6 REPORT—1883. 
ceivable degree is only the development of our original powers, the un- 
folding 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 mathematical 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 geometry, rest on experience; and as regards geo- 
metry, 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 ex- 
perience 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 orderto 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 represen- 
tations, 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 exis- 
tence of such things. I answer that according to every test we have 
of possibility they are not even possible. Their existence, so far as 
