THECON '.HON OF THE INFINITE 
 
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THE 
 
 CONCEPTION OF THE INFINITE, 
 
 AND THE 
 
 SOLUTION OP THE MATHEMATICAL 
 ANTINOMIES: 
 
 A STUDY IN PSYCHOLOGICAL ANALYSIS. 
 
 BY 
 
 GEORGE S. FULLERTON, A.M., B.D., 
 
 ADJUNCT PROFESSOR OF PHILOSOPHY IN THE UNIVERSITY OF PENNSYLVANIA. 
 
 UNIVERSITY 
 
 PHILADELPHIA : 
 
 J. B. LIPPINCOTT COMPANY. 
 
 188 7. 
 
BZ 
 
 Copyright, 1887, by George S. Fdllertok. 
 
 43%ix^ 
 
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PEEFAOE. 
 
 The question treated in this little volume is 
 one of no small interest from several quite 
 different points of view. To one interested in 
 lucid and systematic thinking, the tangle of 
 thought which has always obtained in this 
 corner of the philosophic field cannot but be 
 repulsive and irritating. To be told that of 
 two impossible things one must be true; that 
 of the same two lines one may be looked upon 
 as, at pleasure, equal to, less than, or greater 
 than the other, both remaining unchanged; 
 that Achilles, running rapidly, can never over- 
 take the tortoise, moving slowly ; to be told 
 all this seriously, by men whose calling it is 
 to think and to teach others to think, is well 
 calculated to bring not merely suspicion but 
 contempt upon speculative thought, and de- 
 servedly. Who has not puzzled, on his first 
 
 3 
 
4 PREFACE. 
 
 introduction to Logic, over some of these an- 
 tinomies, and been silenced unconvinced by the 
 practical demonstration, — as by walking, in the 
 case of the argument against motion, — which 
 cuts the knot but does not solve it, leaving in 
 the mind a disagreeable sense that the argu- 
 ment must be wrong somewhere, and yet a 
 consciousness that it certainly seems perfectly 
 sound? When the metaphysician proves to us 
 that a rhinoceros is a mosquito, his chain of 
 reasoning is rendered innocuous by the striking 
 incongruity of the conclusion ; but if we ob- 
 serve no flaw in his reasoning, we cannot help 
 recognizing the perplexing truth that it is the 
 experienced fact alone which has prevented as- 
 sent, and that a precisely similar argument, the 
 conclusion of which cannot be similarly tested, 
 may yet induce assent, though equally errone- 
 ous. If we have no better reason for rejecting 
 an argument, what can be our criterion when 
 we leave the sphere of the immediately palpa- 
 ble? He who has convinced himself that the 
 minute hand of a clock cannot overtake the 
 hour hand, will be enlightened when the clock 
 strikes at noon ; but he who has followed Mr. 
 Spencer into his discussions regarding our no- 
 
PREFACE. 5 
 
 tions of infinite space or time, will be filled 
 with inward dismay if he hang his hope upon 
 any such practical expedient. Civil history can- 
 not be studied in the laboratory, nor erroneous 
 ideas as to infinite space rectified with the aid 
 of the foot-rule. In this sphere, too, the ques- 
 tion of a careful and thorough analysis of our 
 conception of the infinite is of more than a 
 merely intellectual interest, and any erroneous 
 conception which can blossom out into such a 
 development as the " Philosophy of the Condi- 
 tioned," with its implications, has a religious 
 significance which cannot be overlooked. The 
 analysis of this single conception is, moreover, 
 of importance as throwing light upon the pro- 
 cedure of thought in general, and will to many 
 be of more interest in this connection than for 
 its own sake. I have endeavored to write with 
 extreme clearness and simplicity, and to avoid, 
 as much as possible, all issues not directly con- 
 nected with the immediate subject; and whether 
 my discussion meet with assent or dissent, I do 
 not think it will be charged with the obscurity 
 characteristic of discussions upon this much- 
 mooted topic. 
 
 Portions of the book are reprinted, with ad- 
 
6 PREFACE. 
 
 ditions and alterations, from the American Jour- 
 nal of Speculative Philosophy and from the Brit- 
 ish periodical Mind, in which they originally 
 appeared. 
 University of Pennsylvania, December, 1886. 
 
CONTENTS. 
 
 CHAPTEK I. 
 
 PAGE 
 
 Introductory 9 
 
 CHAPTEK II. 
 The Conception not Quantitative . . . .20 
 
 CHAPTEK III. 
 The Antinomies of Hamilton 34 
 
 CHAPTEK IV. 
 Kant, Mill, and Clifford 53 
 
 CHAPTEK V. 
 
 The Conceivable and the Existent . . . 77 
 
 CHAPTEK VI. 
 
 The Conceivability of the Infinite . . .90 
 
THE 
 
 CONCEPTION OF THE INFINITE, 
 
 AND THE 
 
 SOLUTION OF THE MATHEMATICAL 
 ANTINOMIES: 
 
 A STUDY IN PSYCHOLOGICAL ANALYSIS. 
 
 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 
 
 * " De defectu oraculorum," c. 24. 
 
10 THE CONCEPTION OF THE INFINITE. 
 
 when Ave leave the finite and talk of the infinite 
 we are playing with a word, — deceiving our- 
 selves 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 our- 
 selves the negation of this condition, and thus 
 form a negative conception, — i.e., not an affir- 
 mation 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 Cud- 
 worth,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, u Philosophy of the Uncon- 
 ditioned." 
 
 f "Intellectual System,"' chap, v., Andover, 1838, vol. ii. 
 p. 45. 
 
INTRODUCTORY. \\ 
 
 that is such as to which nothing more can pos- 
 sibly 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 progres- 
 sive 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 succes- 
 sively 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. 
 
12 THE CONCEPTION OF THE INFINITE. 
 
 addition of parts, which addition must neces- 
 sarily be itself endless, and therefore the attempt 
 thus to know the infinite futile. Although we 
 are unable to conceive of an absolute commence- 
 ment 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 inconceiv- 
 able. 
 
 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 de- 
 fended* the positions taken by Hamilton; while 
 the same arguments, which are used by Sir 
 
 * See Mr. Mansel's "Philosophy of the Conditioned. 1 
 London: 1866. 
 
INTRODUCTORY. 13 
 
 William Hamilton and his follower, Mr. Man- 
 sel, 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 by Mr. Spencer in support of a 
 philosophy widely different from the Hamil- 
 tonian. 
 
 It would not be difficult to multiply testimo- 
 nies to the inconceivability of the infinite, for 
 the misconception which we find in Plutarch 
 appears and reappears in divers forms in dif- 
 ferent ages and climes, much as the Wandering 
 Jew may be supposed to have presented him- 
 self; 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 by an endless addition of finites — 
 may be not inaptly compared to the threadbare 
 garment on his back. 
 
 It would seem surprising that there could be 
 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 
 
14 THE CONCEPTION OF THE INFINITE. 
 
 appropriate name. But a very little knowledge 
 of psychological processes will make one cogni- 
 zant of the fact that the interpretation of con- 
 sciousness 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 judg- 
 ments of distance are the result of a rapid pro- 
 cess of reasoning, and imply a generalization 
 from past experience. The judgment seems in- 
 stantaneous and intuitive. Those elements which 
 are in fact visual and actually present are not 
 distinguished from those present only by sug- 
 gestion ; 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 gen- 
 eral notion. That men form general notions, or 
 at least represent in mind objects and their re- 
 
INTR OD VCTOR Y. 15 
 
 lations 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 war- 
 fare of Realism, Nominalism, and Conceptual- 
 ism. 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 pres- 
 ent mental representation and an intuition, or 
 presentation, its prototype, to which it is re- 
 ferred ; 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 discre- 
 pancy ? 
 
 Moreover, a clear apprehension of the con- 
 stituent elements of a mental state does not 
 
1(3 THE CONCEPTION OF THE INFINITE. 
 
 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 Nom- 
 inalist, 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 judg- 
 ment of distances by a long training in his vo- 
 cation, may be much more accurate in his judg- 
 ments 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 ana- 
 lytic 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 gen- 
 eral impression conveyed to the mind when the 
 state is called up. 
 
 JSToW, 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 
 
 /3e r ir<u, 
 
INTROD UCTOR Y. 17 
 
 elements with elements somewhat similar, or 
 to imagine the presence of elements closely 
 connected by the laws of association with ele- 
 ments 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 curi- 
 ous 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 contra- 
 dictory to the essential character of the concep- 
 tion. 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 "Wil- 
 liam's Lectures ; but the conception thus called 
 up certainly did not contain the warring ele- 
 
 2* 
 
18 THE CONCEPTION OF THE INFINITE. 
 
 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 prob- 
 ably meant by the infinite just what Hamilton 
 did ; and yet when he tries to prove its con- 
 ceivability, 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 an- 
 alyses 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 em- 
 ployed to eliminate it. I will consider in the 
 remaining chapters of this little book the 
 errors which have arisen from a mistaken no- 
 tion of the content of our conception of the 
 infinite, and show that they all arise from a 
 foreign and contradictory element inadvertently 
 
INTRODUCTORY. 19 
 
 admitted as part of the conception, and that, 
 this element being eliminated, the conception 
 is in no respect inconceivable, nor does it pre- 
 sent any difficulties not presented by any other 
 concept or general notion. 
 
CHAPTER 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 b' 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 
 
THE CONCEPTION NOT QUANTITATIVE. 21 
 
 Much depends upon the answer to this ques- 
 tion, 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 divis- 
 ion 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 num- 
 ber 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 whole, 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 re- 
 
22 THE CONCEPTION OF THE INFINITE. 
 
 garding 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 be- 
 yond or fall within the limits of the other. 
 In other words, we give the difference between 
 the distances included within their respec- 
 tive 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 measure- 
 ment — therefore no comparison — is possible. 
 It is a 'palpable contradiction to compare (i.e., 
 give relations of measurement between the re- 
 spective limits of) two infinites (i.e., things 
 which cannot be measured, as having no 
 limits). 
 
 The terms longer, shorter, and equal, can, there- 
 fore, have no meaning as applied 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 im- 
 
THE CONCEPTION NOT QUANTITATIVE. 23 
 
 possible 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 in- 
 crease 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 with- 
 out 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 pre- 
 cludes the possibility of its increase. 
 
 There are cases, however, in which the error 
 of a wrong conclusion is not so immediately pal- 
 pable 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 di- 
 vided 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 
 
24 THE CONCEPTION OF THE INFINITE. 
 
 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 impossibilit}' of 
 regarding it as a quantitative whole is evident, 
 and the impossibility of increasing or diminish- 
 ing its length, as a w T hole, necessarily follows. 
 All of CB is not greater than all of AB, be- 
 cause 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 com- 
 mon ; 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. 
 
THE CONCEPTION NOT QUANTITATIVE. 25 
 
 greater than the other, from the fact that three 
 of its divisions project heyond the limit of the 
 other, the measurement begins with an im- 
 agined 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 (meas- 
 uring) rule that will not work both ways. 
 The illusion disappears when we 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 mean- 
 ingless one. 
 
 When we turn our attention to the considera- 
 
26 THE CONCEPTION OF THE INFINITE. 
 
 tion of surfaces, we meet with similar misappre- 
 hensions, 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 be- 
 tween 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 
 
THE CONCEPTION NOT QUANTITATIVE. 27 
 
 intersection of two infinite straight lines, must 
 we not conclude that increasing the anscle will 
 increase the area of the surface included be- 
 tween 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 be- 
 tween the parallel lines, we must admit that any 
 number of the divisions between A and B will 
 
28 THE CONCEPTION OF THE INFINITE. 
 
 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 direc- 
 tion, 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 con- 
 sidered 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 
 
THE CONCEPTION NOT QUANTITATIVE. 29 
 
 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 prin- 
 ciple. Our only escape from these warring con- 
 clusions 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 quanti- 
 tative wholes. 
 
 It is clear, therefore, that the true conception 
 of the infinite is not quantitative but qualitative, a 
 
 3* 
 
30 THE CONCEPTION OF THE INFINITE. 
 
 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 pre- 
 dicating 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, ap- 
 plied to infinites, the word has a certain quali- 
 tative 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 quanti- 
 tative 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 
 
THE CONCEPTION NOT QUANTITATIVE. 31 
 
 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 straight- 
 ness, 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 some- 
 thing 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 
 
32 THE CONCEPTION OF THE INFINITE. 
 
 an object recognized as part of a greater object 
 is thereby necessarily recognized as finite. But 
 if the object before the mind is not quantita- 
 tively regarded at all, either as whole or part, 
 our conception may be of the infinite. The 
 plausibility of the objection arises from its con- 
 founding two very different things, the distinc- 
 tion 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 self- 
 contradictory, — 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 
 
THE CONCEPTION NOT QUANTITATIVE. 33 
 
 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 re- 
 mark 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 rea- 
 sonings 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 de- 
 veloped 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 rea- 
 soning in 'which Sir William applies this prin- 
 ciple 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 
 
THE 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 man- 
 ifests. It is plain that space must either be 
 bounded or not bounded. These are contra- 
 dictory alternatives; on the principle of Con- 
 tradiction they cannot both be true, and, on 
 the principle of Excluded Middle, one must 
 be true. This cannot be denied without de- 
 nying the primary laws of intelligence. But 
 though space must be admitted to be neces- 
 sarily 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, be- 
 yond which there is no further space. Every 
 one is conscious that this is impossible. It con- 
 tradicts also the supposition of space as a neces- 
 
36 THE CONCEPTION OF THE INFINITE. 
 
 sary 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 termi- 
 nated 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 power- 
 less 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 in- 
 conceivable, and, could we limit our attention to 
 one alone, we should deem it at once impossible 
 and absurd, and suppose its unknown opposite 
 
THE ANTINOMIES OF HAMILTON. 37 
 
 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." 
 
 It is evident that the difficulties in which Sir 
 William has here involved himself are gratu- 
 itous. The argument used to prove the latter 
 of the contradictories inconceivable breaks down 
 upon a careful examination. 
 
 We may indeed " rise from sphere to sphere 
 in the region of empty space" without trans- 
 cending the finite, and the attempt to thus 
 transcend it is as hopeless as would be the at- 
 tempt of the peacock to escape from his feet by 
 flying; for we cannot arrive at the unlimited 
 while we carry our limits with us. Each suc- 
 cessive stage simply places the limits farther 
 apart, and in no respect helps us to do away 
 with them altogether. 
 
 Such a mode of procedure forcibly reminds 
 
 one of the amusing person in Chamisso's poem, 
 
 who supposed that by quickly turning himself 
 
 around he could cause his queue to hang in 
 
 front : 
 
 4 
 
38 THE CONCEPTION OF THE INFINITE. 
 
 11 Er dreht sich links, er dreht sich rechts, 
 Es thut nichts Gut's, es thut nichts Schlecht's, 
 Der Zopf, der hangt ihm hinten." 
 
