Time: A moving point is a line, a moving line is a plane, a moving plane is a volume... Time is the addition of a scalar (from a field), thus the addition of choice. Time relates affine and projective space. Compare time with space. A moving "center" is a point: the center is what moves, thus what has time.
Function can be partial, whereas a permutation maps completely.
Derangements. Interpret? Not likely...
Symmetry group relates:
Axiom of infinity - can be eliminated - it is unnecessary in "implicit math".
Consider the subsitution q=2 or otherwise introducing 2 into the expansion for Pascal's triangle to get the Pascal triangle for the cube and for the cross-polytope.
The factoring (number of simplexes n choose k - dependent simplex) x (number of flags on k - independent Euclidean) x (number of flags on n-k - independent Euclidean) = (number of flags on n)
What kind of conjugation is that?
Is the fusion of vertices in the demicube related to the fusion of edges of a square to create a torus, or of vertices to create a circle, etc.?
Symmetric group action on an octahedron is marked, 1 and -1, the octahedron itself is unmarked.
A punctured sphere may not distinguish between its inside and outside. And yet if that sphere gets stretched to an infinite plane, then it does distinguish between one side and the other.
Relate triangulated categories with representations of threesome
Kan extension - extending the domain - every concept is a Kan extension https://en.wikipedia.org/wiki/Kan_extension
Long exact sequence from short exact sequence: derived functors.
Express the link between algebra wnd analysis in terms of exact sequences and Kan extensions. Explain why category theory not relevant for analysis.
Gap between structures, within a restructuring, is a "hole", and so methods of homology should be relevant. How does cohomology relate to holes?
If we consider the complement of a topological space, what can we know about it? For example, if it is not connected, then surfaces are orientable.
Consider the connection between walks on trees and Dynkin diagrams, where the latter typically have a distinguished node (the root of the tree) from which we can imagine the tree being "perceived". There can also be double or triple perspectives.
Compare 6 math ways of figuring things out with 6 specifications. Consider how they are related to the 4 geometries. Relate the latter to 4 metalogics. Look at formulas for the 6 specifications and look for a pattern.