Infinity

Aristotle presents many arguments against the existence of an actually infinite body or an actually infinite number of things, all of which ultimately rest on his distinction between an actual and a potential infinite. It is not that infinity in magnitude or multitude is impossible, for Aristotle affirms the infinity of time and insists upon the infinite divisibility of matter. Rather, if an infinite body existed, its infinity would have to be actual, and its actuality would necessarily involve certain determinations, especially those of dimension and place, which are inconsistent with the indeterminacy of the infinite. Similarly, a multitude of coexisting things cannot be infinite, because their coexistence implies that they can be actually numbered, whereas their infinity implies that they are numberless.

The potential infinite, Aristotle writes, "has this mode of existence: one thing is always being taken after another, and each thing that is taken is always finite, but always different." When this takes place in the division of spatial magnitudes, "what is taken persists," but in the succession of times and of generations, "it takes place by the passing away of these in such a way that the source of supply never gives out." Time can be potentially infinite by way of addition because each part passes out of existence in succession. But space, whose parts must coexist, cannot be infinitely extended, for that would require an actually infinite quantity, which Aristotle holds to be impossible. The relation between infinity and continuity, and between the infinite and the indivisible, is treated more fully under the ideas of Quantity and Element.

The essence of the infinite, on Aristotle's account, is not perfection but privation. "It is plain that the infinite is a cause in the sense of matter," he writes. The infinite is what is indeterminate, lacking form and limit. This valuation stands in contrast to the view later developed by Plotinus and adopted by the Christian theologians, according to which infinity, when attributed to God, signifies not indeterminacy but absolute perfection.

Aristotle's distinction between actual and potential infinity provides the framework within which all subsequent discussion of the subject proceeds. Lucretius, asserting an infinite number of atoms in infinite space, and Aquinas, attributing positive infinity to God alone, both define their positions in relation to the Aristotelian analysis.

Lucretius, following the Epicurean physics, asserts that "the first-beginnings of things are infinite" in number, though "the number of shapes is finite." The universe, he argues, is bounded in no direction of its ways, for if it had an edge, nothing would prevent a spear thrown at that edge from passing through, which would prove that there is more beyond. Space extends without limit; atoms fill it in infinite multitude; and worlds like our own are scattered throughout, countless and unending.

The opposition between Lucretius and Aristotle with regard to the divisibility of matter, discussed more fully under the idea of Element, turns on their different conceptions of infinity and continuity. Where Aristotle makes the continuity of matter the condition of its infinite divisibility, Lucretius makes the atom's continuity, its solidity or lack of void, the cause of its indivisibility. The division of matter into smaller and smaller parts finds, on the atomist view, an end in the atomic particles; yet Lucretius also asserts an infinite number of atoms. To contain an infinite number of atoms, an infinite space is required, and this presents no greater difficulty for Lucretius than an infinite time. Aristotle, on the other hand, differentiates between space and time with respect to infinity: time can be potentially infinite because its parts do not coexist, but space, whose parts must coexist, cannot be infinitely extended.

The bearing of Lucretius's cosmology on theology is considerable. An infinite universe of atoms in infinite void leaves no center around which a cosmos is ordered and no boundary within which a creator's plan might be discerned. The relation between this view of the universe and the question of divine providence is treated under the ideas of God and Nature.

Lucretius's assertion of an actually infinite physical universe stands in direct opposition to Aristotle's denial. Newton, while accepting the notion of infinite absolute space, will give it a different significance, treating infinite space not as godless void but as the medium of divine omnipresence.

Plotinus introduces into the discussion of infinity a distinction which, in effect, reverses the Greek valuation of limit and the unlimited. For Aristotle, infinity is essentially privation: the mark of matter's indeterminacy, of what lacks form and measure. Plotinus accepts this as true of quantitative infinity, but he maintains that there is another infinity, above form rather than below it. The One, the first principle from which all things flow, is infinite not because it is formless matter but because it transcends all form. Its infinity is the infinity of power, the inexhaustible source whose generative capacity can never be measured.

For Plato and Aristotle, to be perfect was to be definite, bounded, finished. Plotinus's innovation consists in arguing that the highest reality is beyond definition precisely because any limit would be a deficiency. The One is "not this, not that"; it is infinite in the sense of being beyond every determination. Number, figure, and quantity belong to lower levels of reality. The relation between this conception and the Platonic division, in the Philebus, of goods into the finite and the infinite is relevant here: Plato had treated the infinite as the indeterminate, that which needs the imposition of measure. Plotinus retains this valuation for quantitative infinity but exempts the One from it.

