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Chapter 40: INFINITY

INTRODUCTION

ONE of the persistent questions concerning infinity is whether we can know or comprehend it. Another is whether the infinite exists, and if so, to what kind of thing infinity belongs. It is not surprising, therefore, that the discussion of infinity often borders on the unintelligible.

The idea of infinity, like the idea of eternity, lacks the support of the imagination or of sense-experience. The fact that the infinite cannot be perceived or imagined seems sufficient to lead Hobbes and Berkeley to deny its reality. “Whatsoever we imagine is finite,” writes Hobbes. “Therefore there is no idea, or conception of anything we call infinite. . . . When we say anything is infinite, we signify only that we are not able to conceive the ends and bounds of the thing named, having no conception of the thing, but of our own inability.”

On similar grounds Berkeley rejects the possibility of infinite division. “If I cannot perceive innumerable parts in any finite extension,” he writes, “it is certain that they are not contained in it: but it is evident, that I cannot distinguish innumerable parts in any particular line, surface, or solid, which I either perceive by sense, or figure to myself in my mind; wherefore I conclude that they are not contained in it.”

But for most of the great writers on the subject, the impossibility of representing infinity and eternity to the imagination does not render them inconceivable or meaningless. Yet it does account for the difficulty of grasping their meaning, a difficulty further increased by the fact that, whatever their meaning, infinity and eternity are indefinable. To define the infinite would be to limit—even in thought—the unlimited.

The notion of infinity involves greater perplexities than that of eternity. The meaning of eternity is weighted with the mystery of God, the world, and time. All these affect the conception of infinity; but for the infinite there are also the mysteries of number and of space, of matter and motion. In the sphere of quantity, or of things subject to quantity, infinity is itself the source of mystery, or at least the root of difficulty in analysis. It is the central term in the discussion of the continuous and the indivisible, the nature of series and of limits.

AS INDICATED in the chapter on ETERNITY, that idea in each of its applications seems to have one or the other of two meanings—(1) the meaning in which it signifies infinite time, time without beginning or end, and (2) the meaning in which it signifies the timelessness or immutability of being. Both meanings are negative, so far as our understanding is concerned. Yet what is signified by the second is in itself something positive, at least in the opinion of those who think that to be exempt from change entails having every perfection or being lacking in nothing.

This split in meaning also occurs in the idea of infinity. As applied to being, the term infinite signifies something positive, even though our understanding of what is signified remains negative or, at best, analogical. An infinite being is one which lacks no attribute that can belong to a being. This is the positive condition of absolute perfection. The infinite here still means the unlimited, but that which is unlimited in being has no defect. To lack deficiencies is to be perfect.

It is in this sense that Spinoza defines God as “Being absolutely infinite, that is to say, substance consisting of infinite attributes, each one of which expresses eternal and infinite essence.” Like Spinoza, Aquinas maintains that “besides God nothing can be infinite.” But he 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. This sense of “infinite” corresponds to that meaning of “eternal,” according to which time consists of an endless series of moments, each having a predecessor, each a successor, no matter how far one counts them back into the past or ahead into the future.

But in the field of quantities other than time, the meanings of infinite and eternal part company. There is, of course, some parallelism between infinite space and infinite time, insofar as an infinite extension is one which does not begin at any point or end at any; but the consideration of space and number leads to an aspect of infinity which has no parallel in the consideration of eternity.

“In sizes or numbers,” Pascal writes, “nature has set before man two marvelous infinities. . . . For, from the fact that they can always be increased, it follows absolutely that they can always be decreased. . . . If we can multiply a number up to 100,000 times, say, 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 by changing the integer into a fraction. So that infinite increase includes necessarily infinite division.” As endless addition produces the infinitely large, so endless division produces the infinitesimal or the infinitely small.

A trillion trillion is a finite number, because the addition of a single unit creates a larger number. The fact that the addition of another unit produces a different number indicates that a trillion trillion has a determinate size, which is the same as saying that it is a finite number. An infinite number cannot be increased by addition, for it is constituted—in thought at least—as a number larger than the sum of any two finite numbers; which is another way of saying that it is approached by carrying on the process of addition endlessly. The size of an infinite number is therefore indeterminate.

What Galileo points out about two infinite quantities seems to hold for an infinite and a finite quantity. He asks us to consider the totality of all integers (which is infinite) and the totality of their squares (which is also infinite). On the one hand, there appear to be as many squares as there are integers; on the other hand, the totality of integers includes all the squares. Precisely because “the number of squares is not less than the totality of all numbers, nor the latter greater than the former,” Galileo insists that “the attributes ‘equal,’ ‘greater,’ and ‘less’ are not applicable to infinite, but only to finite quantities.” Nor does the sense in which one finite quantity can be greater or less than another—that is, by a determinate difference between them—apply in the comparison of a finite and an infinite quantity. The latter, being indeterminately large, is indeterminately larger than any finite quantity.

These remarks apply to the infinitely small as well. The infinitesimal is immeasurably small or indeterminately less than any finite fraction, no matter how small, because its own size is indeterminate. The finite fraction, itself a product of division, can be divided again, but if an infinitesimal quantity were capable of further division, it would permit a smaller, and since that smaller quantity would be a determinate fraction of itself, the infinitesimal would have to be determinate in size. Since that is not so, the infinitesimal must be conceived as the indivisible or as the limit approached by carrying on division endlessly.

“Because the hypothesis of indivisibles seems somewhat harsh,” 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. “As there is a limit which the velocity at the end of a motion may attain, but not exceed . . . there is a like limit in all quantities and proportions that begin or cease to be.” Newton warns his reader, therefore, that if he “should happen to mention quantities as least, or evanescent, or ultimate,” 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.”

Later, speaking of quantities which are “variable and indetermined, and increasing or decreasing, as it were, by a continual motion or flux,” he adds: “Take care not to look upon finite quantities as such.” The method of fluxions provides an infinitesimal calculus on the hypothesis of limits rather than of indivisibles.

THROUGH ALL THESE conceptions of infinity—metaphysical, mathematical, and physical—run the paired notions of the unlimited and of limits approached but not attained. The finite is neither unlimited nor does it insensibly approach a limit. There are also the opposite notions of the perfect and the indeterminate. The finite is neither, for it is determinate without being a totality or complete.

Though they have a common thread of meaning, and though each raises similar difficulties for the understanding, the conception of infinity in being or power, and the conception of infinite (or infinitesimal) quantity require separate consideration. The same questions may be asked of each, questions about the existence of the infinite and about our knowledge of it, but the same answers will not be given in each case. There are those who deny the existence of an actually infinite body or an actually infinite number, yet affirm the infinite existence of God. There are those who declare the infinity of matter to be intrinsically unintelligible, but maintain that God, Who is infinite, is intrinsically the most intelligible object. They add, of course, that the infinite being of God cannot be comprehended by our finite intellects.

On each of these points, an opposite view has been taken, but the dispute concerning the infinity of God involves issues other than those which occur in the controversy over the infinite divisibility of matter or the infinity of space and time. It seems advisable, therefore, to deal separately with the problems of infinity as they arise with respect to different objects or occur in different subject matters.

THE CONCEPTION of God, in the words of Anselm, as a being “than which a greater cannot be conceived”—or, in the words of Kant, as an ens realissimum, a most real being—expresses the plenitude of the divine nature and existence. The mediaeval thesis, defended by Descartes, that God’s essence and existence are identical, implies that neither is contracted or determined by the other. The still earlier notion of Aristotle, repeated by Aquinas, that God is pure actuality, carries with it the attributes of completeness or perfection, which are the positive aspects of immutability or incapacity for change. Spinoza’s definition of substance as that which exists, not only in itself, but through itself and by its very nature, entails the autonomy or utter independence of the divine being.

These are so many different ways of stating that God is an infinite being. Both Aquinas and Spinoza make infinity the basis for proving that there can be only one God. When Spinoza argues that “a plurality of substances possessing the same nature is absurd,” he has in mind the identification of infinite substance with 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; but if it were a perfection, one of them would be without it. So it is impossible for many gods to exist”—that is, of course, if infinity is a property of the divine nature. Aquinas makes this condition clear when he goes on to say that “the ancient philosophers, constrained as it were by the truth, when they asserted an infinite principle, asserted likewise that there was only one such principle.”

But while it is impossible for there to be two infinities of being, it is not impossible for there to be two, or more, infinite quantities. One explanation of this difference seems to be the actuality or existence of an infinite being, in contrast to the conceptual character of the infinite objects of mathematics, which are sometimes called “potential infinites” because they are conceived as in an endless process of becoming, or as approaching a limit that is never reached.

When the physical existence of infinite quantities is asserted, as, for example, a universe of infinite extent or an infinite number of atoms, the uniqueness of these actual totalities seems to follow. Two infinite worlds cannot co-exist, though the one world can be infinite in several distinct respects—in space or duration, or in the number of its constituents—even as the infinity of God, according to Spinoza, involves “infinite attributes, each one of which expresses eternal and infinite essence.”

Spinoza’s argument against two actual infinities seems to find confirmation in the position taken by Aquinas that God’s omnipotence does not include the power to create an infinite world. God’s infinity, as we have already noted, follows from the identity of God’s essence and existence. Since a created being has existence added to its essence, Aquinas asserts that “it is against the nature of a created thing to be absolutely infinite. Therefore,” he continues, “as God, although He has infinite power, cannot make a thing to be not made (for this would imply that two contradictories are true at the same time), so likewise He cannot make anything to be absolutely infinite.”

On this view, an infinite world cannot co-exist with an infinite God, if, in their separate existence, one is dependent on the other, as creature upon creator. The infinity of the world or of nature, in Spinoza’s conception, is not separate from the infinity of God, but consists in the infinity of two of God’s attributes—extension and thought.

In our time there has arisen the conception of a finite God—a God who, while the most perfect being, yet is not without capacity for growth or change, a God who is eternal without being immutable. This conception, which in the light of traditional theology appears to be as self-contradictory as a round square, has arisen in response to the difficulties certain critics have found in the traditional doctrine of an infinite being. They point to the difficulty of understanding how finite beings can exist separate from, yet in addition to, an infinite being; they also cite difficulties in the notions of infinite knowledge, infinite power, and infinite goodness.

The infinity of the divine omniscience extends to the possible as well as to the actual. But the possible includes things which are incompatible with one another, things which, in the language of Leibnitz, are not compossible. The incompossible would thus seem to be embraced in the infinite scope of divine thought or knowledge. In the view of one theologian, Nicolas of Cusa, the mystery of God’s infinity is best expressed by affirming that in God all contradictions are somehow reconciled.

