M Theory Lesson 100

On page 127 of their book [1], Lambek and Scott discuss the unifying properties of topos theory, as seen from the perspective of type theory. The three traditional mathematical philosophies in question are intuitionism, Platonism and formalism (in the sense of concrete symbolism). They note that a fourth philosophy is somewhat neglected, namely neologicism: the idea that all mathematics can be formulated in terms of logic. Strict logicism would require deriving the properties of the natural numbers from more foundational principles.

But isn’t that just what the physics has been telling us we must do? Lambek and Scott take the viewpoint that categorical logic treats logic pragmatically, as a part of ordinary mathematics, thus refuting logicism. Should logos theory refuse to accept this point of view? Recall that in M theory any useful logical statement has a physical meaning pertaining to a given experiment and its constraints. The numbers associated with logical statements (or rather diagrams) also take on a physical meaning (loosely speaking, a collection of measurements). Thus both logic and number theory should be derived from the physical principles of measurement, expressed as a higher dimensional topos theory.

There is a chicken-and-egg objection: that the formalism of higher toposes requires the mathematics of logic before acquiring a physical interpretation. However, since physics is undoubtedly guiding the axioms in question, this argument appears weak. Ironically, logicism in M Theory would be the ultimate in reductionism, despite the intuition of emergent schemes on different scales.

[1] J. Lambek and P.J. Scott, Introduction to higher order categorical logic, Cambridge U.P. 1986

5 Responses so far »

  1. 1

    r.mirman said,

    Are you familiar with the proof that the dimension must be 3+1? A lot of what is going on in “physics” is nonsense. If you want information about the books look at the book list given on
    which also has my e-mail address, plus various other comments.

  2. 2

    L. Riofrio said,

    Happy 100th lesson! I hope to see many more of your writings in the future.

  3. 3

    Kea said,

    Hi all. Yes, r.mirman, I’ve seen your site. Personally I think there are several ‘proofs’ that 3 dimensions of space is correct. I prefer a categorical argument myself. I will email you, Caroline.

  4. 4

    CarlBrannen said,

    “The numbers associated with logical statements (or rather diagrams) also take on a physical meaning (loosely speaking, a collection of measurements).”

    Schwinger’s measurement algebra consists of primitive idempotents (projection operators). Is this the connection? If so, maybe I understand more about category theory than I thought.

  5. 5

    Kea said,

    Hi Carl. Well, the connection is really a lot more convoluted than that, but I did find Schwinger’s ideas helpful at one point and they certainly had something to do with me noticing your work in the first place.

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