Category Theory and Metaphysics

March 9, 2011

Posted by David Corfield

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I have been rather remiss, I feel, in promoting some mutual scrutiny between our $n$ -categorical community and that part of the metaphysics community which interests itself in structuralism. Since I heard in the mid-1990s about the metaphysical theory of structural realism put forward by various philosophers of physics, I have thought that category theory should have much to say on the issue. For one thing, a countercharge against those ontic structural realists, who believe that all that science discovers in the world are structures, maintains that the very notion of a relation within a structure involves the notion of relata, things which are being related. Structures must structure some things. Category theoretic understanding ought to have something to say on this matter.

It’s interesting then to see a recent paper by Jonathan Bain – Category-Theoretic Structure and Radical Ontic Structural Realism – which argues that the countercharge can only be made from a set-theoretic perspective.

So there’s one question: If we adopt the nPOV on physics, can we say what we are committing ourselves to the existence of?

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I’ve just had my attention drawn to a related metaphysical question of how to deal with indistinguishables. One proposed solution comes in the shape of quasi-sets. These seem to be an elaborate attempt to graft a notion of indistinguishability onto ZFC with urelements, in order to arrive at collections which are equivalent to other collections when members are replaced by copies of themselves. The intuition here is that a sodium atom is still a sodium atom if we replace one of its electrons by another, and that the particles in a sodium atom form a quasi-set rather than a set.

Setting out from a structural version of set theory, ETCS, where all elements of a set are indistinguishable, we could introduce the notion of different kinds of particle by looking to a similar category of multisets. I see we had some debate at the nLab as to what precisely a multiset should be, and that one side is happy with a version which is equivalent to $Set^2$ , where $2$ is the category with two objects and one non-identity arrow. In other words we have a set where members are labelled by members of a second set.

Second question: Is there a place for multisets in the description of a physical system? Would we have to have some functor from Hilbert spaces to Multisets (leaving aside the QFT issue of the relativity of the number of particles)?

Posted at March 9, 2011 12:07 PM UTC

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Re: Category Theory and Metaphysics

Two years ago I wrote an essay for the FQXI competition on the nature of time, arXiv:0903.1800, where I described a platonic metaphysics which incorporates the idea of time. Roughly, in the platonic world of ideas there are universes and a universe is considered real or physical if there is a passage of time in it (a good analogy is that an abstract universe is like a DVD of a movie, while the real universe corresponds to the DVD being played in a player). The part where the category theory enters is when one tries to describe the mathematical universes, i.e. a universe that corresponds to some theory of everything (e.g. string theory). My idea was to use the objects and relations among them, which I imagined as labeled graphs. The labels where restricted to be finite series of integers, because a TOE is a theory that has to be described by a finite set of symbols on a finite sheet of paper. However, one can imagine more general universes, where the graph labels are infinite series of integers, which would correspond to theories with an infinite number of axioms, which cannot be described by a finite algorithm.
I have always felt that the whole formalism could be given a category theory reformulation, but I do not have the time nor the expertise to do it.

Posted by: Aleksandar Mikovic on March 9, 2011 3:19 PM | Permalink | Reply to this

Re: Category Theory and Metaphysics

You might be interested in Steve Awodey’s paper An answer to G. Hellman’s question “Does category theory provide a framework for mathematical structuralism?”. Spoiler: Steve’s answer is Yes.

Posted by: Bas Spitters on March 9, 2011 9:07 PM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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Yes, it’s a very good paper. He makes a good case for a top-down if-thenism. The bottom-up version translates mathematics into statements like “If this set exists and has these structures and properties, then it has those.” Then you can say these statements have meaning even if the sets don’t exist (metaphysically, not mathematically).

Steve’s if-thenism is more “If something is structured like this, it has that property”. We don’t have to nail down precisely the instantiation of the something.

Posted by: David Corfield on March 11, 2011 12:59 PM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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So there’s one question: If we adopt the nPOV on physics, can we say what we are committing ourselves to the existence of?

I am trying to reply, but I am lacking the right words. Or rather: if I had better words to say it than those of category theory, I would have used them.

If I may draw this analogy trusting that you will not take it as more than an attempt to illustrate: a poet, asked for the meaning of a poem, might similarly reply: if I had other words to say it, I would have used them. The poem is not a riddle from which an answer is to be extracted. The poem is the answer.

Similarly, I think category theory is the way to speak about questions that otherwise make one want to use words like ontic and relata . It is not something that a clearer statement can be extracted from. It is the clearest statement.

The philosophy of physics that I have seen (not much, I should stress!) and found to be contentful and useful is that which I see in Lawvere’s writings: the systematic attempt to find for all concepts in ontology their category theoretic terms.

I feel that the kind of questions that you indicate above may not have answers in every language. They will have answers in category theory, spoken in a way that Lawvere has been demonstrating it can be spoken. I believe that for instance the symposium Sets Within Geometry that you highlighted in your previous posting is, despite its title, aimed at the heart of these kinds of questions.

I also believe though, that the full incantation exists only in the language called $\infty$ -category theory.

Posted by: Urs Schreiber on March 9, 2011 9:47 PM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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That’s a beautiful answer, and it makes me consider whether there was something right about the resistance I’ve felt towards these debates.

