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nLab > Latest Changes: synthetic mathematics

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMay 11th 2014
    • (edited May 11th 2014)

    felt the desire to have an entry on the general idea (if any) of synthetic mathematics, cross-linking with the relevant examples-entries.

    This has much room for being further expanded, of course.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJun 19th 2019

    Since “synthetic homnotopy theory” redirects here, I tried to add some pointers, but just a start:

    Discussion of synthetic homotopy theory (typically understood as homotopy type theory):

    diff, v14, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 25th 2019

    added pointer to

    diff, v16, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJan 6th 2021

    I noticed that the link “synthetic homotopy theory” didn’t go anywhere. Have made it a redirect to synthetic mathematics now, since that is the entry which has a subsection “Synthetic homotopy theory”. Optimally it should be given it’s own entry, though.

    diff, v19, current

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 6th 2021

    Sorry that was me. I was in the process of setting up a new page yesterday, but removing the redirect didn’t generate the usual ? link, and I got called away. I’ll see if I have time today.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJan 6th 2021

    Oh, I see. All the better! Let’s create that page, when anyone finds the time.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 6th 2021

    I’m trying again, have removed the redirect, but synthetic homotopy theory still ends up at synthetic mathematics

    Redirected from “synthetic homotopy theory”

  8. Removed now. Think there is a little bug in such cases, which submitting an edit after the first one usually solves. I’ll try to fix the bug when I get a chance.

    • CommentRowNumber9.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 6th 2021

    Thanks!

  10. There are versions of Tarski’s axioms for synthetic Euclidean geometry that apply for any finite dimension n∈ℕn \in \mathbb{N}.

    Anonymous

    diff, v24, current

    • CommentRowNumber11.
    • CommentAuthormaxsnew
    • CommentTimeMay 18th 2022

    Another recently active area initiated by Jon Sterling. Hopefully he or someone familiar with the topos-theoretic details (maybe me after I do some more reading) can initiate a page.

    diff, v25, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2022

    Just to clarify that the “recently active area” mentioned in #11 is probably the “synthetic Tait computatibility theory” (now requested here)

    • CommentRowNumber13.
    • CommentAuthormaxsnew
    • CommentTimeMay 19th 2022
    Just to be clear, "synthetic Tait computatibility" is not a subfield of "synthetic computability theory", it's a synthetic approach to performing constructions that are called "Tait computability". In particular the toposes involved are not (inherently) related. I've added a little to the idea section of each to say what the toposes involved are.
    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMay 19th 2022

    Just to be clear, “synthetic Tait computatibility” is not a subfield of “synthetic computability theory”

    Would be good to further expand on this at synthetic Tait computability, because this is surprising, given the terminology: I gather then it must be true that also “Tait computability” is not a subfield of “computability”?!

    • CommentRowNumber15.
    • CommentAuthorjonsterling
    • CommentTimeMay 19th 2022

    I’ll try to write something about it soon… It is indeed separate from computability theory; the history is that there is a central technique in CS/Logic called “Tait’s Method of Computability” that I abstracted synthetically. It corresponds to working synthetically in a glued topos, i.e. a topos equipped with a distinguished subterminal.

  16. added link to synthetic guarded domain theory

    Anonymous

    diff, v30, current

    • CommentRowNumber17.
    • CommentAuthorVictor Sannier
    • CommentTimeOct 7th 2024

    Added two philosophical arguments for synthetic mathematics

    diff, v33, current

    • CommentRowNumber18.
    • CommentAuthorvpatryshev
    • CommentTimeDec 7th 2024
    I wonder how come model theory is not even mentioned here? I believe, viewing the issue from the p.o.v. of model theory, everything becomes much clearer. It's also related to this page:
    https://ncatlab.org/nlab/show/computational+trilogy - which also does not seem to be mentioned here.

1 to 18 of 18

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