FWIW, there are no partial orders in the solution I presented.

*Edit:* Well, maybe there are. I guess the partial order in question would have, as elements, categories equipped with a cell complex decomposition, and as relation the notion “is a subcomplex of”.

Hi Urs,

The whole structural induction idea is very new to me and hard to express, but here goes. Mike has suggested in the thread I offered that we can use categorical cell complexes to host something like a partially ordered set of categories. Suppose we have a method of induction-style proof where we prove statements about a structure and then have an induction hypothesis that states that if it is true on $x$ and $x R y$ then it is true for $y$. Next we prove the statement for the minimal elements (…and so on, but it is a bit beyond me so I am at a loss). Thus, we assume we have the partially ordered collection of categories, we have the induction proof method over the partial order, and finally we discover that our collection of partially ordered categories is actually all categories. Maybe it is CAT, the cat of all cats. I don’t know, as I don’t know what categorical cell complexes are. I still have to decide if that is what I am thinking about. Mike seems to have agreed that we might both be thinking of a collection of cats that is, indeed, all cats.

We start there.

Next, we must understand that any statement proved with the inductive proof method is actually a statement about the kinds of structures on which we are basing the partially ordered collection. As we have seen, though, the structures are just categories and nothing more specific or general. Thus, any inductive-type proof (as described) is actually a proof about categories. We then start writing out each of these proofs. After a long time we will start to consider the collection of all such proofs, or statements using that proof method. This collection is, therefore, all statements one might prove about categories. Hence, we are considering a way to present the theory of categories.

]]>Hi Urs,

Some keywords that you mention remind me of things that I am interested in and others here are interested in, but the way you state the above paragraphs sounds weird.

ya, I get that a lot. Again, your patience is appreciated.

I’m visiting Toronto this weekend, but I am mainly looking forward to sitting in cafe’s and doing math! I have Topooses and Local Set Theories with me and Awodey’s small text on Category theory. What a vacation! I just hung a white board in my den (previously a spare bedroom). I want to collect all these ideas that I have been asking about and put them together into something concrete. Let me see if Toronto will inspire me to at least elaborate on my posts.

]]>Ben, what you write above sounds pretty wild. Some keywords that you mention remind me of things that I am interested in and others here are interested in, but the way you state the above paragraphs sounds weird.

There is plenty of discussion here of type theory, physics, foundations, categorfication, foundations of physics, categorification in physics etc. So from the point of view of your inclinations it might be interesting.

If this is to be continued: can we find a way to make this into a more systematic discussion? The first thing would maybe be to break down all these ideas into little bite-sized pieces and see if we can see what it actually is that you are talking about (I currently can’t see it!).

Let’s see, maybe we start with this sentence of yours

Structural inductive type proofs, taken together form a presentation of the theory of categories.

This does not quite parse for me, although very similar-sounding sentences might parse. Maybe we can try to just sort out this piece of your above message, for a start.

So: what do you actually want to say with this sentence?

]]>I guess the only two cents that I would throw in here is to point out my other threads on what I called structural inductionand Bayesian update. If we have an abstraction of structural induction over categories we are allowing for unbounded structure just via the inductive type proof. My approach to physics is to replace spaces (Hilbert spaces, manifolds and sets)with categories themselves. The finite physics approach of Vicary and anyone doing quantum gravity is reflected in my obsession with tiny categories and how they build into other categories. This has inadvertently also solved another problem which I wanted to solve which was to provide an alternate presentation of the theory of categories. Structural inductive type proofs, taken together form a presentation of the theory of categories. I think that some neo realists might object to a vision of physics where the background is so relative: you can only know structure which your apparatus will reflect in terms of IT’S structure preserving maps. Perhaps an objection is that it skirts the issue. Sometimes I think that only causality is real. Then I think of the causally neutral work of Hardy and Spekkens and then feel that causation and correlation are just post hoc conclusions…….but of what?

I am committed to the idea that foundational problems in physics cannot be addressed without addressing the foundations of mathematics. So, while it may look the same: replace Hilbert spaces with categories and start your thinking at the point of tiny structure, I am actually cleaning out the foundation where, in the other case of Hilbert spaces, sets can still generally interfere with problems of interpretation. It works like this: how am I presenting physics?…with Hilbert spaces. How am I presenting Hilbert spaces? With the category of Hilbert spaces. How are you presenting the theory of categories? In set.

]]>Hi I just wanted to thank everyone for participating in this thread. I wish I had lots to add but I am clearly deferring to the folks here as the experts.

]]>What does it mean “correct” ?

Correct observable means: local observable. Or rather: observable that is approximated to any given accuracy by a local observable.

