[FOM] Formalization Thesis
James Hirschorn
James.Hirschorn at univie.ac.at
Fri Jan 11 13:53:56 EST 2008
It has been pointed out in this thread that the FT is highly dependent on the
choice of formal system(s). For example,
Vladimir Sazonov wrote:
> joeshipman at aol.com wrote:
> > I repeat my earlier challenge: can anyone who disputes Chow's
> > Formalization Thesis respond with a SPECIFIC MATHEMATICAL STATEMENT
> > which they are willing to claim is not, despite its expressiblity in
> > English text on the FOM discussion forum, "faithfully representable"
> > or "adequately expressible" as a sentence in the formal system ZFC?
>
> Let me replace "in the formal system ZFC" by "in a formal system".
And went on to give a compelling argument that if we allow "any formal system"
in FT then it becomes a tautology. I find this very feasible, and it seems to
be compatible with the Platonistic viewpoint (although I initially got the
impression that it was not).
I would suggest (perhaps this has already been suggested) FT(ZFC+LC) stating
that every mathematical statement can be faithfully expressed in ZFC+A for
some large cardinal axiom A that is believed to be consistent with ZFC.
There is another possibly relevant issue which apparently has not been
discussed in this thread. This is the duality between "set theoretic
foundations" based on membership and "categorical foundations" based on form.
The duality goes back to Cantor versus Zermelo, with Cantor expounding the
latter. See for example the following paper by Lawvere including the
commentary by McLarty:
http://tac.mta.ca/tac/reprints/articles/11/tr11abs.html
Thus we could consider the dual thesis FT(CAT) stating that all mathematical
statements can be faithfully formalized in category theory, i.e. a formal
category theoretic system. Or perhaps this approach is known to be
equivalent?
James Hirschorn
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