[126]. The whole of Mathematics consists in the organization of a series of aids to the imagination in the process of reasoning.—Whitehead, A. N.
Universal Algebra (Cambridge, 1898), p. 12.
[127]. Pure mathematics consists entirely of such asseverations as that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is of which it is supposed to be true.... If our hypothesis is about anything and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.—Russell, Bertrand.
Recent Work on the Principles of Mathematics, International Monthly, Vol. 4 (1901), p. 84.
[128]. Pure Mathematics is the class of all propositions of the form “p implies q,” where p and q are propositions containing one or more variables, the same in the two propositions, and neither p nor q contains any constants except logical constants. And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as may be involved in the general notion of propositions of the above form. In addition to these, Mathematics uses a notion which is not a constituent of the propositions which it considers—namely, the notion of truth.—Russell, Bertrand.
Principles of Mathematics (Cambridge, 1903), p. 1.
[129]. The object of pure Physic is the unfolding of the laws of the intelligible world; the object of pure Mathematic that of unfolding the laws of human intelligence.—Sylvester, J. J.
On a theorem, connected with Newton’s Rule, etc., Collected Mathematical Papers, Vol. 3, p. 424.
[130]. First of all, we ought to observe, that mathematical propositions, properly so called, are always judgments a priori, and not empirical, because they carry along with them necessity, which can never be deduced from experience. If people should object to this, I am quite willing to confine my statements to pure mathematics, the very concept of which implies that it does not contain empirical, but only pure knowledge a priori.—Kant, Immanuel.