]]>Urs 11:

as we scale this to the limit of infinite size, this changes: the correct algebra to take in the limit has as elements arbitrary but finite formal sums of operators on the lattice sites

What does it mean “correct” ? What is wrong with taking some “completion” ? I mean physically – do we really get new observables if we make it bigger ?

Here are my naive thoughts: In any stage of measurement on such limit observable one has the same meaning and result as one would have from some finite approximation; only infinitely many measurements could see a difference. So I could work with a system with more operators, and for a theoretical physicist thinking on one or another would be the same. I know that from teh point of view of classification etc. one likes to be in a smaller world of allowed models (algebras), but I do not see that this preference in the choice has anything to do with physical requirements.

]]>I would tend to try to understand this question by looking for a logical notion of finite dimensionality, like coherence/properness/compacity.

But you're right, Urs, this looks like a different question. ]]>

I am now getting the impression that all three of us are each talking about a diferent question. :-)

Concerning constructive mathematics in non-finite physics: there is for instance the constructive Gelfand duality theorem and together with the Bohr-topos perspective it allows to speak about all those non-finite QFT systems constructively.

]]>This MO question seems vaguely related to the question of importance of infinite-dimensionality in physics. At least, I think infinite-dimensional things tend to be even harder to work with constructively than they are classically.

]]>I’m not really interested in a philosophical discussion about whether anything in *mathematics* is “truly infinite”. I work with infinite sets all the time and prove existence results about them that have nothing to do with finiteness. My question was whether the infinite-dimensional parts of mathematics really have importance in *physics*.

My opinion is that these types of objects appear because the formalism we use is not fine and rigid enough to have good finiteness hypothesis, but maybe i'm wrong and not enough dynamics oriented. There is some kind of tricky equivariance under R hidden here. For these kinds of objects, you are mostly interested by measure theoretic qualitative results (ergodicity or so).

So from the above general viewpoint, this example falls into the category of measure theoretical dynamical systems, and their qualitative study. What kind of finiteness do we have here? In Lebesgue theory, we mostly use the finiteness of the integral of positive functions, to find good finiteness hypothesis on the objects in play. And then something like the completeness of R or its order completeness, to prove existence results, like Lebesgue's domination theorem.

So if you are interested in the qualitative study of dynamical systems, with main theoretical tool measure theory, i would tend to say that it is also not really infinite dimensional from the doctrinal viewpoint because we use completeness of R as a kind of finiteness/coherence condition to get interesting results.

In fact, i would tend to say that any mathematical theory is finite dimensional in this sense, otherwise it is void. What i find interesting, is to find exactly the finiteness hypothesis we use in the examples we like. For example, you never study an arbitrary real number or an arbitrary measurable set. You are allways interested by a concrete example, and the description of this example has some kind of finiteness property, otherwise you can't do anything.

Maybe i am digressing a bit far from your true interests here... ]]>

What would be a solution to your probleme in this physical 1-dimensional lattice case?

Let’s try to gauge what we are talking about: my comments above were meant as a start of supporting my claim (more details have to wait) that a formalism that only sees finite dimensional Hilbert spaces misses certain physical phenomena which either have no finite approximation, or else for which one needs to take the “thermodynamic limit” of the finite approximations.

The discussion of non-trivial 2d QFTs coming from local nets of type $III$ vN factors (the “infinite” factors) via quantum lattice systems and their uniformly hyperfinite algebras are a genuinely quantum mechanical (as opposed to statistical) example.

All of this deserves to be expanded on, but this is what I was trying to indicate.

]]>This was directed at Mike.

It's just an abstract nonsense remark, nothing deep.

I just mean that to get any existence result, you need a kind of finiteness hypothesis, that one could call ``finite dimensionality''.

For the integers, it depend on how you use them and what you want to prove. If you embed integers in R and use local compactness of R or its completeness to prove existence of a solution to your problem, i tend to think that the system is not really infinite dimensional, since it has this inherent finiteness in its definition (compactness/completeness).

What would be a solution to your probleme in this physical 1-dimensional lattice case? ]]>

Hi,

is the comment directed at me or at Mike? I am not sure I understand.

The set of integers, $\mathbb{Z}$, say appearing in a physical system given by a 1-dimensional lattice, do you count that as a “truly infinite”?

]]>Could you give an example of situation where this finiteness notions (that i would call doctrinal coherence) is not fulfillled by a mathematical theory used in physics, and you truly have an infinite dimensional situation, without any compacity hypothesis for the objects/theories in play? ]]>

I have started *thermodynamic limit* – a stub to serve as a reminder for me when I have more time to come back to this.

I see what you are after. Yes, phase transitions, universal behaviour, scaling laws and such phenomena are not exhibited at any finite lattice stage of the approximation, but only in what technically is called “the thermodynamical limit”.

