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TalkFormMath12pt[1].2.97

TalkFormMath12pt[1].2.97 - 1 THE FORMALIZATION OF...

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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/ February, 1997 I really should be talking to you about more mainstream things like face recognition information retreival passing the Turing test. I have some wonderful code that solves these problems completely, and I want to share it with you now. 0101110101010000001010111010101010101010100100101101000100101 0101011010101001010010000101010101001010101010010111010100101 0101010101010101110001001010101010101001000101010101110101001 0101010010100010001010101010100110101010101010101010101010101 0010000110101010101010101110101010101010101010100100101010101 0010100100000101010011010101011111101100101010101111001010101 0100101010101001010100010101001010101010110010010101010100101 0101010101010010010101010101010010001010101010101010011001010 1011010111010010101010010101010101001010101010101010010101010 1010010100101000001001011001101100101010101010011101010100110 0010101010101010101010010101001010101010101001110110110110101 0101100000101010101010010101011001010101001010010 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?
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2 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? 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. Perhaps one can do a lot here without going too far with convenience; but more convenience than usual seems appropriate. 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 mathematical notation that makes it convenient and readable? These are important matters that have evolved in a certain way - largely not by accident. 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. 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?
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