FOM: use of 'platonism' in f.o.m.
wtait@ix.netcom.com
wtait at ix.netcom.com
Thu Jan 29 22:17:10 EST 1998
I found the passage that I had vaguely remembered from Poincare---in
_Mathematics and Science: Last Essays_, in the paper "Mathematics and
Logic" (p. 73 in my Dover edition):
``But the Cantorians are realists even where mathematical entities are
concerned. These entities seem to them to have an independent existence;
the geometer does not cretae them, he discovers them. These objects
therefore exist so to speak without existing, since they can be reduced
to pure essences. But since, by nature, these objects are infinite in
number, the partisans of mathematical realism are much more infinitist
than the idealists. Infinity to them is no longer a becoming since it
exists before the mind which discovers it. Whether they admit or deny it,
they must therefore believe in the actual infinity.
We recognize in this the theory of ideas of Plato; and it may seem
strange to see Plato classified among the realists. There is nevertheless
nothing more opposed to contemporary idealism than Platonism, even though
this doctrine is also far removed from physical realism.''
The idealists or, as he also calls them, pragmatists seem to include
finistists such as Kronecker and the French intuitionists. One thing that
they reject is impredicative definition (pp. 70-71), which is why I was
interested in finding the reference. In Godel *1933, it is the use of
both excluded middle and impredicative definition that he felt could only
be justified on Platonist grounds (which, at that time---or perhaps in
that sense of `Platonism'--- he found unacceptable). He later recanted in
the case of impredicative definition (e.g. in his paper of Russell).
Bill Tait
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