432 PRESIDENTIAL ADDRESS TO THE [784 
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 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 sight it appears to make out how far the views 
taken by Mill in his System of Logic Ratiocinative and Inductive (9th ed. 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 enquire what is the ground of our belief in axioms, what is the evidence on which 
they rest. I answer, they are experimental truths, generalisations from experience. 
The proposition ‘ 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 enquiring “ 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 ? ” 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 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 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 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 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 
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 truth, it is customary to 
say that the points, lines, circles and squares which are the subjects of geometay 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 howsoever high authority 
this doctrine has been sanctioned, it appears to me psychologically incorrect. The points, 
lines and squares which anyone has in his mind are (as I apprehend) simply copies