The infinity of God’s power, or the divine omnipotence, also raises questions about the possible and the impossible. Is nothing impossible to God or must it be said that there are certain things which not even God can do, such as reverse the order of time or create a world which shall be as infinite and perfect as himself? In the assertion that God cannot do the impossible, Aquinas sees no limitation on God’s power. The impossible, he writes, does not “come under the divine omnipotence, not because of any defect in the power of God, but because it has not the nature of a feasible or possible thing.” For this reason, he claims, “it is better to say that such things cannot be done, than that God cannot do them.” The inability to do the undoable constitutes no violation of infinite power, even as the lack of nothing does not deprive infinite being of anything.

The infinite goodness of God is sometimes set against the fact of evil, or the existence of imperfections, in the created world. This aspect of the problem of evil, like that which concerns man’s freedom to obey or disobey the divine will, cannot be separated from the fundamental mystery of God’s infinity—in power and knowledge as well as in goodness. The problem is considered in the chapter on GOOD AND EVIL. The point there mentioned, that evil is essentially non-being or deprivation of being, leads to one solution of the problem. It accepts the finitude, and consequently the imperfection, of creatures as a necessary consequence of God’s infinity. The best of all possible worlds cannot be infinitely good.

TO MAN ALONE, among all admittedly finite things, has infinity been attributed and even made a distinctive mark of his nature. Does this introduce a new meaning of infinity, neither quantitative nor divine?

It has seldom if ever been questioned that man is finite in being and power. The limits of human capacity for knowledge or achievement are a perennial theme in man’s study of man. Yet it is precisely with regard to capacity that certain writers have intimated man’s infinity.

Pascal, for example, finds the apparent contradictions in human nature intelligible only when man is understood as yearning for or impelled toward the infinite. “We burn with desire,” he says, “to find solid ground and an ultimate sure foundation whereon to build a tower reaching to the Infinite. But our whole groundwork cracks and the earth opens to abysses.” In this fact lies both the grandeur and the misery of man. He aspires to the infinite, yet he is a finite being dissatisfied with his own finitude and frustrated by it.

It is sometimes said that the touch of infinity in man—with the suggestion that it is a touch of madness—consists in his wanting to be God. Those who regard such desire as abnormal or perverse interpret it as a misdirection of man’s natural desire to know God face to face and to be filled with the love of God in the divine presence. But, according to the theory of natural desire, the tendency of each nature is somehow proportionate to its capacity. If man’s restless search for knowledge and happiness can be quieted only by the possession of the infinite truth and goodness which is God, then man’s intellect and will must somehow be as infinite in nature as they are in tendency. Yet that is not an unqualified infinity, for the same theologians who teach that man naturally seeks God also hold that man’s finite intellect cannot comprehend the infinite being of God as God knows Himself. Nor do they think that man’s capacity for knowing and loving God can be fulfilled except in the beatific vision, which is a supernatural gift rather than a natural achievement.

These and related matters are discussed in the chapters on DESIRE and KNOWLEDGE. The great books speak of other objects than God as objects of man’s infinite desire. The appetite for money, for pleasure, or for power seems to be an infinite craving which no finite quantity of these goods ever satisfies. Two comments are made upon this fact, which is so amply evidenced in the human record. One is that man’s infinite lust for worldly goods expresses—even as it conceals—his natural desire for a truly infinite good. The other is that these worldly goods are seductive objects precisely because they are infinite.

Here the word “infinite” is used, not in the sense which signifies perfection, but in the quantitative sense which has the meaning of indetermination. Plato’s division, in the Philebus, of goods into the finite and the infinite separates measured and definite goods from those which need some limitation in quantity. Socrates exemplifies the distinction by reference to the fact that “into the hotter and the colder there enters a more and a less” and since “there is never any end of them . . . they must also be infinite.” In contrast, “when definite quantity is once admitted, there can be no longer a ‘hotter’ or a ‘colder.’” Such things, he says, “which do not admit of more or less” belong “in the class of the limited or finite.”

Following the line of this example, Socrates later distinguishes between infinite and finite pleasures, or pleasures without limit and those which have some intrinsic measure. “Pleasures which are in excess,” he says, “have no measure, but those which are not in excess have measure; the great, the excessive . . . we shall be right in referring to the class of the infinite, and of the more and less,” and “the others we shall refer to the class which has measure.” The fact that the goodness of wealth or of certain pleasures is indeterminate or indefinite makes it necessary to determine or measure the amount of wealth it is good to possess, or the quantity of such pleasure it is good to enjoy.

As in the case of desire, so the human intellect is also said to be infinite in the sense of reaching to an indefinite quantity. On the theory which he holds that the intellect knows by means of universal concepts, Aquinas attributes to the human mind “an infinite power; for it apprehends the universal, which can extend itself to an infinitude of singular things.” Each universal signifies what is common to an indefinitely large class of particular instances.

There is still another sense in which the intellect is said to be infinite, namely, by reason of its having the potentiality to apprehend all knowable things. But this is a relative infinity, as is the corresponding infinity of prime matter, which is conceived as the potentiality for taking on all forms. In both cases, the infinite is qualified by a restriction—on the kind of things knowable to the intellect and the type of forms receivable in matter. The infinity of prime matter—matter totally devoid of form—is also 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.

THE INFINITY OF matter involves different considerations when the problem concerns, not prime matter, but material things—bodies. The question is twofold. Can there be a body of infinite magnitude? Is there an infinite number of bodies? To both questions Aristotle gives the negative answer, while Spinoza seems to answer the first, and Lucretius the second, affirmatively.

Spinoza’s affirmation may be qualified, of course, by his conception of infinite body as an attribute of God. But there is no qualification on Lucretius’ assertion that “the first-beginnings of things are infinite,” unless it is his statement that “the first-beginnings of things have different shapes, but the number of shapes is finite.” It is only the number of atoms which is infinite, not their variety.

Aristotle presents many arguments against the existence of an infinite body or an 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 he affirms the infinity of time and he insists upon the infinite divisibility of matter—but rather that if an infinite body existed its infinity would have to be actual. Its actuality would necessarily involve certain determinations, especially those of dimension and place, which would be inconsistent with the indeterminacy of the infinite. Similarly, a multitude of co-existing things—unlike the moments of time which do not co-exist—cannot be infinite, because their co-existence implies that they can be actually numbered, whereas their infinity implies that they are numberless.

The potential infinite, Aristotle writes, “exhibits itself in different ways—in time, in the generations of man, and in the division of magnitudes. For generally,” he says, “the infinite 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, while in the succession of times and of men, it takes place by the passing away of these in such a way that the source of supply never gives out.”

The opposition between Lucretius and Aristotle with regard to the divisibility of matter is discussed in the chapter on ELEMENT. The notions of infinity and continuity are differently employed on the two sides of the argument. 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. Where Aristotle asserts that at any moment there can be only a finite number of particles in the world because the partition of matter cannot be infinitely carried out short of infinite time, Lucretius, on the contrary, thinks that the division of matter into smaller and smaller parts finds an end in the atomic particles; and yet he also asserts an infinite number of atoms.

To contain an infinite number of atoms, an infinite space is required, according to Lucretius. This presents no greater difficulty for him than an infinite time. Aristotle, on the other hand, differentiates between space and time with respect to infinity. Time can be potentially infinite by way of addition because “each part that is taken passes in succession out of existence.” But though space may be infinitely divisible, it cannot be infinitely extended, for all its parts, unlike those of time, must co-exist. It would therefore have to be an actually, rather than a potentially, infinite quantity, and this Aristotle thinks is impossible.

These and other conflicting views concerning the infinity of space and time 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 do this, Kant sets up what seems to him to be equally strong—or equally inconclusive—arguments for and against the infinity of space and time.

Suppose it be granted, Kant argues on the one hand, that “the world has no beginning in time.” Then it would follow that “up to every given moment in time, an eternity must have elapsed, and therewith passed away an infinite series of successive conditions or states of things in the world.” But since “the infinity of a series consists in the fact that it can never be completed by means of a successive synthesis,” it also “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, Kant argues with what he thinks is equal force, “let it be granted that [the world] has a beginning. A beginning,” he explains, “is an existence which is preceded by a time in which the thing does not exist.” Then, Kant continues, “on the above supposition, it follows that there must have been a time in which the world did not exist, that is, a void time. But in a void time, the origination of a thing is impossible; because no part of any such time contains a distinctive condition of being in preference to that of non-being. . . . Consequently, many series of things may have a beginning in the world, but the world itself cannot have a beginning, and is, therefore, in relation to past time, infinite.”

With regard to the infinity or finitude of space, Kant proceeds similarly. If we suppose space to be infinite, then “the world must be an infinite given total of co-existent things.” But in order to “cogitate the world, which fills all spaces, as a whole, the successive synthesis of the parts of an infinite world must be looked upon as completed; that is to say, an infinite time must be regarded as having elapsed in the enumeration of all co-existing things.” This, Kant argues, “is impossible,” and therefore “an infinite aggregate of actual things cannot be considered as a given whole.” Hence it follows that “the world is, as regards extension in space, not infinite, but enclosed in limits.”

If, however, we suppose “that the world is finite and limited in space, it follows,” according to Kant, “that it must exist in a void space, which is not limited. We should, therefore, meet not only with a relation of things in space, but also a relation of things to space.” But the “relation of the world to a void space is merely a relation to no object” and “such a relation, and consequently the limitation of the world by void space, is nothing.” It follows, therefore, Kant concludes, that “the world, as regards space, is not limited; that is, it is infinite in regard to extension.”

The way in which these opposite arguments nullify each other reveals more than our inability to prove or disprove the infinity of space and time. It shows, in Kant’s theory of human knowledge, that we are “not entitled to make any assertion at all respecting the whole object of experience—the world of sense.”

ONE OTHER PROBLEM of infinity in the sphere of physics receives its initial formulation in one of the great books—in the part of the Dialogues Concerning the Two New Sciences where Galileo discusses the uniform acceleration of a freely falling body. The body which is said to accumulate equal increments of velocity in equal intervals of time is also said to start “from infinite slowness, i.e., from rest.” One of the persons in the dialogue challenges this, saying that “as the instant of starting is more and more nearly approached, the body moves so slowly that, if it kept on moving at this rate, it would not traverse a mile in an hour, or in a day, or in a year, or in a thousand years; indeed, it would not traverse a span in an even greater time; a phenomenon which baffles the imagination, while our senses show us that a heavy falling body suddenly acquires great speed.”

What our senses seem to show us is corrected by an experiment which refines the observation. But this still leaves a purely analytical question. Against the statement that the “velocity can be increased or diminished without limit,” Simplicio points out in the dialogue that “if the number of degrees of greater and greater slowness is limitless, they will never be all exhausted,” and therefore the body will never come to rest when it is slowing down or be able to start to move when it is at rest.

