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wrote about why I like algebra over analysis
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title = 'Rank-Select' | ||
date = 2018-10-16T17:01:00+05:30 | ||
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title = "This Month in Simplexhc" | ||
date = "2017-12-27T17:44:09+05:30" | ||
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title = "Why-I-Like-Algebra" | ||
title = "Why I like algebra over analysis" | ||
date = "2018-09-20T02:41:47+05:30" | ||
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Midnight discussions with Arjun P. | ||
Midnight discussions with my room-mate | ||
[Arjun P](https://researchweb.iiit.ac.in/~arjun.p/). | ||
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Algorithm vs data structure | ||
A proof technique is like an algorith, 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. Meditating on a proof technique is much harder, as is | ||
seeing if a proof techique can be applied in different places. | ||
"Music is art in time. Art is music in space". | ||
This tries to explore what it is about algebra that I find appealing. | ||
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On the other hand, this is probably pointless in combinatorics, because | ||
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 is the probabilistic method. A | ||
more reasonable example is that of the Pigeonhole principle, which still | ||
requires insight to instantiate in practise. | ||
I think the fundamental difference to me comes down to flavour --- | ||
analysis and combinatorial objects feel very "algorithm", while Algebra feels | ||
"data structure". | ||
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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 :) | ||
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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. | ||
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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. | ||
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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. | ||
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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 | ||
practise. | ||
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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 | ||
theorem. | ||
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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_!. | ||
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". | ||
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I'm really glad that this 2 AM discussion allowed me to finally pin down | ||
why I like algebra. |