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THE FORMALIZATION OF MATHEMATICS
by
Harvey M. Friedman
Ohio State University
Department of Mathematics
[email protected]
www.math.ohiostate.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.
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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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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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 Math, Set Theory, Mathematical logic

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