The conception of the infinite, and the solution of the mathematical antinomies: a study in psychological analysis

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Contents

CHAPTEK I.

PAGE

CHAPTEK II.

Chapter I. Introductory

The doctrine that there may be an infinity of worlds, thought Plutarch,* is to be repudiated. Providence could not possibly take charge of so many; "troublesome and boundless infinity" could be grasped by no consciousness.

Plutarch's decision as to the unknowability of the infinite the student of the history of thought will find reiterated by thinkers who agree upon little else than the one point, that

when we leave the finite and talk of the infinite we are playing with a word, — deceiving ourselves into believing that we can know what is in its very nature inconceivable.

The notion of the infinite, it is said,* is a negative one : all our experience of objects being of them as finite, we can think to ourselves the negation of this condition, and thus form a negative conception, — i.e., not an affirmation of a quality or attribute, but a simple denial of a quality known. But to know an object — for example, space — as infinite, this is beyond our power. The human mind is finite; all. which can become an object of consciousness is finite; and "troublesome and boundless infinity" cannot be an object of thought.

" Whereas, for this very reason," said Cudworth, f " because more could be added to the magnitude of the corporeal world infinitely, or without end, therefore it is impossible, that it should ever be positively and actually infinite;

* See Hamilton's Discussions, " Philosophy of the Unconditioned."

f "Intellectual System," chap, v., Andover, 1838, vol. ii. p. 45.

that is such as to which nothing more can possibly be added."

A favorite position : all attempts at cognizing the infinite must result in the indefinite; for the infinite can only be known by the progressive addition of finites, an addition which can be completed only in an infinite time ; in other words, can never be completed. Each stage in the addition gives but the indefinite, and the last stage, the infinite, is, by the terms of the problem, unattainable.

How, indeed, it is asked, could a finite mind know the infinite? "Adequately to know what is infinite is to have infinite knowledge."* We cannot, surely, lay claim to that.

The famous antinomies of the philosopher of Konigsberg have, more than anything else, stirred up the minds of men to a consideration of the problem. The indefinite, said Kant, we may know as a whole, not by passing -successively over each of its parts, but immediately as a unit, by means of its limits ; but in the case of the infinite, since there are no limits, we must arrive at our cognition by a successive

* " The Battle of the Two Philosophies," p. 24.

addition of parts, which addition must necessarily be itself endless, and therefore the attempt thus to know the infinite futile. Although we are unable to conceive of an absolute commencement to time or of an absolute limit to space, we are not on that account justified in assuming either to be infinite, as this we can only know when we have passed successively over all spaces and all times. The whole question is one the decision of which is beyond the scope of human reason; both alternatives are equally inconceivable.

The doctrine of the inconceivability of the infinite is much dwelt upon by Sir William Hamilton as one phase of his cherished theory, the Philosophy of the Conditioned. He argues that space, for example, cannot be cognized as either infinitely extended and infinitely divisible, or, on the other hand, as absolutely limited, whether as a maximum or minimum, just as Kant argued. Mr. Mansel has accepted and defended * the positions taken by Hamilton ; while the same arguments, which are used by Sir

* See Mr. Mansel's "Philosophy of the Conditioned." London: 1866.

William Hamilton and his follower, Mr. Mansel, to elevate faith at the expense of reason or science, are to be found upon the pages of the " First Principles of Philosophy," where they are used hy Mr. Spencer in support of a philosophy widely different from the Hamiltonian.

It would not he difficult to multiply testimonies to the .inconceivability of the infinite, for the misconception which we find in Plutarch appears and reappears in divers forms in different ages and climes, much as the Wandering Jew jaarp be supposed to have presented himself; and, to carry out the simile, the well-worn dress in which the doctrine usually comes to the surface — the statement that we can know the infinite only hy an endless addition of finites — may he not inaptly compared to the threadbare garment on his hack.

It would seem surprising that there could he so universal a misconception with regard to the nature of a conception actually present at some time or other in the mind probably of every man, and in the case of many not unfrequently present; a conception sufficiently familiar and important to be recognized and marked by its appropriate name. But a very little knowledge of psychological processes will make one cognizant of the fact that the interpretation of consciousness is by no means the easy and simple task that by a novice it might be supposed to be.

