ADDRESS. Hi) 
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 measure- 
ment 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 expe- 
rience is not their proper foundation: the mind itself contributes some- 
thing. 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 tie 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 
apse, 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 ‘ Critick 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 prior’, 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 thatin order that sensations may be referred to 
something external, the representation of space must already le at the 
foundation ; and that the external experience is itself first only possible 
by this entation 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 Introduc- 
tory 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 
tsolable 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 con- 