 And how analogous is the condition of one 
 who thinks that the way to reach the infinite is 
 to endlessly continue this hopeless journey be- 
 yond " the universe of matter" to that of the 
 hero as portrayed in the last verse : 
 
 " Und seht, er dreht sich immer noch 
 Und denkt : es hilft am Ende doch — 
 Der Zopf, der hangt ihm hinten." 
 
 It is not by adding space to space that we 
 arrive at the idea of infinite space, and imagi- 
 nation may well sink exhausted in the attempt 
 to find the end of the endless. 
 
 But to think space as infinite it is by no 
 means necessary to take this journey; and, so 
 far from proving that we cannot regard our 
 notion of space as infinite, the failure of any 
 such attempt to know it as a whole is the surest 
 evidence that it is indeed infinite. The latter 
 of the contradictories is thus found to be incon- 
 ceivable only when we suppose a quantitative 
 element in our conception of infinity, and, this 
 error corrected, the antinomy disappears. 
 
THE ANTINOMIES OF HAMILTON. 39 
 
 Sir "William has applied the Law of the Con- 
 ditioned also to the minimum of space: "That 
 the conceivable," he continues, " lies always be- 
 tween two inconceivable extremes is illustrated 
 by every other relation of thought. "We have 
 found the maximum of space incomprehensible; 
 can we comprehend its minimum? This is 
 equally impossible. Here, likewise, we recoil 
 from one inconceivable contradictory only to 
 infringe upon another. Let us take a portion 
 of space, however small ; we can never conceive 
 it as the smallest. It is necessarily extended, 
 and may, consequently, be divided into a half or 
 quarters, and each of these halves or quarters 
 may again be divided into other halves or quar- 
 ters, and this ad infinitum. But if we are unable 
 to construe to our mind the possibility of an 
 absolute minimum of space, we can as little 
 represent to ourselves the possibility of an in- 
 finite divisibility of any extended entity." 
 
 This is at bottom a repetition of the above- 
 mentioned error. Whether we regard space, 
 with the Kantian, as in its nature wholly com- 
 posite, and always capable of further subdivision ; 
 or, with the Berkeleyan, as composed of minima 
 visibilia, themselves not admitting of subdi- 
 
40 THE CONCEPTION OF THE INFINITE. 
 
 vision ; in neither case are we forced into a 
 choice of two inconceivables. The Berkeleyan 
 would claim that the difficulty of conceiving a 
 component part of any extended entity as itself 
 non-extended arises from the fact that the min- 
 imum visibile is represented by the imagination 
 as extended, the notion of extension being car- 
 ried over from those objects to which it right- 
 fully belongs, and is, consequently, not a true 
 minimum visibile. "While the Kantian may main- 
 tain that there is nothing inconceivable in' an 
 infinite series, rightly understood. If we sup- 
 pose a series to be infinite, we cannot, of course, 
 represent it to the mind as a completed whole, 
 but that is unnecessary to — it is incompatible 
 with — our recognition of it as infinite. The an- 
 tinomy, like its predecessor, disappears as soon 
 as it is recognized that there is no quantitative 
 element in our conception of the infinite.* 
 
 * It is odd that the statement that we are unable to con- 
 ceive of any portion of space as the smallest possible, and 
 not itself divisible into spaces, should be so constantly 
 allowed to pass unchallenged. The question is not whether, 
 when we have carried our subdivision of a surface so far 
 that an apparently unextended point of color alone is left, 
 we can in imagination substitute for that point an extended 
 
THE ANTINOMIES OF HAMILTON. 41 
 
 Let us turn now to Sir William's application 
 of the law to our conception of time : 
 
 "In like manner Time; — this is a notion 
 even more universal than space, for while we 
 exempt from occupying space the energies of 
 mind, we are unable to conceive these as 
 not occupying time. Thus, we think every- 
 thing, mental and material, as in time, and out 
 of time we can think nothing. But if we at- 
 tempt to comprehend time, either in whole or in 
 
 surface, and proceed to subdivide that, convinced that any 
 system of relations developed from the latter may lawfully 
 be carried over to all possible future experiences of a simi- 
 lar nature which may be found to be connected with the 
 former ; but the question really is whether we can conceive 
 that identical apparently unextended spot of color divided 
 and subdivided. If we say that we are dividing it in 
 thought, when our division proceeds thus through a rep- 
 resentative, if we insist in applying to the object in the 
 two cases the word "same," we should never forget that 
 we are using this highly ambiguous word "same" in by no 
 means the strictest of its four distinctly different senses, 
 and should be very sure that we are not practising self- 
 deception by juggling with the several meanings of the 
 word. When I say that with the Kantian hypothesis we 
 are not forced to embrace what is inconceivable, I refer 
 only to the point under consideration, the conceivability of 
 an endless series per se. 
 
 4* 
 
 SIT! 
 
 C4LIF0R^ 
 
42 THE CONCEPTION OF THE INFINITE. 
 
 part, we find that thought is hedged in between 
 two incomprehensibles. Let us try the whole. 
 And here let us look back, — let us consider time 
 a "parte ante. And here we may surely flatter 
 ourselves that we shall be able to conceive time 
 as a whole, for here we have the past period 
 bounded by the present; the past cannot, there- 
 fore, be infinite or eternal, for a bounded infi- 
 nite is a contradiction. But we shall deceive 
 ourselves. We are altogether unable to con- 
 ceive time as commencing ; we can easily rep- 
 resent to ourselves time under any relative 
 limitation of commencement and termination, 
 but we are conscious to ourselves of nothing 
 more clearly, than that it would be equally 
 possible to think without thought, as to con- 
 strue to the mind an absolute commencement 
 or an absolute termination of time, — that is, a 
 beginning and an end beyond which time is 
 conceived as non-existent. Goad imagination 
 to the utmost, it still sinks paralyzed within 
 the bounds of time, and time survives as the 
 condition of the thought itself in which we 
 annihilate the universe. On the other hand, 
 the concept of past time as without limit — with- 
 out commencement — is equally impossible. We 
 
THE ANTINOMIES OF HAMILTON. 43 
 
 cannot conceive the infinite regress of time; 
 for such a notion could only be realized by the 
 infinite addition in thought of finite times, and 
 such an addition would itself require an eternity 
 for its accomplishment. If we dream of effect- 
 ing this, we only deceive ourselves by substi- 
 tuting the indefinite for the infinite, than which 
 no two notions can be more opposed. 
 
 " The negation of a commencement of time 
 involves, likewise, the affirmation that an infinite 
 time has, at every moment, already run ; that is, 
 it implies the contradiction that an infinite has 
 been completed. For the same reasons we are 
 unable to conceive an infinite progress of time; 
 while the infinite regress and the infinite prog- 
 ress, taken together, involve the triple contra- 
 diction of an infinite concluded, of an infinite 
 commencing, and of two infinities not exclusive 
 of each other." * 
 
 The statement that past time cannot be re- 
 garded as infinite because limited by the present 
 
 * " Metaphysics," New York, 1880, p. 529. See, also, 
 Spencer's "First Principles," New York, 1875, p. 81: "How 
 self-destructive is the assumption of two or more Infinites 
 is manifest on remembering that such Infinites, by limiting 
 each other, would become finite." 
 
44 THE CONCEPTION OF THE INFINITE. 
 
 is based upon the erroneous supposition that 
 what is limited at one point cannot be infinite. 
 But, as has been shown in the preceding chap- 
 ter, one point is not sufficient to define a line 
 as finite ; and time, which we represent to our- 
 selves under the form of a continuous line, must 
 be regarded as infinite unless limited at two 
 points. Time past and time future are two in- 
 finites, and as such are perfectly conceivable. 
 The difficulty respecting the possibility of two 
 infinites mutually exclusive of each other is a 
 difficulty only under a false conception of the 
 infinites as quantitative wholes, and may easily 
 be made to disappear. 
 
 The assertion, also, that the past cannot be 
 infinite, as " a bounded infinite is a contradic- 
 tion/' may well be scanned. Arguments drawn 
 from the etymological signification of a word 
 are of small value, unless that expresses the 
 true and whole content of the word. That such 
 is not the case here is evident. A line limited 
 at but one point is certainly not finite, for it 
 cannot be regarded as a quantitative whole; 
 cannot be increased, diminished, or compared 
 in length with other lines; in short, is not sub- 
 ject to the conditions of the finite. If, then, for 
 
THE ANTINOMIES OF HAMILTON. 45 
 
 etymological reasons, we exclude it from the 
 class of infinites, we have the finite, the infinite, 
 and a third class, a terthtm quid, which lies be- 
 tween the two, and might be humorously de- 
 scribed as " infinite at one end." But, etymol- 
 ogy aside, there is no difficulty in classing such 
 a line with one that has no limits, for they are 
 subject to precisely the same conditions, and 
 equally distinct from the finite. However, the 
 appellation is a matter of taste ; the thing which 
 it is important to bear in mind is that a line 
 with but one limit can no more be regarded as 
 a quantitative whole than a line absolutely with- 
 out limits ; and, whether we choose to call time 
 past infinite or finite, that we may have a clear 
 knowledge of it as limited only by the present, 
 without attempting to pass over its parts in 
 succession, and thus arrive at the whole. 
 
 The assertion, too, that the negation of a 
 commencement of time " implies the contradic- 
 tion that an infinite has been completed" is 
 misleading. The word " completed" is an un- 
 fortunate one to use in this connection, as it 
 suggests to the mind the idea of progression 
 from a beginning to an end. The denial of a 
 commencement of time does imply that an in- 
 
46 THE CONCEPTION OF THE INFINITE. 
 
 finite is past, but not that it is completed in any 
 such sense as to be enclosed within limits, for 
 it is quite conceivable that it never began, and 
 the present moment is, by supposition, the only 
 limit. 
 
 Removing the misconceptions just noticed, the 
 whole force of the argument for the inconceiv- 
 ability of time as infinite lies, as in the former 
 cases, in the idea that the infinite may only be 
 known by exhausting it in its totality, through 
 the successive addition of its finite parts, and 
 this antinomy also proves to be a gratuitous 
 one. 
 
 Having argued thus far the inconceivability 
 of time as a maximum, Sir William turns, as in 
 his discussion on our knowledge of space, to a 
 consideration of time as a minimum: 
 
 " Now take the parts of time, — a moment, for 
 instance; this we must conceive as either divis- 
 ible to infinity, or that it is made up of certain 
 absolutely smallest parts. One or other of these 
 contradictories must be the case. But each is, 
 to us, equally inconceivable. Time is a pro- 
 tensive quantity, and, consequently, any part of 
 it, however small, cannot, without a contradic- 
 tion, be imagined as not divisible into parts, and 
 
THE ANTINOMIES OF HAMILTON. 47 
 
 these parts into others ad infinitum. But the 
 opposite alternative is equally impossible ; we 
 cannot think this infinite division. One is neces- 
 sarily true ; but neither can be conceived pos- 
 sible. It is on the inability of the mind to con- 
 ceive either the ultimate indivisibility or the 
 endless divisibility of space and time that the 
 arguments of the Eleatic Zeno against the pos- 
 sibility of motion are founded, — arguments 
 which at least show that motion, however cer- 
 tain as a fact, cannot be conceived possible, as 
 it involves a contradiction." * 
 
 With reference to the former of the alterna- 
 tives offered, the Berkeleyan would answer that 
 in the case of " protensive" quantity, as in the 
 case of extensive, the difficulty of conceiving 
 the unit, itself indivisible, lies in the imagina- 
 tion, and may, with precautions, be obviated; 
 while with reference to the latter, the Kantian 
 may answer, as before, that an infinite series is 
 not intrinsically unthinkable. I may remark, 
 en passant, that the idea of the theoretic impos- 
 sibility of motion is a wholly erroneous one, and 
 falls with the errors upon which it is based. 
 
 * " Metaphysics," New York, 1880, pp. 529-30. 
 
48 THE CONCEPTION OF THE INFINITE. 
 
 One turns from an examination of Sir Wil- 
 liam Hamilton's application of the Law of the 
 Conditioned to Space and Time with a convic- 
 tion that if the Philosophy of the Conditioned 
 has no better props to sustain it than these 
 prove to be, it may turn out as insecure an 
 edifice as the house that a certain foolish indi- 
 vidual, according to the parable, founded upon 
 sand. 
 
 It remains to consider a case which ap- 
 parently militates against the theory that an 
 infinite series can never be regarded as a 
 whole. 
 
 Let us suppose a point moving uniformly 
 along the line AB, over the whole of which it 
 can pass in one minute. In J of a minute it 
 will have passed over J of the line; in J of a 
 minute over J more ; in J of a minute J more, 
 etc. When the minute is completed the point 
 will have passed over the whole line. Has it 
 not passed successively over the whole series, 
 thus completing it, and arriving at as its 
 lower limit? And may we not say that the 
 sum of all the terms in the series is equal to 
 the whole line passed over ? Is it not a quan- 
 titative whole? 
 
THE ANTINOMIES OF HAMILTON. 49 
 
 A little reflection will reveal the fallacy in 
 this reasoning. The series is not completed at 
 all, but is truly infinite. It is limited at one 
 poiut by the highest term (J), but is not limited 
 at another point by a lowest term (0) ; for the 
 can only be assumed as a limit to the series by 
 breaking the law of the series, which is that 
 each term shall be half as great as the one pre- 
 ceding. 
 
 We can never, by halving something, arrive 
 at nothing; a division of substance will never 
 give us that which is not substance. The 0, 
 since it does not make one in the series, cannot 
 limit the series. The Kantian may answer the 
 question by reasoning as follows : " The point 
 in question passes over the whole line, not by 
 completing the descending series until it arrives 
 at a lowest term in the simple, and from that 
 passes to zero, but by the successive addition of 
 spaces, which are themselves composites. ' As 
 space is not a composite of substances (and not 
 even of real accidents), if I abstract all compo- 
 sition therein, nothing, not even a point, re- 
 mains ; for a point is possible only as the limit 
 of a space, consequently of a composite. Space 
 and time, therefore, do not consist of simple 
 
 5 
 
50 THE CONCEPTION OF THE INFINITE. 
 
 parts.' * We cannot, therefore, consider any 
 member of the series in question as the smallest 
 possible, nor the zero as a limit to the series; 
 nor can we regard the series as in any sense 
 completed. If we discontinue the subdivision 
 at any point whatever, we may justly say that 
 the foot or yard contains all the terms of the 
 series. But when the point has reached zero it 
 has reached it by breaking the series, not com- 
 pleting it. A completion the law of the series 
 renders impossible." f And the Berkeleyan 
 might answer that a completion of the journey 
 along the line by no means implies the com- 
 
 * " Critique of Pure Reason ;" "Observations on the Sec- 
 ond Antinomy." Ed. Bohn. 
 
 f How the series can be broken by the progressive motion 
 of a point over a line when, by hypothesis, the series is 
 throughout applicable to the line (i.e., the line is infinitely 
 divisible), the Kantian must explain ; the fact remains that 
 neither an infinite nor a finite series can be completed by 
 breaking its law; and unless we claim that half of some- 
 thing, when that something is small enough, is equal to 
 nothing, we cannot bring the zero into the series as a limit. 
 Even could this be done, the desired point would not be 
 established, as the series would then be limited at two points, 
 consequently in no sense infinite, an,d its completion would 
 not be the completion of an infinite series. 
 
