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formofmath_5_21_97

formofmath_5_21_97 - 1 THE FORMALIZATION OF MATHEMATICS by...

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1 THE FORMALIZATION OF MATHEMATICS by Harvey M. Friedman Ohio State University Department of Mathematics [email protected] www.math.ohio-state.edu/~friedman/ May 21, 1997 Can mathematics be formalized? It has been accepted since the early part of the Century that there is no problem formalizing mathematics in standard formal systems of axiomatic set theory. Most people feel that they know as much as they ever want to know about how one can reduce natural numbers, integers, rationals, reals, and complex numbers to sets, and prove all of their basic properties. Furthermore, that this can continue through more and more complicated material, and that there is never a real problem. They are basically correct. However, the formalization of mathematics is extraordinary inconvenient in any of the current formalisms. But why do we care about inconvenience? Put differently, why would anyone want to formalize mathematics, since everybody thinks anybody who cares can? Let me distinguish two concepts of formalization. The first is what I call syntax and semantics of mathematical text. Here there are no proofs. One is only concerned with a completely precise presentation of mathematical information. This is already grossly inconvenient in present formalisms. Why do we want to make this convenient? GENERAL GOALS 1. To obtain detailed information about the logical structure of mathematical concepts. For instance, what are the appropriate measures of the depth or complexity of mathematical concepts? What are the most common forms of assertions? We hope for interesting and surprising information here. 2. To develop a theory of mathematical notation, and notation in general. When how and why do mathematicians break concepts up into simpler ones? What is it about math- ematical notation that makes it convenient and readable? These are important matters that have evolved in a certain way; e.g., consider music notation. 3. To maintain a uniformly constructed database of mathematical information. Such a database would benefit from agreement on notation, and would also help facilitate it.
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2 There could be automatic algorithms for changing notation. Also information retrieval of various kinds seem useful and interesting. The more ambitious concept of formalization includes proofs. These are even much more inconvenient in present formalisms. What is to be gained by making them reasonably convenient? 4. To obtain detailed information about the logical structure of mathematical proofs. For instance, there is a sophisticated area of logic called proof theory, where there is almost no such detailed information. There is a lot of information in logic about unprovability, but virtually nothing about real proofs. What inference rules are really used frequently? Is there a good classification of the levels of triviality?
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