§ Why I like algebra over analysis
Midnight discussions with my room-mate
This tries to explore what it is about algebra that I find appealing.
I think the fundamental difference to me comes down to flavour ---
analysis and combinatorial objects feel very "algorithm", while Algebra feels
To expand on the analogy, a proof technique is like an algorithm, while an
algebraic object is like a data structure. The existence of an algebraic object
allows us to "meditate" on the proof technique as a separate object that does
not move through time. This allows us to "get to know" the algebraic object,
independent of how it's used. So, at least for me, I have a richness of
feeling when it comes to algebra that just doesn't shine through with analysis.
The one exception maybe reading something like "by compactness", which has
been hammered into me by exercises from Munkres :)
Meditating on a proof technique is much harder, since the proof technique
is necessarily intertwined with the problem, unlike a data structure which
to some degree has an independent existence.
This reminds me of the quote: "“Art is how we decorate space;
Music is how we decorate time.”. I'm not sure how to draw out the
tenuous connection I feel, but it's there.
Arjun comes from a background of combinatorics, and my understanding of his
perspective is that each proof is a technique unto itself. Or, perhaps
instantiating the technique for each proof is difficult enough that abstracting
it out is not useful enough in the first place.
A good example of a proof technique that got studied on its own right in
combinatorics is the probabilistic method. A more reasonable example is that of
the Pigeonhole principle, which still requires insight to instantiate in
Not that this does not occur in algebra either, but there is something in
algebra about how just meditating on the definitions. For example,
Whitney trick that got pulled out of the proof of the Whitney embedding
To draw an analogy for the haskellers, it's the same joy of being able to write
down the type of a haskell function and know exactly what it does, enough that
a program can automatically derive the function (djinn). The fact that we know
the object well enough that just writing the type down allows us to infer the
program, makes it beautiful. There's something very elegant about the
minimality that algebra demands. Indeed, this calls back to another quote:
"perfection is achieved not when there is nothing more to add, but when there
is nothing left to take away".
I'm really glad that this 2 AM discussion allowed me to finally pin down
why I like algebra.