On the other hand, even if

The poem is not a riddle from which an answer is to be extracted. The poem is the answer.

it doesn’t stop a huge number of academics believing they are contributing something important by writing lengthy books on, say, The Wasteland. You can even have an online annotated experience of the poem. So, an attempt to render the nPOV on physics more approachable need not be futile, and if one wants to make it approachable for philosophers of science there had better be some effort to address their concerns.

Perhaps I could explain here one of the motivations for the rise of structural realism. Some philosophers had been making the point that when we trace the history of scientific theories we see large shifts in what is taken to exist. They were responding to the realist claim that it would be a miracle if our scientific theories were as successful as they are, and yet their terms denoting unobservable entities not refer. Antirealists pointed to terms, such as caloric and the aether, which appeared in successful theories and yet were later discarded. Partaking in a successful theory is no indication of successful reference; successful reference is no indication of partaking in a successful theory.

Here’s a passage from Larry Laudan’s ‘A Confutation of Convergent Realism’, Philosophy of Science, 48(1), 19-49:

Consider specifically the state of aetherial theories in the 1830s and 1840s. The electrical fluid, a substance which was generally assumed to accumulate on the surface rather than permeate the interstices of bodies, had been utilized to explain inter alia the attraction of oppositely charged bodies, the behavior of the Leyden jar, the similarities between atmospheric and static electricity and many phenomena of current electricity. Within chemistry and heat theory, the caloric aether had been widely utilized since Boerhaave (by, among others, Lavoisier, Laplace, Black, Rumford, Hutton, and Cavendish) to explain everything from the role of heat in chemical reactions to the conduction and radiation of heat and several standard problems of thermometry. Within the theory of light, the optical aether functioned centrally in explanations of reflection, refraction, interference, double refraction, diffraction and polarization. (Of more than passing interest, optical aether theories had also made some very startling predictions, e.g., Fresnel’s prediction of a bright spot at the center of the shadow of a circular disc; a surprising prediction which, when tested, proved correct. If that does not count as empirical success, nothing does!) There were also gravitational (e.g., LeSage’s) and physiological (e.g., Hartley’s) aethers which enjoyed some measure of empirical success. It would be difficult to find a family of theories in this period which were as successful as aether theories; compared to them, 19th century atomism (for instance), a genuinely referring theory (on realist accounts), was a dismal failure. Indeed, on any account of empirical success which I can conceive of, non-referring 19th-century aether theories were more successful than contemporary, referring atomic theories. In this connection, it is worth recalling the remark of the great theoretical physicist, J. C. Maxwell, to the effect that the aether was better confirmed than any other theoretical entity in natural philosophy! (pp. 26-27)

The structural realist response to all this was to say that it’s not the existence of the things mentioned by the theory which explains its success, but the existence of the structures described by the theory. Caloric as a fluid need not exist, but the structure of the heat theory which involved it does correspond, at least approximately, to the structure of that part of the world where it applies. At the level of structure there is convergence, rather than the wild swings of acceptance and rejection of entities.

So that would be an interesting task to tell the history, or expand on the history, of physics from the nPOV.

Posted by: David Corfield on March 10, 2011 9:31 AM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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The structural realist response to all this was to say that it’s not the existence of the things mentioned by the theory which explains its success, but the existence of the structures described by the theory.

That sounds right to me. The term aether had the sole use to put researchers in the psychological condition to feel free to consider the correct structure – wave phenomena without a medium.

What is an electromagnetic wave in vacuum – a sun ray?

Here is the answer in the language of pure category theory:

A space in the sense of physics is an object in a cohesive $\infty$ -topos. A simple example is the line object $\mathbb{A}^1$ .

From these are induced other spaces. For instance the moduli space of all electromagnetic field configurations

$\mathbf{\flat}_{dR}\mathbf{B}^2\mathbb{A}^1 := * \prod_{\mathbf{B}^2 \mathbb{A}^1} Disc \Gamma \mathbf{B}^2 \mathbb{A}^1$

(in the absence of magnetic charge, and where on the right I give the abstract category-theoretic description of this space in terms of the cohesive structure $(\Pi \dashv Disc \dashv \Gamma \dashv coDisc)$ just for the sake of indicating how this arises from intrinsic terms in the language of category theory).

Then if $X$ is the space that we call spacetime, a light-ray on $X$ is a morphism

$F : X \to \mathbf{\flat}_{dR} \mathbf{B}^{2} \mathbb{A}^1 \,.$

This is the structure that just is. Everything else is words.

One can write an exegesis that shows that if we assume smooth cohesion, then $F$ is equivalently what one calls a closed differential 2-form on $X$ . If one unwinds a bit further what this means, one finds that this is equivalently a solution to Maxwell’s equations, describing for instance a ray of sunlight.

Nowadays every child knows that I don’t have to say “aether” to talk about $F$ . But I don’t even have to talk about “electromagnetic waves”, not about “2-forms”, not about de Rham differentials.

All I need to speak about this in full detail and full generality is

the notion of category and topos;
the notion of adjunction.

Now where are the relata ? I don’t know. But I also don’t know where the aether is, and I am sure I don’t need to.

Posted by: Urs Schreiber on March 10, 2011 9:47 PM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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But I also don’t know where the aether is, and I am sure I don’t need to.

