204 



• KNOV\^LEDGE ♦ 



[Sept. 28, 1883. 



THE PHILOSOPHY OF MATHEMATICS.* 



By Pkofessor A. Cayley. 

 IN TWO PARTS.— PART I. 



MATHEMATICS connect themselves on the one side 

 with common life and the physical sciences ; on the 

 other side, with philosophy, in regard to our notions of 

 space and time, and in the questions which have arisen as 

 to the univeisality and necessity of the truths of mathe- 

 matics and the foundation of our knowledge of them. 



As to the former side, I am not making before you a 

 defence of mathematics. Still less would I speak of its 

 utility before, I trusf, a friendly audience, interested or 

 willing to appreciate an interest in mathematics in itself 

 and for its own sake. 



On the other side, the general opinion lias been, and is, 

 that it is indeed by experience that we arrive at the truths 

 of mathematics, but that e.xperience is not their proper 

 foundation ; the mind itself contributes something. But it 

 is maintained by John Stuart Mill that the truths of 

 mathematics, in particular those of geometry, rest on 

 experience ;t and, as regards geometry, the same view is on 

 very different grounds maintained by the mathematician 

 Riemann. 



* Abstract of that portion of Professor Cayley's Address before 

 the British Association at Southport, which relates to the philo- 

 sophy of Mathematics and to certain recent ideas respecting non- 

 Euclidian Geometry and space of more than three dimensions. 



t " It remains to inquire what is the ground of our belief in 

 axioms, what is the evidence on which they rest. I answer, they 

 are experimental truths, genei-alisations from experience. The pro- 

 position ■ Two straight lines cannot enclose a space,' or, in other 

 words, two straight lines which have once met cannot meet again, 

 is an induction from the evidence of our senses." But I cannot 

 help considering a previous argument (p. 259) as very materially 

 modifying this absolute contradiction. After inquiring, " Why are 

 mathematics by almost all philosophers . . . considered to be 

 independent of the evidence of experience and observation, and 

 characterised as systems of necessary truth r " Mill proceeds as 

 follows : — " The answer I conceive to be that this character of 

 necessity ascribed to the truths of mathematics, and even (with 

 some reservations to be hereafter made) the peculiar certainty 

 ascribed to them, is a delusion, in order to sustain which it is 

 necessary to suppose that those truths relate to and express the 

 properties of purely imaginary objects. It is acknowledged that 

 the conclusions of geometi-y are derived partly at least from the 

 so-called definitions, and that these definitions are assumed to 

 be correct representations, as far as they go, of the objects with 

 which geometry is conversant. Now, we have pointed out that 

 from a definition as such, no proposition, unless it be one concern- 

 ing the meaning of a word, can ever follow, and that what appa- 

 rently follows from a definition follows in reality from an implied 

 assumption that there exists a real thing conformable thereto. 

 This assumption in the case of the definitions of geometry is not 

 strictly true ; there exist no real things exactly conformable to the 

 definitions. There exist no real points without magnitude, no 

 lines without breadth, nor perfectly straight, no circles with all 

 their radii exactly equal, nor squares with all their angles perfectly 

 right. It will be said that the assumption does not extend to the 

 actual but only to the possible existence of such things. I answer 

 that, according to every test we have of possibility, they are not 

 even possible. Their existence, so far as we can form any judg- 

 ment, would seem to be inconsistent with the physical constitution 

 of onr planet at least, if not of the universal (sir). To get rid of 

 this difiicnlty and at the same time to save the credit of the sup- 

 posed system of necessary truth, it is customary to say that the 

 points, lines, circles, and squares which are the subjects of geometry 

 exist in our conceptions merely, and are parts of our minds, which 

 minds, by working on their own materials, construct an « priori 

 science the evidence of which is purely mental and has nothing to do 

 with outward experience. By howsoever high authority this doctrine 

 has been sanctioned, it appears to me psychologically incorrect. The 

 points, lines, and squares which any one has in his mind are (as I appre- 

 hend) simply copies of the points, lines, and squares which he has 

 known in his experience. Our idea of a point I apprehend to be 

 simply onr idea of the minimum visible— the small portion of surface 



I think it may be at once conceded that the truths 

 of geometry are truths precisely because they relate to 

 and express the properties of what Mill calls "purely 

 imaginary oVijects." That these objects do not exist in 

 Mill's sense, that they do not exist in nature, may also be 

 granted; that they are "not even possible," if this means 

 not possible in an existing nature, may also be granted. 

