FOM: use of 'platonism' in f.o.m. wtait at
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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