one another in certain of the underived propositions, and, as a
consequence, in certain of the theorems. One can no longer maintain, therefore, that the axioms of geometry are self-evident; for
two such axioms lying in different systems of geometry might be
inconsistent with one another, and two inconsistent statements
could not possibly both be self-evident.As a result of these considerations one can no longer say that
mathematics is strictly a factual science. The important thing
about mathematics is not what it "talks about," but the logical
structure which it exhibits. This led Benjamin Peirce to define
mathematics as "the science which draws necessary conclusions"
-- a definition which suggests that the interest of the mathematician need not be confined to numbers or space, but may extend
to anything which can be organized into a deductive system.
Whenever two propositions are connected by the "if-then" relationship we have the potentialities of a mathematical system.
Mathematics, therefore, is much like a game of checkers; in both
cases we accept certain elements and rules which are completely
arbitrary, and when we "play the game" we attempt to determine
what possible consequences follow from these freely granted assumptions. To ask whether mathematics is true is as absurd as to
ask whether checkers is true. Applied mathematics is, of course,
something quite different. Here the usual conceptions of truth
and falsity apply, and the study is concerned with factual truth
in much the same way that the natural sciences are. A. CORNELIUS BENJAMINSUGGESTED READINGS
| BENJAMIN A. C., An Introduction to the Philosophy of Science. Macmillan Co., 1937.|
|----- "Philosophy of the Sciences", A History of Philosophical Systerns (ed. by
Vergilius Ferm). Philosophical Library, 1950.|
| COHEN M. R., Reason and Nature. Harcourt, Brace & Co., 1931.|
| FEIGL H., and
BRODBECK M. (eds.), Readings in the Philosophy of
Science. Appleton-Century-Crofts, 1953.|
| LENZEN VICTOR, "Philosophy of Science in America", Philosophic
Thought in France and the United States (ed. by
Marvin Farber). University of Buffalo Publ., 1950.|
| MARGENAU HENRY, The Nature of Physical Reality. McGraw-Hill, 1950.|
| MORRIS C., and
CARNAP R. (eds.), International Encyclopedia of
Unified Science. University of Chicago Press, 1938 --.|
Questia, a part of Gale, Cengage Learning. www.questia.com
Book title: American Philosophy.
Contributors: Ralph B. Winn - Editor.
Publisher: Philosophical Library.
Place of publication: New York.
Publication year: 1955.
Page number: 19.
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