FOM12pt[1].5.31.00

FOM12pt[1].5.31.00 - FOUNDATIONS OF MATHEMATICS: PAST,...

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FOUNDATIONS OF MATHEMATICS: PAST, PRESENT, AND FUTURE by Harvey M. Friedman http://www.math.ohio-state.edu/~friedman/ May 31, 2000 1. WHAT IS FOUNDATIONS OF MATHEMATICS? F.o.m. is the exact science of mathematical reasoning, and related aspects of mathematical practice. This includes a number of issues about which we know nearly nothing in the way of exact science. For instance, a mathematician instinctively uses an idea of mathematical naturalness when formulating research questions. But we do not have any idea how to characterize this even in very simple contexts. In f.o.m. we take into account the actual reasoning of actual mathematicians, and the actual development of actual mathematics. The previous paragraph could be viewed as a gross understatement. Why shouldn't we be literally studying the actual reasoning of actual mathematicians, and the actual development of actual mathematics? It turns out, time and time again, in order to make serious progress in f.o.m., we need to take actual reasoning and actual development into account at precisely the proper level. If we take these into account too much, then we are faced with information that is just too difficult to create an exact science around - at least at a given state of development of f.o.m. And if we take these into account too little, our findings will not have the relevance to mathematical practice that could be achieved. This delicate balance between taking mathematical practice into account too much or too litte at a given stage in f.o.m. is difficult and crucial. It also sharply distinguishes f.o.m. from both mathematics and philosophy. In fact, it positions f.o.m. right in between mathematics and philosophy.
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In fact, it is the reason why it is of interest to both mathematicians and philosophers, but also why neither mathematicians nor philosophers are fully comfortable with f.o.m. From the mathematician's point of view, f.o.m. never takes actual mathematics into account enough. Whereas from the philosopher's point of view, f.o.m. takes actual mathematics into account too much, bypassing a number of classical metaphysical issues such as the exact nature of mathematical objects (e.g., what is a number?). For example, a philosopher might be disappointed to see that f.o.m. has nothing new to say about just what a natural number really is. And a mathematician might be disappointed to see that f.o.m. has nothing new to say about just why complex variables is so useful in number theory. Yet I expect that f.o.m. will, some day, say something startling about just what a natural number really is, and also say something startling about just why complex variables is so useful in number theory. It's just that now is not the time where we have any idea how to do this. In fact, this is an example of where interaction between
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.

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FOM12pt[1].5.31.00 - FOUNDATIONS OF MATHEMATICS: PAST,...

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