492 



NATURE 



[Sept. 20, 1883 



cal science. Still less would I speak of this utility before, I 

 trust, a friendly audience, interested or willing to appreciate an 

 interest in mathematics in itself and for its own sake. I would, 

 on the contrary, rather consider the obligations of mathematics 

 to these different subjects as the sources of mathematical theories 

 now as remote from them, and in as different a region of thought 

 — for instance, geometry from the measurement of land, or the 

 Theory of Numbers from arithmetic — -as a river at its mouth is 

 from its mountain source. 



On the other side the general opinion has been and is that it is 

 indeei by experience that we arrive at the truths of mathematic-, 

 but that experience is not their proper foundation : the mind 

 itself contributes something. This is involved in the Platonic 

 theory of reminiscence ; looking at two things, trees or stones or 

 anything else, which seem to us more or less equal, we arrive at 

 the idea of equality : but we must have had this idea of equality 

 before the time when first seeing the two things we were led to 

 regard them as coming up more or less perfectly to this idea of 

 equality ; and the like as regards our idea of the beautiful, and 

 in other cases. 



The same view is expressed in the answer of Leibnitz, the 

 nisi intellectus ipse, to the scholastic dictum, nihil in iniellectu 

 quod 11011 pi ins in sensu : there is nothing in the intellect which 

 was not first in sensation, except (said Leibnitz) the intellect 

 itself. And so again in the "Critick of Pure Reason," Kant's 

 view is that, while there is no doubt but that all our cognition 

 begins with experience, we are nevertheless in possession of cog- 

 nitions a ^ficri, independent, not of this or that experience, but 

 absolutely so of all experience, and in particular that the axioms 

 of mathematics furnish an example of such cognitions a priori. 

 Kant holds further that space is no empirical conception which 

 has been derived from external experiences, but that in order 

 that sensations may be referred to something external, the repre- 

 sentation of space must already lie at the foundation ; and that 

 the external experience is itself first only pos-ible by this repre- 

 sentation of space. And in like manner time is no empirical 

 conception which can be deduced from an experience, but it is a 

 necessary representation lying at the foundation of all intuitions. 

 And so in regard to mathematics, Sir W. R. Hamilton, in an 

 introductory lecture on astronomy (1836), observes: "These 

 purely mathematical sciences of algebra and geometry are 

 sciences of the pure reason, deriving no weight and no assist- 

 ance from experiment, atid i-olated or at least isolable from all 

 outward and accidental phenomena. The idea of order, with its 

 subordinate ideas of number and figure, we must not indeed call 

 innate ideas, if that phrase be defined to imply that all men 

 must possess them with equal clearness and fulness : they are, 

 however, ideas which seem to be si far born with us that the 

 possession of them in any conceivable degree is only the deve- 

 lopment of our original powers, the unfolding of our proper 

 humanity." 



The general question of the ideas of space and time, the 

 axioms and definitions of geometry, the axioms relating to num- 

 ber, and the nature of mathematical reasoning, are fully and 

 ably discussed in Whewell's "Philosophy of the Inductive 

 Sciences" (1S40), which may be regarded as containing an expo- 

 sition of the whole theory. 



But it is maintained by John Stuart Mill that the truths of 

 mathematics, in particular those of geometry, rest on experience ; 

 and, as regards geometry, the same view is on very different 

 grounds maintained by the mathematician Riemann. 



It is not so easy as at first sijbt it appears to make out how 

 far the views taken by Mill in his "System of Logic Katiocina- 

 tive and Inductive" (ninth edition, 1879) are absolutely contra- 

 dict 'ry to those which have been spoken of ; they profe-s to be 

 so; there are m 1st definite assertions (supported by argument), 

 for instance, p. 263 : — " It remains to inquire what is the ground 

 of oar belief in axioms, what is the evidence on which they rest. 

 I answer, they are experimental truths, generalisations from ex- 

 perience. The propoiti .11 'Two straight lines cannot inclose a 

 spice,' or, in other words, two straight lines which have once 

 me; cannot meet again, is an induction from the evidence of our 

 senses." But I cannot help considering a previous argument (p. 

 250) as very materially modifying this absolute contradiction. 

 After inquiring " Why are mathematics by almost all philoso- 

 ph rs . . . considered to be independent of the evidence of 

 experience and observation, and characterised as systems of 

 necessary truth?" Mill proceeds (I quote the whole passage) as 

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

 necessity ascribed to the truths of mathematic, 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 conclu-ions of geometry 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 con- 

 cerning the meaning of a word, can ever follow, and that what 

 apparently 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 geo- 

 metry is not strictly true : there exist no real things exactly con- 

 formable 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 pos- 

 sibility they are not even possible. Their existence, so far as 

 we can form any judgment, would seem to be inconsistent with 

 the physical constitution of our planet at least, if not of the 

 universal [sic]. To get rid of this difficulty, and at the same 

 time to save the credit of the supposed system of necessary 

 truths, it is customary to say that the points, lines, circles, and 

 squares which are the subjects of geometry, exist in our concep- 

 tions merely, and are parts of our minds : which minds, by 

 working on their own materials, construct an a 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 apprehend) simply copies of the points, lines, 

 and squares which he has known in his experience. Our idea 

 of a 1 oint I apprehend to be simply our idea of the minimum 

 visibilr, the small portion of surface which we can see. We 

 can rea'on 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 a perception is present to our senses or a con- 

 ception to our intellects, of att.nding to a part only of that 

 percei tion or con. eption 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 doubt this, we may refer him to 

 Ills own experience. I much question if any one who fancies that 

 he can conceive of a mathematical line thinks so from the 

 evidence of his own consciousness. I suspect it is rather because 

 he suppo es that unless such a perception be possible, mathe- 

 matics could not exist as a science : a supposition which there 

 will be no difficulty in showing to be groundless." 



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 objects" ; 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 

 uvtws oura, in regard to which the corresponding physical objects 

 are as the shadows in the cave ; ard it is only by means of them 

 that we are able to deny the existence of a corresponding physical 

 object ; if there is no conception of straightness, then it is mean- 

 ingless to deny the existence of a perfectly straight line. 



But at any rale the objects of geometrical truth are the so- 

 called imaginary objects of Mill, and the truths of geometry are 

 only true, and ajortiori are only necessarily true, ill regard to 

 the.e so-called imaginary objects; and these objects, points, 

 line-, circles, &c, in the mathematical sense of the terms, have 

 a likeness to and are represented more or less imperfectly, and 

 from a geometer's point of view 1110 matter how imperfectly, by 

 corresponding physical points, line-, circles, &c. I shall have 

 to return to geometry, a d will then speak of Riemann, but I 

 will first refer to another pas-age of the " Logic." 



Speaking of the truths of arithmetic Mill 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 

 w hich 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 assumption may be absolutely true ; one shilling = one 



