[FOM] Alternative Foundations/philosophical

[Joseph Shipman]

The prerequisite I mentioned was enough experience and mathematical intuition to be able to see that interpretability in ZFC (or ZFC+a large cardinal) was possible in principle.

The people who say "it is fine if you are careful" but who work within a formal system that cannot be seen to be interpretable in ZFC+a large cardinal do not qualify and criticism of them is quite appropriate.

However, mathematicians whose formalizations are not inadequate in this sense should not be required to provide the details of a set theoretical interpretation, any more than computer scientists writing about algorithms should be required to provide details of how their algorithms can be implemented by Turing machines.

I think that by now "Zermelo's thesis" has the same professional status as the Church-Turing thesis. Harvey, on the other hand, seems to think not only that some mathematicians are using (in their ordinary work and not metamathematics) formal systems which are not interpretable in ZFC + a large cardinal, but also, and more embarrassingly, that there is genuine disagreement about whether the systems they use are so interpretable.

If Harvey is right, then I still stand by Zermelo's thesis, and declare that the mathematicians who are mistaken about their systems being ZFC+LC-interpretable are unsound and professionally embarrassing in the same way that a computer scientist would be who proposed an "algorithm" that was not actually effective. But I'd prefer to see names named so that the discussion can get into the details.

McLarty seems to have succeeded in repairing the unprofessional neglect of foundations exhibited by the Grothendieck school, so I no longer know of any specific published results where the issue of interpretability in ZFC is a source of contention.

-- JS

Sent from my iPhone

> On Mar 6, 2014, at 6:25 PM, 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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