[FOM] Alternative Foundations/philosophical

[Bas Spitters]

Steve Awodey, Structuralism, invariance, and univalence.
https://www.andrew.cmu.edu/user/awodey/preprints/siu.pdf

Contains answers to many of these questions.


On Thu, Feb 27, 2014 at 5:05 AM, Jay Sulzberger <jays at panix.com> wrote:

>
>
>
> On Tue, 25 Feb 2014, Harvey Friedman <hmflogic at gmail.com> wrote:
>
>  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.
>>
>
> I think the HoTT/Intutionist-Types school would say that their
> foundations are at least as rigorous as any foundations offered
> by the party of Cantor, Dedekind, Frege, Peano, and Hilbert.
> Indeed some would say that HoTT/Intuitionist-Types approach is
> more correct than anything the old incumbents party of
> Cantor, ..., Hilbert has on offer.
>
>
>
>> 2. There is a completely transparent elementary character that is
>> relatively universally understandable.
>>
>
> I have not found the presentation in the HoTT book clear, but I
> have not studied it enough.
>
>
>
>> 3. There is a precious kind of philosophical coherence that transcends
>> mathematics itself.
>>
>
> HoTT school members tend to think their way is superior here.
>
>
>
>> 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.
>>
>
> The HoTT school agrees, and, again, claims their way is better.
>
>
>
>> 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.
>>
>
> Ah, I think the meaning of such "interpretations" is also
> contested.  If one is too easily satisfied, then HoTT folk would,
> I think, complain that the main point is being missed.
>
>
>
>> 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?
>>
>
> Ah, hmmhn, hrrhnmh, I think no.
>
>
>
>> 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
>>
>
> The central propaganda effort of the HoTT movement is to persuade
> people that indeed the notions of "equality" and of "isomorphism"
> of Bourbaki style "structures" are not what most lay members of
> the incumbent party think they are.  As for a good textbook
> presentation, I do not know of any, but I am not in command of
> the literature.
>
> ad comprehensibility: I am encouraged by Robin Adams" paper
> "Pure Type Systems with Judgemental Equality":
>
>   http://www.cs.rhul.ac.uk/~robin/ptseq8.ps.gz
>
> because I can parse every sentence I have read in it.  I have not
> finished reading the paper.
>
> oo--JS.
>
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