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nLab > nLab General Discussions: ludics, locus solum, etc.

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    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 28th 2014

    Rather than derail the thread on particles any further with stuff about Girard’s projects, I thought I’d start a new thread.

    I just discovered by accident an old Café thread started by David C. on Locus Solum. Unfortunately it didn’t get very far. I did relearn that (allegedly) Lawvere once punched out Girard. I won’t relate other gossip I’ve heard about Girard.

    Anyway, if anyone has something more to say (on ludics for example), please be my guest. There’s also a GoI thread here started by Urs.

  2. Back when I was writing my thesis, I found Locus Solum to be helpful, although in retrospect this had a lot more to do with drawing inspiration from Girard’s rhetorical energy than with reusing any of his mathematical development. In particular, one idea which Girard did not invent but which is taken seriously in ludics is the concept of focalization. The “desseins” of ludics are essentially a frightening notation for (cut-free, axiom-free) focalized proofs of (affine) MALL.

    As noted by Tom Hirschowitz in that n-café thread, Girard starts from an “untyped” presentation (i.e., desseins only have an arity rather than a sequent of formulas as a conclusion) because he wants to reconstruct types via a realizability interpretation (as sets of desseins closed under biorthogonality). As you suggested in that thread, though, I don’t think there is anything particularly uncategorical about this, and that the right way to understand it is in fibrational terms.

    Though Girard does not point this out, under Curry-Howard, focalized proofs correspond to programs in continuation-passing style. CPS is a fundamental idea from computer science for analyzing evaluation order, and this gives one formal sense in which ludics is “aware of time” in ways that Girard’s prior semantical investigations of classical linear logic were not—although certainly continuations were already part of the tradition of game semantics (cf. the Appendix entries on “Game semantics”, “Polarity”, and “Time in logic”).

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 28th 2014

    Noam, I was hoping you’d have something to say! I would like to ponder this more, but these seem to be really very helpful comments indeed. Are (at least some of) these comments explored in print?

  4. Todd Trimble wrote:

    Are (at least some of) these comments explored in print?

    Although I prefer that you burn my thesis rather than read it (with a hat tip to Conor McBride), you might enjoy these notes to a talk on “polarity in proof theory and programming” that I gave last summer. It’s very high-level, just trying to point to some of the ideas dancing around “ludics” (in a broad sense) which I think are important. Though the talk does not mention categories, it was informed by long-running work with Paul-André Melliès, especially some ideas about monoidal closed bifibrations which we finally put into print in October but which are still very much in development.

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