The bearing of this reversal on the later theological tradition is considerable. Infinity, in Plotinus's treatment, becomes a positive perfection rather than a privation. The conception of God as infinite in being, power, and knowledge, which Augustine and Aquinas both affirm, rests in large part on this Neoplatonic transformation of what the Greeks had understood by the infinite.

Plotinus transmitted to Christian theology the conception of a positive infinity that belongs to the divine nature. Augustine would develop this conception in the context of divine knowledge, and Aquinas would give it systematic formulation by distinguishing the absolute infinity of God from the merely relative and negative infinity of creatures.

Augustine addresses a question which arises from the conjunction of divine omniscience and infinite number: if God knows all things, and numbers are infinite, does God know an infinite multitude? Aristotle would deny that any such multitude can be actual. But Augustine refuses to limit God's knowledge by the limits of human cognition. "Every infinity is, in a way we cannot express, made finite to God," he writes. What appears endless to us, whether the series of integers, the multitude of future events, or the parts of a continuum, is embraced whole in the single act of divine understanding.

The substance of Augustine's position rests on his doctrine of eternity, discussed more fully under the idea of Eternity. God does not traverse the infinite as a creature would, counting one number after another in succession. He knows all things in a timeless simultaneity. What is successive for finite minds is, for an eternal intellect, present at once. On this view, the Aristotelian distinction between actual and potential infinity applies to the knowledge of temporal creatures but not to the knowledge of God. The infinity of the divine omniscience extends to the possible as well as to the actual, and it embraces things which in the language of Leibniz are incompossible.

The bearing of this on the human condition is that the mind, made in God's image, has a restlessness which points beyond every finite good. Augustine reads this restlessness as evidence that the soul is ordered to the infinite. The appetite for money, for pleasure, or for power, discussed under the idea of Desire, may be an infinite craving which no finite quantity of these goods ever satisfies, but it may also be, as Augustine suggests, a misdirected expression of the natural desire for a truly infinite good.

Augustine's treatment of the relation between divine infinity and human finitude became foundational for the Christian theological tradition. Aquinas would give the distinction more systematic form, arguing from the identity of God's essence and existence to the conclusion that God alone is absolutely infinite, while every created thing has an infinity that is merely relative and negative.

Aquinas distinguishes the absolute or positive sense in which God alone is infinite from the sense of the word in which it can be said that "things other than God can be relatively infinite, but not absolutely infinite." This other meaning, according to Aquinas, is not only relative but negative, for it connotes "something imperfect." It signifies indeterminacy or lack of perfection in being. What Aquinas calls the relative or potential infinite he attributes to matter and to quantities: to bodies, to the magnitudes of space and time, and to number.

God's infinity, Aquinas argues, follows from the identity of His essence and existence. Since the divine being "is not a being received in anything, but He is His own subsistent being," it follows that God Himself is "infinite and perfect." In every creature, existence is received into a limiting essence, so that a stone exists as a stone, a man as a man. Existence is contracted, bounded, determined by what the thing is. But in God, no receiving essence limits the act of being. Both Aquinas and Spinoza, as noted in the discussion under the idea of God, make infinity the basis for proving that there can be only one God. "If many gods existed," Aquinas writes, "they would necessarily differ from each other. Something would therefore belong to one, which did not belong to another. And if this were a privation, one of them would not be absolutely perfect."

Against this absolute infinity, Aquinas sets the merely relative infinities of creation. The infinity of prime matter, which is conceived as the potentiality for taking on all forms, is comparable to the infinity of God in a contrast of extreme opposites: the absolute indeterminacy of pure potentiality on the one hand, the absolute perfection of pure actuality on the other. An actually infinite created world is impossible, Aquinas holds, because "it is against the nature of a created thing to be absolutely infinite." God's omnipotence itself does not extend to making a thing absolutely infinite, for that would entail a contradiction.

Aquinas provides the most systematic medieval treatment of the distinction between the infinity of being and the infinity of quantity. Spinoza's definition of God as "Being absolutely infinite, that is to say, substance consisting of infinite attributes," resembles Aquinas's conception in some respects but differs in making the world's extension itself an attribute of the divine substance, a position Aquinas would reject.

"In sizes or numbers," Pascal writes, "nature has set before man two marvelous infinities." As endless addition produces the infinitely large, so endless division produces the infinitesimal or the infinitely small. "If we can multiply a number up to 100,000 times, we can also take a hundred thousandth part of it by dividing it by the same number we multiply it with, and thus every term of increase will become a term of division." Infinite increase, Pascal insists, includes necessarily infinite division. The two infinities are not separate facts about quantity but two faces of a single structure.