“This would happen,” Salviati answers, “if the moving body were to maintain its speed for any length of time at each degree of velocity, but it merely passes each point without delaying more than an instant, and since each time interval, however small, may be divided into an infinite number of instants, these will always be sufficient to correspond to the infinite degrees of diminished velocity.”

The problem of the infinitesimal velocity provides another illustration of the difference between infinity in the physical and the mathematical orders. Unlike parallel lines in Euclidean geometry, which are lines that remain equidistant from one another when both are prolonged to infinity, an asymptote is a straight line which a curved line continuously approaches but never meets, even when both are infinitely extended. The distance between the curve and its asymptote diminishes to smaller and smaller intervals, but no matter how small they become, the two lines never coincide. The diminishing intervals between the curve and its asymptote are like the diminishing degrees of velocity in a body starting from or coming to rest. But we know that the body does begin or cease to move, and so there is the mysterious jumping of the gap between rest and motion in the physical order, whereas in the mathematical order the limiting point can be forever approached and never reached.

THERE IS ONE other context in which infinity is discussed in the great books.

The logicians treat certain terms and judgments as infinite. Aristotle, for example, regards the negative term—such as not-man or not-white—as indefinite. The indefiniteness of its signification may be seen when such terms are used as subjects of discourse. What is being talked about? The answer must be given, in part at least, in positive terms: not-man represents the whole universe leaving man out, or the totality of everything except man. Thus, in its positive signification, the negative term has a kind of infinity—the infinite totality of subjects diminished by one, the one that is negated.

In his classification of judgments, Kant makes a threefold division of judgments according to quality: the affirmative, the negative, and the infinite. The infinite judgment involves a negative in its construction, but when that negative is given an affirmative interpretation, the infinite significance of the proposition becomes apparent. An example will make this clear.

The proposition this animal is-not white is negative; it simply denies a certain quality of a certain thing. But the proposition this animal is not-white is infinite, for it affirms the negated term, and so places the subject in the infinite class or totality which includes everything except white things. (The position of the hyphen serves to indicate whether the statement shall be construed negatively or affirmatively and infinitely.)

The problems of definition and demonstration are differently solved by logicians according to the way in which they propose to avoid infinite regressions in analysis or reasoning. There would be no end to the process of defining if every term had to be defined before it could be used in the definition of another term. There would be no beginning to the process of proof if, before a proposition could be used as a premise to demonstrate some conclusion, it had itself to be demonstrated as a conclusion from prior premises.

In his essay On Geometrical Demonstration, Pascal refers to the proposal of a plan for defining and proving everything. “Certainly this method would be beautiful,” he says, “but it is absolutely impossible; for it is evident that the first terms we wished to define would presuppose others for their explication, and that similarly the first propositions we wished to prove would suppose others that preceded them, and it is thus clear we should never arrive at the first propositions.”

The chapter on DEFINITION considers the character and choice of the indefinable terms by which an infinite regression is avoided in the elucidation of meanings. The chapters on INDUCTION and PRINCIPLE consider the various sorts of primary propositions—axioms, postulates, assumptions—by which a similar regression is avoided in the process of proof. The chapter on CAUSE deals with the problem of an infinite regression in causes and effects. Here it is appropriate to consider the difference between an infinite series of reasons and an infinite series of causes.

To the extent that both are truly series—the succession of one thing after another—neither seems to be impossible, given infinite time. Those who deny the possibility of an infinite number of causes distinguish between essential and accidental causes, that is, between causes which must co-exist with their effects and causes which can precede their effects, and cease to be before their effects occur. If there were an infinite time, there could be an infinite series of accidental causes. But it may be questioned whether, even granted an infinite time, the relation between the premises and conclusion of reasoning permits an infinite regression. If the truth of a conclusion cannot be known until the truth of its premises is known, then the pursuit of truth may be vitiated by a search ad infinitum.

OUTLINE OF TOPICS

  1. The general theory of infinity 1a. The definite and indefinite: the measured and the indeterminate 1b. The infinite in being and quantity: the actual and potential infinite; the formal and the material infinite

  2. Infinity in the logical order 2a. The infinity of negative and indefinite terms 2b. The distinction between negative and infinite judgments 2c. Infinite regression in analysis and reasoning

  3. The infinite in quantity 3a. Number: the infinite of division and addition 3b. The infinite divisibility of continuous quantities: the infinitesimal; the method of exhaustion and the theory of limits 3c. The infinity of asymptotes and parallels 3d. The infinite extent of space 3e. The infinite duration of time and motion

  4. The infinity of matter 4a. The infinite quantity or extent of matter: the problem of an actually infinite body 4b. The infinite divisibility of matter: the issue concerning atoms 4c. The infinite potentiality of matter: the conception of prime or formless matter

  5. Infinity in the world 5a. The infinite number of things and the infinite number of kinds 5b. The number of causes

  6. The finite and the infinite in the nature of man 6a. The infinity of desire and will: the limits of human capacity 6b. The infinity of the intellect: man’s knowledge of the infinite

  7. The infinity of God 7a. The infinite being or essence of God 7b. The infinite power of God 7c. God’s infinite goodness and love 7d. God’s infinite knowledge

REFERENCES

To find the passages cited, use the numbers in heavy type, which are the volume and page numbers of the passages referred to. For example, in 4 HOMER: Iliad, BK II [265-283] 12d, the number 4 is the number of the volume in the set; the number 12d indicates that the passage is in section d of page 12.

Page Sections: When the text is printed in one column, the letters a and b refer to the upper and lower halves of the page. For example, in 53 JAMES: Psychology, 116a-119b, the passage begins in the upper half of page 116 and ends in the lower half of page 119. When the text is printed in two columns, the letters a and b refer to the upper and lower halves of the left-hand side of the page, the letters c and d to the upper and lower halves of the right-hand side of the page. For example, in 7 PLATO: Symposium, 163b-164c, the passage begins in the lower half of the left-hand side of page 163 and ends in the upper half of the right-hand side of page 164.

Author’s Divisions: One or more of the main divisions of a work (such as PART, BK, CH, SECT) are sometimes included in the reference; line numbers, in brackets, are given in certain cases; e.g., Iliad, BK I [265-283] 12d.

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1. The general theory of infinity

8 ARISTOTLE: Physics, BK III, CH 4-8 280c-286d 12 LUCRETIUS: Nature of Things, BK I [951-1113] 12d-14d 17 PLOTINUS: Sixth Ennead, TR VI, CH 2 311b-c; CH 17-18, 319d-320d 19 AQUINAS: Summa Theologica, PART I, Q 7 31a-34c 20 AQUINAS: Summa Theologica, PART III SUPPL, Q 92, A 1, REP 6, 12 1025c-1032b 28 GALILEO: Two New Sciences, FIRST DAY, 139c-153a passim, esp 144b-145a 30 BACON: Novum Organum, BK I, APH 48 110d-111a 31 DESCARTES: Meditations, III, 86b / Objections and Replies, 112a-c; 123c-d 31 SPINOZA: Ethics, PART I, DEF 2 355a; DEF 6 355b; DEF 8 355c; PROP 7-8 356c-357d 33 PASCAL: Pensées, 72 181a-184b; 231-233, 213b-214a / Geometrical Demonstration, 434b-439b 34 NEWTON: Principles, BK I, LEMMA I-II and SCHOL 25a-32a 35 LOCKE: Human Understanding, BK II, CH XVI, SECT 8-CH XVII, SECT 22 167c-174a 35 BERKELEY: Human Knowledge, SECT 123-132 437c-439c 35 HUME: Human Understanding, SECT XII, DIV 124-125 506a-507a 42 KANT: Pure Reason, 24d; 26d; 130b-149d esp 130b-133c, 136a-137a [Thesis]; 156b-157d 46 HEGEL: Philosophy of Right, ADDITIONS, 17 119a 51 TOLSTOY: War and Peace, EPILOGUE II, 693c-694d

1a. The definite and indefinite: the measured and the indeterminate

7 PLATO: Parmenides, 505c-506d; 510c-511a / Philebus, 609a-639a,c esp 615c-617d 8 ARISTOTLE: Prior Analytics, BK I, CH 13 [32b4-23] 48b-d / Metaphysics, BK I, CH 5 [986a12-b2] 504b-c; CH 6 [987b19-988a16] 505d-506b; CH 8 [989a30-b21] 507c-d; BK V, CH 30 547a-d; BK VII, CH 10 [1036a2-9] 559b-c; BK IX, CH 7 [1049a19-b1] 574d-575a; BK XIII, CH 10 [1087a10-25] 619c 9 ARISTOTLE: Ethics, BK II, CH 6 [1106b28-35] 352b-c 11 NICOMACHUS: Arithmetic, BK I, 811d-812a; 826d-827a 16 KEPLER: Harmonies of the World, 1078a-b 17 PLOTINUS: First Ennead, TR VII, CH 3 28a-c / Second Ennead, TR IV, CH 13-16 55b-57c / Sixth Ennead, TR VI, CH 18, 320c-d 18 AUGUSTINE: Confessions, BK V, par 20 32d-33a / City of God, BK XII, CH 18 354b-d 19 AQUINAS: Summa Theologica, PART I, Q 48, A 1, REP 4 259b-260c; PART I-II, Q 1, A 4 612a-613a; Q 7, A 2, REP 2 652d-653c 31 DESCARTES: Objections and Replies, 112a-b 42 KANT: Pure Reason, 158a-159d; 196c-197a 46 HEGEL: Philosophy of Right, ADDITIONS, 6 117a-b

1b. The infinite in being and quantity: the actual and potential infinite; the formal and the material infinite

