Memorabilia Mathematica

CHAPTER I. Definitions and Object of Mathematics

DEFINITIONS AND OBJECT OF MATHEMATICS

=101.= I think it would be desirable that this form of word [mathematics] should be reserved for the applications of the science, and that we should use mathematic in the singular to denote the science itself, in the same way as we speak of logic, rhetoric, or (own sister to algebra) music.--SYLVESTER, J. J.

Presidential Address to the British Association, Exeter British Association Report (1869); Collected Mathematical Papers, Vol. 2, p. 659.

=102.= ... all the sciences which have for their end investigations concerning order and measure, are related to mathematics, it being of small importance whether this measure be sought in numbers, forms, stars, sounds, or any other object; that, accordingly, there ought to exist a general science which should explain all that can be known about order and measure, considered independently of any application to a particular subject, and that, indeed, this science has its own proper name, consecrated by long usage, to wit, mathematics. And a proof that it far surpasses in facility and importance the sciences which depend upon it is that it embraces at once all the objects to which these are devoted and a great many others besides; ....

--DESCARTES.

Rules for the Direction of the Mind, Philosophy of D. [Torrey] (New York, 1892), p. 72.

=103.= [Mathematics] has for its object the indirect measurement of magnitudes, and it purposes to determine magnitudes by each other, according to the precise relations which exist between them.--COMTE.

Positive Philosophy [Martineau], Bk. 1, chap. 1.

=104.= The business of concrete mathematics is to discover the equations which express the mathematical laws of the phenomenon under consideration; and these equations are the starting-point of the calculus, which must obtain from them certain quantities by means of others.--COMTE.

Positive Philosophy [Martineau], Bk. 1, chap. 2.

=105.= Mathematics is the science of the connection of magnitudes. Magnitude is anything that can be put equal or unequal to another thing. Two things are equal when in every assertion each may be replaced by the other.--GRASSMANN, HERMANN.

Stücke aus dem Lehrbuche der Arithmetik, Werke (Leipzig, 1904), Bd. 2, p. 298.

=106.= Mathematic is either Pure or Mixed: To Pure Mathematic belong those sciences which handle Quantity entirely severed from matter and from axioms of natural philosophy. These are two, Geometry and Arithmetic; the one handling quantity continued, the other dissevered.... Mixed Mathematic has for its subject some axioms and parts of natural philosophy, and considers quantity in so far as it assists to explain, demonstrate and actuate these.

--BACON, FRANCIS.

De Augmentis, Bk. 3; Advancement of Learning, Bk. 2.

=107.= The ideas which these sciences, Geometry, Theoretical Arithmetic and Algebra involve extend to all objects and changes which we observe in the external world; and hence the consideration of mathematical relations forms a large portion of many of the sciences which treat of the phenomena and laws of external nature, as Astronomy, Optics, and Mechanics. Such sciences are hence often termed Mixed Mathematics, the relations of space and number being, in these branches of knowledge, combined with principles collected from special observation; while Geometry, Algebra, and the like subjects, which involve no result of experience, are called Pure Mathematics.--WHEWELL, WILLIAM.

The Philosophy of the Inductive Sciences, Part 1, Bk. 2, chap. I, sect. 4. (London, 1858).

=108.= Higher Mathematics is the art of reasoning about numerical relations between natural phenomena; and the several sections of Higher Mathematics are different modes of viewing these relations.--MELLOR, J. W.

Higher Mathematics for Students of Chemistry and Physics (New York, 1902), Prologue.

=109.= Number, place, and combination ... the three intersecting but distinct spheres of thought to which all mathematical ideas admit of being referred.--SYLVESTER, J. J.

Philosophical Magazine, Vol. 24 (1844), p. 285; Collected Mathematical Papers, Vol. 1, p. 91.

=110.= There are three ruling ideas, three so to say, spheres of thought, which pervade the whole body of mathematical science, to some one or other of which, or to two or all three of them combined, every mathematical truth admits of being referred; these are the three cardinal notions, of Number, Space and Order.

Arithmetic has for its object the properties of number in the abstract. In algebra, viewed as a science of operations, order is the predominating idea. The business of geometry is with the evolution of the properties of space, or of bodies viewed as existing in space.--SYLVESTER, J. J.

_A Probationary Lecture on Geometry, York British Association Report (1844),

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.

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