FOM: message from Hilary Putnam
Vaughan R. Pratt
pratt at cs.Stanford.EDU
Sat Oct 18 21:03:38 EDT 1997
From: Hilary Putnam <hputnam at fas.harvard.edu>
>What makes a concept of mathematics "basic"? I am inclined to think that
>very few concepts are *essential* to mathematics: perhaps only the
>(better: a certain) concept of *proof*. Hilary Putnam
Let us grant for the moment that proof and truth are two essential
concepts of metamathematics. This would then reasonably make them
essential concepts of the mathematics of metamathematics. But I don't
think it is reasonable to jump from there to the statement that they
are essential concepts of mathematics. Why license that passage and
not the passage from "lines are an essential concept of the mathematics
of projective geometry" to "lines are an essential concept of
mathematics"?
As to essential concepts of mathematics, it was presumably possible,
before sets and functions were invented, to do mathematics without
them. But paradoxically their invention seems to have taken that
capacity away from us, and today we seem unable to function without
them. Even category theorists depend on them to define the notion of
category.
One might hope that some enterprising category theorist had developed a
category theory entirely within the internal logic of some closed
category visibly different from Set, e.g. Ab, the closed category of
abelian groups. To my knowledge such an exercise has never been
seriously carried out. Furthermore I believe anyone brave enough to
try this would find it extraordinarily difficult to avoid contamination
from Set leaking into their attempt.
Now if sets are essential to mathematics, surely membership is
necessarily just as essential, since how could one have a coherent
notion of set without a notion of membership?
Well, in practice things haven't turned out the way theory thought they
should. In algebra and topology, if not in number theory, the elements
of sets are anonymous and hence inconsequential. If in answer to the
question "how many groups are there of order 4" you were to reply "they
form a proper class", you would be considered a smart alec.
Algebraists and topologists have no use for the extensionality axiom of
set theory as applied to sets qua carriers, though they certainly
accept it as applied to sets qua subsets, e.g. the open sets of a
topological space.
In today's mathematics, if sets and functions aren't essential then
*nothing* is essential. I am inclined also to recomend not leaving
home without cartesian product. Membership has its uses, but it is not
an essential of mathematics, only of foundations as practiced today in
isolation from real mathematics.
Vaughan Pratt
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