7 PLATO: Parmenides, 495c-497c / Sophist, 571d-573b 8 ARISTOTLE: Physics, BK I, CH 2 [185a28-b4] 260a-b; [185b17-19] 260c; BK III, CH 4-8 280c-286d / Generation and Corruption, BK I, CH 3 [318a13-24] 414c-d / Metaphysics, BK I, CH 5 [987a13-19] 505b; BK IX, CH 6 [1048b9-17] 574a; BK XI, CH 10 [1066a35-b22] 594d-595b; BK XII, CH 7 [1073a3-13] 603a-b 11 NICOMACHUS: Arithmetic, BK I, 812a 17 PLOTINUS: Sixth Ennead, TR VI, CH 17, 319d-320a 18 AUGUSTINE: Confessions, BK VII, par 20-21 49d-50a 19 AQUINAS: Summa Theologica, PART I, Q 7 31a-34c; Q 12, A 1, REP 2 50c-51c; Q 14, A 1, ANS 75d-76c; Q 50, A 2, REP 4 270a-272a; Q 54, A 2, ANS 285d-286c; PART I-II, Q 1, A 4, REP 2 612a-613a; Q 2, A 6, ANS 619d-620d 20 AQUINAS: Summa Theologica, PART III, Q 7, A 12, REP 1 754c-755c; Q 10, A 3, REP 1-2 769d-771b; PART III SUPPL, Q 92, A 1, REP 12 1025c-1032b 28 GALILEO: Two New Sciences, FIRST DAY, 145b-146c; 150d-151c 31 DESCARTES: Meditations, III, 86a-88d / Objections and Replies, 112b; DEF VII 130d 31 SPINOZA: Ethics, PART I, DEF 3, 6 355b; PROP 8, DEMONST and SCHOL 1 356d; PROP 9 357d; PROP 10, SCHOL 358a-b; PROP 13-14 359c-360a; PROP 15, SCHOL-PROP 16 360b-362a; PROP 21-23 364a-365a; PROP 28, DEMONST 365d-366a 33 PASCAL: Pensées, 121 195a / Geometrical Demonstration, 434b-439b 34 NEWTON: Principles, BK III, GENERAL SCHOL, 370a-371a 35 LOCKE: Human Understanding, BK II, CH XII, SECT 4-6 149b-d; CH XIV, SECT 26-27 160c-161a; SECT 30 161c-d; CH XV, SECT 2-4 162c-163b; CH XVI, SECT 8 167c; CH XVII 167d-174a passim, esp SECT 7 169b-c; CH XXIX, SECT 16 237b-238a 35 BERKELEY: Human Knowledge, SECT 123-132 437c-439c 35 HUME: Human Understanding, SECT XII, DIV 124, 506b 42 KANT: Pure Reason, 124d-125b; 130b-133c; 135a-137a,c; 160b-163a / Judgement, 498b-501b 46 HEGEL: Philosophy of Right, INTRO, par 22 17c-d; PART I, par 104—PART II, par 105 39b-40a; ADDITIONS, 17 119a / Philosophy of History, INTRO, 156d-160b; 165a-b 53 JAMES: Psychology, 668a-669a

2. Infinity in the logical order

2a. The infinity of negative and indefinite terms

7 PLATO: Sophist, 571d-573c / Philebus, 615c-616c 8 ARISTOTLE: Interpretation, CH 2 [16a30-33] 25c; CH 3 [16b11-16] 25d; CH 10 [19b5-11] 29d / Prior Analytics, BK I, CH 46 70b-71d 42 KANT: Pure Reason, 158a-159d

2b. The distinction between negative and infinite judgments

8 ARISTOTLE: Categories, CH 10 [12b6-15] 17d-18a / Interpretation, CH 5 [17a8-9] 26b; CH 6 26c-d; CH 10 29d-31c / Prior Analytics, BK I, CH 3 [25a19-26] 40c; CH 46 70b-71d 35 LOCKE: Human Understanding, BK III, CH VIII, SECT 1 283a-b 42 KANT: Pure Reason, 40a-c; 210c-d 46 HEGEL: Philosophy of Right, PART I, par 53 25c-d

2c. Infinite regression in analysis and reasoning

7 PLATO: Parmenides, 489a-d / Theaetetus, 542a-544a esp 543d-544a 8 ARISTOTLE: Posterior Analytics, BK I, CH 3 99b-100a; CH 19-23 111c-116a / Physics, BK I, CH 4 [187a7-14] 262d; CH 6 [189a11-19] 264c / Metaphysics, BK IV, CH 3 [1005b15-34] 524d-525a; CH 4 [1006a5-12] 525a-b; CH 6 [1011a3-14] 530d; BK XI, CH 6 [1063a7-12] 591d / On the Soul, BK I, CH 3 [407a22-30] 636d-637a 9 ARISTOTLE: Ethics, BK I, CH 2 [1094a18-22] 339b; BK III, CH 3 [1112b33-1113a3] 358d-359a 12 EPICTETUS: Discourses, BK I, CH 17, 122d-123a; BK II, CH 20, 164c-d 19 AQUINAS: Summa Theologica, PART I, Q 36, A 3, REP 4 194c-195d; PART I-II, Q 1, A 4, REP 2 612a-613a 20 AQUINAS: Summa Theologica, PART II-II, Q 94, A 2, ANS 221d-223a 25 MONTAIGNE: Essays, 292c-d 30 BACON: Novum Organum, BK I, APH 48 110d-111a 31 DESCARTES: Objections and Replies, 111a-d; 213c-d; 224b,d 33 PASCAL: Geometrical Demonstration, 431b-434a / Arithmetical Triangle, 451b-452a; 458b-459b; 464a-466a 35 LOCKE: Human Understanding, BK II, CH XXI, SECT 23 183b-d; SECT 25 184a-b; BK III, CH IV, SECT 5 260c 42 KANT: Pure Reason, 111d-112d; 115d-119a; 135a-173a esp 158a-159d 53 JAMES: Psychology, 321b [fn 1]; 525a-526b

3. The infinite in quantity

3a. Number: the infinite of division and addition

7 PLATO: Parmenides, 495d-497b 8 ARISTOTLE: Physics, BK III, CH 4 [203b22-24] 281c; [204a7-8] 282a; CH 5 [204b6-9] 282c; CH 6 [206a15] 284b; [206b3-207a14] 284d-285c; CH 7 [207b32-a14] 285d-286a / Metaphysics, BK XI, CH 10 [1066a23-26] 595b 11 EUCLID: Elements, BK IX, PROP 20 183b-184a 11 ARCHIMEDES: Sand-Reckoner, 520a-526b passim 11 NICOMACHUS: Arithmetic, BK I, 812a; BK II, 829b 17 PLOTINUS: Sixth Ennead, TR VI, CH 2 311b-c; CH 17, 319d-320a 18 AUGUSTINE: City of God, BK XII, CH 18 354b-d 19 AQUINAS: Summa Theologica, PART I, Q 7, A 3, REP 3 32c-33c; A 4 33d-34c; PART I-II, Q 1, A 4, REP 2 612a-613a 20 AQUINAS: Summa Theologica, PART III, Q 7, A 12, REP 1 754c-755c; Q 10, A 3, REP 2 769d-771b 28 GALILEO: Two New Sciences, FIRST DAY, 144b-145a 31 DESCARTES: Objections and Replies, 112b 33 PASCAL: Pensées, 121 195a; 233, 213b-214a / Geometrical Demonstration, 434b-439b 35 LOCKE: Human Understanding, BK II, CH XIV, SECT 30 161c-d; CH XVI, SECT 8 167c; CH XVII 167d-174a passim, esp SECT 9 170a-b; CH XXIX, SECT 16, 237b-d 35 BERKELEY: Human Knowledge, SECT 127 438c-d 42 KANT: Pure Reason, 136a-b / Judgement, 498b-d; 499d-500a

3b. The infinite divisibility of continuous quantities: the infinitesimal; the method of exhaustion and the theory of limits

8 ARISTOTLE: Physics, BK III, CH 1 [200b15-19] 278a; CH 4 [203b15-18] 281c; CH 6 [206a8-b33] 284b-285b; CH 7 285d-286c passim; BK V, CH 3 [227a21-34] 308a-b; BK VI, CH 1-2 312b,d-315d; CH 10 [240b8-241a26] 324c-325b / On the Heavens, BK I, CH 1 [268a6-11] 359a; CH 5 [271b8-12] 362d / Generation and Corruption, BK I, CH 2 [317a3-13] 412d-413a / Metaphysics, BK III, CH 4 [1001b7-19] 520b-c; BK V, CH 13 [1020a9-14] 541b; [1020a26-33] 541c 11 EUCLID: Elements, BK V, DEFINITIONS, 4 81a; BK X, PROP 1 191b-192a; BK XII, PROP 2 339a-340b; PROP 5 345b-346b; PROP 10-12 351b-359a; PROP 18 367a-368b 11 ARCHIMEDES: On the Sphere and Cylinder, BK I, ASSUMPTIONS, 5 404b; PROP 13-14 411a-414a; PROP 33-34 424b-427a; PROP 42 431b-432b; PROP 44 433a-b / Measurement of a Circle, 447a-451b passim / On Conoids and Spheroids, PROP 4 459a-460b; PROP 21-22 470a-471b; PROP 25-30 473a-479b / On Spirals, 484b; PROP 18-20 492b-495b; PROP 24-27 496b-500a / On the Equilibrium of Planes, BK I, PROP 6-7 503b-504b; PROP 9 505a-b; PROP 13 507a-508b; BK II, PROP 4 512b-513a / Quadrature of the Parabola, 527a-537b esp 527a-b, PROP 16 533b-534a, PROP 24 537a-b / The Method, 569a-592a 11 NICOMACHUS: Arithmetic, BK I, 811d-812a 16 KEPLER: Epitome, BK V, 973a-975a; 979b-983b passim 19 AQUINAS: Summa Theologica, PART I, Q 7, A 3, REP 3 32c-33c; Q 53, A 1, ANS and REP 1 280d-282a; A 2, ANS and REP 1 282a-283b 20 AQUINAS: Summa Theologica, PART III SUPPL, Q 84, A 3, CONTRARY 985d-989b 28 GALILEO: Two New Sciences, FIRST DAY, 139c-153a passim; SECOND DAY, 193b-194d; THIRD DAY, 201a-202a; 205b-d; 224b-c 31 DESCARTES: Objections and Replies, 112b 31 SPINOZA: Ethics, PART I, PROP 15, SCHOL, 361b-c 33 PASCAL: On the Equilibrium of Liquids, 395a-b / Geometrical Demonstration, 434b-439b 34 NEWTON: Principles, BK I, LEMMA I-II and SCHOL 25a-32a esp LEMMA I 25a, LEMMA II, SCHOL 30b-32a; PROP 73, SCHOL 133b-134a; BK II, LEMMA 2 and SCHOL 168a-170a / Optics, BK III, 542b-543a 35 LOCKE: Human Understanding, BK II, CH XV, SECT 9 164b-d; CH XVI, SECT 4 166a-b; CH XVII, SECT 12 170d; SECT 18 172c-d; CH XXX, SECT 16 237b-238a 35 BERKELEY: Human Knowledge, SECT 98 432a; SECT 123-132 437c-439c 35 HUME: Human Understanding, SECT XII, DIV 124 506a-c 42 KANT: Pure Reason, 70c-d; 161d-163a 45 FOURIER: Theory of Heat, 169a-251b passim, esp 172b, 177a 51 TOLSTOY: War and Peace, BK XI, 469a-d; EPILOGUE II, 695b-c

