Mathematical Physics as Topological Numbers, Types and Quanta by Lawrence B Crowell

Hi Florin,

I am in one sense disposed to Wildberger when it comes to mathematics that is computed or that has a physical meaning. There seems to my mind there are two notions of mathematics with respect to infinity and the continuum. The pure notion, which is the standard approach to mathematics, is in some ways Platonic. I am not particularly anti-Platonic, but the problem I see with Platonism is that it largely has nothing to do with most things you actually want to know. The Platonist type of mathematics, which is embodied by ZF or standard set theory, largely exists on its own, and what is actually calculated in both mathematics and physics is some tiny part of this. I am not really that concerned about the existential aspects of set theory and what might be called Platonism, but I don't think this has what I might call a hard existence. For a thing to have a hard existence it must be computed and there must be a physical way it can be represented. A set, number, function that can't be physiccally represented has a sort of "ghostly" existence at best, and I am not out to exorcize ghosts from mathematics particularly. However, for something to have a hard existence it can't be a ghost, it must have "meat."

The "ghosts" of pure mathematics are useful in some ways, for they allow us to make various arguements so that there "deltas" and "epsilons" that fortunately cancel out and we don't have to actually produce an infinitesimal number in our hands. In that sense these can of course have a utility, but this works only when the system is such that an actual infinity, or infinitesimal, is removed from the answer.

This is in a way not that different from nonlocal hidden variables. Do nonlocal hidden variables exists? Maybe, or for that matter sure; I can arrive at a theory (in fact a vast number of them) of nonlocal hidden variable theory. However, there appears to be a serious obstruction to finding any observable consequence to any hidden variable theory. This obstruction is I think a topological property, and has correspondences in sheaf theory. Classical mechanics is map from the reals to the reals. Quantum mechanics is map from the reals, or really complex or quaterion numbers, to a discrete set of numbers corresponding to eigenvalues. Quantum physics says that we can know all sorts of stuff about those eigenvalues, but we have a very limited contact with the continuum stuff that involves waves and fields. Quantum mechanics then might be telling us much this lesson. Any hidden variable theory is then some set of functions, dynamics and so forth, that tell us how the continuum stuff maps to a discrete set of numbers. An obstruction against this appears to be at least similar to a relationship between mathematics that is "ghostly" and that which has "meat."

I am not an expert on type theory and HOTT, but I have been studying it some. I am not sure about actually using this in physics, but this seems to be a reasonable sort of mathematical foundation for the sort of mathematics that could be relevant to phyhsics in the future. It imposes no notion of infinity, but it treats types as unbounded in their cardinality. There is no fundamental limit to their size, but they must have some sort of index, similar to a homotopy or monodromy, that has to be computed --- it must be inserted into some "register" or "slot." I wrote on this topic because it seemed to be the closest thing that fit the question proposed by FQXI.

I have visited your blog site some, but have not had much time to contribute entries. Maybe I will try to be more diligent on that before long.

Cheers LC

I might take you up on your offer. I probably will not write about the FQXi essay topic, but on a subset of it involved with the Bott periodicity and the large N SU(N) for the structure of an event horizon.

I think this has some bearing on PR boxes. Quantum gravity requires that the field theory be nonlocal. Standard QFT has local field amplitudes with Wightman causality conditions, such as equal time commutators. Nonlocality occurs with the expectations of the operators over the Fock basis which gives quantum waves. Quantum gravity I think involves further violations of inequalities, which PR boxes or nonsignalling conditions are maybe capable of working with. Maybe there are bounds beyond the Tsirelson bound?

If you want I can send to you an article I wrote but have yet to submit for publication that addresses some of this.

Cheers LC

Florin,

I have been either godsmacked or kicked in the face by the mule of stupidity. Your composition approach appears to be a way to look at my large N, or large SU(N) approach to entanglement with black holes. I can send to you if you want a paper I submitted to the GRF essays. The paper I mentioned I would send to you is related to this.

I have been wondering how to get this to work with Jordan algebras, and then looking at your page reminded me of you idea here. It seems perfect. I need an address to send that to.

Cheers LC

I think you are trying to think of things according to some intrinsic "things as they are" sort of perspective. The problem is that we are in the universe, which means we must operate within it, and are thus faced with limits to observability. I think in some ways that is more of a mystical way of knowing than a scientific one.

The lack of observability or knowledge is a sort of topological obstruction. Such obstructions can in their own way tell us things.

Cheers LC

Those limitations are types of obstructions that tell us something entirely new. It also means there is a massive reduction in the number of fundamental degrees of freedom in physics. That is the valuable lesson. For those who bemoan the loss of absolute objective knowledge about the world, such as was lost with quantum mechanics, I can only say that such people have sympathies.

We are no longer in a situation where physics tells us in some intrinsic fashion what the exact nature of systems are. Physics tells us what is observable or measurable; physics does not give a complete "God's eye view" of what nature is. Certainly one problem is that we are in the universe, and we have to us the physical systems in the universe to measure physics. We are not able to make a pure observation that does not disturb a system.

LC

Complex numbers employed in physics have consequences. One is unitarity that is important in quantum physics, another is holomorphy that is important in gauge theory. Quaternions, which are hypercomplex numbers, can be used to derive Maxwell's equations. So these things are useful in making calculations.

LC

Garrison Keillor has his "Guy Noir," who "On the tenth floor of the Atlas building still seeks answers to life's persistent questions." If you have ever listened to his "Prairie Home Companion" you know this well. There are persistent questions, such as "Does God exist," that will probably never be conclusively answered.

The problem is that we transition from physics to metaphysics when we start pondering what the relationship is between mathematics and physics. Smolin has written an essay that I think rather favorably of, but his stance is naturalism, which is not something one can prove. Naturalism is a conjecture about things that is really metaphysics. Platonism, which in some ways is a bit too mystical for my tastes, is also a metaphysics.

What I outline is a mathematics that has what I tend to think of as meat or body. The pure mathematics taught and studied in mathematics departments often involves what might be called "soul." I am not out to disprove or even disavow this soul, but I do tend to think that it has a questionable applicability to modern physics. This is particularly the case with matters of infinity or the continuum. Mathematicians are mostly objectivists who consider their work to involve the discovery of "something," which is not physical. This is an appeal to the existence of this soul, or in its extreme form a Platonic reality. That is fine with me, and I may put on the cap of Platonism when it suits me, and take it off at other times. In my essay I tend to keep the cap off. My essay is largely concerned with what sort of practical aspects of mathematics are likely to impact physics in the next few decades.

I think these more metaphysical questions are not going to be answered, or answered very easily. There is Tegmark's conjecture of the MUH (mathematical universe hypothesis), where in view of Goedel's theorem he replace M with C for computation. This may be the case in some ultimate sense. However, I don't see how this can ever be empirically supported. This may amount to trying to "prove too much." This attempt to get away from dualism between mathematics and physics is a sort of monism. The debate between monism and dualism may never be resolved. The two are sort of the two sides of a Buddhist satori.

LC

Thanks for the words of encouragement. I tried to write the description of Goedel's theorem and related matters in a physical sense, and I wonder if I fell far from the mark on that.

I see that you are in the contest as well. I will try to get to your essay as soon as possible. I have been a bit unable to read many of these the last couple of weeks.

Cheers LC