THE ANTINOMIES OF HAMILTON. 51 
 
 pletion of an infinite series, since relations of 
 quantity may be divorced from all content of 
 actual being, and used purely symbolically. 
 Mathematical reasonings, he would argue, are 
 applicable to space only within the limits of a 
 possible perception ; and the series may be truly 
 infinite, though the line-portions to which some 
 of its members are applied may be limited in 
 number.* 
 
 This problem, it will be seen, is simply the 
 old puzzle of Achilles and the tortoise in a 
 somewhat altered dress ; and the answer which 
 has been given to that puzzle — that the series 
 " runs into infinitesimals which are practical 
 zeros, and, even if theoretically infinite in num- 
 ber, really are all included in that finite length 
 which Achilles will quickly get over"f — is not 
 
 * I have taken up both positions to bring out clearly the 
 fact that the assertion of the conceivability of the infinite 
 is quite independent of metaphysical theories as to the nature 
 of space and time. The two questions are wholly distinct, as 
 I will show in a later chapter. 
 
 f This is the solution of the problem given by the late Pro- 
 fessor Atwater in his little book on the Elements of Logic. 
 The explanation offered could not be much worse. What, 
 we may ask, is meant by a " practical zero," as distinguished 
 from a theoretical? And the assertion that these practical 
 
52 THE CONCEPTION OF THE INFINITE. 
 
 a true answer, since it regards a series, which, 
 by the very terms of its statement, is incapable 
 of completion, as a completed whole and equal 
 to a finite. 
 
 Before leaving this chapter I would remark 
 that it is interesting to notice how wide-spread 
 has been the conviction that the only way to 
 arrive at a cognition of the infinite is to proceed 
 to an endless addition of finites. Hamilton, we 
 have seen, makes much of it. Mansel follows 
 in his footsteps. Mr. Herbert Spencer quotes 
 him with approbation. And Kant, as we shall 
 see in the chapter following, did not escape the 
 snare. " To have actually in the mind the idea 
 of a space infinite," says Locke,* " is to suppose 
 the mind already passed over, and actually to 
 have a view of all those repeated ideas of space 
 which an endless repetition can never totally 
 represent to it; — which carries in it a plain con- 
 tradiction." 
 
 zeros, "even if theoretically infinite in number, really are 
 all included" in a finite length, would seem to draw a dis- 
 tinction between the theoretical and the practical, little to 
 the advantage of the former. It makes theoretical about 
 equivalent to unreal. 
 
 * "Essay concerning Human Understanding," book ii. 
 ch. xvii., I 7. 
 
CHAPTER IV. 
 
 KANT, MILL, AND CLIFFORD. 
 
 It is interesting to notice that the truth that 
 our conception of infinity contains no quanti- 
 tative element has been seen, like Thule, 
 " through the mist" by several acute minds, 
 who have yet not seen the truth with sufficient 
 clearness to escape the common errors arising 
 from the introduction of the contradictory ele- 
 ment into their discussions. Immanuel Kant, 
 although he has based the proof of the thesis of 
 his first antinomy on a false conception of in- 
 finity, and although, after correctly criticising 
 the false conception, he himself lapses into it, 
 yet perceived, and in so many words gave ex- 
 pression to the fact, that the conception of the 
 infinite is qualitative. 
 
 The thesis of the first antinomy maintains 
 that the world has a beginning in time, and is 
 limited with regard to space, both of which 
 propositions are denied in the antithesis. The 
 proofs offered in support of the antithesis may 
 
 5* 53 
 
54 THE CONCEPTION OF THE INFINITE. 
 
 be passed over as extraneous to the subject ; 
 those in support of the thesis I will quote, not 
 for the purpose of again refuting them, for they 
 are identical with those used by Sir William 
 Hamilton in his antinomies, but that I may 
 give the observations appended to them, which 
 are very significant in their contextual connec- 
 tion. The proof proceeds by assuming the 
 truth of the antithesis, and then showing it to 
 be impossible. 
 
 " Granted, that the world has no begin- 
 ning in time; up to every given moment of 
 time an eternity must have elapsed, and there- 
 with passed away an infinite series of suc- 
 cessive conditions or states of things in the 
 world. Now, the infinity of a series consists 
 in the fact that it never can be completed 
 by means of a successive synthesis. It fol- 
 lows that an infinite series, already elapsed, 
 is impossible, and that consequently a begin- 
 ning of the world is a necessary condition of 
 its existence. And this was the first thing to 
 be proved. 
 
 " As regards the second, let us take the oppo- 
 site for granted. In this case the world must 
 be an infinite given total of coexistent things. 
 
KANT, MILL, AND CLIFFORD. 55 
 
 Now, we cannot cogitate the dimensions of a 
 quantity, which is not given within certain lim- 
 its of an intuition, in any other way than by 
 means of the synthesis of its parts,* and the 
 total of such a quantity only by means of a 
 completed synthesis or the repeated addition of 
 unity to itself. Accordingly, 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, which is impossible. For this 
 reason an infinite aggregate of actual things 
 cannot be considered as a given whole, conse- 
 quently, not as a contemporaneously given 
 whole. The world is consequently, as regards 
 extension in space, not infinite, but enclosed in 
 
 * Kant says, in a foot-note, " We may consider an unde- 
 termined quantity as a whole when it is enclosed within 
 limits, although we cannot construct or ascertain its totality 
 by measurement, — that is, by the successive synthesis of its 
 parts. For its limits of themselves determine its complete- 
 ness as a whole.' 1 ' This method being absent in the case of 
 infinites, Kant thinks he can cognize them only by falling 
 back upon the successive synthesis. 
 
56 THE CONCEPTION OF THE INFINITE. 
 
 limits. And this was the second thing to be 
 proved."* 
 
 It will be noticed that in the first part of the 
 proof the word completed (vollendet) is used in 
 the manner before objected to as misleading. 
 When we speak of a 'series as "completed by 
 means of a successive synthesis," we are apt 
 to regard it as a whole, with a beginning as 
 well as an end. Of course, when we are con- 
 sidering time past as limited by the present, 
 such an association must be unfortunate. 
 
 The observations on the thesis are the fol- 
 lowing : 
 
 " In bringing forward these conflicting argu- 
 ments, I have not been on the search for soph- 
 isms for the purpose of availing myself of 
 special pleading, which takes advantage of the 
 carelessness of the opposite party, appeals to a 
 misunderstood statute, and erects its unright- 
 eous claims upon an unfair interpretation. Both 
 proofs originate fairly from the nature of the 
 case, and the advantage presented by the mis- 
 
 * Iramanuel Kant's " Sammtliche Werke," Leipzig, 1867. 
 Dritter Band, s. 304. I have taken the rendering of Meikle- 
 john's excellent translation of the Critique. 
 
KANT, MILL, AND CLIFFORD. 57 
 
 takes of the dogmatists of both parties has been 
 completely set aside. 
 
 " The thesis might also have been unfairly 
 demonstrated by the introduction of an erro- 
 neous conception of the infinity of a given 
 quantity. A quantity is infinite if a greater 
 than itself cannot possibly exist. The quantity 
 is measured by the number of given units — 
 which are taken as a standard — contained in 
 it. Now, no number can be the greatest, be- 
 cause one or more units can always be added. 
 It follows that an infinite given quantity, conse- 
 quently an infinite world (both as regards time 
 and extension), is impossible. It is, therefore, 
 limited in both respects. In this manner I 
 might have conducted my proof; but the con- 
 ception given in it does not agree with the true 
 conception of an infinite whole. In this there 
 is no representation of its quantity ; it is not 
 said how large it is ; consequently its concep- 
 tion is not the conception of a maximum. "We 
 cogitate in it merely its relation to an arbitra- 
 rily assumed unit, in relation to which it is 
 greater than any number. Now, just as the 
 unit which is taken is greater or smaller, the 
 infinite will be greater or smaller; but the in- 
 
58 THE CONCEPTION OF THE INFINITE. 
 
 finity, which consists merely in the relation to 
 this given unit, must remain always the same, 
 although the absolute quantity of the whole is 
 not thereby cognized. 
 
 " The true (transcendental) conception of in- 
 finity is that the successive synthesis of unity in 
 the measurement of a given quantum can never 
 be completed. Hence it follows, without possi- 
 bility of mistake, that an eternity of actual suc- 
 cessive states up to a given (the present) mo- 
 ment cannot have elapsed, and that the world 
 must therefore have a beginning. 
 
 "In regard to the second part of the thesis, 
 the difficulty as to an infinite and yet elapsed 
 series disappears ; for the manifold of a world 
 infinite in extension is contemporaneously given. 
 But, in order to cogitate the total of this mani- 
 fold, as we cannot have the aid of limits consti- 
 tuting by themselves this total in intuition, we 
 are obliged to give some account of our concep- 
 tion, which in this case cannot proceed from 
 the whole to the determined quantity of the 
 parts, but must demonstrate the possibility of a 
 whole by means of a successive synthesis of the 
 parts. But as this synthesis must constitute a 
 series that cannot be completed, it is impossible 
 
KANT, MILL, AND CLIFFORD. 59 
 
 for us to cogitate prior to it, and consequently 
 not by means of it, a totality. For the concep- 
 tion of totality itself is in the present case the 
 representation of a completed synthesis of the 
 parts; and this completion, and consequently its 
 conception, is impossible." 
 
 We here find brought forward a conception 
 of the infinite which is declared faulty; a dec- 
 laration of the point in which it differs from the 
 true conception; and a statement of what, ac- 
 cording to Kant, the true conception really is. 
 The false conception is that "a quantity is in- 
 finite if a greater than itself cannot possibly 
 exist." We may readily see that such a con- 
 ception gives us, not an infinite, but a finite. 
 Not only is the word " greater" inapplicable to 
 infinites, but the very expression, " a quantity is 
 infinite," is absurd, as involving a flat contra- 
 diction. Kant was too keen a thinker not to 
 see that that which admits of an addition ot 
 units, and, consequently, of increase as a whole, 
 cannot be infinite. This does not agree, he says, 
 with the true conception of the infinite, in which 
 "there is no representation of its quantity; it is 
 not said how large it is; consequently its con- 
 ception is not the conception of a maximum." 
 
GO THE CONCEPTION OF THE INFINITE. 
 
 Could there be a clearer recognition of the fact 
 that the conception is not quantitative ? 
 
 But it is evident that Kant did not see the 
 full force and the logical consequences of the 
 statement. In the sentence immediately pre- 
 ceding the one in which he recognizes the quali- 
 tative character of the conception he uses the 
 phrase "an infinite whole;" and in the sentences 
 immediately following he brings forward a con- 
 ception faulty in precisely the same respect as the 
 one criticised : " We cogitate in it merely its rela- 
 tion to an arbitrarily assumed unit, in relation to 
 which it is greater than any number. Now, just 
 as the unit which is taken is greater or smaller, 
 the infinite will be greater or smaller ; but the 
 infinity, which consists merely in the relation to 
 this given unit, must remain always the same, 
 although the absolute quantity of the whole is 
 not thereby cognized." That is, if we designate 
 the infinite by a, the unit by 6, and the infinity 
 (the relation between a and b) by x, we find that 
 a varies as b, and x remains always the same (this 
 can only mean numerically the same). In this 
 case x is simply an indefinite number, and the 
 " absolute quantity of the whole" can certainly 
 be cognized. When we say the infinity remains 
 
KANT, MILL, AND CLIFFORD. Q\ 
 
 always the same, the question naturally arises, 
 The same in what? In amount? If so, we 
 have but the finite. The error is here, perhaps, 
 not quite so palpable, but is just as real as in 
 the case which Kant criticises, and it is of pre- 
 cisely the same nature. Both parts of the proof 
 given in support of the thesis of course fall to 
 the ground when this error is rectified. 
 
 The last two observations are merely a re- 
 statement of the proofs of the thesis. The re- 
 mark made in the last one, that "in order to 
 cogitate the total of this manifold, as we cannot 
 have the aid of limits constituting by themselves 
 this total in intuition, we are obliged to give 
 some account of our conception, which in this 
 case cannot proceed from the whole to the de- 
 termined quantity of the parts, but must dem- 
 onstrate the possibility of a whole by means of 
 a successive synthesis of the parts," will lose 
 all its force if the words " total" and " whole" 
 are abstracted. If this manifold is to be known 
 as a whole, we cannot, of course, arrive at a 
 knowledge of it without cogitating all its parts 
 as contained in it; but since it is impossible 
 that it should be thus known, we may "give 
 some account of our conception" by simply 
 
(52 THE CONCEPTION OF THE INFINITE. 
 
 stating that it is without limit, thus recognizing 
 it as truly infinite. 
 
 It seems odd that Kant should have seen an 
 error, objected to it, and fallen into it on the 
 same page; should have said that there is no 
 representation of its quantity in the conception 
 of the infinite, and then have called it a whole; 
 declared that the true conception does not say 
 how large it is, and yet have affirmed that the 
 infinity is always the same, while the series as a 
 whole varies with the unit. But, with and in 
 spite of all his inconsistencies, it must be allowed 
 that he recognized the truth, however imper- 
 fectly, that our conception of the infinite has, 
 properly speaking, no quantitative element, but 
 is purely qualitative. 
 
 This truth has also been recognized, though 
 less clearly, by John Stuart Mill, in his "Ex- 
 amination of Sir William Hamilton's Philoso- 
 phy," where he maintains, in opposition to Sir 
 William, that the infinite is not inconceivable ; 
 and we can see in the paragraphs which he 
 devotes to the subject that Mill had in mind 
 two 'distinct conceptions, a quantitative and a 
 qualitative. To the former, which he calls the 
 adequate conception, he acknowledges we cannot 
 
KANT, MILL, AND CLIFFORD. (53 
 
 attain ; the latter, which is truly qualitative, 
 though he does not apply to it that name, he 
 claims to be a real conception, implying nothing 
 inconceivable. 
 
 His attempted refutation of Hamilton's argu- 
 ment for the inconceivability of infinite space 
 proceeds as follows:* 
 
 " Our author goes on to repeat the argument 
 used in his reply to Cousin, that Infinite Space 
 is inconceivable, because all the conception we 
 are able to form of it is negative, and a negative 
 conception is the same as no conception. ' The 
 infinite is conceived only by the thinking away 
 of every character by which the finite was con- 
 ceived.' To this assertion I oppose my former 
 reply. Instead of thinking away every char- 
 acter of the finite, we think away only the idea 
 of an end, or a boundary. Sir W. Hamilton's 
 proposition is true of ' The Infinite,' the mean- 
 ingless abstraction, but it is not true of Infinite 
 Space. In irying to form a conception of that, 
 we do not think away its positive characters. 
 "We leave to it the character of space ; all that 
 
 * "Examination of Sir William Hamilton's Philosophy," 
 Boston, 1868, vol. i., pp. 104 et seq. 
 