What should we say about spacetime then? It appears as an object in a cohesive $\infty$ -topos. Should we take it to be an object in a particular cohesive $\infty$ -topos?

Posted by: David Corfield on March 11, 2011 12:51 PM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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This was a very interesting, but possibly provocative, comment.

Maybe it is just words, but how does your “light ray” correspond to what most of us might consider a “light ray”? I think of a light ray as a hypothetical path of a photon, but those words are admittedly vague and hard to define. Maybe more precisely, a light ray is a geodesic.

What you called a “light ray”, unless I’m confused (which is always a high probability), is what I would probably call a solution to Maxwell’s equations, or simply an “electromagnetic wave”.

By calling a “wave” a “ray”, you are (in my mind at least) inviting questions about wave particle duality. What about the photoelectric effect? Can it be described in terms of category theoretic language so simply? What about the double slit diffraction experiment where individual photons are measured on a CCD screen, yet the full diffraction pattern emerges over time?

I would be very interested in learning how the poem of category theory tells these important stories.

Posted by: Eric Forgy on March 12, 2011 12:49 AM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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What should we say about spacetime then? It appears as an object in a cohesive ∞-topos. Should we take it to be an object in a particular cohesive ∞-topos?

I don’t know. But I can tell you some cohesive $\infty$ -toposes that have already been excluded by experiment: it cannot be Disc $\infty$ Grpd and it cannot be ETop $\infty$ Grpd.

The choice Smooth $\infty$ Grpd is one that can probably not be immediately discarded. And it looks like SynthDiff $\infty$ Grpd is relevant more as a convenient context for reasoning about structure in $Smooth \infty Grpd$ than of intrinsic physical relevance. But to the extent that either does apply it is an interesting question whether it needs to be refined to SmoothSuper $\infty$ Grpd to fit experiment. And it seems that in order to correctly describe quantum phenoema, in both cases the derived refinement to DerSmooth $\infty$ Grpd is required.

There is quite a bit still to be worked out to understand the implications of the choice of a given cohesive $\infty$ -topos. Currently we have a fairly good understanding of how all action functionals of quantum field theories of $\infty$ -Chern-Simons theory-type arise intrinsically in any cohesive $\infty$ -topos and how in $Smooth\infty Grpd$ and its refinements they reproduce a good number of known models. What I am now getting into is working out the intrinsic quantization of these, by geometric/deformation quantization of their derived covariant phase spaces. You can maybe see in between the lines already that we are aiming for the quantization to extended QFT (we naturally have the action functional as a higher-gerbe valued assignment in higher codimension and the notion of derived critical locus applies to all of these), of course. Once I understand this better and we actually understand the extended QFTs that exist canonically in any cohesive $\infty$ -topos, there will be much more to be said about which cohesive $\infty$ -topos are in accord with experiments, and which are not.

Posted by: Urs Schreiber on March 12, 2011 9:43 AM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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Interesting. So presumably here, even when you narrow down to a feasible class of cohesive $\infty$ -toposes, or a single one, in which our spacetime can be said to exist, it’s to be done in a top-down way.

So you don’t say something like “Because spacetime is a collection of fundamental entities (points, or whatever) equipped with such and such a structure, it resides in this cohesive $\infty$ -topos.” But rather define the cohesive $\infty$ -topos by its own properties (maybe even within the collection of such toposes), and then say spacetime is an object within it.

Posted by: David Corfield on March 13, 2011 1:48 PM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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it’s to be done in a top-down way.

Maybe, i am not sure if I understand yet what is meant by this. I (still) haven’t tried to read Steve Awodey’s argument yet.

It seems to me that your comments point to the question of how to formalize what it means to have a physical theory .

Somebody hands you a formalism and claims: “This is a physical theory.” You see a bunch of axioms (if your are lucky) and proofs of some implications ( predictions ). Even before asking if the theory matches some experiments, how do you decide if it is correct to call it a theory of physics in the first place? What does it mean to “model spacetime”, for instance?

I’d hope that eventually this is something that has a good answer in terms of topos theory, too, such that I could give a systematic answer to your questions as follows: given a cohesive $\infty$ -topos and some choice of structure inside it (an internal action functional maybe) there is canonically induced an internal structure called a physical theory and this comes with a prescription for what it means to extract an experimental prediction from it.

Certainly the discussion that has been and is revolving around the topos over the semilattice of commutative subalgebras of a given $C^*$ -algebra is aimed, more or less explicitly, towards answering aspects of this question, but I am not sure if the general abstract question has received due attention yet (maybe it has, I may well be ignorant of the relevant developments).

Possibly it makes sense to declare that the following is the bare minimum content of what constitutes a physical theory :

a topos $\mathcal{E}$ , equipped with an object denoted maybe $\mathcal{S}$ or $\mathcal{O}(\mathcal{S})$ and called the space of states or dually the lattice of opens of the space of states , respectively;
a collection of morphisms $P_i : \mathcal{S} \to \Omega$ (or $P_i : \mathcal{O}(\mathcal{S}) \to \Omega$ ) to the subobject classifier, called the predictions of the theory.