 That we cannot " conceive " them depends on the meaning 

 which we attach to the word conceive. I would myself 

 say that the purely imaginary objects are the only realities, 

 the oirwc oiTfi, in regard to which the corresponding phy- 

 sical objects are as the shadows in the cave ; ai d it is only 

 by means of them that we are able to deny the existence 

 of a corresponding physical object. If there is no con- 

 ception of straightness, then it is meaningless to deny the 

 existence of a perfectly straight line. But, at any rate, 

 the objects of geometrical truth are the so-called imaginary 

 objects of Mill, and the truths of geometry are only true, 

 and a furtiori are only necessarily true, in regard to these 

 so-called imaginary objects ; and these objects, points, 

 lines, circles, itc, in the mathematical sense of the terms, 

 have a likeness to, and are represented more or less imper- 

 fectly, and from a geometer's point of view, no matter 

 how imperfectly, by corresponding physical points, lines, 

 circles, etc. 



I shall have to return to geometry, and will then speak 

 of Riemann, but I will first refer to another passage of the 

 " Logic." Speaking of the truths of arithmetic, MiU says 

 (p. 297) that even here there is one hypothetical element : 

 — " In all propositions concerning numbers a condition is 

 implied without which none of them would be true, and 

 that condition is an assumption which may be false. The 

 condition is that 1 = 1; that all the numbers are numbers 

 of the same or of equal units." Here, at least the assump- 

 tion may be absolutely true ; one shilling = one shilling in 

 purchasing power, although they may not be absolutely of 

 the same weight and fineness, but it is hardly necessary ; 

 one coin -f one coin = two coins, even if the one be a 

 shilling and the other a half-crown. In fact, whatever 

 difliculty be raisable as to geometry, it seems to me that no 

 similar difficulty applies to arithmetic ; mathematicians or 

 not, we have each of us, in its most abstract form, the 

 idea of a number ; we can each of us appreciate the truth 

 of a proposition in regard to numbers, and we cannot but 

 see that a truth in regard to numbers is something different 

 in kind from an experimental truth generalised from expe- 

 rience. Compare, for instance, the proposition that the sun, 

 having already risen so many times, will rise to-morrow, 

 and the next day, and the day after that, and so on, and 

 the propo.sition that even and odd numbers succeed each 

 other alternately ad iiifinitum, the latter, at least, seems to 

 have the characters of universality and necessity. Or, 

 again, suppose a proposition observed to hold good for a 

 long series of numbers, 1,000 numbers, 2,000 numbers, as 

 the case may be, this is not only no proof, but it is abso- 

 lutely no evidence, that the proposition is a true proposi- 

 tion holding good for all numbers whatever ; there are in 



which we can see. We can reason about a line as if it had no 

 breadth because we have a power which we can exercise over the 

 operations of our minds — the power, when perception is present to 

 our senses or a conception to our intellects, of attending to a part 

 only of that perception or conception instead of the whole. But 

 we cannot conceive a line without breadth — we can form no mental 

 picture of such a line ; all the lines which we have in our mind are 

 lines possessing breadth. If any one doubts this, we may refer him 

 to his own experience. I much question if any one who fancies 

 that he can conceive of a mathematical line thinks so from the evi- 

 dence of his own consciousness. I suspect it is rather because he 

 supposes that unless such a perception be possible, mathematics 

 could not exist as a science — a supposition which there will be no 

 difficulty in showing to be groundless." 



