TRANSACTIONS OF SECTION A, 429 
that the edifices when they reappear will not be entirely transformed, but will 
still retain something of their historic outlines. It would be contrary to the 
spirit of this Address to undervalue in any way the critical examination and 
revision of principles; we must acknowledge that it tends ultimately to simpli- 
fication, to the clearing up of issues, and the reconciliation of apparent. contradic- 
tions. But it would be a misfortune if this process were to absorb too large a 
share of the attention of mathematicians, or were allowed to set too high a 
standard of logical completeness, In this particular matter of the ‘ arithmetisa- 
tion of Mathematics’ there is, I think, a danger in these respects. As regards the 
latter point, a traveller who refuses to pass over a bridge until he has personally 
tested the soundness of every part of it is not likely to go very far; something 
must be risked, even in Mathematics, It is notorious that even in this realm of 
‘exact’ thought discovery has often been in advance of strict logic, as in the 
theory of imaginaries, for example, and in the whole province of analysis of which 
Fourier’s theorem is the type. And it might even be claimed that the services 
which Geometry has rendered to other sciences have been almost as great in virtue 
of the questions which it implicitly begs as of those which it resolves. 
I would venture, with some trepidation, to go one step further. Mathema- 
ticians love to build on as definite a foundation as possible, and from this point of 
view the notion of the integral number, on which (we are told) the Mathematics 
of the future are to be based, is very attractive. But, as an instrument for the 
study of Nature, is it really more fundamental than the geometrical notions which 
it is to supersede? The accounts of primitive peoples would seem to show that, 
in the generality which is a necessary condition for this purpose, it is in no less 
degree artificial and acquired. Moreover, does not the act of enumeration, as 
applied to actual things, involve the same process of selection and idealisa- 
tion which we have already met with in other cases? As an illustration, suppose 
we were to try to count the number of drops of water inacloud, I am not 
thinking of the mere practical difficulties of enumeration, or even of the more 
pertinent fact that it is hard to say where the cloud begins or ends. Waiving 
these points, it is obvious that there must be transitional stages between a more 
or less dense group of molecules and a drop, and in the case of some of these 
aggregates it would only be by an arbitrary exercise of judgment that they 
would be assigned to one category rather than to the other, In whatever form 
we meet with it, the very notion of counting involves the highly artificial con- 
ception of a number of objects which for some purposes are treated as absolutely 
alike, whilst yet they can be distinguished, 
The net result of the preceding survey is that the systems of Geometry, of 
Mechanics, and even of Arithmetic, on which we base our study of Nature, are all 
contrivances of the same general kind; they consist of series of abstractions and 
conventions devised to represent, or rather to symbolise, what is most interesting 
and most accessible to us in the world of phenomena, And the progress of 
science consists in a great measure in the improvement, the development, and the 
simplification of these artificial conceptions, so that their scope may be wider and 
the representation more complete. The best in this kind are but shadows, but we 
may continually do something to amend them. 
As compared with the older view, the function of physical science is seen to be 
much more modest than was at one time supposed. We no longer hope by levers 
and screws to pluck out the heart of the mystery of the universe. But there are 
compensations, The conception of the physical world as a mechanism, con- 
structed on a rigid mathematical plan, whose most intimate details might possibly 
some day be guessed, was, I think, somewhat depressing. We have been led to 
recognise that the formal and mathematical element is of our own introduction; 
that it is merely the apparatus by which we map out our knowledge, and 
has no more objective reality than the circles of latitude and longitude on the 
sun, A distinguished writer not very long ago speculated on the possibility 
of the scientific mine being worked out within no distant period. Recent dis- 
coveries seem to have put back this possibility indefinitely; and the tendency of 
modern speculation as to the nature of scientific knowledge should be to banish 
