Mathematics
mathematics
(Greek: mathematikos, disposed to learn)
A science consisting of two main divisions: pure mathematics, restricted to the abstract science which by the deductive system examines the conclusions necessarily involved in the idea of numerical and spatial relations, and which includes arithmetic, algebra, geometry, trigonometry, and calculus; and applied mathematics, which treats of the application of the abstract principles of pure mathematics to the concrete, and which comprehends branches of astronomy, physics, etc. The following have distinguished themselves in the fields of pure and applied mathematics.
CATHOLICS
Balzano, Berhard
Binet, Jacques Philippe Marie
Cauchy, Augustin Louis
Clavius, Christopher
Descartes, Rene
Dupin, Charles
Hermite, Charles
Laplace, Pierre Simon
Oresme, Nicole
Pascal, Blaise
Vieta, Francois
OTHER CHRISTIAN MATHEMATICIANS
Barrow, Isaac
Chasles, Michel
Euler, Leonhard
Gauss, Karl
Grassman, Herman
Leibnitz, Gottfried
Pfaff, Johann
Riemann, Georg
Taylor, Booth
New Catholic Dictionary
Fuente: New Catholic Dictionary
Mathematics
The traditional definition of mathematics as “the science of quantity” or “the science of discrete and continuous magnitude” is today inadequate, in that modern mathematics, while clearly in some sense a single connected whole, includes many branches which do not come under this head. Contemporary accounts of the nature of mathematics tend to characterize it rather by its method than by its subject matter.
According to a view which is widely held by mathematicians, it is characteristic of a mathematical discipline that it begins with a set of undefined elements, properties, functions, and relations, and a set of unproved propositions (called axioms or postulates) involving them; and that from these all other propositions (called theorems) of the discipline are to be derived by the methods of formal logic. On its face, as thus stated, this view would identify mathematics with applied logic. It is usually added, however, that the undefined terms, which appear in the role of names of undefined elements, etc., are not really names of particulars at all but are variables, and that the theorems are to be regarded as proved for any values of these variables which render the postulates true. If then each theorem is replaced by the proposition embodying the implication from the conjunction of the postulates to the theorem in question, we have a reduction of mathematics to pure logic. (For a particular example of a set of postulates for a mathematical discipline see the article Arithmetic, foundations of.)
There is also another sense in which it has been held that mathematics is reducible to logic, namely that in the expressions for the postulates of a mathematical discipline the undefined terms are to be given definitions which involve logical terms only, in such a way that postulates and theorems of the discipline thereby become propositions of pure logic, demonstrable on the basis of logical principles only. This view was first taken, as regards arithmetic and analysis, by Frege, and was afterwards adopted by Russell, who extended it to all mathematics.
Both views require for their completion an exact account of the nature of the underlying logic, which, it would seem, can only be made by formalizing this logic as a logistic system (q. v,). Such a formalization of the underlying logic was employed from the beginning by Frege and by Russell, but has come into use in connection with the other — postulational or axiomatic — view only comparatively recently (with, perhaps, a partial exception in the case of Peano).
Hilbert has given a formalization of arithmetic which takes the shape of a logistic system having primitive symbols some of a logical and some of an arithmetical character, so that logic and arithmetic are formalized together without taking logic as prior; similarly also for analysis. This would not of itself be opposed to the Frege-Russell view, since it is to be expected that the choice as to which symbols shall be taken as primitive in the formalization can be made in more than one way. Hilbert, however, took the position that many of the theorems of the system are ideale Aussagen, mere formulas, which are without meaning in themselves but are added to the reale Aussagen or genuinely meaningful formulas in order to avoid formal difficulties otherwise arising. In this respect Hilbert differs sharply from Frege and Russell, who would give a meaning (namely as propositions of logic) to all formulas (sentences) appearing. — Concerning Hilbert’s associated program for a consistency proof see the article Proof theory.
A view of the nature of mathematics which is widely different from any of the above is held by the school of mathematical intuitionism (q. v.). According to this school, mathematics is “identical with the exact part of our thought.” “No science, not even philosophy or logic, can be a presupposition for mathematics. It would be circular to apply any philosophical or logical theorem as a means of proof in mathematics, since such theorems already presuppose for their formulation the construction of mathematical concepts. If mathematics is to be in this sense presupposition-free, then there remains for it no other source than an intuition which presents mathematical concepts and inferences to us as immediately clear. . . . [This intuition] is nothing else than the ability to treat separately certain concepts and inferences which regularly occur in ordinary thinking.” This is quoted in translation from Heyting, who, in the same connection, characterizes the intuitionittic doctrine as asserting the existence of mathematical objects (Gegenstnde), which are immediately grasped by thought, are independent of experience, and give to mathematics more than a mere formal content. But to these mathematical objects no existence is to be ascribed independent of thought. Elsewhere Heyting speaks of a relationship to Kant in the apriority ascribed to the natural numbers, or rather to the underlying ideas of one and the process of adding one and the indefinite repetition of the latter. At least in his earlier writings, Brouwer traces the doctrine of intuitionism directly to Kant. In 1912 he speaks of “abandoning Kant’s apriority of space but adhering the more resolutely to the apriority of time” and in the same paper explicitly reaffirms Kant’s opinion that mathematical judgments are synthetic and a priori.
