[FOM] Alternative Foundations/philosophical

[David Roberts]

Dear Harvey,

How many layers of coding will you accept for a problem and still
consider it as being "in the usual set-theoretic foundations"? In
practice, most people work in a foundation-agnostic way, especially if
they work in so-called 'ordinary mathematics' (for which not the full
strength of ZFC is needed). For sure, one could code work on schematic
homotopy types into ZFC (possibly with an inaccessible, but I strongly
suspect it's not necessary), but actually writing them down in
set-theoretical language *in detail* would be masochistic (reminiscent
of Bourbaki's empty set
http://www.neverendingbooks.org/content/empty-set-according-bourbaki).
>From the point of  view of category theory, almost all of it seems
rather natural and in fact in some ways universal (in the category
theoretic sense: being the optimal/smallest way of getting such a
thing). From the point of view of homotopy type theory, this should be
just a non-standard model of the axioms (this isn't proven, but from
what I understand, it's expected).

This is not a 'simple example', because I'm not playing the 'my
foundation is more convenient than yours' game. At some point work on
problems become so far removed from foundations that the choice of
foundations becomes moot. Do we prefer foundations that are closer to
mathematical practice, or foundations with an economy of symbols and
concepts when written in first-order logic? Pick whichever makes you
sleep at night assured that your work is no more contradictory than
the foundations.

Chasing down an example understandable by an mathematics graduate
which, when written in one foundation is longer than when written in
another foundation is like comparing axiom systems purely on the
number of characters they require when printed on the page. Vaguer
criteria such as "comparatively awkward" are unbecoming of logicians.

Best regards,

David

On 7 March 2014 09:55, Harvey Friedman <hmflogic at gmail.com> wrote:
> 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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