64 THE CONCEPTION OF THE INFINITE. 
 
 belongs to it as space; its three dimensions, with 
 all their geometrical properties. We leave to 
 it also a character which belongs to it as Infi- 
 nite, that of being greater than any other space. 
 If an object which has these well-marked posi- 
 tive attributes is unthinkable, because it has a 
 negative attribute as well, the number of think- 
 able objects must be remarkably small. Nearly 
 all our positive conceptions which are at all 
 complex include negative attributes. I do not 
 mean merely the negatives which are implied in 
 affirmatives, as in saying that snow is white we 
 imply that it is not black, but independent neg- 
 ative attributes superadded to these, and which 
 are so real that they are often the essential char- 
 acters, or differentiae, of classes. Our concep- 
 tion of dumb is of something which cannot 
 speak ; of the brutes, as of creatures which have 
 not reason ; of the mineral kingdom, as the part 
 of nature which has not organization and life; 
 of immortal, as that which never dies. Are all 
 these examples of the Inconceivable ? So false 
 is it that to think a thing under a negation is to 
 think it as unthinkable. 
 
 " In other passages, Sir "W. Hamilton argues 
 that we cannot conceive infinite space, because 
 
KANT, MILL, AND CLIFFORD. 65 
 
 we would require infinite time to do it in. It 
 would of course require infinite time to carry 
 our thoughts in succession over every part of in- 
 finite space. But on how many of our finite 
 conceptions do we think it necessary to perform 
 such an operation ? Let us try the doctrine 
 upon a complex whole, short of infinite, such as 
 the number 695,788. Sir W. Hamilton would 
 not, I suppose, have maintained that this num- 
 ber is inconceivable. How long did he think it 
 would take to go over every separate unit of 
 this whole, so as to obtain a perfect knowledge 
 of that exact sum, as different from all other 
 sums, either greater or less ? Would he have 
 said that we could have no conception of the 
 sum until this process had been gone through ? 
 "We could not indeed have an adequate concep- 
 tion. Accordingly, we never have an adequate 
 conception of any real thing. But we have a 
 real conception of an object if we conceive it by 
 any of its attributes that are sufficient to distin- 
 guish it from all other things. We have a con- 
 ception of any large number when we have 
 conceived it by some one of its modes of com- 
 position, such as that indicated by the position 
 of its digits. We seldom get nearer than this 
 
 6* 
 
QQ THE CONCEPTION OF THE INFINITE. 
 
 to an adequate conception of any large number. 
 But for all intellectual purposes this limited 
 conception is sufficient ; for it not only enables 
 us to avoid confounding the number in our cal- 
 culations with any other numerical whole, — 
 even with those so nearly equal to it that no 
 difference between them would be perceptible 
 by sight or touch, unless the units were drawn 
 up in a manner expressly adapted for displaying 
 it, — but we can also, by means of this attribute 
 of the number, ascertain and add to our concep- 
 tion as many more of its properties as we please. 
 If, then, we can obtain a real conception of a 
 finite whole without going through all its com- 
 ponent parts, why deny us a real conception of 
 an infinite whole because to go through them all 
 is impossible ? Not to mention that even in the 
 case of the finite number, though the units com- 
 posing it are limited, yet, Number being infinite, 
 the possible modes of deriving any given num- 
 ber from other numbers are numerically infinite; 
 and as all these are necessary parts of an ade- 
 quate conception of any number, to render our 
 conception even of this finite whole perfectly 
 adequate would also require an infinite time. 
 " But though our conception of infinite space 
 
KANT, MILL, AND CLIFFORD. 67 
 
 can never be adequate, since we can never ex- 
 haust its parts, the conception, as far as it 
 goes, is a real conception. We completely 
 realize in imagination the various attributes 
 composing it. "We realize it as Space. We 
 realize it as greater than any given space. W r e 
 even realize it as endless, in an intelligible 
 manner, — that is, we clearly represent to our- 
 selves that however much of space has been 
 already explored, and however much more of 
 it we may imagine ourselves to traverse, we are 
 no nearer to the end of it than we were at 
 first time ; however often we repeat the process 
 of imagining distance extending in any direc- 
 tion from us, that process is always susceptible 
 of being carried farther. This conception is 
 both real and perfectly definite. It is not vague 
 and indeterminate as a merely negative notion 
 is. We possess it as completely as we possess 
 any of our clearest conceptions, and we can 
 avail ourselves of it as well for ulterior mental 
 operations. As regards the Extent of Space, 
 therefore, Sir W. Hamilton does not seem to 
 have made out his point; one of the two con- 
 tradictory hypotheses is not inconceivable." 
 One can see from these extracts how the 
 
68 THE CONCEPTION OF THE INFINITE. 
 
 idea of quantity entered into and vitiated 
 Mill's reasonings on the infinite. In arguing 
 that our notion of infinite space is not a purely 
 negative one, he enumerates several positive at- 
 tributes that we leave to the conception when we 
 take away the notion of limits. Among them 
 he places "a character which belongs to it as 
 Infinite, that of being greater than any other 
 space." Evidently the character of being greater 
 than any other space is a quantitative attribute, 
 and can belong only to "a space," — a finite; 
 while space infinite is not, properly speaking, " a 
 space" at all. Since it has no size, it cannot be 
 marked out from other spaces by its size. And 
 Mill has manifestly committed the same error 
 which has misled Sir William Hamilton, whom 
 he criticises, when he says that it would, .of 
 course, require infinite time to carry our thoughts 
 in succession over every part of infinite space. 
 How can we speak of every part of that which 
 is not quantitatively considered? of that which, 
 by its very definition, is incapable of being a 
 whole ? This is precisely Hamilton's error, and 
 the cause of all his difficulties. A little farther 
 on Mill speaks, just as Kant does, of "an infi- 
 nite whole," never noticing the contradiction in 
 
KANT, MILL, AND CLIFFORD. 69 
 
 the adjective; and in the paragraph following he 
 repeats a former blunder in the statement that 
 our knowledge of space can never be adequate, 
 "since we can never exhaust its parts." 
 
 It is strange that side by side with this con- 
 ception of a quantitative infinite, which, to be 
 adequately known, must be known as a whole, 
 we should find a real, though imperfect, anal- 
 ysis of the true conception, and an affirmation 
 of its conceivability. We can conceive infinite 
 space, says Mill ; we can conceive it as space, 
 as greater than any given space, and even as 
 endless, in an intelligible manner. When he 
 comes to describe this intelligible manner of 
 knowing the infinity of space, he uses an un- 
 fortunate phrase, — " we clearly represent to our- 
 selves that however much of space has been 
 already explored, ... we are no nearer to 
 the end of it than we were at first time;" 
 dragging in the idea of a limit, which is also 
 done, as above remarked, by calling it greater 
 than any other space. But with all these con- 
 cessions to the old erroneous doctrine, it cannot 
 be denied that Mill held that we may know In- 
 finite Space in some other way than by a suc- 
 cessive synthesis of finite spaces, and that he 
 
70 THE CONCEPTION OF THE INFINITE. 
 
 attempted an enumeration of the psychical ele- 
 ments comprehended by the conception, leaving 
 out the notion of quantity. This conception, 
 which he calls inadequate, but which he yet 
 insists upon as a conception of the infinite, is 
 qualitative, and harmonizes with the true char- 
 acter of our conception of infinity. 
 
 The last writer to whom I will advert as 
 having had a knowledge, more or less clear, of 
 the true qualitative nature of our conception 
 of infinity, is Professor William Kingdon Clif- 
 ford, by whose untimely death England has lost 
 one of her acutest and most analytic minds. 
 There are in his short paper entitled " Of Boun- 
 daries in General," and devoted to the point, the 
 line, and the surface, some interesting and sig- 
 nificant passages, which I will quote, and upon 
 which I will afterwards comment. The pas- 
 sages are these : 
 
 "Infinite ; it is a dreadful word, I know, until 
 you find out that you are familiar with the 
 thing which it expresses. In this place it 
 means that between any two positions there 
 is some intermediate position; between that 
 and either of the others, again, there is some 
 other intermediate ; and so on without any end. 
 
KANT, MILL, AND CLIFFORD. 71 
 
 Infinite means without any end. If you went 
 on with that work of counting forever, you 
 would never get any farther than the begin- 
 ning of it. At last you would have two po- 
 sitions very close together, but not the same; 
 and the whole process might be gone over 
 again, beginning with those as many times as 
 you like." . . . 
 
 " In fact, when we said that there is an infi- 
 nite number of points in a piece of line-room, 
 we might have said a great deal more. Suppose, 
 for instance, that any one said, ' How many 
 miles is it possible to go up into space V The 
 answer would of course be, ' An infinite num- 
 ber of miles.' (Don't be frightened at this 
 continual occurrence of the word infinite : it 
 still means ' without any end,' and nothing 
 more.) In this case, if you go a mile and count 
 one, then another and count two, and so on, 
 all we mean is that the process would never 
 end. There would still be space left to go up 
 into, however many millions of miles you had 
 counted. But still all those miles would be 
 counted and done with. Your task would 
 have been distinctly begun, and there would be 
 nothing more to say to the miles behind you. 
 
72 THE CONCEPTION OF THE INFINITE. 
 
 But try now to count the points in a piece of 
 line. You count one, two, three, four, a mil- 
 lion points ; and your task is not even begun. 
 The line is all there, exactly as it was before ; 
 absolutely none of it is done with. The mil- 
 lion points take up no more line-room than one 
 point; that is to say, absolutely none at all. 
 When, then, we are talking of the points in a 
 piece of line, we must say not merely that there 
 is a never-ending number of them (which there 
 is), but that they are out of the reach of num- 
 ber altogether. All the points in a line are 
 not, properly speaking, a number of points at 
 all. If we are going to speak about the num- 
 ber of points in a line, we must settle before- 
 hand that we are going to use the word in a 
 new sense, which is not derived from counting, 
 but from this very observation to which wc 
 have applied it. 
 
 "Let us now make use of our idea of a 
 path. When a point moves along a line, we 
 know that between any two positions of it there 
 is an infinite number (in this new sense) of in- 
 termediate positions. That is because the mo- 
 tion is continuous. Each of those positions is 
 where the point was at some instant or other. 
 
KANT, MILL, AND CLIFFORD. 73 
 
 Between the two end positions on the line, the 
 point where the motion began and the point 
 where it stopped, there is no point of the line 
 which does not belong to that series. We have 
 thus an infinite series of successive positions of 
 a continuously moving point, and in that series 
 are included all the points of a certain piece of 
 line-room. May we say, then, that the line is 
 made up of that infinite series of points ? 
 
 " Yes, if we mean no more than that the 
 series makes up the points of the line. But no, 
 if we mean that the line is made up of those 
 points in the same way that it is made up of a 
 great many very small pieces of line. A point 
 is not to be regarded as a part of a line in any 
 sense whatever." 
 
 Evidently Clifford saw more clearly than either 
 Kant or Mill that the notion of quantity is 
 foreign to our conception of infinity. In speak- 
 ing of the infinite extent of space, he does not 
 make the absurd and tautological assertion that 
 it would take an infinite time to exhaust it. He 
 claims that when we call a thing infinite we 
 know very well what it means : it means that it 
 will never end. He never hints at any possi- 
 bility of knowing the infinite as a whole, recog- 
 
74 THE CONCEPTION OF THE INFINITE. 
 
 nizing very clearly that it cannot be a whole. 
 And he has used the word number in two senses 
 (which a careful reading of the extracts quoted 
 will show to be a quantitative and a qualitative 
 sense), to mark the difference between numbers 
 regarded as constituent parts of wholes or sums 
 total, and number regarded as unlimited units, 
 having no relation to wholes of any sort, — a dis- 
 tinction which precisely corresponds to the dis- 
 tinction which I have drawn in a preceding 
 chapter between the quantitative and the quali- 
 tative uses of the word all, as applied to the 
 members of a finite and an infinite series. 
 
 But it is also evident that Clifford did not see 
 the full significance and value of the truth which 
 he recognized. This is clear from his contrast- 
 ing the relation of linear miles to infinite exten- 
 sion with the relation of mathematical points to 
 a line, and indicating that in the latter case 
 there is something peculiarly hopeless in the at- 
 tempt to complete the line by the addition of 
 such points. " You count one, two, three, four, 
 a million points, and your task is not even 
 begun. The line is all there, exactly as it was 
 before ; absolutely none of it is done with." So 
 that an addition of points, however long con- 
 
KANT, MILL, AND CLIFFORD. 75 
 
 tinued, has no tendency to exhaust and complete 
 the line. But in the case of infinite space, says 
 Clifford, if you go a mile and count one, another 
 and count two, and so on, all those miles would 
 be counted and done with. " Your task would 
 have been distinctly begun, and there would be 
 nothing more to say to the miles behind you." 
 But in reality, so far as they touch the question 
 under discussion, the two cases are precisely 
 similar. The points have not even begun to 
 exhaust the line, because their addition has no 
 tendency to exhaust it; their number has noth- 
 ing to do with it.* And, similarly, one may 
 justly hold that the miles counted have not even 
 begun to exhaust the infinite space, since the 
 increasing of the distance between limits has no 
 tendency to make them approach the limits of 
 that which is without limit ; since the adding of 
 quantities has no tendency to make a sum equal to 
 that which is not a quantity, and cannot be 
 equal to anything ; and since adding mile to 
 mile has no tendency to successively exhaust the 
 parts of that which is not a whole, and can have 
 
 * I speak here, of course, from Clifford's stand-point, 
 assuming the infinite divisibility of a line. 
 
76 THE CONCEPTION OF THE INFINITE. 
 
 no parts. The relation between a mile or a 
 thousand miles and infinite extension is no 
 closer than that which Clifford conceived be- 
 tween a point or a thousand points and a given 
 line. 
 
 One cannot but see, therefore, that Clifford 
 has not grasped the true nature of our concep- 
 tion of infinity in all its consequences so thor- 
 oughly as he might have grasped it. It will be 
 seen, however, that he has discussed it more 
 satisfactorily than either of the writers before 
 cited. 
 
CHAPTER V. 
 
 THE CONCEIVABLE AND THE EXISTENT. 
 
 There are two quite distinct questions to be 
 considered in a discussion of the infinite : the 
 first is, " Can an infinite object be conceived ?" 
 and the second, " Does any infinite object really 
 exist V 3 
 
 Now, one may answer the former of these ques- 
 tions in the affirmative, and yet not be com- 
 mitted to a similar answer to the latter. The 
 two propositions, "An infinite object is conceiv- 
 able" and "An infinite object exists" are dif- 
 ferent in nature, and when the former is affirmed 
 the latter still calls for proof. One may hold 
 that space is subjective, a mere abstraction from 
 experience of extended objects, and that, conse- 
 quently, space, together with the imaginary line 
 in space, exists only as it is produced in thought; 
 and, though he may on that account deny that 
 that line, as an imagined object, or any actually 
 existent line is infinite, or that any line could 
 possibly be made infinite, he may yet claim that 
 
 he can conceive an infinite line. 
 
 7* 77 
 
78 THE CONCEPTION OF THE INFINITE. 
 
 If he belong to one school of thought he will 
 not only claim that he can conceive space in- 
 finite, but will assume on a priori grounds the 
 infinity of space as actually existent. If he be an 
 adherent of another school he may hold that the 
 proposition " space is infinite" is incapable of 
 proof, and that it can never be maintained ; but 
 he will not on that account deny that he can con- 
 ceive infinite space. One may maintain that our 
 assent to the former proposition is conditioned 
 on our assent to the latter; that if the infinite 
 be so unattainable and even contradictory a con- 
 ception as Sir William Hamilton has held, we 
 would have no reason to believe the existence of 
 any infinite object either possible or actual ; but 
 certainly no one will hold that the first of the 
 two propositions is so based upon the second as 
 to necessarily stand or fall with it. The fact 
 that I can imagine 
 
 "... the Cannibals, that each other eat, 
 The Anthropophagi, and men whose heads 
 Do grow beneath their shoulders," 
 
 does not prove such objects to have real exist- 
 ence. If men were only able to represent in the 
 imagination what has its actual prototype in 
 nature, how would we account for 
 
THE CONCEIVABLE AND THE EXISTENT. 79 
 
 " . . . the pert Fairies and the dapper Elves," 
 
 and the cloud of unreal creatures with which 
 the teeming poetic imagination of mankind has 
 peopled the world from earliest ages ? Where 
 would be the dragon, the basilisk, the roc? 
 Where the valley of diamonds and the palace of 
 Aladdin ? The fact that I can conceive such does 
 not prove that in the whole realm of nature such 
 objects may be found. And, similarly, the fact 
 that no objects of a certain kind have an actual 
 existence is no proof that the various qualities 
 representing them may not be found grouped in 
 mind, as a result of the constructive imagination. 
 I may examine in detail all actual horses and all 
 actual men, and conclude that the qualities be- 
 longing to each animal are distinct and sepa- 
 rate ; but I cannot assure myself that some one 
 will not form in his mind the notion of an ani- 
 mal which combines the two sets of qualities, — a 
 centaur. All our ideas are not derived in their 
 ultimate shapes from direct intuition of objects, 
 but are a result of mental processes, which recast 
 and variously combine the material furnished. 
 Though we may not be able to add a new 
 element to those simple factors of our experi- 
 ence primarily given, yet it cannot be denied that 
 
80 THE CONCEPTION OF THE INFINITE. 
 
 we may so transpose or combine, add or elimi- 
 nate, some of those elements, as to obtain a prod- 
 uct consisting of elements which are not found 
 thus combined in any single object as it is pre- 
 sented to us in nature. When it is asked, there- 
 fore, whether any given conception be conceiv- 
 able, it is not necessary to an affirmative answer 
 that any existent object be proved to correspond 
 to the conception. It is enough if the qualities 
 connotated by the name be shown to be such as 
 may be mentally put together, and are not mutu- 
 ally repugnant or contradictory. We find thus 
 that the answer to the first question put forward 
 at the beginning of the chapter does not depend 
 on the answer to the second, and may be true 
 though that be false. When it is asked whether 
 the answer to the second question may not be 
 independent of the answer to the first, the an- 
 swer is not so clearly affirmative ; and although 
 this point is not directly connected with the sub- 
 ject under discussion, I may say, a proptos of thv 
 antagonism which Hamilton exhibits between 
 Faith and Knowledge, that, were the things of 
 which I have just spoken so defined that I could 
 in no conceivable manner put together in mind 
 the contradictory qualities attributed to them, I 
 
THE CONCEIVABLE AND THE EXISTENT. gl 
 
 should be doing rashly to assume their ex- 
 istence, however veracious the witness upon 
 whose word their acceptance might depend. It 
 may be a question whether any force can be 
 brought to bear upon a man sufficiently strong 
 to lead him to believe in the actual existence of 
 an inconceivable object; that is, of that which 
 cannot become an object of thought, nor, conse- 
 quently, of belief. However, the point in which 
 we are now chiefly concerned is this : that there 
 are a vast number of things that Ave have no 
 reason to believe in as actually existent, but 
 which we nevertheless conceive with perfect 
 clearness. 
 
 Again. When we put forward the proposi- 
 tion that an infinite object is conceivable; 
 unless we examine carefully this term used, — 
 conceivable, — and find out whether it has one 
 meaning or two, we may involve ourselves in 
 serious error. If the word has two meanings, 
 or is in common use made to stand for two 
 distinct mental operations ; when we prove that 
 an infinite object is inconceivable in one ac- 
 ceptation of the term, we do not necessarily 
 prove that it is inconceivable in the other 
 sense. If we suppose that in its first meaning 
 
82 THE CONCEPTION OF THE INFINITE. 
 
 the word conceivable is synonymous with im- 
 aginable, and that, when we eliminate one or 
 more of the qualities which must be present 
 to make an object imaginable, we may still in 
 some way mentally grasp the remaining quali- 
 ties, this second operation we may call con- 
 ceiving in the second and narrower sense of the 
 term. The proof that there is such a mental 
 operation I will reserve for the next chapter. 
 
 Let it be marked, therefore, that we have to 
 consider three distinct propositions, which de- 
 mand as many distinct kinds of proof, when 
 we assert, in the first place, that an infinite ob- 
 ject is conceivable, and in the second, that an 
 infinite object exists. 
 
 If I assert, then, that I can conceive space 
 to be infinite, the proof should consist in an 
 analysis of the qualities represented by the 
 word, and an examination of their mutual con- 
 sistency or repugnancy. If one element com- 
 prehended in the conception that I have in 
 mind be the notion of a certain quantity, then 
 I must in some way mentally grasp that quan- 
 tity and connect it with the other elements in 
 the conception, as I do in the representation 
 in imagination of finite objects. If there be in 
 
THE CONCEIVABLE AND THE EXISTENT 83 
 
 the conception no quantitative element, all that 
 is necessary is the ability to grasp in thought 
 certain qualitative elements, and this would fall 
 under the second and narrower sense of the 
 term conception. In either case the proposi- 
 tion depends on no fact of actual being, and 
 may be established without any reference to 
 such. 
 
 But the proposition " space is infinite" de- 
 mands in the way of proof what the previous 
 propositions can dispense with. Admitting my 
 ability to conceive infinite space, I may yet 
 doubt its existence,* which doubt I may pro- 
 ceed to remove by proof. I may assume on 
 a priori grounds the existence of infinite space, 
 as a postulate of Reason; or I may deny the 
 possibility of establishing the proposition a 
 priori and be forced to fall back upon obser- 
 vation, which proceeds by adding space to 
 space. In the latter case, although I may ad- 
 mit that the conception of the infinite is qual- 
 itative, and means in the case in point only 
 that a progression along a given line may be 
 
 * I purposely avoid all consideration of metaphysical argu- 
 ments drawn from theories as to the nature of space. 
 
84 THE CONCEPTION OF THE INFINITE. 
 
 endless, yet the only apparent possibility of es- 
 tablishing this fact with reference to any given 
 line lies in a continued addition of parts. That 
 is, though the conception itself may be quali- 
 tative, and contain no reference to a whole and 
 parts, the only way in which it seems possible 
 to prove any actual object infinite is by em- 
 ploying the ideas of quantity and totality in 
 adding part to part, which ideas are in fact 
 contradictory to ideas already contained in the 
 conception. 
 
 Now, it is evident that observation, adding 
 part to part, could never attain the infinite, 
 for the infinite time in which it is said to be 
 possible to accomplish this is no quantity of 
 time at all, and the phrase merely expresses in 
 a disagreeably tautological and roundabout way 
 that it can never be. But in spite of this, 
 since our experience in going from part to 
 part seems to be analogous to what would be 
 our experience of the object if it were, indeed, 
 infinite; and since there seems to be no other 
 way of proving from experience its infinity, we 
 are apt to imagine that by this process we are' 
 somehow proving the infinity of an object, or 
 are at least in the way to prove it. Evidently 
 
THE CONCEIVABLE AND THE EXISTENT. 85 
 
 this feeling was in the mind of Kant when he 
 put forward the first of his antinomies ; and it 
 is also evident that it exerted its influence on 
 the mind of Hamilton, leading him to " rise 
 from sphere to sphere in the region of empty 
 space" in the fruitless endeavor to exhaust in- 
 finite space. The same feeling it was that 
 caused Clifford to affirm that, when we have 
 counted one, two, three, four miles up into 
 space, although we must admit that the attempt 
 can never give us infinite space, we may yet 
 regard our task as " distinctly begun." The 
 attempt must fail, he thought, hut this is the 
 only way even to make the attempt. 
 
 Now, if Sir William Hamilton had set before 
 himself the task of proving that we can never 
 know space to be infinite, his argument would 
 have been to the point. If he denied the 
 possibility of knowing space to be infinite a 
 ■priori, no method of proving it such was left 
 except the method of observation. He could 
 not, of course, reasonably hold that the em- 
 pirical method is a true method of proving 
 anything to be infinite, but he might prove 
 with the arguments that he has used for an- 
 other purpose that the only even apparent 
 
 8 
 
86 THE CONCEPTION OF THE INFINITE. 
 
 method of proving space infinite is merely ap- 
 parent, and that consequently we can never 
 know space to be infinite. 
 
 Or if Sir William had set before himself the 
 task of proving that we cannot imagine or call 
 up before the mind an exact and complete rep- 
 resentation of an infinite line, his arguments 
 would not have been aside from the point. 
 The imagination can picture but a small ex- 
 tent of line at any one time ; this portion must 
 consequently be limited ; each portion succes- 
 sively added in imagination must also have its 
 limits; and we have no escape from the yery 
 difficulty which we have before met in the at- 
 tempt to know an object as infinite, — the im- 
 possibility of getting rid of the limits altogether. 
 Whether we conceive our conception of an in- 
 finite line to be quantitative or qualitative, 
 there is an equal impossibility of getting rid 
 of the quantitative in trying empirically to 
 prove a line infinite and in imagining an infi- 
 nite line. 
 
 Sir William accepted this argument as a proof 
 of the inconceivability of the infinite. Now, an 
 argument which will prove any infinite object 
 unknowable as an actually existing thing will 
 
THE CONCEIVABLE AND THE EXISTENT. 87 
 
 prove an infinite object unimaginable only on 
 the supposition that the element which has 
 caused the impossibility in the former case be 
 also present in the latter. An argument which 
 would prove that I cannot know a line to be in- 
 finite, because I cannot in the attempt to prove 
 it such get rid of the element of quantity, would 
 prove that I cannot imagine an infinite line, pro- 
 vided that the attempt to imagine such a line 
 must be similar in kind to the former process, 
 and must meet the same difficulty with the 
 quantitative element. And, similarly, an argu- 
 ment which would prove an infinite line un- 
 knowable and unimaginable on account of the 
 difficulty occasioned by this quantitative element, 
 would prove an infinite line inconceivable (in the 
 narrower sense of the term) only if the element 
 is necessarily present in conceiving (in the nar- 
 rower sense) an infinite line that has caused the 
 trouble in the former cases. 
 
 But if we can prove that, when the notion of 
 quantity is eliminated from a certain complex of 
 psychic elements, the remaining elements can 
 be in some way grasped in mind, and are thus 
 grasped in certain actual and not infrequent 
 mental operations, in this case the proofs, which 
 
88 THE CONCEPTION OF THE INFINITE. 
 
 might be suitable when applied to the former 
 propositions, can have no force whatever. 
 
 When we come in fact to make a more com- 
 plete analysis of the elements comprehended in 
 our conception of the infinite, we will see that 
 the arguments which have been so often ad- 
 vanced to prove the infinite inconceivable, and 
 regarded as so unanswerable, are really not to 
 the point at all, but may be classed under the 
 old logical fallacy of ignoratio elenchi. Sir Wil- 
 liam Hamilton proves inconceivable something 
 which is not at all supposed by the phrase " an 
 infinite line," and which, in fact, contains an 
 element flatly contradictory to one of those in- 
 dicated by the phrase. 
 
 In view of the foregoing, the exhibition of the 
 danger in which we stand of confounding three 
 distinct propositions and their appropriate proofs, 
 the two points which I would insist upon are 
 these : (1) That the conceivability of the infinite, 
 in the narrower sense of the term, is quite dis- 
 tinct from a knowledge of its existence, or from 
 its imaginability, and it is with the possibility of 
 the first alone that we have to do ; and (2) that, 
 whatever be the metaphysical tenets embraced 
 by one as to the nature of Space and Time, or 
 
THE CONCEIVABLE AND THE EXISTENT. 89 
 
 of those universal laws of Being known by one 
 school as Rational Intuitions and by an opposing 
 school as Highest Generalizations from Expe- 
 rience, — that these tenets, however they may 
 lead to the acceptance or rejection of the propo- 
 sition which affirms the existence of an infi- 
 nite, can yet not affect the proposition which 
 affirms its conceivability. 
 
 8* 
 
CHAPTER VI. 
 
 THE CONCEIVABILITY OF THE INFINITE. 
 
 When we analyze the mental state in which 
 we have reference to an infinite — let us take, 
 for example, an infinite line — we find the fol- 
 lowing elements : in the first place, there are 
 present the usual qualities of a line; for the 
 fact of our conceiving it as without limits need 
 not alter any of its usual qualities, any more 
 than the fact of our being unable to see the 
 ends of a telegraph wire need force us to deny 
 that it is a wire of a certain diameter, mate- 
 rial, or color; and, in the second place, there 
 is present the notion that, however far we may 
 go in thought, we will find a continuation of 
 the line. In other words, there is the notion of 
 unlimited ])ossibility of quantity, — a notion which, 
 be it marked, is strictly qualitative. That 
 there is no quantitative element present has 
 been insisted upon in previous chapters. But 
 quantity in general, not this or that quantity, 
 
 is as much a qualitative notion as color or form; 
 90 
 
THE CON CEIV ABILITY OF THE INFINITE. 91 
 
 and in defining the second element present in 
 our conception of an infinite line, I have used 
 the word advisedly to bring out what is a dis- 
 tinctive characteristic of the conception. The 
 word infinite does not denote a quantity, but 
 it has reference to quantity, and it cannot, in 
 accordance with its derivation and true signifi- 
 cation, be rightly applied to what is incapable 
 of being quantitatively considered. My objec- 
 tion to the usage of the word infinite, by some 
 who recognize that the conception for which 
 it stands is qualitative, is that they overlook 
 the distinctive characteristic of this conception, 
 which marks it out from other qualitative con- 
 ceptions, — that is, its necessary reference to 
 quantity, though not itself quantitative. If, by 
 that process of abstraction which takes place 
 when I compare objects similar in some of 
 their qualities, I fix my attention upon the 
 other qualities of any finite line, disregarding 
 its length, and leaving out of view for the time 
 being its limits, my conception is qualitative; 
 and yet it is not the conception of an infinite 
 line. In this case, so far from affirming infi- 
 nite length, I do not think of length at all. 
 But, in the case of an infinite line, I add to 
 
92 THE CONCEPTION OF THE INFINITE. 
 
 the former complex of qualities a new quality, 
 possibility of quantity in general, not this or 
 that quantity. When I try to bring before my 
 mind the notion of an infinite line, what I am 
 distinctly conscious of is this : I represent in 
 imagination a line of indefinite length, and 
 then run mentally along the line representing 
 additional line-portions, — a proceeding which 
 would of itself of course give me only the 
 finite ; but what makes my conception distinc- 
 tively of the infinite is that, in this progression 
 or notion of continued increase, I fix my at- 
 tention upon the progression itself, and elimi- 
 nate by abstraction the limits to which such a 
 progression is subject. I do not, be it marked, 
 merely fix my attention upon the other quali- 
 ties of a given line, abstracting from the notion 
 of limits ; but I have in mind a progression, a 
 possibility of ever-increasing quantity, and I 
 abstract from the limits of this progression. 
 The two conceptions are distinctly different, 
 although both are qualitative, and they should 
 not be confounded with one another. 
 