Man stands suspended between these two infinities: the infinite universe that exceeds his comprehension and the infinitesimal divisibility of matter that recedes beneath his senses. In his essay On Geometrical Demonstration, Pascal makes this the foundation of a theory about the limits of definition and proof. "The first terms we wished to define would presuppose others for their explication, and similarly the first propositions we wished to prove would suppose others that preceded them." An infinite regression is avoided, as discussed under the idea of Principle, only by accepting certain terms as undefined and certain propositions as undemonstrated. The parallel with the problem of infinite regression in causes, treated under the idea of Cause, is evident.

The bearing of these two infinities on the human condition is, for Pascal, a matter of the gravest importance. Man is "a nothing in comparison with the infinite, an all in comparison with the nothing, a mean between nothing and all." The recognition of this disproportion is, on Pascal's view, evidence at once of human misery and of a kind of grandeur: only a being whose nature reaches toward the infinite could recognize its own finitude as a deficiency. The connection between this restlessness and the desire for God is discussed more fully under the idea of Desire.

Pascal's treatment of infinity differs from the metaphysical analyses of Aristotle and Aquinas in that it is directed primarily toward the human significance of the concept. The two infinities become, in his hands, an occasion for reflecting on the limits of human knowledge and the nature of man's relation to God.

Newton treats infinity in two quite different though related contexts: in the doctrine of absolute space and time, and in the method of fluxions. Absolute space, in Newton's conception, exists "in its own nature, without relation to anything external," and extends without limit in every direction. Absolute time "flows equably" from infinite past to infinite future. These are not features of human perception but of reality itself, the immovable frame within which all motion and change occur. Where Aristotle had denied that space could be actually infinite because its parts must coexist, Newton simply asserts it as a physical postulate.

For the infinitely small, Newton proposes an analysis in terms of what he calls "nascent and evanescent quantities," or quantities just beginning to be more than nothing or just at the point at which they vanish into nothing. "Because the hypothesis of indivisibles seems somewhat harsh," he reframes the infinitesimal as a limit. "If I should happen to mention quantities as least, or evanescent, or ultimate," Newton writes, the reader "is not to suppose that quantities of any determinate magnitude are meant, but such as are conceived to be always diminished without end." The method of fluxions, discussed under the idea of Mathematics, thus provides an infinitesimal calculus on the hypothesis of limits rather than of indivisibles. The distinction between approaching a limit and attaining it corresponds to the difference between infinity in the physical and the mathematical orders.

The theological bearing of Newton's infinite space has often been noted. Newton appears to treat absolute infinite space as the medium of divine omnipresence, so that God's infinity is, in a sense, made extensive in the infinity of space and the infinity of time. The relation between this conception and the metaphysical infinity attributed to God by Aquinas and Spinoza raises the question of whether quantitative infinity can be an attribute of the divine nature or belongs, as the medieval tradition held, to a different order of being.

Kant will challenge Newton's assumption that infinite absolute space and time are features of reality in itself, arguing instead that they are forms of human sensibility, conditions under which anything can appear to experience. On this view, Newton's infinities are not discoveries about the cosmos but preconditions of how any finite mind must perceive it.

Leibniz maintains, against the Aristotelian tradition, that actual infinity is not only possible but realized in the world. The world contains infinitely many substances or monads, each of which mirrors the whole universe from its own point of view. "Each portion of matter may be conceived as a garden full of plants, and as a pond full of fish. But each branch of the plant, each member of the animal, is still such a garden or such a pond." The actual infinite, far from being a contradiction, is for Leibniz the density of the real.

Leibniz's position rests on the conjunction of his mathematical and metaphysical work. The infinitesimal calculus, developed independently of Newton, treats the infinitely small as a legitimate object of mathematical reasoning, though Leibniz himself sometimes hedges on whether infinitesimals are actual quantities or useful fictions. His monadology extends the infinite into metaphysics: because God, in choosing the best of all possible worlds, maximizes the variety of essences, the world must contain infinitely many perspectives. Leibniz distinguishes the true infinite, which is God, from the merely unlimited extension of the world and from what he calls the syncategorematic infinite of mathematics, but he does not deny actual infinity to any of them. The question of whether the physical existence of infinite quantities is possible, which Aristotle had answered negatively and Spinoza had connected to the attributes of God, receives a different treatment in Leibniz.

