WHY DOES THE INSTITUTE CARE ABOUT CATEGORY THEORY IN MATHEMATICS?
Because it’s the closest to a data representation of the neural relations in language. That said if we wrote in math or even symbols we’d find fewer followers than we do with operational language. 😉
Via Michael:
Category theory was created in the 1950s to half-resolve the problems of set theory (axiom of choice, axiom of infninity) early on in 20th century. (CD: see my work on intuitionism and operationalism)
For Mathematics
https://t.co/CgDTImh50e
For Programmers
https://t.co/Tv1pRl5VDM
The core insight of category theory is that most math structures “are the same in some way” (CD: There exists a pattern of consistent relations between them.)
If you have these FOUR structures, you’re a Category
1. Every category starts as a SET
2. Then that SET gets MORPHISMS
(state transitions, references, maps; not quite “operations” because many math state transitions are maps not operations) (CD: unlike applied mathematics, pure mathematics can drop the time and sequence dimensions and transform instantly, so both dimensions are irrelevant.)
3. There must be an IDENTITY operation (a return to self map) (CD: reversibility. no information loss.)
4. Maps must COMPOSE. (The formal terminology is that the associative law holds) (CD: in our work composability refers to fitness within the grammatical rules of disambiguation.)
Examples:
– Sets are categories, trivial ones
– Groups
– Rings
– Topological spaces
– Vector spaces (the objects are basis vectors, the morphisms are matrices that change basis)
What results when you climb to the top of category mountain?
You start with large chalkboards full of dots and arrows between them. (Directed graphs, or “Quills”).
The graphs (their vertices as objects, and arrows as maps) form higher dimensional shapes (triangles, squares, pentagons, “associahedrons”). (cd: I love the term “associahedrons” because this is what operational LANGUAGE produces. And this is why I originally started the foundations course with geometry before I realized it was too complex for the students.)
When you take a FUNCTOR (which maps one entire Category to another Category), you’re now drawing NEW arrows, one for each functorial pair. (CD:A set of transformations)
What happens is that if you follow these lines, the higher dimensional subshapes you found earlier MAY be preserved in some form as well.
So if a square structure the original category becomes a triangle in the new category: well, that means something. (Means we lost structure when applying the functor. The functor is called “forgetful”). So category B is the same as A, minus a few structures that B has
As such…
You get to compare environments (categories) in which to “do math”.