[FOM] Alternative Foundations/philosophical

[Harvey Friedman]

This concerns the thread:

Friedman wrote:

>* 2. In conversations about these matters, I have always asked for the
*>* SIMPLEST IMPORTANT situation where the usual set theoretic foundations
*>* is impossible to work with in a practical sense. I say that the burden
*>* would then be on me to show how the usual foundations can be readily
*>* adapted to do at least as good a job as alternatives. In each case, I
*>* never got a sufficiently clear account of such a SIMPLE IMPORTANT
*>* situation so that I could dig my teeth into how it could be properly
*>* handled in the usual foundations.
*
The word "impossible" is a strong one.  What is the SIMPLEST IMPORTANT
instance of a problem that is impossible to handle practically using
probability distributions instead of random variables, or impossible to
handle practically using ideal class groups instead of adeles or ideles,
or impossible to handle practically using classical analysis instead of
nonstandard analysis?  I think the answer in each case is, there isn't
one.  Mathematical practice is a vague and amorphous thing that isn't
subject to the well-ordering principle.  There's no sharp boundary between
what is convenient and what is cumbersome.  Nevertheless there can be
heaps of things that are handled better using new foundations even if we
can't specify the minimum size of a heap.

Tim

***************

The phrase I used "where the usual set theoretic foundations is
impossible to work with in a practical sense" is basically taken from
conversations with an expert. I am also interested in any formulation
such as

"A PARTICULARLY SIMPLE (VERY SIMPLE, SIMPLE) situation where the usual
set theoretic foundations is extremely awkward (awkward, comparatively
awkward) to work with in a practical sense".

As I wrote earlier, the challenge would be how to adapt the usual set
theoretic foundations to handle such situations, and evaluate (the
naturalness, clarity, simplicity, general applicability of) that
adaptation. There are already lots of interesting and nonobvious
adaptations of the usual set theoretic foundations for various
situations that greatly increase the practicality. E.g., abstraction
notations (more than just the usual set abstraction), abbreviations,
partially defined terms, etcetera. Without these, the actual
formalization of actual mathematics is "impossible in a practical
sense", recognizing that impossible is a strong word. We can replace
the phrase "impossible in a practical sense" with many other similar
phrases.

With regard to unlimited category theory, it certainly has been my
impression that many (some) people have used it without any
justification, and believe that "it doesn't require any justification,
since there is no problem with it if you are careful". Perhaps this is
the "liberation movement in action". Well, there is the rather thought
provoking http://www.cs.nyu.edu/pipermail/fom/2014-March/017889.html .
An interesting question to me is whether one can still assert "...
there is no problem with it if you are careful". My general feeling is
that this question could stimulate more and more powerful forms of
http://www.cs.nyu.edu/pipermail/fom/2014-March/017889.html applying to
yet wider contexts where the path to inconsistency is even more
immediate. Also, perhaps there are results to the effect that
convincing inconsistencies do not arise in many other wide varieties
of contexts - and seeing what is really driving such inconsistencies
in general terms.

Harvey Friedman
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