3c. The infinity of asymptotes and parallels

11 EUCLID: Elements, BK I, DEFINITIONS, 23 2a; POSTULATES, 5 2a 11 APOLLONIUS: Conics, BK II, PROP 14-15 691b-692b; PROP 17 693b-694a 31 DESCARTES: Geometry, BK II, 306b-307a 34 NEWTON: Principles, BK I, LEMMA 18, SCHOL 57a-b; LEMMA 22, 65b; PROP 27, SCHOL, 69b-70a 52 DOSTOEVSKY: The Brothers Karamazov, BK V, 120d-121c

3d. The infinite extent of space

8 ARISTOTLE: Physics, BK III, CH 5 [205a10-206a8] 283b-284b; BK IV, CH 5 [212b11-21] 292a / On the Heavens, BK I, CH 5-7 362c-367b / Metaphysics, BK XI, CH 10 [1066b22-1067a38] 595b-596a 12 LUCRETIUS: Nature of Things, BK I [951-1113] 12d-14d esp [951-1007] 12d-13b; BK II [89-94] 16a-b; [294-307] 18d-19a; [1048-1066] 28b-c 18 AUGUSTINE: City of God, BK XI, CH 5 324d-325c 19 AQUINAS: Summa Theologica, PART I, Q 23, A 7, ANS 138d-140a; Q 46, A 1, REP 8 250a-252d; Q 66, A 4, REP 5 348d-349d 23 HOBBES: Leviathan, PART I, 54b-c; PART II, 162b-c; PART IV, 271b 30 BACON: Novum Organum, BK I, APH 48 110d-111a 31 DESCARTES: Objections and Replies, 112b 33 PASCAL: Pensées, 121 195a; 205-206 211a / Geometrical Demonstration, 434b-439b passim 34 NEWTON: Principles, DEFINITIONS, SCHOL, 8b-11a / Optics, BK III, 542b-543a 35 LOCKE: Human Understanding, BK II, CH XIII, SECT 4 149b; SECT 21 152d-153b; CH XIV, SECT 26 160c-d; CH XV 162b-165c passim; CH XVI, SECT 8 167c; CH XVII 167d-174a passim, esp SECT 3-4 168b-d, SECT 11 170c 35 BERKELEY: Human Knowledge, SECT 117 436a 42 KANT: Pure Reason, 24c-d; 28d-29c; 135a-137a,c; 152a-d; 160b-163a / Judgement, 501a-b 51 TOLSTOY: War and Peace, EPILOGUE I, 693c-694a 53 JAMES: Psychology, 631a

3e. The infinite duration of time and motion

7 PLATO: Phaedrus, 124b-c / Timaeus, 450c-451a; 460c-d 8 ARISTOTLE: Topics, BK I, CH 11 [104b13-18] 148a-b / Physics, BK III, CH 7 [207b21-27] 286b; CH 8 [208a20-23] 286d; BK IV, CH 13 [222a29-32] 302b; BK VI, CH 2 [233a13-b16] 315a-c; CH 10 [241a26-b20] 325b-d; BK VIII, CH 1-2 334a-337b; CH 6 344b-346b passim; CH 8-9 348b-353b / On the Heavens, BK I, CH 2 [269a2-10] 360c-d; CH 3 [270a1-24] 361c-362a; BK I, CH 9 [279a12]-BK II, CH 1 [284b6] 370b-376a passim; BK II, CH 6 379c-380c / Generation and Corruption, BK II, CH 10-11 437d-441a,c / Meteorology, BK I, CH 14 [353a15-19] 459c / Metaphysics, BK IX, CH 8 [1050b20-28] 576c-d; BK XI, CH 10 [1067a33-38] 596a; BK XII, CH 6 [1071b3]-CH 7 [1072b14] 601b-602d; CH 7 [1073a5-11] 603b; CH 8 [1073a24-34] 603c 9 ARISTOTLE: Motion of Animals, CH 6 [700a29-701a7] 236a-b 12 LUCRETIUS: Nature of Things, BK I [215-264] 3d-4b; [483-634] 7a-8d; [988-1051] 13b-14a; BK II [80-141] 16a-d; [294-307] 18d-19a; [569-580] 22b; [1048-1066] 28b-c; [1105-1147] 29a-c; BK V [170-194] 63b-c; [376-379] 66a; [416-431] 66c-d; [1204-1217] 76d-77a 12 AURELIUS: Meditations, BK V, SECT 13 271b; SECT 23 272b; BK VI, SECT 15 275a-b; BK IX, SECT 28 293d-294a; BK X, SECT 7 297b-c; SECT 27 299d 16 PTOLEMY: Almagest, BK XIII, 429a-b 16 KEPLER: Epitome, BK IV, 888b-891a 17 PLOTINUS: Second Ennead, TR I, CH 1-5 35a-37c / Third Ennead, TR VII 119b-129a passim / Fourth Ennead, TR IV, CH 8 161d-162d 18 AUGUSTINE: Confessions, BK VII, par 21 49d-50a; BK XI, par 12-17 92b-93c; par 40 98d-99a / City of God, BK XI, CH 4-6 324a-325d; BK XII, CH 12-20 349b-357a 19 AQUINAS: Summa Theologica, PART I, Q 10, A 2, REP 2 41d-42c; A 4, ANS 43b-44b; A 5, ANS 44b-45c; Q 14, A 12, ANS 85d-86d; Q 46 250a-255d; Q 61, A 2 315c-316a; Q 66, A 4 348d-349d; Q 75, A 1, REP 1 378b-379c 20 AQUINAS: Summa Theologica, PART III SUPPL, Q 77, A 2, ANS and REP 1 945a-946b; Q 91, A 2 1017c-1020c 21 DANTE: Divine Comedy, PARADISE, I [73-81] 107a 23 HOBBES: Leviathan, PART I, 50a 25 MONTAIGNE: Essays, 293d-294a 28 GALILEO: Two New Sciences, FOURTH DAY, 245b-d 30 BACON: Novum Organum, BK I, APH 48 110d-111a; BK II, APH 35, 163a-b; APH 48, 186b-d 31 DESCARTES: Rules, XII, 27b-c 33 PASCAL: Pensées, 121 195a; 205-206 211a; 231-232 213b / Geometrical Demonstration, 434b-439b passim 34 NEWTON: Principles, DEF III 5b; DEFINITIONS, SCHOL, 8b; LAW I 14a / Optics, BK III, 540a-541b 35 LOCKE: Human Understanding, BK II, CH XIV, SECT 26-31 160c-162a; CH XV, SECT 3-5 162d-163c; SECT 11-12 165a-c; CH XVI, SECT 8 167c; CH XVII 167d-174a passim, esp SECT 5 168d-169a, SECT 10 170b-c, SECT 16 172a-b; CH XXIX, SECT 15 237a; SECT 16, 237d-238a 36 STERNE: Tristram Shandy, 292a-293b 42 KANT: Pure Reason, 26d; 27b-c; 130b-133c; 135a-137a,c; 152c; 160b-161d 51 TOLSTOY: War and Peace, EPILOGUE II, 693c-694a 53 JAMES: Psychology, 882a

4. The infinity of matter

4a. The infinite quantity or extent of matter: the problem of an actually infinite body

8 ARISTOTLE: Physics, BK I, CH 4 [187b14-22] 262d; BK III, CH 4 [203b15-29] 281c-d; CH 5 [204b35-206a8] 282b-284b / On the Heavens, BK I, CH 5-7 362c-367b / Metaphysics, BK XII, CH 7 [1073a3-13] 603a-b 12 LUCRETIUS: Nature of Things, BK I [1008-1051] 13c-14a; [1094-1113] 14c-d; BK II [478-568] 21a-22b; [1048-1089] 28b-d 17 PLOTINUS: Second Ennead, TR IV, CH 7, 52b-c 19 AQUINAS: Summa Theologica, PART I, Q 7, A 3 32c-33c; Q 47, A 3, REP 2 258c-259a 31 SPINOZA: Ethics, PART I, PROP 13 359c-d; PROP 14, COROL 2 360a; PROP 15, SCHOL 360b-361d; PART II, PROP 2 374a 35 LOCKE: Human Understanding, BK II, CH XIII, SECT 21-22 152d-153c; CH XXVII, SECT 27 211a-b 35 BERKELEY: Human Knowledge, SECT 47 421c-422a

4b. The infinite divisibility of matter: the issue concerning atoms

8 ARISTOTLE: Physics, BK III, CH 6-7 284b-286c; BK VI, CH 4 316d-318a / On the Heavens, BK III, CH 6 [304b23-305a9] 396a-b; BK IV, CH 6 [313b6-22] 405a,c / Generation and Corruption, BK I, CH 2 [315b25-317a17] 411b-413a; CH 8 423b-425d; CH 9 [327a1-23] 426a-c; CH 10 [327b34-328a18] 427b-c / Metaphysics, BK VII, CH 13 [1039a2-11] 562d; BK X, CH 1 [1053a21-24] 579d / On Sense and the Sensible, CH 6 [445b4-446a20] 683b-684c; CH 7 [449a21-30] 688d-689a 12 LUCRETIUS: Nature of Things, BK I [483-634] 7a-8d 16 COPERNICUS: Revolutions of the Heavenly Spheres, BK I, 517a-b 17 PLOTINUS: Second Ennead, TR IV, CH 7 52a-c / Fourth Ennead, TR I, CH 1, 139d 19 AQUINAS: Summa Theologica, PART I, Q 3, A 1, ANS 14b-15b; Q 7, A 3, REP 3 32c-33c; A 4, ANS and REP 1 33d-34c; Q 50, A 2, ANS 270a-272a 28 GALILEO: Two New Sciences, FIRST DAY, 147d-148b 30 BACON: Novum Organum, BK II, APH 8 140b 31 DESCARTES: Objections and Replies, 112b 34 NEWTON: Principles, BK III, RULE III 270b-271a / Optics, BK III, 541b 35 LOCKE: Human Understanding, BK II, CH XVII, SECT 12 170d; CH XXIII, SECT 31 212b-c; CH XXIX, SECT 16, 237b-d; BK IV, CH X, SECT 10, 351d-352a 35 BERKELEY: Human Knowledge, SECT 47 421c-422a 42 KANT: Pure Reason, 100c-d; 131c; 137a-140c; 152a-d; 161d-163a 43 THE FEDERALIST: NUMBER 31, 103d 45 FARADAY: Researches in Electricity, 850b,d-855a,c

4c. The infinite potentiality of matter: the conception of prime or formless matter

