October 5, 1883.] 



SCIENCE. 



479 



equal clearness and fulness: they are, however, ideas 

 which seem to be so 'far born with us that tlie posses- 

 sion of them in any conceivable degree is only the 

 development 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 number, and the nature of niatlieraalical 

 reasoning, are fully and ably discussed in Whewell's 

 "Philosophy of the inductive sciences" (1840), which 

 may be regarded as containing an exposition of the 

 whole theory. 



But it is maintained by John Stuart Mill that the 

 truths of mathematics, in particular those of geome- 

 try, 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 sight it appears, to make 

 out how far the views taken by Mill in his 'System 

 of logic ratiocinative and inductive' (nintli edi- 

 tion, 1879) are absolutely contradictory to those which 

 have been spoken of. They profess to be so. There are 

 most definite assertions (supported by argument) : for 

 instance, p. 263, "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, generalizations from experience. The propo- 

 sition ' 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 evi- 

 dence of our senses." But I cannot help considering 

 a previous argument (p. 250) as very materially modi- 

 fying this absolute contradiction. After inquiring, 

 " Why are mathematics by almost all philosophers 

 . . . considered to be independent of the evidence 

 of experience and observation, and characterized as 

 systems of necessary truth ?" Mill proceeds (I quote 

 the whole passage) as follows: "The answer I con- 

 ceive to be, that this character of necessity ascriljed 

 to the truths of mathematics, and even (with some 

 reservations to be hereafter made) the peculiar cer- 

 tainty 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 con- 

 clusions 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 con- 

 versant. Now, we have pointed out, that, from a 

 definition as such, no proposition, unless it be one 

 concerning 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 conformal)le 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 per- 

 fectly right. It will be said that the assumption does 

 not extend to the actual, but only to the possible, ex- 



istence 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 

 judgment, would seem to be inconsistent with the 

 physical constitution of our planet at least. If not of 

 the universal [Mc]. To get lid of this difliculty, 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 conceptions 

 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 

 liowsoever high authority this doctrine has been 

 sanctioned, it appears to me psychologically incor- 

 rect. 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 lias known 

 in his experience. Our idea of a point I appreliend 

 to be simply our idea of the minimum visibile, the 

 small portion of surface 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 a percep- 

 tion is present to our senses, or a conception to our 

 intellects, of attending to a part only of that percep- 

 tion 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 liave in our mind are lines possessing breadth. If 

 any one doubt this, we may refer him to his own ex- 

 perience. 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 supposes, that, unless 

 such a perception be possible, mathematics could not 

 exist as a science, — a supposition which there will be 

 no difliculty 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 imaginaiy objects.' Th.at these objects do 

 not exist in Mill's sense, that they do not exist in 

 n.iture, may also be granted. That they ,are ' not 

 even possible,' if this means not possible in an ex- 

 isting nature, may also be granted. That we cannot 

 'conceive' tliem depends on the meaning which we 

 attach to the word ' conceive.' I would myself say 

 that the purely imaginary objects are the only reali- 

 ties, the iiTuf oiTQ, ill regard to which the correspond- 

 ing physical objects are as the shadows in the cave; 

 .and it is only by means of them that we are able to 

 deny the existence of a corresponding physical ob- 

 ject. If there is no conception of straiglitness, 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 Im.iginary objects of Mill; and the 

 truths of geometry are only true, and a fortiori are 

 only necessarily true, in regard to the<e so-called 

 imaginary objects. And these objects, points, lines, 

 circles, etc., in the mathematical sense of the terms. 