This is a good question to have a decent nLab entry about. I’ll try to look into it. Not right now, but maybe in a few days.

]]>What is the *physical* meaning of infinite dimensionality? I believe that the *mathematics* of infinite dimensionality is structurally different from that of finite dimensions, but can you say anything to convince me that it isn’t just an artifact of the mathematics we’ve chosen, that there are real observed physical phenomena that can’t be explained by a finite-dimensional model?

Another comment on the idea that finite systems approximate infinite systems well enough:

I once had a discussion about this with Alain Connes, concerning the physical relevance of his result about outer automorphisms of type III factors. He amplified the following point, which I think is central:

Naively it looks as if the Schrödinger-Heisenberg evolution equation says that time propagation in QFT is an *inner* algebra automorphism, because one writes

for $H$ the Hamiltonian. But a careful formalization for the case of quantum field theory shows that this is not so. Starting for instance with a lattice approximation of your quantum field, say a $n \times n$ lattice as for the Ising model, where a field operator $A$ as above acts on the Hilbert space $\oplus_{0 \leq i, j \leq n } \mathcal{H}_0$ for $\mathcal{H}_0$ the (finite dimensional) space of states at one one lattice site, the Hamiltonian is in typical cases of the form

$H = \sum_{i,j} H_{i,j} + interaction terms$being the sum of Hamiltonians acting on each lattice site, and similarly some interaction terms relating neighbouring lattice sites.

So for finite $n$ here, time evolution is indeed an inner automorphism.

However, as we scale this to the limit of infinite size, this changes: the correct algebra to take in the limit has as elements arbitrary but finite formal sums of operators on the lattice sites. So in the limit the Hamiltonian is no longer in fact an element of the algebra. And hence the Schrödinger/Heisenberg time evolution equation becomes an outer automorphism and as such structurally quite different from the finite-dimensional case.

]]>Hi Mike,

what the Coecke-program exploits is, at its heart, the FQFT formulation of the Schrödinger picture of quantum mechanics, where we consider state spaces given by objects in a suitable tensor category and “evolution”, hence operations on them by morphisms there.

What the Bohr-topos program is based on is the dual AQFT formulation of the Heisenberg picture, where one considers algebras of observables instead.

There is a crucial structural difference between finite and infinite degrees of freedom:

In the FQFT picture this has been made most explicit maybe by Stolz-Teichner in their program. Their work is today probably still the only one that systemtically considers FQFT in the non-topological case, the one of actual physical interest. They find that the asymmetry introduced in the infinite-dimensional case, where there is an evaluation map $V \otimes V^* \to k$ but no longer a unit map $k \to V \otimes V^*$ is a genuine structural property and not something to be discussed away. More recently, following an observation (yet another observation) by Segal they have refined this further, finding that the formalism of rigged Hilbert spaces, which is the one relevant in physics (that’s where the Dirac deltas etc live) is actually induced from the representation theory of such cobordism categories without full duals.

In the dual AQFT picture one can similarly see in the 2d case nicely that the “finite dimensional” case of algebras of observables of type II von Neumann factors is uninteresting, with all the interesting physics only appearing in the “infinite dimensional” case of type III factors.

]]>in the Coecke program state spaces are finite dimensional.

This is an interesting question! My impression has generally been that when physicists work with infinite dimensional spaces, they generally like to pretend that they are finite dimensional as much as possible, or at least to do things with them that are only formally justifiable for finite-dimensional spaces, like use delta-functions as an orthonormal basis or integrate over spaces of paths. So I had sort of internalized a model where physicists are formally using very large finite-dimensional spaces (maybe space and time are discrete at the planck scale), but approximate them by infinite-dimensional spaces because continuous calculus is a super convenient tool.

these are not alternatives, but are complementary

Can you expand on that?

]]>the infinity of the universe

This has nothing to do with the infinite extension, or not, of the universe!

The space of states already of a single particle on the interval $[0,1]$ is not finite dimensional.

The only finite dimensional state spaces are those of topological theories and tensor factors of infinite-dimensional spaces that reflect discrete properties. For instance the space of states of the electron on the interval is the above infinite dimensional space tensored with $\mathbb{C}^2$. For quantum computation one uses only that finite $\mathbb{C}^2$-factor and ignores the rest. This is good for computation, but not for reflecting the fundamental nature of reality.

hear more of your thoughts on this

I haven’t looked at the articles yet that you mention above. On the issue of the Coecke-school approach compared to the Bohr topos-approach: these are not alternatives, but are complementary. I am not really into quantum logic, so I can’t tell you which one you’d want if you are interested in that. I do know however that the Bohr-topos approach can describe actual quantum field theory, a student of mine discussed that here.

]]>