Trying to collect my thoughts...
Solution.
One of the things I was exploring in Python was about how I used the adjacency matrix to develop my presentation. I started wondering if there was a connection between my
ones_vec and the universal quantifier. Since each node in the graph corresponds to a one-hot vector, I thought maybe this vector could stand in for “all points”.One of the interesting things I observed is that the
ones_vec took twelve steps for \(Y^\beta\) to repeat but only five steps for \(X^\alpha\text{.}\) This was strange because I expected the product of a two-cycle and three-cycle to take a minimum of six steps. I think the reason this is shorter has to do with how multiplication by my ones_vec contains less information than the exponents of the matrix itself.The other idea I found myself playing with was the idea that my previous bijection \(\mathbb{N} \leftrightarrows \mathbb{N} \times \mathbb{N}\text{,}\) using a diagonal based ordering, wasn’t necessarily “unique”. By using the uniqueness of prime factorizations, however, it should be possible to build a unique map \(\mathbb{N} \leftrightarrows \mathbb{N}^{m}\) for arbitrarily large \(m\text{.}\) Using factors of two, I was able to build a second bijection \(\mathbb{N} \leftrightarrows \mathbb{N} \times \mathbb{N}\text{.}\) The only troublesome part of this process was getting the domain and codomain to align properly.
With these bijections \(\mathbb{N} \leftrightarrows \mathbb{N} \times \mathbb{N}\) comes this idea of “uplifting an endomap”. My idea is that if I can enumerate the points of \(X\) with \(n \in \mathbb{N}\text{,}\) then each arrow of the endomap \(X^\alpha\) can be mapped to a number in \(\mathbb{N}\) that expresses the source-target relationship in \(\mathbb{N} \times \mathbb{N}\text{.}\) This only works on endomaps because of the one-to-one relationship, but maybe I can use the prime factorization to extend this idea to an arbitrary graph somehow.
I’ve also found myself thinking about the gender problems again this week. In particular, I’m wondering if the real reason for the use of the word “gender” is really just a surface level similarity between the chromosome pairs
XX and XY with maps defined over domain and codomains specifed by \(X \rightarrow X\) and \(X \rightarrow Y\text{.}\) This is consistent with my hypotheses about my two missing maps from Session 15 being the result of improperly accounting for the case where \(X\) and \(Y\) are potentially the same object.In the same vein, I’m also wondering if I’ve been thinking about the binary relationship of \(B_1,B_2\) wrong. I’ve been thinking of these two objects like \(\{\text{true},\text{false}\}\) values, but maybe the binary operation I should be paying attention to is “set membership” itself. If I have some map \(X \xrightarrow{f} Y\text{,}\) maybe there’s some way to check if \(Y \subset X\) using the properties of the sum \(X+Y\text{?}\)