Then if we have a way to identify generalized elements $\psi : \Psi \to \mathcal{S}$ (or $\psi : \Psi \to \mathcal{O}(\mathcal{S})$ ) with observable reality, such that the pairing

$P(\psi) : \Psi \stackrel{\psi}{\to} \mathcal{S} \stackrel{P}{\to} \Omega$

can be interpreted as the truth value of finding the prediction $P$ realized by measurement on the state $\psi$ , we could have the right to call “ $(\mathcal{E}, \mathcal{S}, \{P_i\}_i)$ ” a physical theory .

The discussion of the semilattice of commutative subalgebras of the algebra of bounded operators on a Hilbert spaces revolves around realizing this for a generic system of quantum mechanics. Or rather, as is briefly remarked in the introduction of Heunen-Landsman-Spitters, the constructions considered in that context would also serve to provide the above formal data for any copresheaf of algebras of observables as it appears in the AQFT-formulation of quantum field theory. Details remain to be worked out.

If I’d allow myself to speculate on the basis of these observations – more in order to indicate how the argument might proceed at all than to claim that this is how it will proceed – I might suggest that an answer to your question might be along the following lines:

given a cohesive $\infty$ -topos $\mathbf{H}$ , there exists canonically internal structures that desrve to be called extended action functionals. Pick one such that for an object $X$ in $\mathbf{H}$ it can be interpreted as an action functional $\exp(i S)$ for fields on $X$ . Then construct from this a (bare bones) internal physical theory along the above lines as follows: for each morphism $\Sigma \hookrightarrow X$ we can restrict the action functional to fields on $\Sigma$ , form canonically the (derived) covariant phase space $hofib( d S|_\Sigma : Fields(\Sigma) \to Phases)$ , which is canonically equipped with suitable structure that induces a noncommutative algebra of functions $A_\Sigma$ on it. As $\Sigma$ varies and probes all of $X$ , these arrange into a suitable co-presheaf of algebras on $X$ , or something similar. This is equivalently an algebra internal to copresheaves (or similar). Apply to that the Butterfield-Isham-Döring/Heunen-Landsman-Spitters construction to produce the corresponding (internal?) topos $\mathcal{E}$ with object $\mathcal{S}$ (or $\mathcal{O}(\mathcal{S})$ ) and propositions $\{P_i\}$ . Imagine that you can figure out a sensible way to match the internal truth values of propositions about states defined this way to observations made in real-world experiment.

Then, finally, if these propositions match the experiments to a good degree, we would be entitled to say that $X \in \mathbf{H}$ is a good model for spacetime.

At least that would be a sketch of a possible road-map towards this goal. Among the many open questions here I should mention that if that’s about right at all, then at best for something like an effective field theory that does come with a fixed spacetime background in the first place.

But I’d rather stop here for the moment…

Posted by: Urs Schreiber on March 14, 2011 11:24 AM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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the question of how to formalize what it means to have a physical theory… I’d hope that eventually this is something that has a good answer in terms of topos theory, too

I’m not really following this discusssion, but I have to say that that really sounds ridiculous to me, and the sort of thing that gives category theorists a bad name. Topos theory is wonderful and all, but not everything in the world is topos theory!!

Somebody hands you a formalism and claims: “This is a physical theory.” You see a bunch of axioms (if you are lucky) and proofs of some implications ( predictions ). Even before asking if the theory matches some experiments, how do you decide if it is correct to call it a theory of physics in the first place?

If something makes predictions about the results of experiments, then in what sense would it not make sense to call it a physical theory?

Posted by: Mike Shulman on March 14, 2011 4:45 PM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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You probably shouldn’t go wandering into the physics common room and declare that all anyone should ever study are the morphisms of a cohesive $\infty$ -topos (possible storyline for The Big Bang Theory?), but isn’t there a point to thinking this way sometimes, and taking it seriously? Imagine wandering into the Mathematics common room in, say, 1880 and declaring that everyone is really studying sets.

Thinking in such an abstract way about physics has the added excitement that one has to consider the relationship between world, model, experiment and our information state. E.g., there’s something worth pondering about what part of physical theory concerns information processing.

Posted by: David Corfield on March 14, 2011 5:24 PM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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I’m not really following this discusssion, but I have to say that that really sounds ridiculous

That’s a great lead-in. ;-)

not everything in the world is topos theory!!

Luckily we are not discussing everything .

What are we discussing?

David mentioned various discussions and schools among philosophers of physics, who try to come to grips with what it means to have a theory of reality.

Was the aether a physical theory that has been proven wrong by experiment, as usually considered nowadays? Or was it actually a successful physical theory, as Maxwell and his contemporaries thought, since in fact it did seem to make a wealth of predictions, some of them rather subtle? Or was it actually no physical theory at all but just a word that happened to make practitioners study the correct theory in terms of the correct structures (wave phenomena without a medium), as the school of structural realism asserts? But can we have structures in reality without objects that are being structured?

Probably everybody is free to regard these questions as quite irrelevant for everything but, well, philosophical debates. But maybe on the $n$ -category café, where we allow ourselves to find it interesting to consider questions such as homotopy in logic that the average working mathematician finds quite ridiculous, we may also allow ourselves to find interest in discussion of the relation between logic and scientific formalism on the one hand and that evasive thing called reality on the other.