There are but few who have an analytical knowledge of even the most common of their mental operations ; and the average man is very literally incapable of telling what may at any moment be passing in his mind. It is a matter of surprise, for instance, to one unaccustomed to psychological analysis, to learn that distance is not directly perceived by vision, but that judgments of distance are the result of a rapid process of reasoning, and imply a generalization from past experience. The judgment seems instantaneous and intuitive. Those elements which are in fact visual and actually present are not distinguished from those present only by suggestion ; that is, present in the imagination. The mental state is grasped as a unit, and for practical purposes no analysis into its elements is necessary. So it is with the concept, or general notion. That men form general notions, or at least represent in mind objects and their relations in some way different from that in which simple intuitions are singly represented, all are ready to acknowledge. But that the psychical elements concerned in this act are not clearly apprehended is evident from the common warfare of Realism, Nominalism, and Conceptualism. So is it, again, in the case of memory. The act is usually described as if there were present in the mind the two elements of a present mental representation and an intuition, or presentation, its prototype, to' which it is referred ; while the picture before the mind is but one, as an examination of consciousness during the act will very readily show. And the same truth is illustrated in innumerable ways by the debates and disputes of philosophers, past and present, as to mind and the faculties in which it is manifested. As a typical instance may be given the analysis of consciousness left us by Mr. John Stuart Mill as compared with that presented in Sir William Hamilton's Lectures. "Was not each describing what was present in his own mind? Whence the discrepancy ?

Moreover, a clear apprehension of the constituent elements of a mental state does not always seem to be necessary, from a practical point of view, to enable one to use that state as a unit, with substantial accuracy. The Nominalist, the Realist, and the Conceptualist all speak the same language, refer to the same objects, and, in the use of the complex mental phenomenon which they so variously analyze, are at one. A seaman, practised in the judgment of distances by a long training in his vocation, may be much more accurate in his judgments of distance than the accomplished author of the "New Theory of Vision" himself, though he may be quite ignorant of the mental process by which he arrives at his conclusions. The process itself is not directly affected by an analytic consciousness of the steps of the process, nor is its result. The mental state is in most cases recognized only as a unit, since it is as a unit that it is useful to the individual ; and the name by which it is known expresses the general impression conveyed to the mind when the state is called up.

Now, it is manifest, that when one begins to analyze this vague and indefinitely grasped total, and to separate it into its elements, it is quite possible for him to confound some of the elements with elements somewhat similar, or to imagine the presence of elements closely connected by the laws of association with elements really present; in short, to find what is not there, and what, when in practice he uses the word indicating the state as a unit, he never means to express by it. And, in view of this fact, we may see how it is possible for the curious error regarding the inconceivability of the infinite, which has been adverted to, to have arisen, and to have held its place in the writings of philosophers. The word has always been .used, and the ideas for which it stands have often been in men's minds ; but in attempting to explicate the conception, we find that almost all place among the qualities it connotes a notion drawn from finites, and which is contradictory to the essential character of the conception. Few men have talked more about the infinite than Sir William Hamilton ; and it is probable that every time he used the word it called up very much the same mental state in his mind as that which arose in the mind of Mr. John Stuart Mill when he read Sir William's Lectures ; but the conception thus called up certainly did not contain the warring ele-

2* ments which Sir William finds in it when he undertakes to prove it unthinkable. And when Mr. Mill criticises Sir William Hamilton, and declares the infinite not inconceivable, he probably meant by the infinite just what Hamilton did ; and yet when he tries to prove its conceivability, he finds in it what was not really contained in the mind of either when the word was used.

The problem is simply one of psychological analysis, — a question of what may be the true content of a complex mental state ; and the fact that such an analysis may be incorrectly made need not surprise us, since erroneous analyses are only too common. What may well create surprise, however, is that the difficulties into which this erroneous analysis has led those guilty of it, the antinomies of which it is the source, have not called attention to the error, and that a rigorous analysis has not been employed to eliminate it. I will consider in the remaining chapters of this little book the errors which have arisen from a mistaken notion of the content of our conception of the infinite, and show that they all arise from a foreign and contradictory element inadvertently

admitted as part of the conception, and that, this element being eliminated, the conception is in no respect inconceivable, nor does it present any difficulties not presented by any other concept or general notion.

CHAPTEB II.

THE CONCEPTION NOT QUANTITATIVE.

We will suppose two parallel straight lines, A and B, unlimited in extent, and intersected by perpendiculars, ab, a'b', a"b", etc., drawn at equal distances from each other.

b V b"

It is evident that each division upon A is equal

to its corresponding division upon B, and the

sum of any number of divisions upon A will

equal the sum of a similar number upon B.

