Epistemology
Introduction E9F5FC Questions FFFFC0 Software 
See: Yoneda lemma, Yoneda lemma slides, Foursome, Math notebook, Category theory, Adjunction, Walks, Univalence axiom
Writing up my understanding of Yoneda's lemma in terms of the four levels of knowledge: Whether, What, How, Why.
Understanding
Knowing is understanding scope. Knowing is conditional understanding which varies in scope. The four levels of knowledge are four levels of separating scope, which is to say, understanding scope.
Natural Isomorphism Consequences Example: Preorders Robert Harper: I find it helpful to consider a degenerate form of the Yoneda Lemma, for preorders, in which case it says that {$x\leq y$} iff {$(\forall z)\, z\leq x \Rightarrow z\leq y.$} From left to right use transitivity of the preorder; from right to left, instantiate z to x, and use reflexivity. The full form of the Lemma is a generalization of this simple, but surprisingly useful, observation. {$A\leq B$} iff {$(\forall C)\, C\leq A \Rightarrow C\leq B.$} Example: Cayley's theorem The Fundamental Theorem of Computation
Automata Hierarchy
{$\frac{1}{r^2}$} laws The {$\frac{1}{r^2}$} laws are said to arise from
Entropy
Global quantum Calculate on the basis of the global quantum
Experiment and measurement An experimental measurement is a very special situation because it is extreme by design. It is connecting the global quantum with the local quantum. Thus it is opening up a world in the classical domain where different resolutions are entertainable. Building knowledge Foursome  Yoneda lemma
Universality Quanta magazine has published a series of interesting articles about the https://www.quantamagazine.org/search?q[s]=universality  universality principle in statistics, and also the TracyWidom distribution including this short video. I happened to learn about this just after my interpretation of the Yoneda lemma. I will explain the possible link that I perceive between the two. The Yoneda Lemma establishes a natural isomorphism which says that we can look at functors from a category (C) to (Set) in two different ways. In my understanding, on the one hand, we can think of them as a calculation taking place in its own system, in which case we just look at the network of morphisms. On the other hand, we can think of them as a calculation taking place within a subsystem of a larger system, in which case we need to have initial states and final states, which is to say, we need to have objects. What is the value of having two points of view when they are naturally isomorphic? Well, apparently, those different points of view can matter, as follows. From the network of morphisms point of view, the network grows as (n2), whereas from the systems and subsystems of objects point of view, the network grows as (n). Simply put, the number of possible binary relationships is quadratic to the number of states or nodes or objects. So as a system grows, it can be burdensome to maintain the quadratic growth, and if at all possible, one might shift perspectives to express the same system in terms of linear growth. Which is to say, if we have a cost to our perspectives, then we may shift them, yielding a phase transition. This is to say that the two points of view may have real implications. In other words, they may be real. Of course, cognitively, they are real, but they may be real not just cognitively. That makes our thinking about these things all the more interesting. It seems possible that the TracyWidom distribution models precisely such a shift. On the one side it goes down (e−N) and on the other side it goes down (e−N2), and apparently it models this phase transition. The asymmetry of the statistical curve reflects the nature of the two phases. Because of mutual interactions between the components, the energy of the system in the strongcoupling phase on the left is proportional to ( N^2 ). Meanwhile, in the weakcoupling phase on the right, the energy depends only on the number of individual components, ( N ). “Whenever you have a strongly coupled phase and a weakly coupled phase, TracyWidom is the connecting crossover function between the two phases,” Majumdar said. At the Far Ends of a New Universal Law Furthermore, the identity morphisms in the Yoneda Lemma could be crucial for such transitions. Imagine a system that is a collection of interrelated actions (arrows) which satisfy a composition law, but without any objects. And imagine that the system keeps growing through the addition of new actions. The burden of keeping track of everything will grow. Now some of those actions may selfcompose, which is to say, they may loop upon themselves. And some of those selflooping actions may have no effect on any other actions, upon composition, which is to say, they are identity actions, "do nothing" actions. Then what could happen is that those identity actions could anchor and define a subsystem. What does such a definition mean? It means that a phase transition takes place, where many of those identity actions, and all the related actions, may form a coherent subsystem, disjoint from the main system, that can "drop out" of it. And their dropping out is simply a matter of a shift in perspective. They can still be thought of as in the main system, but they can also be thought of as a selfstanding subsystem which is incorporated into the main system, and they can also be thought of as their very own system. The Yoneda Lemma, in my interpretation, is the framework that allows for these shifts and ambiguities in perspective. Which is to say, the Yoneda Lemma may be precisely the conceptual framework that grounds such a shift in perspective by which a growing system comes to be made up of subsystems. As I read a bit about random matrices, which is related to this, I saw that such concepts are important in the study of topological insulators. There is a nice series of videos and lecture notes on that, Course on topology in condensed matter. I learned that "zero energy excitations" are crucial in this subject. When there are no such zero energy excitations, then certain systems aren't able to transform into other systems, and thus we get classes of systems. Well, not knowing anything, such "zero energy excitations" bring to mind the "do nothing" actions. Another physical idea in all of this is that objects and arrows are distinguished in the way that fermions and bosons are. Two objects cannot be in the same place, and the same is true for fermions. But two arrows can be in the same place, and the same is true for bosons. I mean to say that two arrows can start at the same object and end at the same object but be entirely different arrows. So a shift from an "object point of view" to an "arrow point of view" may be like a shift from "fermion statistics" to "boson statistics". These mathematical connections are very interesting to me because the topological insulators are classified by Bott periodicity, which I think is related to an eightcycle of conceptual frameworks (divisions of everything) that is central in my philosophy. Also, they relate to Dysons's Threefold Way, symmetry classes of random matrices (complex Hermitian, real symmetric, or quaternion selfdual) that I think relate to the classical Lie groups, which I think are related to the cognitive frameworks that I study, including the foursome: WhetherWhatHowWhy. I think the Yoneda Lemma is modeling the relationship between a system and a subsystem. That is an asymmetric relationship and so it involves an asymmetric phase transition. From my studies of the ways of figuring things out in math and physics, there is what happens before we have a system, and what happens once we have a system. And at the crucial point in between, we have in physics the establishment of a subsystem within a system (as with Faraday's pail). And in math, we have a symmetry group. Well, the Yoneda Lemma is a generalization of Cayley's theorem, whereby every group can be thought of as a permutation group. And here we see the alternatives  we can keep track of a group using (n×n) matrices, or we can allow it to keep track of itself, however it likes, using its own (n) elements. The TracyWidom distribution deals with the largest eigenvalue for a random matrix. The universality principle is an apparently related but slightly different matter of the spacing between all of the eigenvalues. I just want to add that such a spacing also seems like a very important matter in emerging systems. In our universe, it is surprising and very important for physicists that the forces in nature have their effects at different orders of magnitude, because otherwise it would be impossible for us to tease them apart and make sense of any of it. Similarly, architect Christopher Alexander, in his 15 principles of life, notes the importance of Levels of Scale, typically of size 3, which are crucial so that we have gaps between the levels. Apparently, these could be mechanisms for assuring such gaps in levels of scale. I want to conclude with the following idea from Silicon Valley on the value of networks. A single fax machine is worthless. It becomes valuable when there is a second fax machine. And with each new fax machine, the value of the network grows by that machine's possible relations with all the other machines. Which is to say, the value of the network grows quadratically. Of course, the flipside is that the cost of a network can grow quadratically, and that can drive the appearance of subsystems within the system. Learning from Videos Bartosz video
Proof of Yoneda lemma