7 PLATO: Timaeus, 455c-458b 8 ARISTOTLE: Physics, BK I, CH 7 [191a7-12] 266d; CH 9 [192a25-33] 268c / Generation and Corruption, BK II, CH 1 [328b32-329a24] 428b,d-429a; CH 5 [332a19-26] 432c-d / On the Soul, BK III, CH 5 [430a10-17] 662c 17 PLOTINUS: Second Ennead, TR IV, CH 1-3 50a-d; CH 6 51d-52a / Third Ennead, TR VI, CH 7-19 110d-119a 18 AUGUSTINE: Confessions, BK XII, par 3-6 99d-100c; par 9, 101c; par 14-16 102b-103a; par 22 104a-b; par 24-26 104c-105b; par 28-31 105c-107a; par 40 109b-110a; BK XIII, par 48 124a 19 AQUINAS: Summa Theologica, PART I, Q 7, A 1, ANS and REP 2 31a-d; A 2, ANS and REP 3 31d-32c; A 3, REP 4 32c-33c; A 4, ANS 33d-34c; Q 66, A 1-2 343d-347b 28 HARVEY: On Animal Generation, 494a-b

5. Infinity in the world

5a. The infinite number of things and the infinite number of kinds

7 PLATO: Sophist, 573a-b 8 ARISTOTLE: Sophistical Refutations, CH 1 [165a5-13] 227b-c / Physics, BK I, CH 2 [184b15-24] 259b-c; CH 4 [187b25-188a18] 262b-263c; CH 6 [189a11-20] 264c; BK III, CH 4 [203b15-29] 281c-d / Metaphysics, BK I, CH 3 [984a11-17] 502b; BK III, CH 4 [999a24-29] 518a 9 ARISTOTLE: Generation of Animals, BK I, CH 1 [715b2-17] 255c-d / Rhetoric, BK I, CH 2 [1356b28-35] 596b-c 12 LUCRETIUS: Nature of Things, BK I [1008-1051] 13c-14a; BK II [478-568] 21a-22b; [1048-1089] 28b-d 17 PLOTINUS: Sixth Ennead, TR I, CH 1 252a-c 19 AQUINAS: Summa Theologica, PART I, Q 7, A 4 33d-34c; Q 23, A 7 138d-140a; Q 46, A 2, REP 8 253a-255a; Q 50, A 3 272a-273b 31 SPINOZA: Ethics, PART I, PROP 16 362a; PROP 21-23 364a-365a 33 PASCAL: Pensées, 121 195a 42 KANT: Pure Reason, 193a-200c esp 197b-198a 53 JAMES: Psychology, 873a-b

5b. The number of causes

8 ARISTOTLE: Physics, BK VII, CH 1 326a-327b / Generation and Corruption, BK II, CH 11 [337a26-29] 440a / Metaphysics, BK II, CH 2 512b-513b; BK VII, CH 6 [1031b28-1032a4] 554d; CH 8 [1033b2-5] 556c / On the Soul, BK III, CH 2 [425b11-17] 657d 19 AQUINAS: Summa Theologica, PART I, Q 2, A 3, ANS 12c-14a; Q 75, A 1, REP 1 378b-379c; PART I-II, Q 1, A 4, ANS and REP 2-3 612a-613a 30 BACON: Advancement of Learning, 43a-d / Novum Organum, BK I, APH 48 110d-111a 31 DESCARTES: Meditations, III, 87c-88a / Objections and Replies, 111a-d; 213c-d 31 SPINOZA: Ethics, PART I, APPENDIX, 370c-371b 42 KANT: Pure Reason, 130b-133c; 140b,d-143a; 152a-153a; 164a-171a / Judgement, 556b-c; 577c-578a 51 TOLSTOY: War and Peace, EPILOGUE I, 693c-694a

6. The finite and the infinite in the nature of man

6a. The infinity of desire and will: the limits of human capacity

OLD TESTAMENT: Psalms, 42; 63—(D) Psalms, 41; 62 / Proverbs, 27:20 / Ecclesiastes, 4:7-8; 5:10-12; 6:7—(D) Ecclesiastes, 4:7-8; 5:9-11; 6:7 / Isaiah, 14:12-14—(D) Isaias, 14:12-14 / Habakkuk, 2:5—(D) Habacuc, 2:5 APOCRYPHA: Ecclesiasticus, 10:9; 11:10; 14:9; 23:16-17; 31:1-11—(D) OT, Ecclesiasticus, 10:10; 11:10; 14:9; 23:21-24; 31:1-11 / Baruch, 3:16-19—(D) OT, Baruch, 3:16-19 NEW TESTAMENT: John, 4:13-14 / I Timothy, 6:10 5 EURIPIDES: Phoenician Maidens [499-525] 382b-c 5 ARISTOPHANES: Plutus [143-197] 630d-631b 7 PLATO: Gorgias, 275d-277b / Republic, BK VIII, 412a / Philebus, 609a-639a,c / Laws, BK VIII, 733b-d; BK IX, 751b-d 8 ARISTOTLE: Metaphysics, BK II, CH 2 [994b9-16] 512d-513a 9 ARISTOTLE: Ethics, BK I, CH 2 [1094a18-22] 339b; BK III, CH 12 [1119a6-12] 366c / Politics, BK I, CH 8 [1256b26]-CH 9 [1258a18] 450c-452b; BK II, CH 7 [1267a42-b5] 463b 12 LUCRETIUS: Nature of Things, BK III [59-93] 30d-31b; [931-977] 42a-c; [1003-1010] 43a; [1076-1084] 44a; BK IV [1097-1120] 58c-d; BK V [1405-1435] 79b-d; BK VI [1-34] 80a-c 12 EPICTETUS: Discourses, BK II, CH 9 184c-185d; BK IV, CH 4-5 225a-230b; CH 9 237d-238d 14 PLUTARCH: Pyrrhus, 319b-321a / Caius Marius, 353d-354a,c / Pompey, 525a-b; 533a-c / Caesar, 599b-c / Cicero, 706b-c 18 AUGUSTINE: Confessions, BK I, par 1-6 1a-2c; par 19 5d; BK II, par 2-4 9b-d; BK IV, par 15-19 23a-24b; BK VI, par 26 42d-43a; BK XI, par 1-4 89b-90b; BK XII, par 10 101c 19 AQUINAS: Summa Theologica, PART I, Q 63, A 3 327b-328b; PART I-II, Q 1, A 4 612a-613a; Q 2, A 1, REP 3 615d-616c; Q 30, A 4 751c-752b 20 AQUINAS: Summa Theologica, PART II-II, Q 27, A 6 524c-525c; Q 28, A 3 528d-529c 22 CHAUCER: Troilus and Cressida, BK I, STANZA 58-66 8b-9b / Wife of Bath’s Prologue [5953-5960] 262a 23 HOBBES: Leviathan, PART I, 76c-d 25 MONTAIGNE: Essays, 123d; 149b-d; 224d-225a; 350d-354b; 429a-b; 489b-c; 503b-d 26 SHAKESPEARE: Julius Caesar, ACT I, SC II [10-34] 574c-d; ACT III, SC II 583c-586c 27 SHAKESPEARE: Troilus and Cressida, ACT I, SC I [109-124] 109b; ACT III, SC II [65-107] 120d-121b / Macbeth, ACT I, SC V 288a-d; SC VII 289b-290b 31 DESCARTES: Meditations, IV 89a-93a passim, esp 90b-91b 33 PASCAL: Pensées, 109 193b-194a; 125-183 195b-204b; 425 243b-244b 35 LOCKE: Human Understanding, BK II, CH XXI, SECT 45 189b-d 36 STERNE: Tristram Shandy, 236b-238a 39 SMITH: Wealth of Nations, BK I, 71b-d 42 KANT: Fund. Prin. Metaphysic of Morals, 256c-257c / Practical Reason, 298d-300a / Judgement, 584d-585c; 586d-587a 43 MILL: Utilitarianism, 462c-463a 46 HEGEL: Philosophy of Right, INTRO, par 5-10 13a-15a; par 13-14 15c-16a; par 22-27 17c-18d; PART I, par 35 21a-b; PART II, par 105 40a; ADDITIONS, 6 117a-b; 11 118a; 17 119a; 22 120c-d; 26 121a-b; 118 136a-b 47 GOETHE: Faust, esp PART I [3217-3250] 79a-b, PART II [11,239-258] 273b-274a 50 MARX: Capital, 62a-b; 71d-72c; 292c-295a 51 TOLSTOY: War and Peace, EPILOGUE I, 671d-672a; EPILOGUE II, 692c-694d 54 FREUD: Civilization and Its Discontents, 800d-801a