Since, therefore, each division upon the one line

has its corresponding division upon the other,"

will not the equation hold .good when all the

divisions are considered ? That is, will not the

sum of all the divisions upon A be equal to the

sum of all the divisions upon B ? And will not

the sum of all the divisions on both lines be

equal to twice the sum of all the divisions on

either? Must we not here regard one infinite

as greater than another ? 20

Much depends upon the answer to this question, as it will reveal very clearly the content which one attributes to his conception of the infinite. To the giving of the wrong answer may be traced that misconception of the true nature of infinity which has been such a fruitful source of supposed antinomies. Broadly stated, the question is, Can infinites be regarded as comparable with each other, as greater or less than, or equal to, each other ? Let us consider the case of the parallel lines.

It is true that we must consider each division on the one line equal to each division on the other ; and taking any number of divisions on the one, and adding them to an equal number of divisions on the other, we obtain a sum equal to twice the number of given divisions on either. But when we say " all the divisions on the one are equal to all the divisions on the other," we speak of the lines as quantitative wholes, and introduce an error with the word all.

To conceive of a thing as a ivhole, we must assign to it limits. In saying " the whole" of any object, we refer to„ those limits beyond which there is none of that object. In regarding any object as a quantitative whole, we necessarily think it as finite.

"When we compare one line with another, and declare its extent greater or less than that of the other, we mean that, when the one is applied to the other, its limits extend beyond or fall within the limits of the other. In other words, we give the difference between the distances included within their respective limits. Measuring is merely giving the distance between limits. "When two lines are infinite, we have no point to measure from, and no point to measure to, and no measurement — therefore no comparison — is possible. It is a palpable contradiction to compare (i.e., give relations of measurement between the respective limits of) two infinites (i.e., things which cannot be measured, as having no limits).

The terms longer, shorter, and equal, can, therefore, have no meaning as aj?plied to infinite lines. They can be used only in speaking of the finite. We cannot, then, say that one infinite is greater or less than another, and just as little can we say that all infinites are equal; for any such proposition, however possible in words, is impossible in thought, and is an attempt to join contradictory notions.

In such cases as the above, where the lines are nowhere limited, the impossibility of an increase in length may be clearly seen. A line can only be lengthened by. adding to it at its extremities, and it is impossible that a line without ends should be added to. If one holds that the sum of two such lines is greater than either line separately, he simply states that that may be increased, the very conception of which precludes the possibility of its increase.

There are cases, however, in which the error of a wrong conclusion is not so immediately palpable as in the case just stated; for example, the case of a line limited at but one point. Suppose the line AB limited only at the point A.

Continue the line to C. If now the line be divided by points placed at equal distances from each other, into equal divisions, AC containing three such divisions, will not the whole line CB be greater by three divisions than the whole line AB ?

AB is limited at A ; consequently there is nothing to prevent our adding to it at its one extremity. Does it not seem natural to assume that in thus adding we increase the sum total of the line ? "We have gone through the same process as that by which we increase finite lines.

When we recollect, however, that the line AB is limited only at one point, and is not, therefore, as a line, defined (for two points are necessary to define a line), the impossibility of regarding "it as a quantitative whole is evident, and the impossibility of increasing or diminishing its length, as a whole, necessarily follows. All of CB is. not greater than all of AB, because the word all (in its quantitative sense*) cannot be applied to either.

Suppose we attempt a comparison of the two lines. Let CB be superposed upon AB in such a manner that C will fall upon A. The two lines will then have the one limit in common ; but one limit does not furnish data for lineal comparison, and no judgment can be formed as to the comparative length of the two lines. Where the one line is regarded as

* This distinction will be noticed later.

greater than the other, from the fact that three of its divisions project beyond the limit of the other, the measurement begins with an imagined point infinity, which is regarded as a common limit of the other extremities of the two lines, and concludes with the limits at A and C.

The error of such an attempt at measurement is clearly revealed by beginning to measure at A and C. It may be here justly remarked that we have before us a concrete instance of the truth of the old adage, that is a poor (measuring) rule that will not work both ways. The illusion disappears when Ave begin to measure at the given and only limits.

Now, drawing the necessary inference from the foregoing, we may answer the question whether a line altogether without limits is not greater than a line limited at but one point, by saying that the very nature of the conceptions precludes the possibility of the words greater or less being applied to either ; that neither of the lines can be regarded as a quantitative whole ; and that, consequently, the question is a meaningless one.

