[FOM] Foundations Crucial
Colin McLarty
colin.mclarty at case.edu
Sun Mar 2 18:48:41 EST 2014
I have not written on this thread until now as I'm busily preparing for
events this month at Warwick (
http://www2.warwick.ac.uk/fac/sci/maths/research/events/2013-2014/nonsymp/fermat/).
But I ll say just a word:
On Fri, Feb 28, 2014 at 1:21 PM, Harvey Friedman <hmflogic at gmail.com> wrote:
> I gave an example when I mentioned naive category theory as a form of unacceptable "liberation of mathematicians". The use of "the category of all categories" has nothing to do with "making an error". The issue is far deeper and far more subtle. .........
>
> Doesn't recent work indicate that using the category of all categories in
> a category theoretic environment is just as seriously flawed as was using
> naive set theory, without its fundamental accompanying foundational work?
>
It has been known since Eilenberg and MacLane created category theory,
decades before categorical foundations were conceived, that using a
"category of all categories" is as problematic as using a "set of all
sets." Indeed using a "category of all groups" or "all topological
spaces" etc (in a purportedly absolute sense) is as problematic.
You can do these things rigorously if you accept the kind of non-naive
restrictions on definitions also found in NF and other set theories with a
set of all sets. I find that not attractive. Being utterly naive about
them is unacceptable.
But I do not think you will find anyone who has ever been that naive about
it.
best, Colin
> Does the way you and Conway refer to "liberation" minimize the great
> fundamental importance of fully rigorous foundations? I would concede that
> it should not be a requirement that every mathematician do foundations of
> mathematics. But every mathematician should recognize its great fundamental
> importance in all context, the great general intellectual need for it, the
> great insights one gets from it, and what greatly importance things are
> missing when one does not have it.
>
> Of course, a critical challenge for foundations is to develop deep
> foundations of subjects other than mathematics that in any way compare to
> what we have for the usual foundations of mathematics. I predict that we
> will see advances along these lines for statistics and physical science
> comparable to the usual foundations of mathematics by 2100. The general
> intellectual excitement will be comparable to the Goedel era in foundations
> of mathematics.
>
> Harvey Friedman
>
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