 The question, therefore, whether I can con- 
 ceive an infinite line is identical with the ques- 
 tion whether I can mentally grasp the various 
 
THE CO NCEIV 'ABILITY OF THE INFINITE. 93 
 
 qualities of a line, and the notion of a contin- 
 ual increase of such a line, without including 
 the notion of limits ; and it will he seen that 
 this question is simply one of the phases of 
 the broader question which is concerned with 
 the possibility of the concept or general notion. 
 A certain complex of qualities being necessary 
 to the existence of a given object in nature, or 
 to its subjective existence as represented in the 
 imagination, is there any mental operation by 
 which we may grasp some of these qualities, 
 to the exclusion of others, and convey to our 
 own and other minds by the use of the word 
 which stands for this new complex a distinct 
 meaning? 
 
 There have been held with reference to this 
 problem, as is well known, three opinions : the 
 doctrine of the Realists, that general ideas 
 have corresponding to them a counterpart Re- 
 ality, — a doctrine which may be passed over as^ 
 now abandoned, though its effects make them- 
 selves felt in many directions; the doctrine of 
 the Conceptualists, that, although general ideas 
 cannot exist in Nature, nor be represented in 
 the Imagination, yet they have a true mental 
 existence, and are the result of a distinct men- 
 
94 THE CONCEPTION OF THE INFINITE. 
 
 tal operation ; and the doctrine of the Nomi- 
 nalists, that the only generality that has a 
 separate existence, subjective or objective, is 
 the Name, which may be applied indifferently 
 to many similar objects. 
 
 The Conceptualist may hold that it is possi- 
 ble, unless the words include repugnant ele- 
 ments, to conceive an infinite line, — that is, to 
 grasp in mind a certain complex of psychic 
 elements which are yet incapable of being 
 pictured in the imagination as an infinite line. 
 To think, in the sense of to form such a con- 
 cept, is to him something other than to im- 
 agine. What cannot be imagined may yet be 
 thought. The word man, which we define as 
 comprehending the elements of rationality and 
 animality, he claims, does not in the least in- 
 clude all those other qualities which must be 
 combined with these two before we can picture 
 in the imagination or know as existing any 
 given man. If we select the two qualities in 
 which all the objects of a class resemble each 
 other, and give to these two a special name, 
 have we not brought them into consciousness 
 in some way in which we have not the other 
 qualities ? 
 
THE CONCEIVABILITV OF THE INFINITE. 95 
 
 And when we turn to the Nominalist, it 
 would not be hard to show that, although his 
 doctrines, if taken in strictness, would deny 
 the possibility of the mental operation by which 
 we arrive at the concept, and consequently of 
 the operation by which we may grasp in thought 
 the various elements implied in the phrase " an 
 infinite line," yet one may find in his teachings 
 by implication ample justification for assuming 
 its possible and actual existence. I will take 
 some extracts from four well-known Nominal- 
 ists to show how palpable is the fact stated, 
 and I will first quote from Berkeley, Locke's 
 great opponent on the subject of the Abstract 
 Idea : 
 
 "Whether others have this wonderful fac- 
 ulty of abstracting their ideas, they best can 
 tell ; for myself, I find, indeed, I have indeed 
 a faculty of imagining or representing to my- 
 self the ideas of those particular things I have 
 perceived, and of variously compounding and 
 dividing them. I can imagine a man with two 
 heads, or the upper parts of a man joined to 
 the body of a horse. I can consider the hand, 
 the eye, the nose, each by itself abstracted or 
 separated from the rest of the body. But, 
 
96 THE CONCEPTION OF THE INFINITE. 
 
 then, whatever hand or eye I imagine, it must 
 have some particular shape and color. Like- 
 wise the idea of man that I frame to myself 
 must be either of a white, or a black, or a 
 tawny, a straight, or a crooked, a tall, or a 
 low, or a middle-sized man. I cannot by any 
 effort of thought conceive the abstract idea 
 above described. And it is equally impossible 
 for me to form the abstract idea of motion 
 distinct from the body moving, and which is 
 neither swift nor slow, curvilinear nor recti- 
 linear; and the like may be said of all other 
 abstract general ideas whatsoever. To be plain, 
 I own myself able to abstract in one sense, as 
 when I consider some particular parts or quali- 
 ties separated from others, with which, though 
 they are united in some object, yet it is pos- 
 sible they may really exist without them. But 
 I deny that I can abstract from one another, 
 or conceive separately, those qualities which it 
 is impossible should exist so separated, or that 
 I can frame a general notion by abstracting 
 from particulars in the manner aforesaid, which 
 last are the two proper acceptations of abstrac- 
 tion. And there is ground to think most men 
 will acknowledge themselves to be in my case. 
 
THE CONCEIV ABILITY OF THE INFINITE. 97 
 
 The generality of men, which are simple and 
 illiterate, never pretend to abstract notions. It 
 is said they are difficult, and not to be attained 
 without pains and study; we may, therefore, 
 reasonably conclude that if such there be, they 
 are confined only to the learned."* 
 
 So much for Berkeley's position with respect 
 to the abstract notion. But mark the conces- 
 sions which he is forced to make in a later 
 section : 
 
 " But here it will be demanded, how we can 
 know any proposition to be true of all partic- 
 ular triangles except we have first seen it 
 demonstrated of the abstract idea of a triangle 
 which equally agrees to all? For, because a 
 property may be demonstrated to agree to 
 some one particular triangle, it will not thence 
 follow that it equally belongs to any other 
 triangle, which in all respects is not the same 
 with it. For example, having demonstrated 
 that the three angles of an isosceles rectangular 
 triangle are equal to two right ones, I cannot, 
 therefore, conclude this affection agrees to all 
 
 * "Principles of Human Knowledge." Introduction, Sect. 
 10. Works, ed. Fraser, vol. i. pp. 141, 142. 
 
98 THE CONCEPTION OF THE INFINITE. 
 
 other triangles which have neither a right 
 angle nor two equal sides. It seems, there- 
 fore, that to he certain this proposition is uni- 
 versally true, we must either make a particu- 
 lar demonstration for every particular triangle, 
 which is impossible, or once for all demonstrate 
 it of the abstract idea of a triangle, in which 
 all the particulars do indifferently partake, and 
 by which they are all equally represented. To 
 which I answer, that though the idea I have 
 in view whilst I make the demonstration be, 
 for instance, that of an isosceles rectangular 
 triangle whose sides are of a determinate length, 
 I may nevertheless be certain it extends to all 
 other rectilinear triangles, of what sort or 
 bigness soever. And that because neither the 
 right angle, nor the equality, nor determinate 
 length of the sides are at all concerned in the 
 demonstration. It is true the diagram I have 
 in view includes all these particulars, but then 
 there is not the least mention made of them 
 in the proof of the proposition. It is not said 
 the three angles are equal to two right ones, 
 because one of them is a right angle, or be- 
 cause the sides comprehending it are of the 
 same length. Which sufficiently shows that the 
 
THE CO NCEIV ABILITY OF THE INFINITE. 99 
 
 right angle might have been oblique and the 
 sides unequal, and for all that the demonstra- 
 tion have held good. And for this reason it is 
 that I conclude that to be true of any obliquan- 
 gular or scalenon which I have demonstrated 
 of a particular right-angled equicrural triangle, 
 and not because I demonstrated the proposi- 
 tion of the abstract idea of a triangle. And 
 here it must be acknowledged that a man may 
 consider a figure merely as triangular without 
 attending to the particular qualities of the 
 angles or relations of the sides. So far he 
 may abstract, but this will never prove that 
 he can frame an abstract, general, inconsistent 
 idea of a triangle. In like manner we may 
 consider Peter so far forth as man, or so far 
 forth as animal, without framing the fore- 
 mentioned abstract idea either of man or of 
 animal, inasmuch as all that is perceived is 
 not considered."* 
 
 In the former of the two extracts Berkeley 
 has declared himself able to abstract only so 
 far that he can represent to himself in im- 
 agination what can exist separately in nature. 
 
 - " Principles." Introduction, Section 16. 
 
100 THE CONCEPTION OF THE INFINITE. 
 
 He denies that he can conceive separately those 
 qualities which it is impossible should exist 
 separately. But when he supposes an objector 
 to ask how it is possible for something proved 
 to be true of a particular triangle, to be known 
 to be true of all triangles, he answers that it 
 is seen that neither the right angle, nor the 
 equality, nor the determinate length of the 
 sides are at all concerned in the demonstra- 
 tion. In other words, he admits that, so far 
 as that demonstration goes, we have to do only 
 with those elements in which all triangles agree. 
 And if we can reason about certain elements to 
 the exclusion of others ; if we can see that cer- 
 tain objects are alike in certain elements and 
 unlike in the others ; if we can give a name to 
 objects simply to express the presence of these 
 same elements, however the elements accom- 
 panying them may vary, then surely the ele- 
 ments of the concept have been before the 
 mind in some way in which the others have 
 not, and have been grasped together. 
 
 Berkeley frankly admits as much in the con- 
 cluding sentences of the latter extract, sen- 
 tences which were added twenty-four years 
 after the first publication of the essay, when 
 
THE CON CEIV ABILITY OF THE INFINITE. 101 
 
 mature reflection, we may suppose, had brought 
 him to see that on his previous principles, strictly 
 held, all comparison of objects differing in any 
 of their qualities would be impossible. If we 
 can consider a figure merely as triangular, with- 
 out attending to the particular qualities of the 
 angles or relations of the sides, then we can 
 in some sort divorce the elements included 
 under the general word triangle from the ac- 
 companying elements and consider them sepa- 
 rately. In those last few sentences Berkeley 
 admits all that a reasonable Conceptualist would 
 care to prove, and the words "abstract idea," 
 as there used, are equivalent to " object of the 
 imagination," a something which is not implied 
 in the formation of the abstract or general 
 notion. 
 
 Every one, Nominalist or Conceptualist, must 
 acknoAvledge that we can compare objects and 
 recognize them as like or unlike, — not merely 
 like or unlike as wholes, but in this or that 
 element; like in length, unlike in breadth; like 
 in color, unlike in shape. Now, no one claims 
 that we can call into clear consciousness the 
 element of length alone, and picture it divorced 
 of breadth and color ; but when we recognize 
 
 9* 
 
102 THE CONCEPTION OF THE INFINITE. 
 
 two objects as like in length and unlike in 
 breadth, the elements must in some way have 
 been present in mind separately, so as to be 
 recognized as length and breadth. If one object 
 that what is present in consciousness must ipso 
 facto be perceived, and that we cannot perceive 
 length as a factor by itself, nor recall in memory 
 any perception of such a factor during the act 
 of comparison, I answer that what is in con- 
 sciousness is by no means necessarily in a clear 
 analytic consciousness, and that we may by a 
 process of deductive reasoning be sure that 
 certain elements are present as factors in a 
 given mental state, while we are yet quite un- 
 able to call these elements into a clear analytic 
 consciousness, separated from certain other 
 elements bound to them by long association 
 and habit. As an instance, I refer to vision. 
 That distance is itself unperceivable by sight 
 we must admit. That judgments of distance 
 are a result of reasoning from an observed 
 constant connection of certain visual with cer- 
 tain other elements, may be satisfactorily es- 
 tablished when the above proposition is ad- 
 mitted. But to call into clear consciousness 
 by itself the purely visual sensation, which 
 
THE CONCEIV ABILITY OF THE INFINITE. 103 
 
 forms the basis of the judgment, is altogether 
 impossible. That it is a factor, and an im- 
 portant factor, in the complex consciousness 
 which we have at the time, we know, and yet 
 its presence, as a single and distinct element, 
 is capable of being only deductively known. 
 Notice a further point which is worthy of re- 
 mark. If we vary the purely visual element, 
 allowing all the other elements to remain the 
 same, — that is, if we change the color of the 
 object, but do not change in any respect the 
 form or size of the image on the retina, — a 
 difference is at once remarked, and the change 
 of color recognized. But the difference is not 
 recognized as a difference between two purely 
 visual sensations when the result of the actual 
 comparison comes into clear consciousness, but 
 as a difference in one of their elements between 
 two complexes or wholes. That is to say, the 
 two visual sensations cannot be brought into 
 clear consciousness and compared with each 
 other alone, but only as they are connected 
 with certain other elements in complexes or 
 wholes; it is the presence of two or more such 
 wholes, which we wish to compare, that pri- 
 marily impels to the narrowing of the atten- 
 
104 THE CONCEPTION OF THE INFINITE. 
 
 tion to the single similar or dissimilar elements. 
 This point is specially worthy of remark, as 
 there is something closely analogous to this in 
 the formation of the concept in general, and 
 this special case may help to throw light upon 
 all cases in which that which cannot be im- 
 agined is yet thought. 
 
 When I form the concept of length by com- 
 paring two objects in length and affirming 
 agreement, and then recognizing as a distinct 
 element that in which they agree, I certainly 
 do not compare the objects simply as wholes, 
 but compare the lengths; and I must surely 
 have had these elements in mind in some way 
 in which I had not the other elements which 
 go to make up the object. Whether I can call 
 into clear consciousness the psychic elements 
 present during the operation or not, it does 
 not much matter. I evidently have specialized, 
 selected some elements from among others, and 
 compared length with length, as element with 
 element. The name which we give to such 
 resemblances is the name representing a gen- 
 eral or abstract idea. Whether the possibility 
 of thus comparing single elements may not be 
 always conditioned by the presence of two or 
 
THE CONCEIVABILITY OF THE INFINITE. 105 
 
 more objects or complexes in which the ele- 
 ments are present I will consider later. 
 