When we turn our attention to the considera- 3 tion of surfaces, we meet with similar misapprehensions, and arising from the same cause. It is asserted, for example, that if we suppose three parallel straight lines infinite in extent, one of which, A, is separated by a distance of two metres from the middle line B, while the other, C, is distant from it but one metre, we must conclude that the surface included between A and B is double the surface included between B and C.

Let the lines be intersected by perpendiculars one metre apart. Do we not find that each metre in the surface contained between B and C has corresponding to it two metres in the other surface ? And is not this proportion the same for twenty or for two hundred as for one, and quite independent of number? If, then, the proportion will hold good for any number, why will it not hold good when all the divisions are considered ?

Or if we consider the angle formed by the intersection of two infinite straight lines, must we not conclude that increasing the angle will increase the area of the surface included between the lines forming the angle ? And that diminishing the angle will diminish the area of the included surface ? Let A and B be two infinite straight lines intersecting at C ;

and let A'C be an infinite straight line, making a smaller angle with CB than is made by AC. Must we not affirm that the surface ACB is greater than the surface A'CB, and that it is equal to the sum of A'CB and ACA' ? The answer to be given in these cases is evidently the answer which has already been given. The quantitative relations of equality and inequality certainly hold good for all quantities; and taking the case of the surfaces included between the parallel lines, we must admit that any number of the divisions between A and B will have double the area of a similar number of the divisions between B and C ; but when we speak of all the divisions, we do not refer to any number; we do not express by the word a quantity; and where the notion of quantity is wanting, manifestly, quantitative relations will not hold. So in the case of the increased or diminished angle.

It is unnecessary to multiply instances, as the principle is in all cases the same. In general,, wherever the limit is removed in any one direction, whether in the case of lines, of surfaces, or of solids, the object can no longer be regarded as a quantitative whole, and is not to be considered finite.

It remains to consider a class of cases of an apparently different nature. It is argued, for example, that an infinite series of dollars will exceed in value an infinite series of cents ; that, where the unit differs, the difference will extend to the series in its totality. It is easy to show the error of such a position by showing what the assertion necessarily involves.

Suppose that, instead of counting one cent in the one series to each dollar in the other, we vary our mode of procedure by counting one hundred cents in the one to each dollar in the other. . It is true that the one series is exhausted one hundred times as rapidly as the other ; but since they are both infinite (will never end), we may continue this forever (to infinity), and the two series will have equal values. Or we may count two hundred cents in the one to each dollar in the other, or three hundred, or four hundred; so that the same series will be equal to, or twice, thrice, or four times as great as another ; its value depending merely on the mode of reckoning.

If it is just to conclude that an infinite series of dollars is one hundred times as great in value as an infinite series of cents, we must also accept the conclusions arrived at by the other modes of reckoning; all are based on the same principle. Our only escape- from these warring conclusions is to declare the principle underlying all the modes of reckoning an erroneous one. The error lies in regarding these infinite series as in any way capable of being compared with each other; in looking upon them as quantitative wholes.

It is clear, therefore, that the true conception of the infinite is not quantitative but qualitative, a

3* fact which has been very generally overlooked, and with disastrous consequences much the same in all cases.

I have said that the word " whole," as predicating totality, cannot be applied to infinites, as naturally follows from the qualitative nature of our conception of the infinite; but the use of the word is really unavoidable, and, when it is used, it should be borne in mind that, applied to infinites, the word has a certain qualitative sense quite different from that in which it is applied to finites. If I were to speak of " all (possible) men" as distinguished from " all (possible) angels," I should in no respect limit the infinite possible number of either (the phrase " infinite number," though, strictly considered, incorrect, may be understood in a qualitative sense as expressing unlimited units), and my conceptions would be not quantitative but qualitative ; for the mind would be occupied, not with the number of objects, but with certain conditions which any object must satisfy to fall within the one class or the other.

If, on the other hand, I speak of " all (actual) men," I may regard them as a definite known or unknown number, and my conception may be quantitative.

Similarly, if, in speaking of an infinite line, I should say, " the line AB is in all its parts a straight line," I could only mean that one of the general conditions of the line is straightness, and that, whatever part of it may be thought of, it must agree with this condition. The " all" is in such a case equivalent to "any," not. to "every," and the two meanings may easily be confounded. Indeed, there is something misleading in the very expression, " an infinite line," for the unreflective mind is apt to regard the object to which it is applied as a unit, a whole ; it is very necessary in using the phrase to keep in mind its true meaning.