I had suggested that these kinds of questions can be usefully considered and can have useful implications if one manages to formalize them sufficiently. Because the problem with the discussion of the aether , to stay within the example, is that it is just a word and that it is hard to say what of the sentence “Space is homogeneously filled with a substance called aether that may be excited by and transmit waves.” is supposed to be a theory . I don’t know what the aether is, but if I can formalize what the theory that goes by its name predicts, then I don’t need to.

To illustrate this, I pointed out that a large chunk of standard physics theory about space, time, electromagnetism and these things (for example) can be discussed fully formally in the context of cohesive $\infty$ -toposes, since in there I have a languge that has just the right expressiveness to speak of the geometric structures involved in these concepts.

But then David kept pushing on and asked questions that I took to be after the very identification not just of general structures in physics, but of the explicit reality that we do observe. Suppose we do agree that $\infty$ -topos theory gives an efficient means to speak of geometric structures such as involved in our current best theories in theoretical physics, how do we go one step further and promote such formalism to an actual physical theory ? How would be identify the $\infty$ -toposes that support objects that come close to being good models of the spacetime that we observe around us?

Again, it is easy to regard such questions as irrelevant for most every purpose. But I think on a blog sub-titled on Math, Physics and Philosophy , if this discussion is ridiculous, then the entire blog is. I could turn this around: how can we have so much discussion about the very foundation of logic, without any discussion as to how it connects to David’s interest on the philosophy of science. It seems to me that somewhere at the very bottom of it all, these two topics certainly must touch and certainly deserve to be discussed.

In view of this, i want to share an observation: I think I do know well what kind of worries made your alarm bells ring: too many physicists have wildly played around with mathematical formalism, trying to see if it matches their needs in theory-building, in a way that one would rather not watch in public and whose failure, while maybe useful (we learn from failures) we’d rather not see publically, for the sheer naïvety and incoherence that it displays.

On the other hand, what also frequently raises strong emotions within me is the observation that mathematicians have the trait of watching with such contempt those attempts, instead of realizing that it is, for the sake of humanity, at them just as much as on every other suitably educated person to bang their head against these riddles. Among two persons, one making a fool of himself for trying what needs to be tried, the other just raising his eyebrows with his hands in his pockets, I’d rather keep with the former.

Therefore I replied to David by doing two things: I mentioned the only attempt that I am aware of for formalizing what it means to have a formalism that makes statements about fundamental physical reality, and I indicated a rough sketch of how I could imagine this could be connected to the previous discussion.

That one attempt that I mean is the re-formulation of quantum logic in terms of internal logic in a topos.

You may think that it is all clear what it means to have a physical theory : well, it’s something that makes a prediction about experimental outcomes, so what? But greater minds than both of us found that in view of the quantum nature of our best theories, it is in fact far from clear what this means, and that something needs to be solved here.

Again, certainly one can easily declare this kind of question as irrelevant and move on to do whateverone does in daily life. But I don’t think it is ridiculous to ask this question, nor - given that it is a question about how logic meets reality - that it is ridiculous to try to answer it by invoking tools of topos theory.

John von Neumann thought there was something to be understood here. He suggested that what is called quantum logic is needed to make sense of the nature as a theory of a theory of fundamental physics. Various arguments have been put forward that suggest that, however, the original approach of von Neumann is unsatisfactory and fails to accomplish what one would hope it accomplishes, and that instead the correct way to understand the theory-nature of theories of quantum mechanics is to understand them in terms of the internal logic of a suitable topos.

Maybe I’ll write out an $n$ Lab entry with history and details on this, but for the moment I trust that you can accept that such arguments have been put forward and that if one accepts any relevance of such foundational or philosophical questions at all, then these are arguments that need to be considered seriously, even if they happen to be in fact misled in some way (because it will be interesting to see in which way so).

But even without knowing any details of these arguments: I am surprised to see a topos theorist not exclaim Well, obviously! when faced with the suggestion that a system of non-standard logic might have to be understood as governed by the internal logic of some topos. How can it be that you find that a ridiculous idea? Wouldn’t it be ridiculous not to consider this? For maybe it’s wrong, but certainly we’d be interested in seeing why it’s wrong?

What I can imagine is that you find ridiculous the ways in wich people – who on average are not full-time topos-theorists but try to go beyond their primary area of expertise since they see the need – go about trying to solve this riddle. I know of topos theorists who find the idea that in order to understand the foundational logic nature of a system of quantum mechanics one should pass to the presheaf topos over a lattice of subalgebras a bad idea for the sole reason that “presheaf toposes are so simple”. That may well be, and quite possibly if toposes have any role at all to play in understanding the nature of quantum physics, it may well be that the present attempts to do it with presheaf toposes will turn out to be way too naive – but I am sure glad somebody is trying that possibility first.

There was a time at the beginning of the last century when people said that applying group theory in physics was ridiculous.

