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Resource for the mappings between areas of math and their applications?

by elityre1 min read30th Mar 20203 comments

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One of the things that I want to do while I'm in self-quarantine is to learn more math.

One thing that would be really helpful for me is a mapping between areas of math and the technical subfields that use that math.

I'd want to be able to go in both directions.

From math to application: It's been recommended to me (and it fits my experience so far), that whenever I'm learning new math, I should learn an application of that math in parallel. For instance, when studying calculus, study mechanics at the same time. The applied field gives motivation and grounding to the math. It would be really good if I could take any given math course, and quickly get a list of all the fields that make use of it.

From technical field to mathematical prerequisite: There are lots of bits of science that I am interested in getting a technical understanding of (areas of neuroscience, and psychology, and machine learning, and physics, and economics). But very often I don't know where to start. I know that I am missing some of the prerequisites that these field's methods depend on. But I don't usually know what those prerequisites are. It would be helpful if I could take an area that I am interested in, and quickly backtrack to the math it is using.

Is there any existing resource that captures this information?

If not, how could I go about this, in each case?

Thanks,

Eli

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My sense of this mapping is that it only exists for a small subset of mathematics. It might be the case that you're only interested in learning math insofar as it has practical applications (relevant to some technical field), but most higher-level mathematics (topology, analysis, algebra, logic) have an unclear relation to technical fields, i.e. there are fields that use concepts drawn from those subject areas but have no 1-1 connection as there might be for mechanics/calculus.

I'm not aware of any such mapping, but just off the top of my head:

economics: calculus, optimization problems, probability theory

computer science: computability theory, complexity theory

physics: calculus, differential equations

It might also be useful to go through a course catalog to familiarize yourself with the existence of various technical fields/mathematics. You might be able to explicitly trace dependencies to figure out what the connected fields are, but I don't expect this to work in general. Caltech's course catalog can be located: catalog.caltech.edu/documents/3199/caltech_catalog-1819.pdf

My sense is that if you want to learn math so you can do stuff with it, you should learn slightly more than the minimum amount required to be able to do what you want. Anything else is probably wasted effort. I consider most of the value of the amount of math I've learned to be in making my brain into a different shape and giving me more abstractions than any concrete knowledge I now possess; it's not clear to me what I can do now that I couldn't do before, but it is clear that I can think about problems from more angles.

If it is interesting to you, the level hierarchy for mathematics in my brain looks something like:

Level 1: calculus, probability/statistics, linear algebra, number theory

Level 1.5: differential equations [technically a subfield of calculus, but it's usually split off because it involves a bunch of techniques] [aka an entire class on "how to solve fewer differential equations using more time than Mathematica"]

Level 2: abstract algebra, analysis [with probability as a subfield], topology, logic + intuitive set theory (model theory?) [technically, this contains computability theory and complexity theory as a subfield, but it's pretty independent] [technically, category theory also lives here]

Level 3: algebraic topology, algebraic number theory, descriptive set theory, even abstracter algebra, topology but like weirder, logic but you chart the implications of various assumptions in excruciating detail, <things that combine 2 or more things for level 2>, <things that build on level 2 things more deeply>

Level 4: this is too far above my current level to even figure out what's happening. I have a fairly strong belief that the value of math starts to rapidly diminish the higher the "level" and that this level is almost completely aesthetic.

This was a helpful answer. Thank you!

I will go against the advice that you were offered. Especially early on I think trying to understand applications can be a bit of a trap. Either the application is so simple it can be explained without the math[e.g. twisting a factory band into a Mobius strip to make the band wear on both sides, square-cube law, logistic curves in epidemics] or the details are actually quite complicated, which may obscure one's understanding of what is the actual generalizable math concept and what is specific to this problem.

The prototypical application of calculus is Newton's work on astronomy & mechanics. This is a typical case of the latter.

That said, I suppose you've heard of

[1] 3Blue1Brown https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw