FOUNDATIONS OF MATHEMATICS: PAST, PRESENT, AND FUTURE
by
Harvey M. Friedman
http://www.math.ohiostate.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
f.o.m. people and mathematicians and philosophers is not only
valuable, but crucial. F.o.m. can serve effectively as a
middle man (woman) between the two fields. Can you imagine
much fruitful discussion between mathematicians and
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 Fall '08
 JOSHUA
 Math, Computer Science, Science, Mathematical logic, f.o.m.

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