=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._