The prophetic promise of category theory

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When it comes to  creating the final theory of everything, physicists have an ever broadening (and bewildering) choice of mathematical tricks with which to tackle the mysteries of the universe.

A couple of years ago, random matrix theory cropped up as a potential framework for a new kind of science. And a fascinating idea it is too.

Talking of a new kind of science, what  cellular automatons can’t do, isn’t worth knowing, or so we’re told.

Then there is the non-compact real form of the E8 Lie algebra, a surfer’s dream of a final theory.

Today there’s a new kid on the block called category theory, a kind of stripped down, souped up group theory that has been taking mathematics by storm since it was invented in the 1940s.

It’s chief claim is that it has become a hugely powerful tool for unifying concepts in mathematics and so is obviously going to do the same for physics. (That kind of reasoning may not be as madcap as it sounds).

Chief among the category theory evangelists is Bob Croeke at the University of Oxford who today publishes a kind of idiot’s physicists guide to category theory, to give any interested parties a taste for the field.

“Category theory should become a part of the daily practice of the physicist,” argues Croeke. “The reason for this is not that category theory is a better way of doing mathematics, but that monoidal categories constitute the actual algebra of practicing physics.”

For an old dog, this will surely be a trick too far. But for any whipper snappers out there, it looks intriguing, no?

Ref: arxiv.org/abs/0808.1032: Introducing Categories to the Practicing Physicist

12 Responses to “The prophetic promise of category theory”

  1. Peter Morgan says:

    Bob begins with “Consider a physical system of type A (e.g. a qubit, or two qubits, or an electron, or classical measurement data)”. I am worried, however, that although classical particle systems can be separated from the rest of the world, classical random fields and quantum systems cannot. For quantum fields, for example, the Reeh-Schlieder theorem makes any region of space-time not separable from the rest of the universe, which presumably includes the experimental apparatus that confines it. The Copenhagen interpretation also makes a whole experimental apparatus essentially holistic, as does Bell in his Physics Today article “Against Measurement”. On rather too many approaches for this paper to be able to take it as obvious, we cannot talk about a “physical system” separately from the experimental apparatus we use, as a matter both of practice and of principle.

    Category theory is useful as a mathematical abstraction almost whatever conceptual starting point we take in Physics, but it is problematic for this paper that the very first step taken talks about a “physical system” when that concept is quite tendentious.

  2. ctthy says:

    Peter Morgan: it depends on how you look at it, really. One way of looking at the “point” of category theory is to view it as a robust formalism for working with “structured sets*” as integral wholes (ie, you are discussing properties of objects-and-their-relations all at once)…by itself this doesn’t give you anything useful, but it does offer the potential for a more-rigorous way of dealing with “inextricable / inseparable” objects.

    Viz, define a “physical system” to be a category C and a functor F from some “universe” category U; you can then bake-in any systemic dependencies by asserting F must preserve such-and-such property in U.

    Such an approach isn’t that different (in concept) from the algebra-of-observables approach to QM, but generalized to arbitrary kinds of things.

  3. James MacFarlane says:

    It’s chief claim is that it…

    That’s “its” to those of use who are friends of the English language, not “it’s”.

  4. Zephir says:

    By Aether Wave Theory the universality for physical and matheatical systems is simply given by the fact, the number theory depends on existence of countable colliding entities, i.e. the particles. Without such particles formal math would have nothing to describe – it means, the math depends on physics, not vice-versa.

    Many objects in the Nature (for example the ripples at the water surface) aren’t directly countable – so they cannot be handled with math by deterministic way at all, while still being perfectly real.

  5. Zephir says:

    Mr. Garett’s E8 group model can be derived from Aether theory as well. Lie group is not just void geometrical structure. It’s root system is describing the tightest structure of kissing hyperspheres (“unparticles”), where the kissing points are sitting at the centers of another hyperspheres, recursively. The Aether Wave Theory proposes at least two dual ways, how to interpret such structure.

    The cosmological one is maybe easier to realize: it considers, the current Universe generation is formed by interior of giant dense collapsar, which behaves like black hole from outer perspective. This collapse was followed by phase transition, which proceeded like crystallization from over-saturated solution by avalanche-like mechanism. During this, the approximately spherical zones of condensing false vacuum have intersect mutually, and from these places the another vacuum condensation has started (a sort of nucleation effect). We can observe the residuum of these zones as a dark matter streaks. The dodecahedron structure of these zones should corresponds the E8 group geometry, as being observed from inside.

    The second interpretation of E8 is relevant for Planck scale, i.e. for outer perspective. The dense interior of black hole is forming the physical vacuum, which is filled by spongy system of density fluctuations, similar to nested foam. Such structure has even a behavior of soap foam, because it gets more dense after introducing of energy by the same way, like soap shaken inside of closed vessel. Such behavior leads to the quantum behavior of vacuum and particle-wave duality. Every energy wave, exchanged between pair of particles (i.e. density fluctuations of foam) is behaving like less or more dense blob of foam, i.e. like gauge boson particle. Every boson can exchange its energy with another particles, including other gauge bosons, thus forming the another generation of interacalated particles.

    Therefore the E8 Lie group solves the trivial question: which structure should have the tightest lattice of particles, exchanged/formed by another particles? And such question has even perfect meaning from classical physics point of view! Such question has a perfect meaning in theory, describing the most dense structure of inertial particles, which we can even imagine, i.e. the interior of black hole.

  6. Blake Stacey says:

    Um, becoming “part of the daily practice of the physicist” does not automatically imply “being a tool to invent a Theory of Everything”. In fact, a tool which is only useful in a regime where all fundamental forces are unified will be irrelevant to many, if not most, practicing physicists.

    (And let’s not get started on Wolfram. Dude: Don’t. Go. There.)

  7. John Baez says:

    I can derive Aether Wave Theory and that surfer dude’s E8 theory from my theory, so I win.

  8. Zephir says:

    It can be really great, so go ahead – this is approach, the contemporary physics requires by now. But the purpose isn’t to win, but to bring a more convincing and simple explanation of reality for the rest of people without additional requirements to believe in something, ad-hoced postulates in particular. By such way, the proper theory balances the belief and its causal predictions by the same way, like the Aether particles are balancing the longitudinal and transversal waves in their mutual interactions.

  9. Zephir says:

    Our understanding of reality is strictly transversal wave based, so we are used to define reality as a most effective way for transversal wave spreading, i.e. spreading along surfaces, string, branes. During condensation of particle gas we are prefer to see the formation of aggregates and categories in it, the hierarchy and categorization is the result of our choice of (p)referential energy spreading.

    But from general perspective, here are no categories, aggregates or fluctuations. The energy/information spreads through system by uniform way, independent to any particular geometry. The AWT is cognitively neutral, it enables to describe the Universe from fractal geometry perspective, based on rational number of dimensions.

  10. Ragtime says:

    Zephur. This is just a string of words with no sense!

  11. Zephir says:

    The longitudinal waves spreads by Huyghens principle, it means every space-time event starts an infinite number of another space-time events, so it cannot serve for causual/deterministic information spreading. Instead of this is the manifestation of quantum uncertainty.

    We can discuss these aspects of AWT here, for example, on public forum:

    http://www.scienceagogo.com/forum/ubbthreads.php?ubb=showflat&Number=27167