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nLab > Latest Changes: BHK interpretation

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 18th 2023

    fixed the pointer to:

    diff, v16, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 18th 2023

    I have similarly boosted the list of “original references”:

    However, albeit just from scanning through these: NONE of these seems to have a recognizable statement of the BHK interpretation, not even remotely (?) They would all fit well as references at intuitionism, where they have been missing. (Will copy them over now….)

    The first statement of the so-called “BHK interpretation” that I have found so far is in

    and Troelstra there does not attribute the idea to anyone, he states it as if its his original insight (“Let me first present… ” and then “We shall try to make these explanations more precise…” )

    Later in

    Troelstra does name the very same idea now after “BHK”, but he says “Brouwer-Heyting-Kreisel” instead of “Brouwer-Heyting-Kolomogorov”. (!?)

    (Indeed, Kreisel’s work is the only one that Troelstra substantially referred to in the previous 1969 article).

    diff, v17, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 18th 2023
    • (edited Feb 18th 2023)

    made explicit the relevant section number in

    Interestingly, Girard does not speak of “Brouwer-Heyting-Kolmogorov” either, instead he says: “Heyting semantics”.

    So I have rewritten the entry’s original paragraph on attribution (now a new subsection: here).

    It would be good to track down who first spoke of the “Brouwer-Heyting-Kolmogrov interpretation”, thereby misquoting Troelstra (apart from taking away his due credit).

    diff, v17, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeFeb 18th 2023

    am renaming the entry from “Brouwer-Heyting-Kolmogorov interpretation” to “BHK interpretation” (short of renaming it to “Troelstra interpretation”)

    diff, v17, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeFeb 18th 2023

    fixed the previously broken link to

    • Martin Escardo, Chuangjie Xu, The inconsistency of a Brouwerian continuity principle with the Curry–Howard interpretation, 13th International Conference on Typed Lambda Calculi and Applications (TLCA 2015), Leibniz International Proceedings in Informatics (LIPIcs) 38 (2015) [doi:10.4230/LIPIcs.TLCA.2015.153, pdf]

    and added the missing publication data

    diff, v17, current

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeFeb 18th 2023

    Oh, now I see where Kolmogorov comes in: here. Will edit accordingly…

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeFeb 18th 2023

    Have now:

    rewritten and considerably expanded the Idea-section (here)

    added a list of scans of historical versions of the statement (here)

    further fine-tuned the discussion of attribution (here)

    diff, v18, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeFeb 19th 2023

    Added pointer to

    and added a screenshot of what exactly this says in its section 7.1.1.

    This seems to be the first reference where Heyting really states the content of what came to be known as the BHK interpretation of the logical connectives (24 years after Kolmogorov). While beginning in the 1930s Heyting speaks about a “proof interpretation” he does not seem to make the more concrete until this 1956 book (as far as I can see, so far).

    diff, v21, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeFeb 19th 2023

    further adjusted/expanded the version in the Attribution-section (here)

    also added pointer to

    qwhich seems to be the origin of switching from “Brouwer-Kolmogorov-Kreisel” to “Brouwer-Kolmogorov-Kolmogorov” (!)

    diff, v21, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeFeb 22nd 2023
    • (edited Feb 22nd 2023)

    just for completeness, I have now added (here) also a scan of the statement according to Troelstra & van Dalen (1988)

    diff, v23, current

1 to 10 of 10

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