FOM: Millenium Conference
Ara R. Aslyan
arik at ipia.sci.am
Tue Jan 18 06:48:56 EST 2000
On Fri, 14 Jan 2000, Mark Steiner wrote:
> As far as general intellectual interest of "core" mathematics, versus
> philosophy, history, and foundations. Harvey asked me my view of their
> view. There is no way really to know without an in-depth interview, but
> my guess is (as I already said) that they would say that mathematics has
> more intellectual interest than the history, philosophy, or foundations
> of the field, if only because these other fields draw their intellectual
> interest from core mathematics (of course, this is not an airtight
> argument, as I pointed out above).
In order to express my disagreement with such a general point of
view, let me consider the following example.
Arithmetics is a study of various properties of natural numbers
0,0',0'', etc. while foudational studies (in Hilbert sense)
roughly speaking concerned with the methods and principles being used in
arithmetics, more precisely with the study of formal systems (PA,PRA,Z_o
etc.). As it is well known much of arithmetics could be derived in this
formal systems, using only a small part of proof theoretic strength of
them. As this commented by P. Bernays any consistency proof for
this formal systems would be a consistency proof "in advance" for a rather
large possibilities than is needed to carry of an ordinary arithmetics.
As it follows from Goedels results no such proof is possible within this
systems (of course for the fixed formal system itself).
Now the question is how the general intellectual interest of a
subject to be "mesured". One way to do this (for me) to mesure the
g.i.i. by the complxity of subject matter. With this respect, it is clear
(at least for me) from the mentioned above that f.o.m. concerned with
rather complicated issuers, and uses more complicated technics than
is needed in ordinarry arithmetics.
It would be interesting to hear your concept of g.i.i..
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