Posted by: Urs Schreiber on March 14, 2011 7:27 PM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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I think you misunderstood what I meant. I wasn’t saying anything about irrelevance, or practical applications. I’m perfectly happy with exploring new ways to think about physical theories, although I personally am not very interested in these philosophical debates. But the idea of “formalizing what it means to have a physical theory” sounds to me limitative rather than expansive. That’s the only thing I thought was ridiculous. Indeed it’s the very fact that

in view of the quantum nature of our best theories, it is in fact far from clear what this means, and that something needs to be solved here

which makes me think that we should be open about what constitutes a “physical theory”. If someone hands us a formalism, as you said, calling it a physical theory and making predictions, and if those predictions match experiments in some way, then on what grounds could we wonder “whether it is correct to call it a theory of physics”? Isn’t that more like rejecting quantum mechanics because “God doesn’t play dice”?

I am surprised to see a topos theorist not exclaim Well, obviously! when faced with the suggestion that a system of non-standard logic might have to be understood as governed by the internal logic of some topos.

As long as you continue to include the word might, I’m with you. But a topos theorist should also be aware that topos logic is only one particular type of non-standard logic. I haven’t really seen convincing evidence yet that topos-logic is similar enough to quantum-logic that it’s useful or enlightening to try to model the one with the other.

Posted by: Mike Shulman on March 14, 2011 8:24 PM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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Given that we have in parts switched sub-topic to the discussion of the virtues or not of the Isham-Döring/Heunen-Landsman-Spitters proposal for interpreting a kind of quantum logic internal to a suitable topos, I’d like to use the occasion to maybe have some more discussion on this. For my own sake I want eventually to spell out what has been achieved, and what not and make up my mind.

I can’t tonight, since I need to stop working now. But here is a rough indication of what I currently seem to understand:

The basic observation is that every non-commutative algebra naturally induces a commutative algebra internal to the presheaf topos over its own poset of commutative subalgebras.

This is pretty much a tautology. But maybe a useful one. What needs to be evaluated is the claim that the internal perspective of this particular topos is useful.

Now HLS point out/claim that in that particular topos, the ordinary internal measure theory produces externally precisely all the intricacies of quantum states on noncommutative algebras. The general claim is that the “classical theory” internal to this topos is precisely the “quantum theory” of the algebra externally. Here the quotation marks are to indicate that this requires some qualification, which I will not go into right now.

Is this right in this generality? Is this useful? I am thinking: if indeed this is so, at least it answers positively an old question of whether quantum theory can be understood as classical theory in a suitable topos. I am not really sure yet what to do with this fact, but it sounds like a good fact to know.

Some information about the algebra is lost by passing to its posite of commutative subalgebras. Is that a problem? Is that good? Harding and Döring point out that it is precisely the Jordan algebra structure of the algebra that is remembered. This seems noteworthy: as John B. was amplifying in his series on the Three-fold way, the Jordan algebra structure is precisely what counts in the formalization of quantum observables in terms of convex cones that is popular in quantum computing. This seems to support the idea that passing to the topos over the posite of commutative subalgebra is precisely the right thing to do.

Or is it? Mike, could you maybe comment on what makes you be dismissive of this proposal? I’d like to know. It seems to me that something interesting is going on, but that more insights would be useful, to see what all this is supposed to be telling us.

Posted by: Urs Schreiber on March 14, 2011 10:52 PM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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I didn’t mean to be “dismissive” of it in the sense of definitely thinking it’s not valuable. I just mean I haven’t yet been convinced that it is valuable. I’m glad someone is thinking about it in case it does turn out to be valuable. As you say, what needs to be evaluated is the claim that this perspective is useful. I haven’t been following the developments closely, but I’ve heard a few conference talks, and they didn’t leave me thinking that any usefulness of this perspective has been demonstrated yet.

In particular, I’m somewhat worried about the loss of information in passing to the poset of commutative subalgebras, but I probably don’t really know enough quantum mechanics to be qualified to worry. The monoidal-categories graphical-calculus approach to quantum mechanics and information theory that Bob Coecke and his group have been developing seems to me like a more faithful category-theoretic description of the “quantum world.” Cf. the Rosetta stone and all that.

I’d be more excited about topos-theoretic quantum mechanics if I could see a translation of some nontrivial bit of “classical” quantum mechanics into this language. Can you solve the harmonic oscillator? Predict the results of any experiments?

Posted by: Mike Shulman on March 15, 2011 1:01 AM | Permalink | Reply to this

Re: Category Theory and Metaphysics

Agreed, we lack worked examples.
(I believe the same holds for old-style quantum logic.)

However, John Harding and Andreas Doering show in Abelian subalgebras and the Jordan structure of a von Neumann algebra that we can “almost” reconstruct the von Neumann algebra from its commutative subalgebras. Hence, there is good hope that not too much is lost in the transition to the topos.

Bas

Posted by: Bas Spitters on March 15, 2011 10:14 AM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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In particular, I’m somewhat worried about the loss of information in passing to the poset of commutative subalgebras

As I said, it is interesting to note that Harding and Döring show that the isomorphism classes of posets of commutative subalgebras of $A$ correspond to the isomorphism class of the Jordan algebra structure of $A$ . In view of the relation between Jordan algebras and convex cones of observables that John was reviewing lately, it seems as if this can be read as saying that passage to the poset of commutative subalgebras forgets precisely that part of the information in $A$ that is redundant for the description of the quantum theory.

I need to think more about this, but this seems to be an interesting hint to me.