It may be objected to what I have here brought forward that any theory which denies that we have knowledge of the infinite as a whole may justly be called agnostic. If we do not know the infinite as a whole, do we not know only its parts, which are finite ? And have we any true knowledge of the infinite at all?

I answer, the conception of a part is, as well as the conception of a whole, quantitative, and an object recognized as part of a greater object is thereby necessarily recognized as finite, But if the object before the mind is not quantitatively regarded at all, either as whole or part, our conception may be of the infinite. The plausibility of the objection arises from its confounding two very different things, the distinction between which will be more clearly drawn in the last chapter of this monograph.

But as a preliminary answer to the objection, I may say that the assertion that we do not know the infinite as a whole is by no means equivalent to the assertion that we do not know the infinite. We do not know the moon as square, but that would scarcely prove that we have no knowledge of the moon, since the notion of squareness forms no part of a true knowledge of that object. Just as little is the quantitative conception of totality necessary to a knowledge of the infinite.

It is not agnosticism to declare the mind unable to think that which is in its nature selfcontradictory, — to define an object as infinite and then think it as limited ; while, on the other hand, any theory which maintains that we may know as a whole that which, in its very conception, precludes the possibility of its being so considered, may be accused of the direst agnosticism, as discrediting a fundamental law of thought, the law of non-contradiction. The theory attacked may as a last resource avail itself of the old argumentum ad hominem, and remark in pointed terms that the kettle is not as black as some other vessels in the speculative kitchen.

Chapter III. The Antinomies Of Hamilton

The evils resulting from overlooking the fact that the conception of the infinite is qualitative are evident when we examine some of the reasonings based upon the supposition that the conception contains a quantitative element.

And first I will examine, so far as it touches the point in question, that agnostic theory developed by Sir William Hamilton under the name of the Philosophy of the Conditioned, the fundamental principle of which is that " all that is conceivable in thought lies between two extremes, which, as contradictory of each other, cannot both be true, but of which, as mutual contradictories, one must."* The reasoning in which Sir William applies this principle to our knowledge of space is worthy of attention. "... We conceive space," he says, — " we cannot but conceive space. I admit, therefore, that Space, indefinitely, is a positive

* " Metaphysics," New York, 1880, p. 527. 34

TEE ANTINOMIES OF HAMILTON. 35

and necessary form_ of thought. But when philosophers convert the fact, that we cannot but think space, or, to express it differently, that we are unable to imagine anything out of space, — when philosophers, I say, convert this fact with the assertion that we have a notion, a positive notion, of absolute or of infinite space, they assume not only what is not contained in the phenomenon, nay, they assume what is the very reverse of what the phenomenon manifests. It is plain that space must either be bounded or not bounded.. These are contradictory alternatives; on the principle of Contradiction they cannot both be true, and, on the principle of Excluded Middle, one must be true. This cannot be denied without denying the primary laws of intelligence. But though space must be admitted to be necessarily either finite or infinite, we are able to conceive the possibility neither of its finitude nor of its infinity.

" We are altogether unable to conceive space as bounded, — as finite ; that is, as a whole, beyond which there is no further space. Every one is conscious that this is impossible. It contradicts also the supposition of space as a necessary notion ; for if we could imagine space as a terminated sphere, and that sphere not itself enclosed in a surrounding space, we should not be obliged to think everything in space; and, on the contrary, if we did imagine this terminated sphere as itself in space, in that case we should not have actually conceived all space as a bounded whole. The one contradictory is thus found inconceivable ; we cannot conceive space as positively limited.

" On the other hand, we are equally powerless to realize in thought the possibility of the opposite contradictory; we cannot conceive space as infinite, as without limits. You may launch out in thought beyond the solar walk, you may transcend in fancy even the universe of matter, and rise from sphere to sphere in the region of empty space, until imagination sinks exhausted; with all this, what have you done ? You have never gone beyond the finite ; you have attained, at best, only to the indefinite, and the indefinite, however expanded, is still always the finite. . . . Now, then, both contradictories are equally inconceivable, and, could we limit our attention to one alone, we should deem it at once impossible and absurd, and suppose its unknown opposite as necessarily true. But as we not only can, but are, constrained to consider both, we find that both are equally incomprehensible; and yet, though unable to view either as possible, we are forced by a higher law to admit that one, but one only, is necessary."