it has been robust to any and all examples of the actualities examined by human beings (even after the approximate 2,300 years since Aristotle first proposed these considerations), and remains an indispensable notion. Moreover, the span of such circumscription makes the argument against infinity as substantive move even further beyond reach (perhaps even all the way to futility) when considerations such as being and nonbeing and the ongoing passing of time are laid to account. 

Infinity must be conceivable, for the absence of infinity is inconceivable. Yet the presence of infinity as an actual characteristic or feature of that which exists is likewise inconceivable. The arbitration of the infinite provided by potentiality and actuality allows for both the sense in which the infinite is and the sense in which it is not for, as we have seen, both are necessary to conceive of what is and the continuity of changes and emergence in relation to what is. Thus, while infinity exists as a conceivable attribute of something/anything, in terms of addition and/or division, and is logically necessary for understanding change, it remains in the realm of potential and is absent in the actual trajectories of change as it becomes realized in time and space - the infinite becomes finite as the potential becomes actual. 

Footnotes

^[1] According to modern definitions, infinitude represents a state of being unbounded by number or extent, such as in distance or quantity (e.g., Hanks, 2003, 722).

^[2] For this reason, Aristotle declares that “...the infinite is either in addition or in division,” and that “Even Plato... made two infinites, because there seems to be an excess and a going-to-infinity both in extent and in reduction...” (Ackrill, 1987; Physics, 206^b, 27-29). All manifestations of unbounded behavior can be understood as either increasing/ongoing and therefore being added or decreasing/diverging and therefore being divided. 

^[3] In the case of addition, a simpler argument can be made: e.g., a number can be added to any sequence as a further unit without ever reaching a point at which the ability to add a number is exhausted: “...in the case of time and the race of men the things taken cease to be, yet so that the series does not give out” (Ackrill, 1987; Physics, 206^b, 1-2). 

^[4] There are myriad numerical problems with infinity that have been identified. However, to the extent of my knowledge, most if not all of these problems are based in the problem presented by the notion of infinity, which excludes itself from the set of all definite numbers and is tractable only in its conceptual relation to the number system. This is a relevant point to the current exegesis because what is can be extensively modeled and explicated through numerical computation (some even claim that all facets of reality including consciousness can eventually be quantified exhaustively and without remainder).

^[5] Probability theory may suggest that any possibility, no matter how inconceivable, is possible, though it may be highly unlikely that it will ever become actual. 

References

Ackrill, J.L. (Ed.). (1987). A new Aristotle reader. Princeton, New Jersey: Princeton University Press.

Hanks , P. (Ed.). (2003). Oxford english reference dictionary. Oxford, NY: Oxford University Press.