6b. The infinity of the intellect: man’s knowledge of the infinite

OLD TESTAMENT: Exodus, 33:12-33 / Job, 11:7-9; 26:14; 28; 36:26; 38-41 / Proverbs, 20:24; 25:3 / Ecclesiastes, 3:11; 8:17; 11:5 APOCRYPHA: Wisdom of Solomon, 9:13-16—(D) OT, Book of Wisdom, 9:13-16 / Ecclesiasticus, 1:2; 18:4-7—(D) OT, Ecclesiasticus, 1:2; 18:2-6 NEW TESTAMENT: John, 1:18 / Romans, 11:33-34 / I Corinthians, 2:16 / I Timothy, 6:15-16 7 PLATO: Philebus, 610d-613a 8 ARISTOTLE: Physics, BK I, CH 4 [187b7-14] 262d; CH 6 [189a11-19] 264c; BK III, CH 6 [207a21-31] 285c-d / Metaphysics, BK II, CH 2 [994b17-30] 513a-b; BK III, CH 4 [999a24-29] 518a; BK XII, CH 7 [1072b14-29] 602d-603a; BK XIII, CH 10 [1087a10-25] 619c / On the Soul, BK I, CH 3 [407a22-30] 636d-637a; BK III, CH 4 [429a19-28] 661c; CH 5 [430a10-17] 662c 9 ARISTOTLE: Rhetoric, BK I, CH 2 [1356b28-35] 596b-c 11 NICOMACHUS: Arithmetic, BK I, 812a 16 KEPLER: Harmonies of the World, 1080a 17 PLOTINUS: Sixth Ennead, TR VI, CH 3, 312a-b 18 AUGUSTINE: Confessions, BK V, par 20 32d-33a; BK VII, par 20-21 49d-50a; BK XII, par 3-6 99d-100c 19 AQUINAS: Summa Theologica, PART I, Q 1, A 1 3b-4a; Q 7, A 2, REP 2 31d-32c; Q 12-13 50b-75b; Q 14, A 1, ANS 75d-76c; A 12 85d-86d; Q 28, A 4, REP 2 160c-161d; Q 32, A 1 175d-178a; Q 54, A 2, ANS 285d-286c; Q 79, A 2, ANS 414d-416a; Q 86, A 2 462a-463a; Q 87, A 3, REP 2 467b-468a; Q 88, A 1, ANS 469a-471c; A 3 472c-473a; Q 91, A 3, REP 2 486b-487d; Q 94, A 1, ANS 501d-503a; PART I-II, Q 2, A 6, ANS 619d-620d; Q 30, A 4, ANS and REP 2 751c-752b 20 AQUINAS: Summa Theologica, PART I-II, Q 68, A 2, ANS 89c-90c; PART III, Q 10, A 3 769d-771b; PART III SUPPL, Q 92 1025b-1037c esp A 1, ANS and REP 6, 12 1025c-1032b 21 DANTE: Divine Comedy, PURGATORY, III [37-45] 56b; PARADISE, IV [115]-V [12] 111d-112b; XIX [22-99] 135b-136a; XXI [73-102] 139a-b 23 HOBBES: Leviathan, PART I, 54b-c; PART III, 162c-d; PART IV, 262b 25 MONTAIGNE: Essays, 238c-239b; 251c-257d 28 HARVEY: On Animal Generation, 494a-b 31 DESCARTES: Meditations, 72a-b; III, 86b-87a; IV 89a-93a passim / Objections and Replies, 112a-d; 121d-122b; 169a; 211c-d; 212c-213a; 213d-214a 31 SPINOZA: Ethics, PART I, PROP 30 366c-d 33 PASCAL: Pensées, 72 181a-184b; 231-233 213b-216a / Geometrical Demonstration, 435a-b 35 LOCKE: Human Understanding, BK II, CH XIII, SECT 4 149b; CH XIV, SECT 26-31 160c-162a; CH XV, SECT 2-3 162c-d; SECT 12 165b-c; CH XVI, SECT 8 167c; CH XVII 167d-174a esp SECT 15 171b-172a; CH XX, SECT 9 202c-203a; CH XXIII, SECT 31 212b-c; SECT 33-37 212d-214b passim; CH XXIX, SECT 15-16 237a-238a; BK IV, CH X, SECT 19, 354b-c 35 BERKELEY: Human Knowledge, INTRO, SECT 2 405b 40 GIBBON: Decline and Fall, 308d 42 KANT: Practical Reason, 291a-292a / Judgement, 495a-498b; 501a-b 46 HEGEL: Philosophy of Right, INTRO, par 13-14 15c-16a; PART III, par 185 64b-d; par 358 113c / Philosophy of History, INTRO, 186a; PART I, 270d-271c, 278a-c; PART III, 304c-306b 47 GOETHE: Faust, PART I [1810-1815] 43a; PART II [11,441-452] 278b 51 TOLSTOY: War and Peace, BK I, 50b-c; BK II, 156d; 162b-164a,c; BK XV, 631a-c; EPILOGUE II, 693c-694d 53 JAMES: Psychology, 392a; 631a

7. The infinity of God

17 PLOTINUS: Fourth Ennead, TR III, CH 8, 146b-c; TR IV, CH 9 162d-163a / Fifth Ennead, TR V, CH 10-11 233b-234a / Sixth Ennead, TR V, CH 4 306d-307a 18 AUGUSTINE: Confessions, BK I, par 1-6 1a-2c; BK VI, par 4 36a-b; BK VII, par 1 43b-c; par 7 45a-d; BK XIII, par 12 113b-d / City of God, BK V, CH 9-10 213b-216c 19 AQUINAS: Summa Theologica, PART I, Q 7 31a-34c; Q 54, A 2, ANS 285d-286c; PART I-II, Q 1, A 4, REP 1 612a-613a 21 DANTE: Divine Comedy, PURGATORY, III [37-45] 56b; PARADISE, XXXIII 156b-157d 23 HOBBES: Leviathan, PART I, 54b-c; PART II, 162a-d 25 MONTAIGNE: Essays, 251c-257d 31 DESCARTES: Discourse, PART IV, 52b-d / Meditations, III, 86a / Objections and Replies, 122a-b; 123c-d; DEF VII 130d 31 SPINOZA: Ethics, PART I, DEF 6 355b; PROP 10-11 358a-359b; PROP 13 359c-d; PROP 16 362a; PART II, PROP 7 375a-c 32 MILTON: Paradise Lost, BK III [372-415] 143b-144b; BK VIII [411-421] 241a 33 PASCAL: Pensées, 231-233 213b-216a 34 NEWTON: Principles, BK III, GENERAL SCHOL, 370a-371a 35 LOCKE: Human Understanding, BK II, CH XVII, SECT 1 167d-168a; CH XXIII, SECT 34-35 213a-c; BK III, CH VI, SECT 11-12 271b-272b passim 40 GIBBON: Decline and Fall, 81b 41 GIBBON: Decline and Fall, 230a-b 42 KANT: Pure Reason, 188c; 192c-d / Practical Reason, 344b-c / Judgement, 590b-d; 592a-c 51 TOLSTOY: War and Peace, BK XV, 631a-c

7a. The infinite being or essence of God

OLD TESTAMENT: Exodus, 3:14 / I Kings, 8:27—(D) III Kings, 8:27 / Job, 11:7-9 / Psalms, 9:7; 45:6; 48:8-14; 102:12; 136; 139:7-10—(D) Psalms, 9:8; 44:7; 47:9-15; 101:13; 135; 138:7-10 / Isaiah, 41:4; 43:10-11; 44:6; 48:12—(D) Isaias, 41:4; 43:10-11; 44:6; 48:12 / Daniel, 4:34—(D) Daniel, 4:31 NEW TESTAMENT: John, 8:58 18 AUGUSTINE: Confessions, BK I, par 10 3b-c; BK VII, par 20-21 49d-50a; BK XIII, par 4 111c / City of God, BK XII, CH 2 343c-d / Christian Doctrine, BK I, CH 32 633c-d 19 AQUINAS: Summa Theologica, PART I, Q 7, A 1-2 31a-32c; Q 14, A 1, ANS 75d-76c; A 3 77d-78b; Q 54, A 2, ANS 285d-286c; Q 75, A 5, REP 4 382a-383b; Q 79, A 2, ANS 414d-416a; Q 86, A 2, REP 1 462a-463a 20 AQUINAS: Summa Theologica, PART II-II, Q 2, A 1, ANS 710a-711c; PART III SUPPL, Q 92, A 1, REP 6, 12 1025c-1032b 25 MONTAIGNE: Essays, 293d-294a 31 DESCARTES: Discourse, PART IV, 52a-d / Meditations, 72a-b; III, 86a / Objections and Replies, 108b-114c passim 31 SPINOZA: Ethics, PART I 355a-372d esp DEF I 355a, DEF 3, 6 355b, AXIOM I-II 355c-d, PROP 1-17 355d-363c, PROP 20-21 363d-364c, PROP 23 364d-365a, PROP 25 365b, PROP 29-31 366b-367a; PART II, PROP 1 373d-374a 33 PASCAL: Pensées, 469 256a 35 LOCKE: Human Understanding, BK II, CH XV, SECT 2-4 162c-163b; SECT 12 165b-c; CH XVII, SECT 1 167d-168a; SECT 16-17 172a-c; SECT 20 172d-173c; CH XXIII, SECT 33-35 212d-213c; BK III, CH VI, SECT 11-12 271b-272b 35 BERKELEY: Human Knowledge, SECT 117 436a 42 KANT: Pure Reason, 201b-c; 205a-b 46 HEGEL: Philosophy of History, INTRO, 156d-160b; 165a-b; PART III, 305c-306c

7b. The infinite power of God

OLD TESTAMENT: Genesis, 18:14 / Exodus, 15:18 / Deuteronomy, 10:14; 32:39 / I Samuel, 2:6-8—(D) I Kings, 2:6-8 / Job, 12:14-15; 38-41 / Psalms, 62:11; 114; 148—(D) Psalms, 61:12; 113:1-8; 148 / Isaiah, 40:22-31; 44:24-46:13—(D) Isaias, 40:22-31; 44:24-46:12 / Jeremiah, 32:27—(D) Jeremias, 32:27 / Daniel, 4:34-35—(D) Daniel, 4:31-32 APOCRYPHA: Judith, 16:13-17—(D) OT, Judith, 16:15-21 / Rest of Esther, 13:9-11—(D) OT, Esther, 13:9-11 / Wisdom of Solomon, 11:17-26; 12:12-18—(D) OT, Book of Wisdom, 11:18-27; 12:12-18 / Ecclesiasticus, 18:1-7—(D) OT, Ecclesiasticus, 18:1-6 / II Maccabees, 8:18—(D) OT, II Machabees, 8:18 NEW TESTAMENT: Matthew, 3:9; 19:26 / Mark, 10:27 / Luke, 1:37 / Revelation, 19:6—(D) Apocalypse, 19:6 8 ARISTOTLE: Metaphysics, BK XII, CH 7 [1073a3-11] 603a-b 17 PLOTINUS: Fifth Ennead, TR V, CH 10 233b-c 18 AUGUSTINE: Confessions, BK I, par 12 4a; BK V, par 20 32d-33a; BK VIII, par 6-7 44d-45d / City of God, BK V, CH 10 215c-216c; BK VII, CH 30 261b-d; BK XIV, CH 27 396c-397a; BK XXI, CH 5-8 563d-568d 19 AQUINAS: Summa Theologica, PART I, Q 25, A 2-3 144c-147a; Q 45, A 5, REP 3 245c-247a; Q 65, A 3, REP 3 341c-342b; Q 92, A 2, REP 2 489d-490c; PART I-II, Q 1, A 4, REP 1 612a-613a 20 AQUINAS: Summa Theologica, PART I-II, Q 113, A 9 368d-369c; PART II-II, Q 23, A 2, REP 3 483d-484d; PART III, Q 10, A 3, REP 3 769d-771b 31 DESCARTES: Meditations, III, 86a / Objections and Replies, 229c-d 31 SPINOZA: Ethics, PART I, DEF 3-6 355b; DEF 8 355c; PROP 1-8 355d-357d; PROP 11-17 358b-363c; PROP 20-25 363d-365b; PROP 32-35 367a-369a; APPENDIX 369b-372d; PART II, PROP 3 374a-c 32 MILTON: Paradise Lost, BK II [119-225] 113b-116a; BK III [372-415] 143b-144b / Samson Agonistes [300-329] 346a-b 33 PASCAL: Pensées, 654 292b 34 NEWTON: Principles, BK III, GENERAL SCHOL, 370a 35 LOCKE: Human Understanding, BK II, CH XV, SECT 12 165b-c; CH XVII, SECT 1 167d-168a; CH XXIII, SECT 34-35 213a-c passim; BK III, CH VI, SECT 11-12 271b-272b passim 35 BERKELEY: Human Knowledge, SECT 152 443c-d 36 STERNE: Tristram Shandy, 334a-b 37 FIELDING: Tom Jones, 186c-d 42 KANT: Pure Reason, 180b-c; 181b; 192c-d; 201b-c / Practical Reason, 352a-c / Judgement, 502d-503a; 504b-d; 592a-c; 600d-601c 46 HEGEL: Philosophy of History, INTRO, 156d-160b 51 TOLSTOY: War and Peace, EPILOGUE II, 684b-d

