The Category Enriched over the Category of Finite Sets, The Finitely Triangulated Manifold and the Magnitude of a Finite Category

I missed this bit the first time I listened to this talk: at 9:02 The magnitude of a finite category is the Euler characteristic of its classifying space and the magnitude of the poset of simplices of a finitely triangulated manifold is the Euler characteristic of the manifold. It's quite hard to find out what the classifying space of a category is: see What Does the Classifying Space of a Category Classify? by Michael Weiss. See also Terence Tao Formalising Riemann-Stieltjes Integrals in Lean Mathlib where I wrote "... an Abstract Simplicial Complex which satisfies the Augmentation Property is a Matroid, and a Finite Simple Matroid is a Geometric Lattice. Then Simplicial sets are used to define quasi-categories, a basic notion of higher category theory. A construction analogous to that of simplicial sets can be carried out in any category, not just in the category of sets, yielding the notion of simplicial objects."

48:35 Diversity measures are measures of the entropy of probability distributions on a metric space.

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The idea I have is that in physics we abstract a manifold of events we call space-time and that this process is based on empirical observation of the physical world around us (subjective direct experience) and the interpretations of maps from that abstract manifold to our immediate surroundings. The source of these maps is nothing more than our interpretation of the symbolic descriptions of the maps of others, and the result of that process of interpretation is a new symbolic description of a map from the abstract manifold of space-time for others to interpret. So in the process of doing this our direct experience is conditioned by the knowledge we have (or which we think we have) of the universal physical laws. The manifold we call space-time is a finite partially ordered set of discrete events, each of which must have some substantive physical record in the immediate experience of some conscious subject who can communicate a symbolic representation of that event to some other conscious subject capable of interpreting it as a substantive physical record of that event.

For example: there was an event which occurred around 1689 when Isaac Newton wrote Newton's Scholium on Time, Space, Place and Motion in which he tried to explain this problem. As I write I only know of this because there is in my immediate experience a representation of the Stanford Encyclopedia of Philosophy page written by Robert Rynasiewicz, which claims it is based on a 1934 publication by rev. Florian Cajori, Berkeley, CA of the text translated from Latin by Andrew Motte in 1729. But neither my location, nor that of the reader of this sentence, relative to Newton's in 1689 are relevant because these are all abstract representations of the event. In that sense the events in space-time thus represented are absolute.

The same holds for the events in the absolute space Newton described with his equations of motion. These events concerning the dropping of objects and their subsequent motion under the gravitational force are described in terms of maps from the abstract space of a large class of actual events which Newton described in the language of mathematics, using variables to represent distances and times. The distances and times are assumed to have been measured from certain definite points established in the process of interpreting the mathematical expression of the laws as they apply to some definite material situation which is the direct subjective experience of the person carrying out the experiment Newton describes. So now we can characterise a universal physical law as one which is in some sense natural and arises from the translations between maps from the space of direct subjective experience to the abstract space-time of events described by those maps. The naturality is that the translations between maps preserve the invariants of the abstract manifold. And what are the invariants of the manifold? Well they are just the things translations between maps preserve! We simply can't do any better than that if we insist that our knowledge be based on empirical observation. But I think in reality very few scientists are so insistent, otherwise they would not be so surprised at the "unreasonable effectiveness of mathematics in the physical sciences". See Two Talks on Cosmology.

I wrote above that our direct experience is conditioned by the knowledge we have of the universal physical laws. This is because as conscious rational agents we act on the physical world to bring about events and it is our knowledge of the universal laws that gives our actions conscious intent: we act with intent just because we believe that such actions will inevitably produce outcomes according to universal laws. If we are rational agents then we also observe the consequences of our actions and when they are different to the ones we intended we can learn something from the experience if we understand the specific conditions which lead to an outcome different from that we were led to expect by the universal laws. So the state of our collective knowledge inevitably has effects on how we experience the world, and in fact much of what we experience is in some way a consequence of what we know.

See Can Nano-materials Push off the Vacuum?.

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But it is not Humanity alone who has this ability to abstract regularity from abstraction. It is in a sense the closest idea we have of an ultimate universal law, and if so, then it will be observed everywhere in Nature.