 Hume warmly applauds the position taken 
 by Berkeley with reference to the abstract 
 idea, calling it " one of the greatest and most 
 valuable discoveries that have been made of 
 late years in the republic of letters," and he 
 undertakes to confirm it with proofs that he 
 hopes will put it " beyond all doubt and con- 
 troversy." For the same purpose for which 
 I quoted the two extracts from Berkeley, I 
 will quote the last part of the section which 
 he devotes to the establishment of this posi- 
 tion : 
 
 " It is certain that the mind would never 
 have dreamed of distinguishing a figure from 
 the body figured, as being in reality neither 
 distinguishable, nor different, nor separable, did 
 it not observe that even in this simplicity there 
 might be contained many different resemblances 
 and relations. Thus, when a globe of white 
 marble is presented, we receive only the im- 
 pression of a white color disposed in a certain 
 form, nor are we able to separate and distin- 
 guish the color from the form. But observing 
 afterwards a o-lobe of black marble and a cube 
 
106 THE CONCEPTION OF THE INFINITE. 
 
 of white, and comparing them with our former 
 object, we find two separate resemblances in 
 what formerly seemed, and really is, perfectly 
 inseparable. After a little more practice of this 
 kind we begin to distinguish the figure from 
 the color by a distinction of reason, — that is, we 
 consider the figure and color together, since 
 they are, in effect, the same and undistinguish- 
 able, but still view them in different aspects, ac- 
 cording to the resemblances of which they are 
 susceptible. When we would consider only the 
 figure of the globe of white marble, we form in 
 reality an idea both of the figure and color, 
 but tacitly carry our eye to its resemblance 
 with the globe of black marble; and in the 
 same manner, when we would consider its 
 color only, we turn our view to its resemblance 
 with the cube of white marble. By this means 
 we accompany our ideas with a kind of reflec- 
 tion, of which custom renders us, in a great 
 measure, insensible. A person who desires us 
 to consider the figure of a globe of white mar- 
 ble without thinking on its color, desires an 
 impossibility; but his meaning is that we 
 should consider the color and figure together, 
 but still keep in our eye the resemblance to 
 
THE CON CEIV ABILITY OF THE INFINITE. J 07 
 
 the globe of black marble, or that to any other 
 globe of whatever color or substance."* 
 
 It is not hard to see that we cannot distin- 
 guish in a body figured " many different re- 
 semblances and relations" without bringing the 
 resembling elements in some sense singly into 
 thought; if the mental complex which we call 
 an object were an indissoluble unit, we could 
 affirm a general likeness or unlikeness between 
 it and other objects, but we could not affirm 
 that the resemblance lay in the figure or color. 
 If, as Hume asserts, the figure and color " are, 
 in effect, the same and undistinguishable," why 
 do we find the one susceptible of the one class 
 of resemblances and the other of another class ? 
 If we take the words literally, should not the 
 figure, viewed in one aspect, be susceptible of 
 resemblances of figure, and viewed in another 
 of color ? And, similarly, if the color is one 
 with the figure, — the same and undistinguisha- 
 ble, — should not the color, viewed in one aspect, 
 be susceptible of resemblances of color, and 
 viewed in another of figure ? Hume's admis- 
 
 *" Treatise of Human Nature," bk. i. Sect. 7. Works, 
 Boston, 1854, vol. i. p. 42. 
 
108 THE CONCEPTION OF THE INFINITE. 
 
 sion that the two elements are known as giving 
 different resemblances, in itself refutes his pre- 
 vious assertion that they are undistinguishahle. 
 If color he recognized as like color, and figure 
 like figure, the two qualities are distinguished 
 as different, and are in reality separately 
 grasped. 
 
 I will now take a passage from Mr. J. S. 
 Mill's " Examination of Sir William Hamil- 
 ton's Philosophy" : 
 
 " The formation, therefore, of a Concept 
 does not consist in separating the attributes 
 which are said to compose it from all other 
 attributes of the same object, and enabling us 
 to conceive those attributes, disjoined from 
 any others. We neither conceive them, nor 
 think them, nor cognize them in any way as a 
 thing apart, but solely as forming, in combina- 
 tion with numerous other attributes, the idea 
 of an individual object. But, though thinking 
 them only as part of a larger agglomeration, 
 we have the power of fixing our attention on 
 them to the neglect of the other attributes 
 with which we think them combined. While 
 the concentration of attention actually lasts, 
 if it is sufficiently intense, we may be tempo- 
 
THE C0NCE1VAB1L1TY OF THE INFINITE. 109 
 
 rarity unconscious of any of the other attri- 
 butes, and may really, for a brief interval, 
 have nothing present to our mind but the at- 
 tributes constituent of the concept. In gen- 
 eral, however, the attention is not so com- 
 pletely exclusive as this; it leaves room in 
 consciousness for other elements of the con- 
 crete idea; though of these the consciousness 
 is faint in proportion to the energy of the con- 
 centrative effort, and the moment the attention 
 relaxes, if the same concrete idea continues to 
 be contemplated, its other constituents come 
 out into consciousness. General concepts, there- 
 fore, we have, properly speaking, none; we 
 have only complex ideas of objects in the con- 
 crete ; but we are able to attend exclusively to 
 certain parts of the concrete idea ; and by that 
 exclusive attention we enable those parts to de- 
 termine exclusively the course of our thoughts 
 as subsequently called up by association, and 
 are in a condition to carry on a train of medi- 
 tation or reasoning relating to those parts only, 
 exactly as if we were able to conceive them 
 separately from the rest."* 
 
 * " Examination of Sir William Hamilton's Philosophy," 
 
 vol. li. p. 64. Boston, 1868. 
 
 10 
 
HO THE CONCEPTION OF THE INFINITE. 
 
 This passage is so clearly in harmony with 
 the views of the Conceptualist, as I have por- 
 trayed them, that it seems scarcely necessary 
 to comment upon it. But I cannot resist the 
 temptation to delay for a moment over an in- 
 consistency into which Mill was forced by his 
 attempt to recognize, though a Nominalist, a 
 truth which the Nominalist, pure and simple, 
 cannot recognize. The formation of a Concept, 
 he insists, does not consist in " separating the 
 attributes said to compose it from all other 
 attributes of the same object, and enabling us 
 to conceive those attributes, disjoined from any 
 others." This position lie emphasizes by the 
 further affirmation that " we neither conceive 
 them, nor think them, nor cognize them in any 
 way as a thing apart, but solely as forming, in 
 combination with numerous other attributes, 
 the idea of an individual object." These sen- 
 tences are certainly unequivocal : they con- 
 tain an emphatic assertion of the nominalistic 
 doctrine. 
 
 But, side by side with such statements, we 
 find it asserted that we may fix the attention 
 upon the attributes constituent of the concept, 
 to the neglect of the other attributes of the 
 
THE CO NCFAV ABILITY OF THE INFINITE. H\ 
 
 object, and that while the concentration of at- 
 tention actually lasts, if it is sufficiently intense, 
 " we may be temporarily unconscious of any of 
 the other attributes, and may really, for a brief 
 interval, have nothing present to our mind but 
 the attributes constituent of the concept." 
 Surely, if the only elements before the mind 
 are those constituent of the concept; if we 
 may be conscious of these, even for a brief 
 interval, and conscious of these alone; surely 
 in such a case we conceive, or think, or in some 
 way cognize the attributes forming the concept 
 as separate and apart, and not for the time 
 being, in combination with numerous other 
 attributes. Mr. Mill goes even further in the 
 above admission than most of us would care to 
 follow him. In speaking as he does of the 
 process, and not distinguishing between the 
 imagining of an object and the knowing of one 
 or more of its isolated qualities, he clearly in- 
 timates, although he does not distinctly say, 
 that the elements before the mind during the 
 formation or use of a concept are in conscious- 
 ness in the same way that the whole complex or 
 object may be in consciousness. But, to recur 
 to the before-mentioned analogy of the purely 
 
112 THE CONCEPTION OF THE INFINITE. 
 
 visual element in vision, Ave know that, although 
 we may so concentrate the attention as to dis- 
 tinguish the hlue color of one object from the 
 red color of another, and so must have com- 
 pared in some rapid manner these purely visual 
 sensations, yet when we try to call into clear 
 consciousness the mere sensation of color, we 
 cannot do it without imagining the color as 
 on a surface, or combined with psychic ele- 
 ments not purely visual. That is to say, the 
 single and separate sensations cannot be called 
 into a clear consciousness, and their presence 
 when we use the concept, or have occasion to 
 compare them singly with each other, is some- 
 thing quite distinct and different from the pres- 
 ence in consciousness of the complex which is 
 knowable as an object. And such would seem 
 also to be the case wherever we call before 
 the mind the single psychic elements which can 
 yet not be represented alone in the imagination. 
 The element must have been grasped separately, 
 but it can be brought into a clear consciousness 
 only in combination. 
 
 If now we recognize in each of two objects 
 presented to us a certain quality or complex 
 of qualities upon which we can 1ix the atten- 
 
THE CON CFAV ABILITY OF THE INFINITE. \\% 
 
 tion, and if we discover that, so far as these 
 qualities go, there is an undistinguishable sim- 
 ilarity in the objects, the differences arising al- 
 together from other qualities, why may we not 
 call the complex of qualities in point a general 
 notion or general idea f Of course, whether we 
 should call the qualities in the two instances 
 the same, even if they were undistinguishably 
 similar, would depend on our use of the word 
 same,* and our ideas of what constitutes same- 
 ness or identity; but I can see no objection to 
 using the words " general notion" to indicate 
 the fact that a certain complex of qualities is 
 to be found in many different combinations 
 with other qualities. Should it still be insisted 
 that, since we cannot bring separately into 
 clear consciousness these elements of objects 
 known, we have no reason to assume that 
 we actually conceive them or think them sep- 
 arately, I will not quarrel over the use of 
 a word, but will simply state that I find the 
 word " conceive" a useful one to express that 
 concentration of the attention upon certain 
 
 * I have pointed out before the fact that the word 
 'same" is commonly used in four quite distinct senses. 
 10* 
 
114 THE CONCEPTION OF THE INFINITE. 
 
 qualities of an object which takes place when 
 objects are compared, and which eliminates 
 from consciousness, or at least subordinates 
 all other qualities of the objects; and I will so 
 use the word, applying it to an operation the 
 existence of which Mr. Mill has in so many 
 words admitted. 
 
 The last author whom I will quote is Mr. 
 Bain. I will take some passages from the 
 chapter on abstraction in his compendium of 
 psychology and ethics, where he supports the 
 Nominalistic doctrine : 
 
 "We are able to attend to the points of 
 agreement of resembling things, and to neg- 
 lect the points of difference, as when we think 
 of the light of luminous bodies, or the round- 
 ness of round bodies. This power is named 
 Abstraction. 
 
 " It is a fact that w T e can direct our attention 
 or our thoughts to the points of agreement 
 of bodies that agree. We can think of the 
 light of the heavenly bodies, and make asser- 
 tions, and draw inferences respecting it. So 
 we can think of the roundness of spherical 
 bodies, and discard the consideration of their 
 color and size. In such an object as the full 
 
THE CONCEIV 'ABILITY OF THE INFINITE. \\§ 
 
 moon we can concentrate our regards upon its 
 luminous character, wherein it agrees with one 
 class of objects, or upon its figure, wherein it 
 agrees with another class of objects. We can 
 think of the taste of a strawberry, either as 
 agreeing with other tastes or as agreeing with 
 pleasures generally. . . ." 
 
 " Every Concrete thing falls into as many 
 classes as it has attributes; to refer it to one 
 of these classes, and to think of the corre- 
 sponding attribute, are one mental operation. 
 
 " When a concrete thing before the view re- 
 calls others agreeing in a certain point, our at- 
 tention is awake upon that point; when the 
 moon recalls other luminous bodies, we are 
 thinking of its light; when it recalls other 
 round bodies, we are thinking of its roundness. 
 The two operations are not different but iden- 
 tical . 
 
 " On this supposition, to abstract, or to think 
 of a property in the abstract, is to classify 
 under some one head. To abstract the prop- 
 erty of transparency from water is to recall, 
 at the instance of water, window-glass, crystal, 
 air, &c. ; to abstract its liquidity is to recall 
 milk, vinegar, melted butter, mercury, &c. ; to 
 
116 THE CONCEPTION OF THE INFINITE. 
 
 abstract its weight is to bring it into compari- 
 son with other kinds of gravitating matter. 
 
 " Hence abstraction does not properly con- 
 sist in the mental separation of one property 
 of a thing from the other properties, as in 
 thinking of the roundness of the moon apart 
 from its luminosity and apparent magnitude. 
 Such a separation is impracticable ; no one can 
 think of a circle without color and a definite 
 size. All the purposes of the abstract idea 
 are served by conceiving a concrete thing in 
 company with others resembling it in the at- 
 tribute in question, and by affirming nothing 
 of the one concrete but w T hat is true of all 
 those others." 
 
 ... u In abstract reasoning, therefore, we 
 are not so much engaged with any single 
 thing as with a class of things. When we 
 are discussing government, we commonly have 
 in view a number of governments alternately 
 thought of; if we notice in any one gov- 
 ernment a certain feature, we run over the 
 rest in our mind, to see if the same feature is 
 present in all. There is no such thing as an 
 idea of government in the abstract; there is 
 only possible a comparison of governments in 
 
THE CONCEIVABILITY OF THE INFINITE. H7 
 
 the concrete ; the abstraction is the likeness or 
 community of the individuals."* 
 
 It will be noticed that throughout this ex- 
 tract Mr. Bain does not distinguish between 
 the elements of an act which come out into a 
 clear consciousness and the elements which do 
 not so come out, but are necessary to the possi- 
 bility of the operation. When Mr. Bain says, 
 for instance, that there is no such thing as the 
 idea of government in the abstract, but that 
 we can compare governments in the concrete, 
 and recognize the likeness of the individuals, 
 it is perfectly true that all that we are clearly 
 conscious of is several individual objects and 
 a similarity between them ; but when we come 
 to analyze this recognition of a similarity, it 
 will be s£en that the elements which are known 
 as similar are quite incapable, by themselves, 
 of forming a concrete object, and yet they are 
 distinguished by the mind from the dissimi- 
 lar elements; they must, therefore, have been 
 in some sort grasped separately, though they 
 cannot separately be brought into a clear con- 
 sciousness. Whether, during this rapid act of 
 
 Mental and Moral Science," London, 1868, pp. 176-78. 
 