The doctrine that the concepts of mathematics are empirical and the postulates elementary experimental truths has been held in various forms (either for all mathematics, or specially for geometry) by J. S. Mill, H. Helmholtz, M. Pasch, and others. However, the usual contemporary view, especially among mathematicians, is that the propositions of mathematics say nothing about empirical reality. Even in the case of applied geometry, it is held, the geometry is used to organize physical measurement, but does not receive an interpretation under which its propositions become unqualifiedly experimental or empirical in character; a particular system of geometry, applied in a particular way, may be wrong (and demonstrably wrong by experiment), but there is not, in significant cases, a unique geometry which, when applied in the particular way, is right.
M. Bocher,
The fundamental conceptions and methods of mathematics, Bulletin of the American Mathematical Society, vol. 11 (1904), pp. 115-135.
J. W. Young,
Lectures on Fundamental Concepts of Algebra and Geometry, New York, 1911.
Veblen and Young,
Projecthe Geometry, vol. 1, 1910 (see the Introduction).
C. I. Keyser,
Doctrinal functions, The Journal of Philosophy, vol. 15 (1918), pp. 262-261.
G. Frege,
Die Grundlagen der Arithmetik, Breslau, 1884; reprinted, Breslau, 1934.
G. Frege,
Grundgesetze der Arithmetik, vol. 1, Jena, 1893, and vol. 2, Jena, 1903.
B. Russell,
The Principles of Mathematics, Cambridge, England, 1903; 2nd edn., London, 1937, and New York, 1938.
B. Russell,
Introduction to Mathematical Philosophy, London, 1919. – —
R. Carnap,
Die logizistische Grundlegung der Mathematik, Erkenntnis, vol. 2 (1931), pp. 91-105, 141-144, 145.
A. Heyting,
Die intuitionistische Grundlegung der Mathematik. ibid., pp. 106-115.
J. v. Neumann,
Die formalistische Grundlegung der Mathematik, ibid., pp. 116-121, 114-145, 146, 148.
R. Carnap,
The Logical Syntax of Language, New York and London, 1937.
L. E. J. Brouwer,
Intuitionisme en Formalisme, Groningen, 1912 ; reprinted in Wiskunde, Waarheid, Werkelijkheid, Groningen, 1919; English translation by A. Dresden, Bulletin of the American Mathematical Society, vol 20 (1913) pp. 81-96.
H. Weil,
Die heutige Erkenntnislage in der Mathematik, Symposion, vol. 1 (1926), Pp. 1-32.
D. Hilbert,
Die Grundlagen der Mathematik, Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Universitt, vol. 6 ( 1928), pp. 65-85 ; reprinted in Hilbert’s Grundlagen der Geometrie, 7th edn.
A. Heyting,
Mathematische Grundlagenforschung, Intuitionismus, Beweistheorie, Berlin, 1934.
H. Poincare,
The Foundations of Science, English translation by G. B. Hilsted, New York, 1913.
E. Nagel,
The formation of modern conceptions of formal logic tn the development of geometry, Osiris, vol. 7 (1939), pp. 142-224.
A. N. Whitehead.
An Introduction to Mathematics, London, 1911, and New York, 1911.
G. H. Hardy,
A Mathematician’s Apology, London, 1940.
Histories
Moritz Cantor,
Vorlesungen ber Geschichte der Mathematik, 4 vols., Leipzig, 1880-1908; 4th edn., Leipzig, 1921.
Florian Cajori,
A History of Mathematics, 2nd edn., New York and London, 1922.
A History of Elementary Mathematics, revised edn., New York and London, 1917.
A History of Mathematical Notations, 2 vols., Chicago, 1928-1929.
D. E. Smith,
A Source Book in Mathematics, New York and London, 1929.
T. L. Heath,
A History of Greek Mathematics, 2 vols., Oxford, 1921.
Felix Klein,
Vorlesungen ber die Entwicklung der Mathematik im 19. Jahrhundert, 2 vols., Berlin, 1926-1927.
J. L. Coolidge,
A History of Geometrical Methods, New York, 1940.