Hi, I'm on my third attempt trying to make sense of category theory.
It's a very abstract way of looking at all of math in terms of "objects"
(like sets, groups, vectors spaces, etc.), "arrows" between them (also
known as morphisms), "functors" that map categories (objects & arrows)
into other categories (objects & arrows), "natural transformations" that
map functors into functors, and so on.
I've been watching a series of 10 minute video lessons on that by the
"Catsters".
https://www.youtube.com/user/TheCatsters
They were done eight years ago by two friends, both personable. Eugenia
Chang is delightful in her over-the-top enthusiasm and dynamic humor.
I'm happy that she's really blossomed, both as a high powered category
theorist but also as a champion for popularizing math. I appreciate her
very helpful talk on the the periodic table in higher order category theory:
https://www.youtube.com/watch?v=lJGUMlgCxz8
Then I learned that she had a great 5-minute television appearance on
Stephen Colbert's Late Show:
https://www.youtube.com/watch?v=mA402F5K47o
She talked about her new book which links math and cooking:
"How to Bake Pi: An Edible Exploration of Mathematics"
https://www.amazon.com/How-Bake-Pi-Exploration-Mathematics/dp/0465097677
There's a very nice article about her in the New York Times:
http://www.nytimes.com/2016/05/03/science/eugenia-cheng-math-how-to-bake-pi.html
And she gave a nice Ted-x talk:
https://www.youtube.com/watch?v=CfdFw3hXkf0
She's currently at the Chicago Art Institute as a Resident scientist.
As for myself, I've been studying a lot of math. I'm starting to think
that behind all of math there must be a set of ways by which we create
abstractions. Basically, if we have several examples, (it's not clear
if one example is enough), then we can identify abstract qualities that
they all satisfy. Then we can discard the examples, think in terms of
the abstract qualities, and define abstract entities that satisfy them.
And then we can continue this process in various (but I think, quite
limited) ways. The upshot is that abstraction can grow and grow.
Whereas we have a belief that abstraction will help us get to the root
of unity. Instead, we have all of this math that gets generated,
increasingly abstract and difficult to follow.
It's a bit like my experience as a database developer. On the one hand,
there's a belief that if I think through a system that is general
enough, then it will serve for all possible applications. But actually
that approach tends towards "overengineering". Instead, it's probably
best to find a balance where, say, half of the energy goes into
designing the overall system and half of the energy gets spent on
dealing with exceptions.
I believe that if we thinking about what is going on here, then we can
actually describe exactly how this dilemma unfolds. And we can define it
rigorously and apply it practically by choosing an optimal point between
overengineering and underengineering. In math, that would mean finding
contexts that are not too specific and not too general. Of course, math
education can serve our entire lives or just the moment at hand, and it
is an art to balance the two.
In summary, in order to encompass all of math and discover its unity, I
think I need to watch how math discovery unfolds, especially through
abstraction, and show how that process accounts for all of math.
Andrius
Andrius Kulikauskas
m...@ms.lt
+370 607 27 665