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

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

Geometry is important, but I think it is built up from quantum entanglement. Of course there are a number of papers in this contest which purport that Bell's theorem, entanglement and the quantum physics of nonlocal correlations is wrong. One is wise to not bite on this. Quantum mechanics is one of the most experimentally tested areas of physics. So far all of the strange consequences of quantum physics hold up, this is even if they appear to be so utterly bizarre and upside down.

Quantum states entangle with a black hole. The observer on the outside easily loses information on the exact black hole horizon state her part of the EPR pair is entangled with. This is a form of entropy; the entropy illustrates the lack of knowledge. The horizon area of the black hole is then a direct measure of entropy, where any quantum bit in our outside world entangled with the black hole is entangled with a Planck unit of area on the stretched horizon. It is difficult to localize that of course. If one tries to find which region of the horizon your EPR part of the pair is entangled with the Heisenberg microscope argument tells us the other part of the EPR pair will be sent into a huge uncertainty in position. Hence one is not able to recover this information.

The horizon of a black hole is then built up from entanglement. Further, the null boundaries of spacetime contain the holographic information of the entire region. We then have the physics of space or spacetime really being built up from entanglement of quantum states. Geometry, at least geometry used to model physics, is then an emergent property.

LC

As I remember Penrose and his triangle in "Road to Reality" had consciousness or mind with physical reality and mathematics as a triality. The I recall that he had mathematics as the foundation of physics, and physics the foundation of mind, and to complete the cycle mind gave conscious recognition to mathematics.

Platonism is not something I take that strongly. I will put on the "hat of Platonism" when it suits me, and at other times I will not. I discuss largely the aspect of mathematics that I think has "meat," while a lot of mathematics involving infinity and infinitesimals is what I call "soul." I am not out to deny the "soul," but I do think the "meat" has a more direct connection with physical reality.

LC

Thanks for the positive word. I wrote the bulk of this essay up in a single day. I spent another day correcting it. I tried to keep the mathematics somewhat "physical," in that the discussion on incompleteness and numbers is oriented towards what one might actually encounter in computing these things.

I read your essay early on, and as with most essays I have read I have yet to assign a score to it. So far you are running at the top. I will probably have to re-read yours. I have been a bit busy and unable to attend to this contest that much.

Cheers LC

The connection with the Bott periodicity of large N, or SU(N), entanglements with associahedra is with the some sort of projection of this thing. The Stasheff polytope K_5 has 14 vertices, and this might be seen as composed of two copies of 7 elements mapped to the Fano plane, with each of these "sevens" associated with a projective point "в€ћ", or as associated with eight elements by the Hopf fibration. So the associahedra might be some sort of "double cover" with the Bott periodicity on A&D type Lie groups.

I think to construct such operads one has to work more from the ground up. Spacetime as a monoid, groupoid or so called magma might lead to the sort of spacetime algebra that leads to the sort of category theory. The set of possible operations would then form these "trees" and are associated with certain homotopies and categories. My essay discusses this in an informal sense, and homotopy as a route to category theory with monad/monoid structure.

I can discuss this more tomorrow or later. I see that you have an essay as well, which I will try to get to in the coming days. These are coming in faster than I can read them.

Cheers LC