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BRITISH ASSOCIATION, SEPTEMBER 1883. 
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may accompany a life of virtue and justice, and to show that, independently of all 
these, justice was a thing desirable in itself and for its own sake—not by speaking 
to you of the utility of mathematics in any of the questions of common life or of 
physical science. Still less would I speak of this utility before, I trust, a friendly 
audience, interested or willing to appreciate an interest in mathematics in itself and 
for its own sake. I would, on the contrary, rather consider the obligations of 
mathematics to these different subjects as the sources of mathematical theories now 
as remote from them, and in as different a region of thought—for instance, geometry 
from the measurement of land, or the Theory of Numbers from arithmetic—as a 
river at its mouth is from its mountain source. 
On the other side, the general opinion has been and is that it is indeed by 
experience that we arrive at the truths of mathematics, but that experience is not 
their proper foundation: the mind itself contributes something. This is involved in 
the Platonic theory of reminiscence; looking at two things, trees or stones or anything 
else, which seem to us more or less equal, we arrive at the idea of equality: but 
we must have had this idea of equality before the time when first seeing the two 
things we were led to regard them as coming up more or less perfectly to this idea 
of equality; and the like as regards our idea of the beautiful, and in other cases. 
The same view is expressed in the answer of Leibnitz, the nisi intellectus ipse, 
to the scholastic dictum, nihil in intellectu quod non prius in sensu: there is nothing in 
the intellect which was not first in sensation, except (said Leibnitz) the intellect 
itself. And so again in the Critiok of Pure Reason, Kant’s view is that while there is 
no doubt but that all our cognition begins with experience, we are nevertheless in 
possession of cognitions a priori, independent, not of this or that experience, but 
absolutely so of all experience, and in particular that the axioms of mathematics 
furnish an example of such cognitions a priori. Kant holds further that space is no 
empirical conception which has been derived from external experiences, but that in 
order that sensations may be referred to something external, the representation of 
space must already lie at the foundation; and that the external experience is itself 
first only possible by this representation of space. And in like manner time is no 
empirical conception which can be deduced from an experience, but it is a necessary 
representation lying at the foundation of all intuitions. 
And so in regard to mathematics, Sir W. R. Hamilton, in an Introductory Lecture 
on Astronomy (1836), observes: “ These purely mathematical sciences of algebra and 
geometry are sciences of the pure reason, deriving no weight and no assistance from 
experiment, and isolated or at least isolable from all outward and accidental phenomena. 
The idea of order with its subordinate ideas of number and figure, we must not indeed 
call innate ideas, if that phrase be defined to imply that all men must possess them 
with equal clearness and fulness: they are, however, ideas which seem to be so far born 
with us that the possession of them in any conceivable degree is only the development 
of our original powers, the unfolding 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