FOM: Re: Categorical foundations
Colin McLarty
cxm7 at po.cwru.edu
Fri Jan 23 15:25:39 EST 1998
>Translated into the category vs set discussion:
>Let both, proponents of sets and proponents of
>categories, fix one system each which they think suits
>their needs for formalizing mathematical reasoning.
Cateporical foundations as a whole is about exploring foundations
for a great many different, and mutually inconsistent, theories: a category
of linear spaces and transformations cannot also be a category of
didfferentiable spaces, and neither can be a category of recursive sets.
Schlottmann's proposal is reasonable, though for comparing the
particular subject of categorical set theory with ZF.
>Say, the system of set theory is ZFC, the system for
>category theory is XYZ (I apologize for being not familiar
>enough with category theory to make a definite proposal
>here). Then, set theorists translate XYZ into ZFC,
>categorists translate ZFC into XYZ. After this, write
>a textbook and simply let mathematicians do their work
>in whatever of the two systems they like and see which
>system survives.
The translations are familiar. A warning here: Do not attempt to
understand the translation between categorical set theory and set theories
related to ZF without first understanding each of categorical set theory and
ZF. The translations can be found in Johnstone TOPOS THEORY or Mac Lane and
Moerdijk SHEAVES IN GEOMETRY AND LOGIC.
>If both systems stand the test in everyday life, then,
>obviously, none of them is more convenient for f.o.m.
The test of everyday life applies to everything, whether we want it
to or not. There would be no use me pushing a fraudulent view here since if
my theory is incomprehensible, burdensome, and dishonest, then anyone I
convince will simply go on to fail in that everyday life we all live.
Neither will name calling change the outcome by much.
Your proposal is fine. And it has been done.
Colin
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