Jetbundle - Groups, Monoids, Homomorphisms and Vibes, ...

Here's the whole blackboard:

I had to time-travel to get that picture, so please look at it!

My comment:

This is great material you're presenting, but the tech you're using is several steps back from a chalkboard or a pen and a piece of paper. I need to see what's been written to be able to refer back to definitions when you use them later, and I can't do that without rewinding the video. You've serialised a manifold isomorphic to R^3!  I have a problem with the bit right at the beginning though. It's not clear to me what is S^2 and what is Q. Clearly the points q_1 and q_2 are on S^2, and q then seems to be path of points on S^2 and t_1 and t_2 are on the real line? So the path function q picks out for each t in the interval [t_1,t_2] a single point in the general configuration space Q. So what we are trying to ascertain is whether there is some sort of canonical representation of the dynamics of the system that fixes the trajectories it can take through the configuration space Q. So this canonical state is what is called the phase of the system and it is a point in the phase space. Then for the dynamics of the system to be an objective property of the system itself (and not dependent on the description we give of its motion through some local reference space) we need to show that however we construct this phase pace, the individual steps in the construction process each preserve the accidents of the local reference space descriptions, things like the co-ordinate systems we use, their origins and the units of measurement etc. That way we can be sure that when we describe a physical in some local space, using local laboratory units and co-ordinate systems, we are able to map that local description to one in the canonical phase space and identify it with a class of such systems which are essentially the same system and ex hypothesi they will behave in the same way. So that is the sense in which we think that "Nature picks out the paths which preserve these invariants"? Personally, I think that how Nature does this is none of our goddamned business, but we should definitely be careful about how it is that we know what Nature is doing! 🙂  

See Charlotte Moser on Forecasting.

The reason he chooses a particular topology for the configuration space is that in some systems the paths are not reversible "in place". This is called hysteresis and it means that when a system moves from q_1 to q_2 in the configuration space it follows one path q, say, but when it returns it follows another, q', say, from q_2 to q_1. So the kinematics of the system are not microscopically reversible but the dynamical evolution is reversible. Then we have to ask how it is that these two paths are in any sense "the same physical state" in the phase space.

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Considerations such as these seem to be what led people to study infinity category theory. 

 

See The Category Enriched over the Category of Finite Sets, The Finitely Triangulated Manifold and the Magnitude of a Finite Category and About Logic - Interview with Dana Scott.

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