7c. God’s infinite goodness and love

OLD TESTAMENT: Isaiah, 43:1-4; 46:3-4; 49:14-15; 63:8-9—(D) Isaias, 43:1-4; 46:3-4; 49:14-15; 63:8-9 / Jeremiah, 31 esp 31:3—(D) Jeremias, 31 esp 31:3 APOCRYPHA: Wisdom of Solomon, 11:23-26; 16:20-29—(D) OT, Book of Wisdom, 11:24-27; 16:20-29 NEW TESTAMENT: Matthew, 19:16-17 / John, 3:16; 13:31-17:26 / Romans, 8:35-39 / Galatians, 2:20 / Ephesians, 3:14-21 / I John, 3:1 17 PLOTINUS: Fifth Ennead, TR V, CH 10-11 233b-234a 18 AUGUSTINE: Confessions, BK V, par 20 32d-33a; BK XIII, par 2-5 111a-d / City of God, BK XI, CH 10 327d-328d; BK XII, CH 1-3 342b,d-344b 19 AQUINAS: Summa Theologica, PART I, Q 2, A 3, REP 1 12c-14a; Q 4, A 1 20d-21b; Q 6, A 1-3 28b-30b; Q 11, A 3, ANS 49a-c; Q 54, A 2, ANS 285d-286c; PART I-II, Q 1, A 4, REP 1 612a-613a; Q 5, A 2, REP 3 637c-638a 20 AQUINAS: Summa Theologica, PART I-II, Q 64, A 4, REP 3 69b-70a; PART II-II, Q 23, A 2, REP 3 483d-484d; PART III, Q 1, A 1 701d-703a 21 DANTE: Divine Comedy, PURGATORY, XV [40-81] 75d-76a; PARADISE, XXIX [127-145] 151c-d 31 DESCARTES: Objections and Replies, 229c-d 31 SPINOZA: Ethics, PART V, PROP 35-36 460d-461c 32 MILTON: Paradise Lost, BK VII [139-173] 220a-221a 33 PASCAL: Pensées, 580 276b 35 LOCKE: Human Understanding, BK II, CH XVII, SECT 1 167d-168a; BK III, CH VI, SECT 11-12 271b-272b passim 37 FIELDING: Tom Jones, 186c-d 42 KANT: Fund. Prin. Metaphysic of Morals, 263a-b / Practical Reason, 325d-326a; 345a-c; 351b-352c / Judgement, 592a-c 44 BOSWELL: Johnson, 539d-540a 52 DOSTOEVSKY: The Brothers Karamazov, BK II, 24a-c

7d. God’s infinite knowledge

OLD TESTAMENT: I Samuel, 2:3—(D) I Kings, 2:3 / Job, 11:5-11; 34:21-25; 36:4; 37:16 / Psalms, 94:7-12; 100:5; 117:2; 119:142,160; 139; 146:6; 147:5—(D) Psalms, 93:7-12; 99:5; 116:2; 118:142,160; 138; 145:6-7; 146:5 / Proverbs, 15:3,11 / Isaiah, 29:15-16; 40:28—(D) Isaias, 29:15-16; 40:28 APOCRYPHA: Rest of Esther, 13:12—(D) OT, Esther, 13:12 / Ecclesiasticus, 15:18-19; 16:17-20; 17:15,17,19-20; 23:18-20; 39:19-21; 42:18-21—(D) OT, Ecclesiasticus, 15:19-20; 16:16-20; 17:13-17; 23:25-29; 39:24-25; 42:18-22 / Susanna, 42-43—(D) OT, Daniel, 13:42-43 NEW TESTAMENT: Romans, 11:33-36 / I Corinthians, 1:17-2:16 / Ephesians, 3:10 / Colossians, 2:2-3 / Hebrews, 4:13 / I John, 3:20 17 PLOTINUS: Sixth Ennead, TR II, CH 21 279b-280a 18 AUGUSTINE: Confessions, BK III, par 10 15b-d; BK V, par 5 28b-c / City of God, BK V, CH 9-10 213b-216c; BK XI, CH 10, 328c-d; CH 21 333a-d; BK XII, CH 17-18 353a-354d / Christian Doctrine, BK I, CH 8 626c-627a 19 AQUINAS: Summa Theologica, PART I, Q 14, A 1, ANS 75d-76c; A 3 77d-78b; A 12 85d-86d; Q 16, A 5, ANS 97c-98b; Q 18, A 4 107d-108c; Q 54, A 2, ANS 285d-286c; Q 79, A 2, ANS 414d-416a; PART I-II, Q 1, A 4, REP 1 612a-613a 21 DANTE: Divine Comedy, PARADISE, IV [115]-V [12] 111d-112b; XIX [22-99] 135b-136a; XXI [73-105] 139a-b 30 BACON: Advancement of Learning, 98d-99b 31 DESCARTES: Objections and Replies, 122a-b 31 SPINOZA: Ethics, PART I, PROP 21 364a-c; PART II, PROP 1 373d-374a; PROP 3-4 374a-c 32 MILTON: Paradise Lost, BK II [188-193] 115b; BK III [56-134] 136b-138a; BK VIII [109-130] 219b-220a; BK VIII [412-436] 241a-b 35 LOCKE: Human Understanding, BK II, CH XV, SECT 12 165b-c; CH XVII, SECT 1 167d-168a; CH XXIII, SECT 34-35 213a-c passim; BK III, CH VI, SECT 11-12 271b-272b passim 42 KANT: Practical Reason, 303b-304a; 325d-326a; 344a-348a; 351b-352c / Judgement, 590b-d; 592a-c

CROSS-REFERENCES

For:

  • The distinction between the potential and actual infinite, and the infinite of division and addition, see QUANTITY 7.
  • Other discussions of the infinity of space, time, and motion, see ASTRONOMY 8c(1); CHANGE 13; ETERNITY 2; SPACE 3a; TIME 2b; WORLD 4a; and for the conception of eternity as infinite time, see ETERNITY 1; TIME 2.
  • The issue concerning the existence of atoms or the infinite divisibility of matter, see ELEMENT 5b; ONE AND MANY 3a(3).
  • The problem of an infinite regression in causes, see CAUSE 1b, 7b; CHANCE 14; PRINCIPLE 1b; and for the related problem of infinite regression in definition and reasoning, see DEFINITION 1c; PRINCIPLE 3a(3); REASONING 5b(1).
  • The treatment of the infinite and the infinitesimal in mathematics, see MATHEMATICS 4d; QUANTITY 2, 3a, 3c, 4c, 7.
  • The special logical sense in which judgments are called “infinite,” see JUDGMENT 6b.
  • The conception of the human intellect and of prime matter as having comparable types of infinity, see MIND 2b.
  • Another discussion of the finite and the infinite in relation to human nature, see MAN 10d, 13.
  • The special consideration of infinity in relation to human desire, see DESIRE 7-7a(3); and for the special consideration of the limits of human knowledge, see KNOWLEDGE 5a-5a(6).
  • The problem of our knowledge of the infinite, see KNOWLEDGE 5a(4).
  • The infinity of God, and of God’s knowledge, power, and goodness, see BEING 7b(4); GOD 4e-4f, 5c, 5f; GOOD AND EVIL 2; KNOWLEDGE 7a; LIBERTY 5d; MIND 10f; NATURE 1b; TRUTH 2d; WILL 4b; WORLD 3a.

ADDITIONAL READINGS

Listed below are works not included in Great Books of the Western World, but relevant to the idea and topics with which this chapter deals. These works are divided into two groups.

I. Works by authors represented in this collection II. Works by authors not represented in this collection.

For the date, place, and other facts concerning the publication of the works cited, consult the Bibliography of Additional Readings which follows the last chapter of The Great Ideas.

I.

DESCARTES. The Principles of Philosophy, PART I, 26-27 SPINOZA. Correspondence, XII NEWTON. The Method of Fluxions and Infinite Series BERKELEY. The Analyst ———. A Defence of Free Thinking in Mathematics HUME. A Treatise of Human Nature, BK I, PART II, SECT I-II KANT. De Mundi Sensibilis (Inaugural Dissertation) ———. Prolegomena to Any Future Metaphysic, par 50-54 HEGEL. Science of Logic, VOL I, BK I, SECT I, CH 2 (B, C); SECT II, CH 2 (C) W. JAMES. Some Problems of Philosophy, CH 10-11

II.

PROCLUS. The Elements of Theology, (E) ANSELM OF CANTERBURY. Monologium ———. Proslogium MAIMONIDES. The Guide for the Perplexed, PART I, CH 73 DUNS SCOTUS. Opus Oxoniense, BK I, DIST 13 ———. Reportata Parisiensia CRESCAS. Or Adonai, PROPOSITIONS 1-3 NICOLAS OF CUSA. De Docta Ignorantia BRUNO. De l’infinito, universo e mondi ———. De Immenso et Innumerabilibus DONNE. Lovers Infinitenesse SUAREZ. Disputationes Metaphysicae, XXVIII, XXX (2), XXXV (3), XLI (5) LEIBNITZ. New Essays Concerning Human Understanding, BK II, CH 17 ———. Correspondence with Clarke VOLTAIRE. “Infinity,” in A Philosophical Dictionary ———. The Ignorant Philosopher, CH 18 BOLZANO. Paradoxien des Unendlichen WHEWELL. On the Philosophy of Discovery, CH 26 STALLO. Concepts and Theories of Modern Physics, CH 13 BRADLEY. The Principles of Logic, Terminal Essays, VI FULLERTON. The Conception of the Infinite, and the Solution of the Mathematical Antinomies BOSANQUET. Logic, VOL I, CH 4 C. S. PEIRCE. Collected Papers, VOL VI, par 112-126 COUTURAT. De l’infini mathématique CANTOR. Contributions to the Founding of the Theory of Transfinite Numbers ROYCE. The World and the Individual, SERIES I; Supplementary Essay (4) PEANO. Formulaire de mathématiques E. W. HOBSON. The Theory of Functions of a Real Variable and the Theory of Fourier’s Series POINCARÉ. Science and Method, BK II, CH 3 WHITEHEAD and RUSSELL. Principia Mathematica, PART III, SECT C; PART V, SECT E B. RUSSELL. Principles of Mathematics, CH 13, 17, 23, 37-43 ———. Our Knowledge of the External World, V-VII ———. Introduction to Mathematical Philosophy, CH 3, 9, 13 DE SITTER. Kosmos (The Expanding Universe) WEYL. The Open World, LECT I