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Outlines of a Formalist Philosophy of Mathematics by Haskell B. Curry (Eds.)

By Haskell B. Curry (Eds.)

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Then (- at 1 is a constant. constant and a = b then b is a constant. 3. Rules for polynomials (poly). a. Every constant is a poly. b. If a and b are poly. so is a + b. c. If a and b are poly. so is a x b. d. , so is - a. II. One binary predicate, =. IlIA. Axioms. 1. A set of axiom schemes for a commutative ring with unit, without the existential assumptions as to 0, I, and - a, whose existence is already guaranteed. 2. c. , a x a-I = 1. B. Rules of Procedure. 1. a = b ~ b = a. 2. a = b & b = c ~ a = c.

Ax! ax! ax! ax! x z. 4 It does not express an elementary proposition, but a metaproposition. ) 5 These can, of course, be translated into the syntax language. FORMAL SYSTEMS AND SYNTAX 49 be granted. But one can be formal without being syntactical, moreover the insistence on a syntactical point of view does not solve all the problems. To sum this all up: - the syntax of a language is essentially a formal system represented in a certain way. This representation is an importaht and fruitful one. One of its achievements is that it enables us to think of a formal system as something very concrete without losing sight of abstractness, and so incidentally to show that we do not need to presuppose mystical entities of a logical or other idealistic kind in order to be formal.

In extenso then - assuming that we know what the terms are the predicates are uniquely defined by the primitive frame. In regard to the terms, however, the situation is entirely different. We have seen that concerning the tokens the primitive frame says nothing whatever. We can therefore take for those tokens any objects we please, and similarly we can take for operators any ways of combining these objects which have the requisite formal properties 1. We can, for instance 2, take the 1 The only formal property of any significance is the number of objects to be combined by each operator.

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