The monoidal-categories graphical-calculus approach to quantum mechanics and information theory that Bob Coecke and his group have been developing seems to me like a more faithful category-theoretic description

This is not a valid dichotomy, I would think. Bob Coecke et al. proceed via the FQFT Schrödinger picture by regarding morphisms between spaces of states in a monoidal category (or higher category) (and in fact as far as I can see they restrict to finite dimensional spaces of states, which is quite a restriction everywhere except in quantum computing). Dual to that is the AQFT perspective that instead describes copresheaves of algebras of observables, the Heisenberg picture. Both pictures are supposed to be dual to each other, even though this has been fully established only in special classes of cases.

But Isham-Döring/Heunen-Landsman-Spitters try to go beyond both of that towards answering the question: what does it mean that either of these structures – a monoidal functor, a copresheaf of algebras – defines a physical theory .

To illustrate this point with a somewhat superficial but hopefully useful example: suppose tomorrow somebody claims that he has found a symmetric monoidal $(\infty,4)$ -category and a fully dualizable object inside it such that the 4d TQFT associated to this data (corresponding to the Coecke et al. perspective) by the cobordism hypothesis-theorem is the theory of quantum gravity that describes our reality?

How do you check this claim? How do you extract from some object in some 4-category something like, say, the observed value of the cosmological constant?

Probably a lot can be said in reply to this question in a by hand sort of way, but I do think it is interesting, at least in a thread like David started here, to ask if we have a systematic and formal way to say what it means to exatract from a formalism that claims to encode a quantum theory something that we can formally agree on does so.

I think trying to formalize subtle, previously philosophical, questions is fruitful. You said above

But the idea of “formalizing what it means to have a physical theory” sounds to me limitative rather than expansive.

But I think you would agree that formalizing the idea what it means to have geometry (Euclidean, Riemannian) or notably what it means to have logic was a step that allowed people to go considerably beyond what could be achieved without such a formalization, instead of having been limitative.

Posted by: Urs Schreiber on March 15, 2011 9:09 AM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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Both pictures are supposed to be dual to each other…

Is that ‘duality’ to be read in a category theoretic sense?

Posted by: David Corfield on March 15, 2011 10:22 AM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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Both pictures are supposed to be dual to each other…

Is that ‘duality’ to be read in a category theoretic sense?

Unfortunately I only meant duality in the vague sense of: one can pass back and forth between the two pictures without losing information. I have only a vague idea of the fully systematic statement that might be possible here. Just recently in the context of another thread Liang Kong wrote me an email saying that he has some ideas on this. And I can imagine some other people who probably do, too and there is a bunch of partial results in the literature.

Hm, I guess now I should really list and spell out some details instead of being so mysterious. Remind me later, I don’t have time right now.

Posted by: Urs Schreiber on March 16, 2011 11:22 AM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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Here is a result akin to that of Harding-Döring, that indicates that passing to the poset of commutative subalgebras of a noncommutative $C^*$ -algebra retains the right information as far as quantum theory is concerned:

Benno van den Berg, Chris Heunen, Noncommutativity as a colimit (pdf)

it is observed that

there is a functor from all $C^\ast$ -algebras to their partial commutative $C^\ast$ -algebras of normal elements;
every partial $C^\ast$ -algebra is the colimit over its poset of finitely generated commutative total $C^\ast$ -algebras.

The first step can be seen as picking out the physically relevant information, the second step says that this information is that which is encoded in that poset of commutative subalgebras.

Posted by: Urs Schreiber on March 15, 2011 10:06 AM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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Okay, those are definitely some reasons to hope that not much is lost. Now maybe we (or, rather, someone) can work some examples!

Bas wrote:

I believe the same holds for old-style quantum logic.

Yes, I’ve never been too convinced by old-style “quantum logic” either. (-:

And Urs wrote:

How do you extract from some object in some 4-category something like, say, the observed value of the cosmological constant?

Isn’t the onus for telling you how to do that on the person who is describing what he calls a “physical theory”?

I’m not against formalizing a certain type of physical theory one is interested in studying, but I think it’s hubris to think that one could formalize all physical theories in the same way. Imagine if someone had tried to formalize all physical theory before quantum mechanics; surely they would have required it to make deterministic predictions? Big advances have come in the past by changing our idea of what it is that we’re looking for.

Posted by: Mike Shulman on March 15, 2011 4:22 PM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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It seems that yesterday Bas and I found a useful statement about the “Bohrification” of AQFT nets of observables that might be useful for the discussion here. But I guess currently it’s top secret. At least I have started an $n$ Lab entry Bohrification. (Bas, please expand or let me know if this is urgently missing some data points).

Posted by: Urs Schreiber on March 16, 2011 11:12 AM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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Imagine if someone had tried to formalize all physical theory before quantum mechanics; surely they would have required it to make deterministic predictions? Big advances have come in the past by changing our idea of what it is that we’re looking for.

I don’t see a problem with attempts to circumscribe existing forms of theory, even if we should expect that future forms will look rather different. The very process can help bring about new insights which may generate new forms of theory.

It’s plausible that set theoretic reductionism played an important part in pushing mathematics forward, even if we see it now as conceptually limited. Indeed, one could argue that it helped reveal the limitations of the informal intuition it formalised.

