[FOM] Alternative Foundations/philosophical

[Harvey Friedman]

I agree with a great deal of what Chow has just said. However, I have some
problem with a full endorsement of the following from his message:
\
"I am sympathetic to John Conway's "Mathematicians' Liberation Movement"

(which of course has also been discussed before on FOM), which basically
says that we're mature enough now to be able to pick whatever foundations
we find convenient, knowing that we can always, in principle, translate
between any two if them if we really want to.

In other words, if a proponent of a new system claims certain advantages
over the old system, I do not think the reaction should be to get all
defensive and say, "But I can do that with the old system too!"  Instead,
one should open-mindedly explore if the new system helps foster new ideas
that advance the field.  The sooner we abandon childish turf wars, the
faster mathematics (and the foundations of mathematics) will advance."

There are two aspects of the usual foundations that are generally
accepted (or are they generally accepted?).

1. There is a crucial kind of absolute rigor in the presentation.

2. There is a completely transparent elementary character that is
relatively universally understandable.

3. There is a precious kind of philosophical coherence that transcends
mathematics itself.

4. It has been used in order to address the obvious great fundamental
methodological issues, the most well known of which concern whether or
not propositions can be proved or refuted - both generally and
specifically.

A certain amount of this would also be a priori clear for an
alternative foundation if that alternative foundation was in an
appropriate sense interpretable in the usual foundation. However, such
an interpretation is generally not nearly enough to ensure 2.

How do 1-4 fare with alternative foundations?

With regard to the "liberation Movement", if one is concerned with
fully complete rigorous presentations, then has history shown that
generally speaking one either doesn't have this at all, or one has it
done incorrectly, replete with inconsistencies?

Isn't an example of this kind of thing, the idea of using general category
theory as an alternative foundation, with the "liberated" use of things
like the category of all categories? Hasn't that been recently shown to
lead to convincing inconsistencies within the usual mindset of general
category theory?

Also, has there been a philosophically coherent presentation of altered
notions of equality? If so, the FOM would benefit greatly from seeing this
discussed.

Harvey Friedman
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