4.26 REPORT—1904, 
tions of geometry, on the logic of the most rudimentary arithmetical processes, as 
well as of the more artificial operations of the Calculus. These discussions are 
legitimate and inevitable, and have led to some results which are now widely 
accepted. Although they were carried on to a great extent independently, the 
questions involved will, I think, be found to be ultimately very closely connected. 
Their common nexus is, perhaps, to be traced in the physiological ideas of which 
Helmholtz was the most conspicuous exponent. To many minds such discussions 
are repellent, in that they seem to venture on the uncertain ground of philosophy. 
But, as a matter of fact, the current views on these subjects have been arrived 
at by men who have gone to work in their own way, often in entire ignorance 
of what philosophers have thought on such subjects. It may be maintained, 
indeed, that the mathematician or the physicist, as such, has no special concern 
with philosophy, any more than the engineer or the geographer. Nor, although 
this is a matter for their own judgment, would it appear that philosophers 
have very much to gain by a special study of the methods of mathematical or 
physical reasoning, since the problems with which they are chiefly concerned are 
presented to them in a much less artificial form in the circumstances of ordinary 
life. As regards the present topic I would put the matter in this way, that 
between Mathematics and Physics on the one hand and Philosophy on the other 
there lies an undefined borderland, and that the mathematician has been engaged 
in setting things in order, as he is entitled to do, on his own side of the boundary. 
From this point of view, it would be of interest to trace in detail the 
relationships of the three currents of speculation which have been referred to. 
At one time I was tempted to take this as the subject of my Address; but, 
although I still think the enterprise a possible one, I have been forced to recognise 
that it demands a better equipment than I can pretend to. I can only 
venture to put before you some of my tangled thoughts on the matter, trusting 
that some future occupant of this Chair may be induced to take up the question 
and treat it in a more illuminating manner. 
If we look back for a moment to the views currently entertained not so very 
long ago by mathematicians and physicists, we shall find, I think, that the preva- 
lent conception of the world was that it was constructed on some sort of absolute 
geometrical plan, and that the changes in it proceeded according to precise laws; 
that, although the principles of mechanics might be imperfectly stated in our 
text-books, at all events such principles existed, and were ascertainable, and, when 
properly formulated, would possess the definiteness and precision which were held 
to characterise, say, the postulates of Euclid. Some writers have maintained, 
indeed, that the principles in question were finally laid down by Newton, and have 
occasionally used language which suggests that any fuller understanding of them 
was a mere matter of interpretation of the text. But, as Hertz has remarked, 
most of the great writers on Dynamics betray, involuntarily, a certain malaise 
when explaining the principles, and hurry over this part of their task as quickly 
as is consistent with dignity. They are not really at their ease until, having 
established their equations somehow, they can proceed to build securely on these. 
This has led some people to the view that the laws of Nature are merely a system 
of differential equations; it may be remarked in passing that this is very much 
the position in which we actually stand in some of the more recent theories of 
Electricity. As regards Dynamics, when once the critical movement had set in, 
it was easy to show that one presentation after another was logically defective 
and confused ; and no satisfactory standpoint was reached until it was recognised 
that in the classical Dynamics we do not deal immediately with real bodies at 
all, but with certain conventional and highly idealised representations of them, 
which we combine according to arbitrary rules, in the hope that if these rules be 
judiciously framed the varying combinations will image to us what is of most 
interest in some of the simpler and more important phenomena. The changed 
point of view is often associated with the publication of Kirchhoff’s lectures on 
Mechanics in 1876, where it is laid down in the opening sentence that the problem 
of Mechanics is to describe the motions which occur in Nature completely and in 
the simplest manner. This statement must not be taken too literally; at all 
