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Atrium > Mathematics, Physics & Philosophy: HoTT in polymath project?

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 26th 2018

    Siddhartha Gadgil reporting on recent progress in the polymath project on homogeneous length functions (here) claims at 37:54-38:20 that “on the conceptual level it is crucially based on homotopy type theory”.

    That would be an interesting confluence of communities.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeJul 26th 2018

    That would be really interesting, but is a bit hard to believe. Does he give any indication of what he means by that?

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 26th 2018

    There’s some indication here.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJul 26th 2018

    Possibly it would be good to hear Tobias Fritz about this, who has been contributing here a fair bit and who was part of that Polymath project, too.

    • CommentRowNumber5.
    • CommentAuthorMike Shulman
    • CommentTimeJul 26th 2018

    Thanks David. That comment makes it sound as though there is actually nothing to do with HoTT, only with computer formalization of proofs in general.

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 27th 2018

    You can find out much more about what he’s thinking from here and this presentation.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJul 27th 2018
    • (edited Jul 27th 2018)
    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 27th 2018

    Motivation seems well-aligned with Voevodsky about the need for a language closer to real mathematics, as in this quote

    “ The roadblock that prevented generations of interested mathematicians and computer scientists from solving the problem of computer verification of mathematical reasoning was the unpreparedness of foundations of mathematics for the requirements of this task.”

    “ Formulating mathematical reasoning in a language precise enough for a computer to follow meant using a foundational system of mathematics not as a standard of consistency applied only to establish a few fundamental theorems, but as a tool that can be employed in everyday mathematical work. ”

    Perhaps not so oriented to higher homotopy then.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJul 27th 2018
    • (edited Jul 27th 2018)

    From the above pages it is clear that they are using (a new implementation of?) some kind of dependent type theory with identity types. Maybe it’s actually MLTT, at least they claim they can “import” code from Lean.

    I don’t see any mentioning of univalence, or function extensionality or even of any subtlety related to identity types.

    Ah, and I just got a reply from Siddhartha Gadgil, here:

    I don’t think there was anything specifically homotopic, but since I learnt the type theory from the HoTT book (and am a topologist) there may have been some impact on the way of looking at type theory.

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