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president’s address— section a. 
Mathematics, Sir "Win. Hamilton, had such an influence on Maxwell? 
Was it not that their thoughts moved in parallel though distant chan- 
nels? Kant is, I believe, held by many metaphysicians to be their 
greatest modern master. Does anybody deny that his power was 
largely owing to his scientific and especially his mathematical attain- 
ments? Professor Tait has accused the great mathematician, Sir 
Wm. llo wan Hamilton, of having a metaphysical twist. 
The mention of Kant at once suggests the consideration of 
another of the foundations of mathematics — our spatial conceptions. 
If there are forms of thought, if there are boundaries to the region in 
which by our mind’s constitution we must think, if there are funda- 
mental instincts whose validity we dare not presume to impugn 
though we cannot trace them to a foundation in experience, surely ail 
these receive illustration in our geometrical conceptions, Kant, if I 
understand (second-hand) aright, held that our mind is so constituted 
that a conception not bounded by the rules of Euclidean space is an 
impossibility. But it is notorious among mathematicians to-day that, 
so far from this being the case, it is now a definite physical problem 
to determine experimentally whether or not Euclid’s ideas of space 
are the correct ones. If the very basis of all mathematics is thus 
subject to revision at any moment, surely we cannot hold that the 
science is differentiated in kind from other sciences by reason of the 
absolute certainty of premises. 
The same aspect of the subject could be illustrated from the 
history of the acceptance by mathematicians of the ideas of positive 
and negative quantities, of continuous and discontinuous quantities, and 
of the imaginary. By some it is thought that the imaginary is now 
miscalled. Have we not found a real operator that possesses all the 
properties of the imaginary ? Undoubtedly we have, so far as the 
imaginary appears in ordinary algebra. But no sooner do we examine 
this operator than we find the old imaginary appearing in the pro- 
cesses of the new real operator. And the worst of it is that it is as 
imaginary as ever. 
Many volumes might be — perhaps have been, were they all 
collected — profitably written on the metaphysics involved in the three 
laws of motion. Perhaps, if physicists were better trained meta- 
physicians, we should cease to see a curious circularity of definition in 
the subject of heat. Absolute temperature is defined in terms of 
quantity of heat. But the meaning of absolute temperature is tacitly 
assumed in defining what is meant by quantity of heat. 
Surely it is not the mathematician who should cast the first stone 
at the metaphysician, nor the latter who should perform a similar office 
for the former. Let each perform his allotted task for the common 
weal, and not waste precious energy in mutual vituperation. 
But metaphysical thought is not the only activity of the mind 
whose relations to mathematical thought are by most totally misappre- 
hended. Is it a daring thing to mention mathematics and emotion in 
the same breath ? I am afraid that most people believe them to be as 
far asunder as the poles. But why ? Joy, despair, a trembling 
excitement in the presence of some new vast truth, a sense of 
exquisite beauty, all these sometimes to an intense degree are the lot 
of the mathematician. You think the mathematician has no sense of 
beauty in bis work. The veriest tyro will he able to set you right. 
