description of the general scope of modern Pure Mathematics—I will not call it a definition—would be to say that it deals with form, in a very general sense of the term; this would include algebraic form, functional relationship, the relations of order in any ordered set of entities such as numbers, and the analysis of the peculiarities of form of groups of operations.—Hobson, E. W.

Presidential Address British Association for the Advancement of Science (1910); Nature, Vol. 84, p. 287.

119. The ideal of mathematics should be to erect a calculus to facilitate reasoning in connection with every province of thought, or of external experience, in which the succession of thoughts, or of events can be definitely ascertained and precisely stated. So that all serious thought which is not philosophy, or inductive reasoning, or imaginative literature, shall be mathematics developed by means of a calculus.—Whitehead, A. N.

Universal Algebra (Cambridge, 1898), Preface.

120. Mathematics is the science which draws necessary conclusions.—Peirce, Benjamin.

Linear Associative Algebra, American Journal of Mathematics, Vol. 4 (1881), p. 97.

121. Mathematics is the universal art apodictic.—Smith, W. B.

Quoted by Keyser, C. J. in Lectures on Science, Philosophy and Art (New York, 1908), p. 13.

122. Mathematics in its widest signification is the development of all types of formal, necessary, deductive reasoning.—Whitehead, A. N.

Universal Algebra (Cambridge, 1898), Preface, p. vi.

123. Mathematics in general is fundamentally the science of self-evident things.—Klein, Felix.

Anwendung der Differential- und Integralrechnung auf Geometrie (Leipzig, 1902), p. 26.

124. A mathematical science is any body of propositions which is capable of an abstract formulation and arrangement in such a way that every proposition of the set after a certain one is a formal logical consequence of some or all the preceding propositions. Mathematics consists of all such mathematical sciences.—Young, Charles Wesley.

Fundamental Concepts of Algebra and Geometry (New York, 1911), p. 222.

125. Pure mathematics is a collection of hypothetical, deductive theories, each consisting of a definite system of primitive, undefined, concepts or symbols and primitive, unproved, but self-consistent assumptions (commonly called axioms) together with their logically deducible consequences following by rigidly deductive processes without appeal to intuition.—Fitch, G. D.

The Fourth Dimension simply Explained (New York, 1910), p. 58.

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.

Critique of Pure Reason [Müller], (New York, 1900), p. 720.

131. Mathematics, the science of the ideal, becomes the means of investigating, understanding and making known the world of the real. The complex is expressed in terms of the simple. From one point of view mathematics may be defined as the science of successive substitutions of simpler concepts for more complex....—White, William F.

A Scrap-book of Elementary Mathematics, (Chicago, 1908), p. 215.

132. The critical mathematician has abandoned the search for truth. He no longer flatters himself that his propositions are or can be known to him or to any other human being to be true; and he contents himself with aiming at the correct, or the consistent. The distinction is not annulled nor even blurred by the reflection that consistency contains immanently a kind of truth. He is not absolutely certain, but he believes profoundly that it is possible to find various sets of a few propositions each such that the propositions of each set are compatible, that the propositions of each such set imply other propositions, and that the latter can be deduced from the former with certainty. That is to say, he believes that there are systems of coherent or consistent propositions, and he regards it his business to discover such systems. Any such system is a branch of mathematics.—Keyser, C. J.

Science, New Series, Vol. 35, p. 107.

133. [Mathematics is]