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BanffLect022407 - 1 CONCEPT CALCULUS by Harvey M Friedman...

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1 CONCEPT CALCULUS by Harvey M. Friedman Mathematical Methods in Philosophy Banff Research Center University Professor Ohio State University February 21, 2007 1. THE GENERAL NATURE OF CONCEPT CALCULUS. Concept Calculus is a new mathematical/philosophical program of wide scope. The development of Concept Calculus began in Summer, 2006. Concept Calculus promises to connect mathematics, philosophy, and commonsense thinking in a radically new way. Advances in Concept Calculus are made through rigorous mathematical findings, and promise to be of immediate and growing interest to philosophers. Developments in Concept Calculus generally consist of the following. a. An identification of a few related concepts from commonsense thinking. In the various developments, the choice of these concepts will vary greatly. In fact, all concepts from ordinary language are prime targets. b. Formulation of a variety of fundamental principles involving these concepts. These various principles may have various degrees of plausibility, and may even be incompatible with each other. There may be no agreement among philosophers as to just which principles to accept. Concept Calculus is concerned only with logical structure. c. Formulation of a variety of systems of such fundamental principles in b. These systems generally combine several such fundamental principles in some attractive way. d. An analysis of the “interpretation power” of these resulting systems.
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2 e. In particular, for each of the resulting systems, a determination of whether they interpret mathematics – as formalized by ZFC. A system T having interpretation power at least that of mathematics (ZFC) has special significance. This means that if T is without contradiction then mathematics (ZFC) is without contradiction. I.e., we have a consistency proof for mathematics (ZFC) relative to that of T. Furthermore, relative consistency proofs arising this way are generally very finitary. 2. INTERPRETATION POWER. The interest of Concept Calculus rests considerably on the significance of interpretation power. Interpretability between formal systems was first precisely defined by Tarski. We define this for systems S,T in first order predicate calculus with equality. S,T may have completely different symbols. An interpretation of S in T consists of i. A one place relation de-fined in T which is meant to carve out the domain of objects that S is referring to, from the point of view of T. ii. A definition of the constants, relations, and functions in the language of S by formulas in the language of T, whose free variables are restricted to the domain of objects that S is refer-ring to (in the sense of i). iii. It is required that every axiom of S, when translated into the language of T by means of i,ii, becomes a theorem of T.
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