[FOM] the amazing language of set theory

[katzmik at macs.biu.ac.il]

Bill, there is certainly a sense in which set-theoretic foundations have
contributed to the solution of problems, just like Descartes' work. Namely
infinitesimals were first justified rigorously in the 20th century in the
context of set-theoretic foundations (Robinson actually chose to do this in
the framework of the theory of types but the set-theoretic foundation is
closely related to this). As you know infinitesimals have been part of
analysis since the 17th century and their successful formalisation is a
contribution to solving problems and not merely the creation of "ONE
framework" which is in the end an elusive goal as the development of 20th
century mathematics showed.

MK

On Sat, March 26, 2016 07:59, WILLIAM TAIT wrote:
> Martin,
>
> I’m not sure that the comparison with Descartes is sound. The advantage gained
> by Cartesian geometry was in ease of solving problems. No one would claim that
> translation of a problem—e.g.of analysis---into one in set theory generally
> makes it easier to solve. (The opponents of ‘set theoretic foundations’ often
> point to just this fact.) A real contribution of set theoretic foundations is
> that it provides ONE framework for all problems and the reasoning about their
> solutions. In each case, one can assume that the objects in question are in
> some universe of sets. The coherence of the statement of the problem and of
> the reasoning about it then becomes a question about the coherence of
> reasoning in one theory, namely set theory (together with whatever set
> theoretic assumptions the problem demands).
>
> I want to point out that this point of view is not an endorsement of the view
> that ‘everything in math is a set’. [As Dedekind (sort of) said: it is
> meaningless to ask whether 0 is an element of 1.]
>
> But I also want to point out that the very virtues of the categorical /
> type-theoretical foundations, which I completely understand, preclude this
> particular virtue of set theoretic foundations.
>
> I suggest for this reason that many current arguments  about set-theoretic
> foundations versus (one’s favorite version of ) type theory just spin their
> wheels. Its nice that both are around.
>
> Bill
>
> On Mar 25, 2016, at 6:39 PM, Martin Davis <martin at eipye.com> wrote:
>>
>>
>> Descartes revolutionized mathematics by showing that geometry could be
>> reinterpreted in the language of algebra. This was not always the most
>> appropriate language for actually dealing with some specific geometric
>> problem. But of course that was not the point. To take just one example: The
>> calculus of Leibniz and Newton made area and volume calculations that had
>> required artful innovations by such as Eudoxus, Archimedes, Cavalieri, and
>> Torricelli, into student exercises. Of course, having the equations of the
>> relevant curves and surfaces was crucial.
>>
>> The somewhat analogous twentieth century discovery that all of mathematics
>> can be reinterpreted into statements in the language of set theory (just one
>> binary relation, usually written as the Greek letter "epsilon") has
>> revolutionized not mathematical practice, but rather foundational studies.
>> It became clear that Zermelo's axioms suffice for the great bulk of ordinary
>> mathematics, and, when suitably augmented to form our familiar ZFC, for
>> transfinite mathematics as well.
>> Then techniques developed by Gödel and Cohen made it possible to see that it
>> had been impossible to resolve various problems that had been around for a
>> long time, only because the ZFC axioms were insufficient for their proofs.
>> Many other fascinating foundational results have been obtained.
>>
>> In the ongoing discussion here on FOM, I don't think the distinction has
>> always been made between a foundation suitable as the starting point of
>> metamathematical investigations, and one suited for use in a computerized
>> proof verification system for use by mathematicians in certifying the
>> correctness of their work. The tradeoffs inevitably involved in the latter
>> case seem to me to make it to a considerable extent, an engineering
>> decision. Harvey Friedman advocates using set theory itself, of course not
>> in the raw form presented above, but nicely "sugared" to make it pleasant
>> for working mathematicians. It may not be amiss to mention that my late
>> colleague Jack Schwartz had developed such a system. The Italian logicians
>> Domenico Cantone, Eugenio Omodeo, and Alberto Policriti have been carrying
>> the development further.
>>
>> Martin
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