The bearing of this on the principle of sufficient reason, discussed under the idea of Cause, is that if the world is actually infinite in its composition, the chain of sufficient reasons for any particular state of affairs must terminate outside the series, in a necessary being whose choice is the ground of contingent infinity. Only an infinite creation, on Leibniz's view, could adequately express an infinite creator.

Leibniz's affirmation of actual infinity in the created world represents a departure from both the Aristotelian denial and the Thomistic restriction of absolute infinity to God. Kant, examining the conflicting views of Aristotle, Leibniz, and Newton concerning the infinity of space and time, will argue that the question cannot be resolved by proof or argument, and that the antinomies which result from the attempt reveal the limits of theoretical reason.

The conflicting views concerning the infinity of space and time, advanced by Aristotle, Lucretius, Leibniz, and Newton, appear in Kant's statement of the first cosmological antinomy. His intention is not to resolve the issues but to show that they cannot be resolved by proof or argument. To this end, Kant constructs what seem to him to be equally strong or equally inconclusive arguments for and against the infinity of the world.

Suppose, Kant argues on the one hand, that "the world has no beginning in time." Then "up to every given moment in time, an eternity must have elapsed, and therewith passed away an infinite series of successive conditions." But since "the infinity of a series consists in the fact that it can never be completed by means of a successive synthesis," it follows that "an infinite series already elapsed is impossible, and that consequently a beginning of the world is a necessary condition of its existence." On the other hand, if we grant that the world has a beginning, then there must have been a time in which the world did not exist, a void time in which "no part of any such time contains a distinctive condition of being in preference to that of non-being." From this it follows that "the world itself cannot have a beginning, and is, therefore, in relation to past time, infinite." With regard to space, Kant proceeds similarly, constructing arguments of apparently equal force for the finitude and the infinity of the world's spatial extent.

The way in which these opposite arguments nullify each other reveals, in Kant's theory of human knowledge, that we are "not entitled to make any assertion at all respecting the whole object of experience." Space and time, on Kant's view, are not things in themselves but forms of human sensibility, conditions under which anything can appear to us. The question "is the world finite or infinite?" confuses a regulative idea, the notion of a series that can always be extended, with a constitutive claim about a completed totality.

Kant's treatment of the cosmological antinomies, which extend also to the infinite divisibility of matter and the existence of a necessary being, represents an attempt to show that reason cannot decide these questions theoretically. Hegel will accept Kant's diagnosis that the traditional conception of infinity leads to contradiction, but will reverse the conclusion: contradiction, for Hegel, is not a sign of reason's failure but the very movement by which thought advances beyond the finite.

Hegel accepts Kant's observation that the traditional conception of infinity leads to contradiction, but he reverses the conclusion. Contradiction, in Hegel's view, is not a sign of reason's failure but the very movement by which thought advances. The antinomies are not barriers at which reason must halt but transitions through which it passes to a higher comprehension. Hegel distinguishes what he calls the "bad infinite" from the "true infinite." The bad infinite is the endless series: one finite thing after another, without end, infinity conceived merely as the negation of the finite, always standing outside it and always presupposing the very limit it claims to overcome.

The true infinite, by contrast, has a different logical structure. If the infinite is simply the not-finite, then it is bounded by the finite and therefore itself finite. Real infinity, Hegel argues, must contain its other within itself. It is the finite returned to itself through its own self-negation, the process by which limit is posited, transcended, and preserved in a higher unity. The image Hegel employs is the circle, in contrast to the endless line: in the circle, every point is both terminus and beginning, and the finite is not abandoned but taken up. Spirit, history, the concept itself are, on this view, infinite in a sense that the quantitative infinity of mathematics and physics cannot capture. The relation between this conception and the broader question of whether the real is rational is discussed under the ideas of Philosophy and Being.

The bearing of Hegel's position on the possibility of metaphysics after Kant is considerable. Hegel argues that reason can know the infinite, but only by refusing to place it beyond the finite as an unreachable beyond. The infinite is the self-development of finite moments into a whole that comprehends them. This is neither the Aristotelian potential infinite nor the Leibnizian actual infinity of monads, but an infinity that is concrete, present, and self-conscious.

Hegel's distinction between the bad and the true infinite represents an attempt to move beyond the alternatives that had structured the discussion since Aristotle: the potential infinite of process and the actual infinite of completed totality. Whether this third conception resolves or merely redefines the traditional difficulties is a question on which subsequent philosophy has been divided.