Posted by: David Corfield on March 16, 2011 1:07 PM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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I don’t see a problem with attempts to circumscribe existing forms of theory, even if we should expect that future forms will look rather different. The very process can help bring about new insights which may generate new forms of theory.

That’s what I am thinking, too..

As you pointed out in this thread, philosophers of science are already (already? or since thousands of years?) discussing what it means to have a theory of reality (does it need relata ? , can it do with just relations ? and what does that mean? ). I think: if we do it at all, let’s try to make it productive and constructive by trying to formalize the entities under discussion as much as possible. That will make the discussions much more interesting.

Similarly, when Mike says it is hubris to try to formalize what it means to have a theory of reality, I am reminded of all the physicists that already speak of having (or not) a theory of everything . I have seen a conference talk where the speaker said something like: “Now that we have convinced ourselves that we have found the theory of quantum gravity, it turns out the remaining biggest problem is how to convince the other community that indeed we do.” If we have such fights at all, then I’d rather have we’d try to formalize as much as possible, so that there is a chance that some insight of intrinsic value is gained in the discussion.

Finally, I find it curious that Mike seems to take the historical step from classical to quantum mechanics as an indication that formalization of what it means to have a theory of reality has already been proven to be hopeless in the past: because on the contrary the current attempts such as Bohrification are driven by the observation that the opposite conclusion seems to be valid:

if classical phyicists had tried to abstract as much as possible the bare minimum structure necessary to speak of pure and non-pure states in classical mechanics, they might have found that all they did with ordinary phase spaces could be done with any regular locale in any ambient Grothendieck topos. That is sufficient to have a good theory of valuations that encode pure and mixed states in classical mechanics.

The interest in this topos-theoretic formulation of the kinematical part of physical theory arises precisely from the recent insight that by simply switching the ambient topos, this same statement applies to mixed and pure states in quantum mechanics.

Therefore I think if people had tried harder to formalize what it means to have a physical theory, the chance that they would have been taken by utter surprise by the existence of quantum mechanics might have been slightly decreased. At least they might have been aware that the bare minimum structure which is actually required to make sense of the symbols they put to paper allowed for considerably more general physics than classical physics.

Posted by: Urs Schreiber on March 16, 2011 1:40 PM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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I don’t see a problem with attempts to circumscribe existing forms of theory, even if we should expect that future forms will look rather different.

That’s exactly what I was saying. I just think we should be explicit that what we’re circumscribing is existing forms of theory. Probably all I was taking issue with is Urs’ choice of wording in a single comment, so that this discussion is rapidly becoming pointless. (-:

Similarly, when Mike says it is hubris to try to formalize what it means to have a theory of reality, I am reminded of all the physicists that already speak of having (or not) a theory of everything.

Yes, I think that’s hubris too. (-:

Of course in hindsight it is often easy to say “if people had thought in thus-and-such a way, then they would have realized thus-and-so, and the initially surprising development of so-and-such would have been obvious.” But I don’t really find it likely that if people in 1900 had actually tried to formalize what it meant to have a “theory of reality,” they would have come up with regular locales in ambient Grothendieck toposes, since Grothendieck wasn’t born until 1928. (This is, of course, not to cast any aspersions on the value of such observations as ahistorical motivation for current work.) I am reminded of the following quote from Milan Kundera:

Man proceeds in the fog. But when he looks back to judge people of the past, he sees no fog on their path. From his present, which was their faraway future, their path looks perfectly clear to him, good visibility all the way. Looking back, he sees the people proceeding, he sees their mistakes, but not the fog. And yet all of them… were walking in fog, and one might wonder: who is more blind?

Posted by: Mike Shulman on March 16, 2011 7:55 PM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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Probably all I was taking issue with is Urs’ choice of wording in a single comment, so that this discussion is rapidly becoming pointless.

I guess we have figured out that the crass disagreement that seemed to be there for a moment is not in fact present.

But I do like the general topic of the discussion, and am hoping that every once in a while we can come back to this together.

Posted by: Urs Schreiber on March 16, 2011 9:25 PM | Permalink | Reply to this

Re: Category Theory and Metaphysics

On the ontological issues, you might want to look at the not-so-recent paper of O’Leary-Hawthorne and Cortens, “Ontological nihilism”, Philosophical Studies 79 (1995) 143-165. The title is a bit sensational; what they propose is not to deny that anything exists, but that objects in a mildly technical sense exist. What remains, you might think, must be structure, but since the proposal never touches on transitive verbs, it’s not clear what they would do with relations.

Another more recent attempt to do without objects is Ladyman and Ross, Every thing must go. They propose a new version of structural realism which they think is consistent with mathematical structuralism.

Posted by: Dennis Des Chene on March 10, 2011 6:47 PM | Permalink | Reply to this

Re: Category Theory and Metaphysics

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Everything Must Go

I’ve read a chunk of that. It’s very opposed to what is called ‘armchair metaphysics’, i.e., metaphysics which doesn’t take due account of physics’ best account of the world.

James Ladyman also wrote the Stanford article on structural realism. There’s not a whiff of category theory there, so perhaps one could pass on a similar charge that the best account (the nPOV) isn’t being taken into account.

Posted by: David Corfield on March 11, 2011 12:29 PM | Permalink | Reply to this

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March 9, 2011