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Published online by Cambridge University Press: 14 March 2022
We have undertaken to discuss the nature of mathematical systems, the way in which they are discovered, and the uses to which they are put in the empirical sciences. The empirical sciences employ mathematical systems in framing final formulations, but adopt mathematical techniques long before reaching that stage. It is a prerequisite that the mathematics they employ has been developed separately and within its own domain. We shall return to the relation between mathematics and the empirical sciences before we are done, but in the meanwhile we shall be constrained to begin by discussing the situation in mathematics in isolation from any empirical considerations.
1 Raymond L. Wilder, Introduction to the Foundations of Mathematics (New York 1952, John Wiley and Sons), p. 43.
2 “The General and Logical Theory of Automata” in Cerebral Mechanisms in Behavior edited by Lloyd A. Jeffress (New York 1951, John Wiley & Sons), p. 16.
3 Collected Papers of Charles Sanders Peirce (Cambridge 1931–36, Harvard University Press), ed. Hartshorne and Weiss, 5.579. See also the references in James Feibleman, Introduction to Peirce's Philosophy (New York 1946, Harper), pp. 139–40.
4 Col. Pap., 3.527.
5 The theory of the syllogism takes up certain important distinctions, as for instance that between the deduction of a universal proposition and a particular one, i.e. one containing an existential quantifier; deduction from one proposition as well as from two; etc.
6 Introduction to a Form of General Analysis (New Haven 1910, Yale University Press), p. 1.
7 Science and the Modern World, p. 68.
8 Louis O. Kattsoff, A Philosophy of Mathematics (Ames, Iowa 1948, Iowa State College Press), p. 230.
9 J. H. Curtiss, “A Federal Program in Applied Mathematics” in Science, Vol. 107 (1948), p. 257.
10 Op. cit., loc. cit.
11 For a good short account, see von Neumann, in Cerebral Mechanisms in Behavior, p. 26.
12 J. Bronowski, “The Logic of Experiment” in Nature, Vol. 171 (1953), p. 194.
