FOM: Re: Towards a Consensus?
cxm7 at po.cwru.edu
Mon Feb 9 06:41:31 EST 1998
Reply to message from sriis at fields.utoronto.ca of Fri, 06 Feb
One of Soren's remarks helped me a lot in understanding
>ZFC has a commitment to ONE universe. Topos theory is about many
>universes - HOW CAN YOU AVOID RELATIVISM if you do not single out
>at least a class of universes which for example gives the correct results
>in number theory??
Of course most sciences (physics, chemistry, biology, for
example) survive competing conceptions of their basic problems. In
historical fact, so does mathematics. But Soren helped me see that
if your sole handle on mathematics is one foundational axiom set,
then you will necessarily see any proposed alternative as
abandoning all rationality--as "relativism".
Gauss and Riemann did math with no set theoretic conception
of the continuum. Dedekind and Cantor did set theory with no
conception of the iterative heirarchy of sets. Most mathematicians
today work with only the vaguest notion of foundational issues. I
avoid relativism by looking to this large and well established
(though suspiciously "list 2") body of work.
> HOW CAN YOU AVOID BEING UNSCIENTIFIC if you are
>not aiming for the MAXIMAL deductive strength (by for example accepting
>only intuitionistic arguments)??
Here is a confusion between deductive strength of logics
(classical versus intuitionist) and of theories. Every classical
theory is also "intuitionistic" (as that term is used today, though
not as used by Brouwer). You just take the law of excluded middle
as an axiom scheme.
It is "unscientific" to talk about consistent axiomatic
theories of maximal deductive strength. We have known since Goedel
there are none.
The best established route to greater deductive strength
today is large cardinal axioms. Certainly Soren does not mean that
the sole aim of "scientific" fom will be ever larger cardinal
axioms, or ever new Goedel sentences for ZF+(all the Goedel
sentences so far).
Soren does not mean "maximal deductive strength" at
all in the quite simple, literal sense of the term. He says:
>Remark: An important issue (which I only adsressed briefly) is the issue
>of "deductive strength". I think this involves much more than just
>deductive comparison between systems of set theory ... I think we see
>the ideal of aiming for the highest deductive strength also in weak systems
>a la Van Dreis! ..All this is a different discussion...
A different discussion indeed. It might even include weak
systems a la Grothendieck or Lawvere.
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