118 THE CONCEPTION OF THE INFINITE. 
 
 concentration and comparison, the other ele- 
 ments which go to form the object actually 
 disappear from consciousness, or are only dimly 
 perceived, as Mr. Mill suggests that they are in 
 most cases, does not affect the peculiar char- 
 acter of the act. When I compare in height 
 two trees which I see side by side in the dis- 
 tant landscape before me, I am perhaps con- 
 scious of several objects in their immediate 
 vicinity in a dim and indefinite way, but the 
 two objects compared are present in con- 
 sciousness in a manner very different, and are 
 grasped, so to speak, separately. And when I 
 fix my attention upon the height of the two 
 trees, finding them similar or dissimilar in this 
 one element, we have every reason to suppose 
 that something very analogous takes place, 
 and that this one element is present in con- 
 sciousness in some way quite different from 
 the others, and is grasped separately for the 
 time being. Were it not so, we could not 
 say the trees are alike in height, but different 
 in contour or color of foliage. We are justi- 
 fied in assuming that when we recognize two 
 trees as like in height, but not in color, we 
 have compared height with height and color 
 
THE CON CEIV ABILITY OF THE INFINITE. H9 
 
 with color, and not merely compared the one 
 object as an undistinguishable complex of qual- 
 ities with another object as another undis- 
 tinguishable complex. 
 
 As I have before said, the name which we 
 choose to apply to this operation is of little 
 consequence; the point chiefly to be borne in 
 mind is that we have here an operation differ- 
 ing from ordinary imagination, in that it takes 
 cognizance of certain psychic elements which 
 can yet not be called into clear consciousness 
 by themselves as a mental picture. Whether 
 the two operations completely differ in their 
 ultimate nature is another question. When 
 the Conceptualist asserts that though he can- 
 not imagine length apart from breadth or 
 color, yet he can conceive or think it, he 
 merely marks by a distinct name his recogni- 
 tion of an operation different from imagina- 
 tion, and which is implied in all comparison 
 of objects. What may be the peculiar psychic 
 elements present in the operation he does not 
 necessarily know, nor express when he uses the 
 name.* 
 
 * Kant seems to have despaired of the possibility of ever 
 making this analysis: " Dieser Schematismus unseres Ver- 
 
120 THE CONCEPTION OF THE INFINITE. 
 
 Arguing from the analogy of the purely 
 visual element in vision, one might conclude 
 that what is actually present in consciousness 
 in comparing lengths, for instance, is the dis- 
 tinctive element which is present in combina- 
 tion with other elements (and, consequently, in 
 a modified form) in all our experience of ex- 
 tended objects, but which, in the act of com- 
 paring two objects, may be brought into suf- 
 ficient prominence to be considered, for the 
 moment, alone, and alone compared with its 
 kind. When we make the attempt to call it 
 into clear consciousness, the element appears as 
 modified by, and in combination with, others ; 
 but it is not improbable that, in the act of 
 comparison, it obtains in its pure state suffi- 
 cient recognition to make possible a comparison 
 with a similar element also in its pure state. 
 However, whether we can describe just what 
 
 standes, in Ansehung der Erscheinungen und ihrer blossen 
 Form, ist eine verborgene Kunst in den Tiefen der mensch- 
 lichen Seele, deren wahre Handgriffe wir der Natur 
 schwerlich jemals abrathen und sie unverdeckt vor Augen 
 legen werden" (" Kritik, von dem Schematismus der 
 reinen Verstandesbegriffe"), but he did not doubt the 
 operation. 
 
THE CON CFAV ABILITY OF THE INFINITE. 121 
 
 is present during the act or not, we may be 
 sure that a mental separation of two objects 
 into their elements is necessary in order to 
 a recognition of them as in some points simi- 
 lar and in some dissimilar. 
 
 Hume has asserted that if we knew but the 
 one object, and had no other objects with 
 which to compare it, we would never distin- 
 guish between the several elements composing 
 it; and the same thought is made prominent 
 by Mr. Bain when he states that to refer an 
 object to a class of other objects, and to think 
 of the corresponding attribute, are but one 
 act. Mr. Bain's statement is, of course, some- 
 what inaccurate, as it confounds two very dif- 
 ferent parts of the one operation ; but the point 
 upon which both of these writers insist, — 
 namely, that a comparison of objects is neces- 
 sary to an analysis of the objects into their 
 resemblances, — what should be called their ul- 
 timate elements, — is worthy of attention. In 
 all probability, were it not for a comparison 
 of objects, a constant experience of groups of 
 psychic elements containing likenesses and un- 
 likenesses, we should never analyze the groups 
 
 aud make prominent single elements, separated 
 
 11 
 
122 THE CONCEPTION OF THE INFINITE. 
 
 from their accompaniments. And since the 
 single elements do not themselves come out 
 into a clear analytic consciousness, it is not easy 
 to see how we could be sure that they had been 
 separately grasped, if we could not infer their 
 presence from the possibility of comparison and 
 classification of objects. That this analysis im- 
 plies the presence of two objects, is necessarily 
 a classification, after the analytic habit has been 
 formed, as Mr. Bain insists, is not so clear ; but 
 that, as a preliminary to the act of concentra- 
 tion by which we form the concept, we call 
 into consciousness at least one concrete object, 
 I think, cannot be doubted ; and this fact might 
 easily mislead one into taking the Nominalistic 
 position that a recognition of particular objects 
 expresses the whole process. Particulars must 
 be present of course, if they are to be compared; 
 but, when compared, they are not compared as 
 wholes. 
 
 In view of the foregoing, I would, therefore, 
 regard the fact as beyond all doubt, that there 
 are mental operations differing distinctly from 
 imagination, in that certain elements, of which 
 we have usually, as single elements, no ana- 
 lytic consciousness, but which are merged with 
 
THE CONCEIVABILITY OF THE INFINITE. 123 
 
 others into an indivisible whole, are brought for 
 the time being into such prominence as to be 
 compared individually with similar elements, and 
 recognized as like or unlike. It does not follow 
 that we may have a clear consciousness of the 
 steps in the rapid process in which this compari- 
 son takes place, or clearly recognize the nature 
 of the elements compared ; but from the fact of 
 the comparison, about which there can be no 
 doubt, we may be very sure that the operation 
 in question has taken place. 
 
 Now, when we return to the particular con- 
 ception which we have been considering, — that 
 of an infinite line, — we find it merely a con- 
 crete instance of this general truth, which all 
 must either explicitly or implicitly admit. As 
 I have said, the elements constituent of this 
 conception are the usual qualitative attributes 
 of a line and the notion of continued pro- 
 gression, of unlimited possibility of quantity. 
 These elements may be brought into mind, to 
 the exclusion of the notion of limits, which 
 are yet present in all imagined lines and in 
 all intuitions of lines in nature, by employing 
 the process usual in forming a concept. When 
 I think of an infinite line, I first represent to 
 
124 THE CONCEPTION OF THE INFINITE. 
 
 myself a line of some indefinite length, and 
 I then run mentally along this line, adding 
 new portions, — that is, I successively think 
 several increasing lengths. I have now before 
 my mind what Hume and Bain insist upon 
 and make so prominent in forming the con- 
 cept, several concrete objects similar in some 
 of their qualities. Having mentally passed 
 over several of these line portions, I then fix 
 my attention, not upon that in which they 
 differ, — the quantitative element, — but upon 
 that in which they resemble, the usual quali- 
 tative attributes of a line, and the notion of 
 increase or progression, which is common to 
 all. This is precisely what I do in forming 
 the concept man or animal. The concrete ob- 
 jects are compared, their differences eliminated 
 by abstraction, and their likenesses grasped to- 
 gether under a distinctive name. Or I may 
 select one of the qualities in which objects 
 agree, and consider it alone, as when I com- 
 pare men of the same age and color, only as 
 to their height, and pronounce them equal in 
 height. If this be possible, if, in using the 
 word man, I can distinguish between that in 
 which men agree and that in which they clisa- 
 
THE CONCEIVABILITY OF THE INFINITE. 125 
 
 gree, and if it be further possible for me to fix 
 my attention upon one of the points in which 
 they agree, to the exclusion of others, then it 
 is possible to abstract from the particular quan- 
 tities or amounts of several lines present in im- 
 agination, and think onlv of a constant increase 
 or progression. That both the one and the 
 other are not only possible but actual opera- 
 tions, is proved beyond possibility of doubt by 
 our constant comparison of objects, our use of 
 general language, our frequent use of the word 
 infinite, to indicate what is clearly distinguished, 
 readily defined, and conveys a distinct meaning 
 to speaker and hearer. 
 
 One point I will here remark upon before 
 passing, and that is the distinction sometimes 
 drawn between the abstract and the general 
 notion, a point at which I have a few pages 
 before briefly hinted. Admitting that Hume 
 was right in saying that had we not had oc- 
 casion to compare two objects we should never 
 have analyzed either into its elements, the ques- 
 tion naturally arises, whether, after we have 
 formed this habit of analysis by comparison, we 
 may not by mere effort of will fix the attention 
 
 upon one element of a single object, without any 
 11* 
 
126 THE CONCEPTION OF THE INFINITE. 
 
 reference to its occurrence in another object? 
 To take an example, can I not, in imagining a 
 window or a door, fix my attention upon its 
 length, without thinking of the length of any- 
 thing else, or comparing the object with any 
 other extended object? 
 
 The question cannot be answered off-hand, 
 as by saying that in recognizing my perception 
 as a perception of length — in using the word 
 length — I necessarily class the object with 
 other long objects; for it is at least thinkable 
 that I may have so associated the word with 
 this peculiar element as to have it suggested 
 by the presence of the element, and still may 
 not be conscious of other combinations in 
 which the element occurs. It is, I think, 
 highly probable, however, that when we con- 
 centrate attention upon one element of an 
 object, there is a more or less dim and vague 
 reference to other objects, and that there is a 
 rapid comparison ; but this fact must be proved 
 by an interrogation of consciousness during the 
 act, and upon this point I will not insist. If 
 it be allowed that the one element may be 
 known without reference to its occurrence in 
 two or more objects, we have what may justly 
 
THE CONCEIV r ABILITY OF THE INFINITE. 127 
 
 be called the abstract notion, as distinguished 
 from the element recognized as present in sev- 
 eral combinations, in which latter case we may 
 call it the general notion. And even if we 
 deny that the abstraction is possible except 
 there be two or more objects present in mind, 
 and a comparison of them, yet it must be ac- 
 knowledged that the prominence of these ob- 
 jects in consciousness varies greatly, and, ac- 
 cordingly, we may have either the intension or 
 the extension of the concept most prominently 
 before the mind. If we are concerned, not so 
 much with the combinations in which the ele- 
 ment occurs as with the element itself, we 
 may call our notion abstract; if we have prom- 
 inently in mind the number of occurrences, we 
 may call it general. In either case the dis- 
 tinction between the abstract and the general 
 notion is a legitimate one. 
 
 This point is not one directly connected 
 with the subject with which I am concerned, 
 — the conceivability of the infinite, — but is one 
 too interesting to be overlooked in any exami- 
 nation into the nature of the concept; and, 
 indeed, it can scarcely be considered totally 
 foreign to the subject in hand, as the operation 
 
128 THE CONCEPTION OF THE INFINITE. 
 
 of forming a concept, and the act of conceiving 
 an infinite, are not different in their nature, and 
 may be viewed in the same aspects. And to 
 the objection which may be made to my class- 
 ing the notion of this or that particular in- 
 finite line with the concept or general notion, 
 as I have done throughout this chapter, the 
 objector taking the ground that the individual 
 or the intuition is something quite different 
 and distinct from the concept, — to this objec- 
 tion I answer that the notion of any particular 
 infinite line is not a complete intuition, in that 
 one of the elements of the intuition is elimi- 
 nated by abstraction; and that when, in the for- 
 mation of any concept, we fix the attention 
 upon certain elements of an intuition to the 
 exclusion of others, we have in mind, so to 
 speak, a constituent part of an intuition : the 
 fact that we recognize its similarity, or, if we 
 so choose to use the word, its sameness with 
 parts of other intuitions, does not alter the 
 individual character of the elements which we 
 actually have in mind. The operation of form- 
 ing a concept and the operation of conceiving 
 an infinite line are in nature identical. 
 
 It seems impossible that any one, having 
 
THE CONCEIVABILITY OF THE INFINITE. 129 
 
 reflected upon the fact of his constantly grasp- 
 ing in concepts elements which can yet not be 
 separately imagined, and having, after an anal- 
 ysis of what is in his mind when he calls up 
 the notion of an infinite, discerned the iden- 
 tity of this latter operation with the former, it 
 seems impossible that such an one should hold 
 an infinite line, or infinite time or space to be 
 inconceivable. But, being loath to give up his 
 former position, such a man will probably put 
 forward in a new form an objection upon which 
 I have already commented. We have heard 
 him object, " If we do not know the infinite 
 as a whole, do we not know only its parts, 
 which are finite?" And now we will hear 
 him object that, " Even if it be true that we 
 can grasp in thought the notion of progression, 
 and the notion of a line in general, this will 
 give us no knowledge of an infinite line, but 
 will give us only the elements of an incomplete 
 image, which cannot be called distinctly before 
 consciousness, and therefore cannot be known 
 as an object at all." If, however, one feel 
 himself aggrieved because he cannot represent 
 to himself, endowed with all the qualities neces- 
 sary to an object of the imagination, that which 
 
130 THE CONCEPTION OF THE INFINITE. 
 
 he has already defined as wanting some of 
 those qualities, he will be unreasonable enough 
 to think it ground for complaint that he can- 
 not in thought make parallel lines meet, or 
 imagine a triangle with four sides. The word 
 infinite means devoid of limits, and it neces- 
 sarily follows that an infinite line cannot be 
 known as a quantity, consequently not as a 
 whole. Every object which is seen or imag- 
 ined has necessarily limits, definite or indefi- 
 nite: an infinite line, as infinite, cannot become 
 an object of the imagination. But from this it 
 by no means follows that I cannot call a par- 
 ticular line infinite, provided I have some proof 
 of the fact other than its conceivability, and 
 that I cannot know my conception to be in 
 harmony with the reality. Suppose that, either 
 from testimony or by means of some a priori 
 chain of reasoning, I have good reason to be- 
 lieve a given line endless, I can conceive the 
 line as without end, and I may know my con- 
 ception, although it does not represent the 
 total content of my consciousness when at any 
 moment I gaze upon this or that part of the 
 line, to be a true and real conception, and in 
 harmony with my experience as I progres- 
 
THE CON CEIV ABILITY OF THE INFINITE. 131 
 
 sively pass over the line ; and I may be cer- 
 tain that, however long my experience may 
 continue, it will yet not prove incompatible 
 with the conception I have formed. In this 
 sense, and in this sense alone, is any infinite 
 object conceivable, and there is no other con- 
 ceivable way in which we could conceive it. 
 An infinite object which could be known as a 
 whole is not even an object of thought, for 
 the elements indicated by the words cannot be 
 so put together as to express a meaning. But 
 the conception of the infinite, as I have de- 
 fined it, contains in it nothing either contra- 
 dictory or beyond the grasp of the human 
 mind, and is, indeed, a very common concep- 
 tion, as is evidenced by use of the word infi- 
 nite in literature, ancient and modern, to say 
 nothing of the constant occurrence of the word 
 in the debates of those very philosophers who 
 find the conception such a stone of stumbling. 
 And that the conception is a real one, having 
 a real consonance with experience, those of us 
 who hold to the Christian doctrine of Immor- 
 tality will not be slow to maintain. 
 
 THE END. 
 
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