Part 2; Collected Mathematical Papers,

Vol. 2, p. 5._

=111.= The object of pure mathematics is those relations which may be conceptually established among any conceived elements whatsoever by assuming them contained in some ordered manifold; the law of order of this manifold must be subject to our choice; the latter is the case in both of the only conceivable kinds of manifolds, in the discrete as well as in the continuous.

--PAPPERITZ, E.

_Über das System der rein mathematischen Wissenschaften, Jahresbericht der Deutschen Mathematiker-Vereinigung, Bd. 1, p. 36._

=112.= Pure mathematics is not concerned with magnitude. It is merely the doctrine of notation of relatively ordered thought operations which have become mechanical.--NOVALIS.

_Schriften (Berlin, 1901), Zweiter Teil, p. 282._

=113.= Any conception which is definitely and completely determined by means of a finite number of specifications, say by assigning a finite number of elements, is a mathematical conception. Mathematics has for its function to develop the consequences involved in the definition of a group of mathematical conceptions. Interdependence and mutual logical consistency among the members of the group are postulated, otherwise the group would either have to be treated as several distinct groups, or would lie beyond the sphere of mathematics.--CHRYSTAL, GEORGE.

_Encyclopedia Britannica (9th edition), Article “Mathematics.”_

=114.= The purely formal sciences, logic and mathematics, deal with those relations which are, or can be, independent of the particular content or the substance of objects. To mathematics in particular fall those relations between objects which involve the concepts of magnitude, of measure and of number.--HANKEL, HERMANN.

_Theorie der Complexen Zahlensysteme, (Leipzig, 1867), p. 1._

=115.= _Quantity is that which is operated with according to fixed mutually consistent laws._ Both operator and operand must derive their meaning from the laws of operation. In the case of ordinary algebra these are the three laws already indicated [the commutative, associative, and distributive laws], in the algebra of quaternions the same save the law of commutation for multiplication and division, and so on. It may be questioned whether this definition is sufficient, and it may be objected that it is vague; but the reader will do well to reflect that any definition must include the linear algebras of Peirce, the algebra of logic, and others that may be easily imagined, although they have not yet been developed. This general definition of quantity enables us to see how operators may be treated as quantities, and thus to understand the rationale of the so called symbolical methods.--CHRYSTAL, GEORGE.

_Encyclopedia Britannica (9th edition), Article “Mathematics.”_

=116.= Mathematics--in a strict sense--is the abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations.

--MURRAY, J. A. H.

_A New English Dictionary._

=117.= Everything that the greatest minds of all times have accomplished toward the _comprehension of forms_ by means of concepts is gathered into one great science, _mathematics_.

--HERBART, J. F.

_Pestalozzi’s Idee eines A B C der Anschauung, Werke [Kehrbach], (Langensalza, 1890), Bd. 1, p. 163._

=118.= Perhaps the least inadequate 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] the study of ideal constructions (often applicable to real problems), and the discovery thereby of relations between the parts of these constructions, before unknown.--PEIRCE, C. S.

_Century Dictionary, Article “Mathematics.”_

=134.= Mathematics is that form of intelligence in which we bring the objects of the phenomenal world under the control of the conception of quantity. [Provisional definition.]--HOWISON, G. H.

_The Departments of Mathematics, and their Mutual Relations; Journal of Speculative Philosophy, Vol. 5, p. 164._

=135.= Mathematics is the science of the functional laws and transformations which enable us to convert figured extension and rated motion into number.--HOWISON, G. H.

_The Departments of Mathematics, and their Mutual Relations; Journal of Speculative Philosophy, Vol. 5, p. 170._