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I am wondering how someone that is not exclusively interested in mathematics (but nevertheless aims to become a decent applied mathematician), but for example, a theoretical computer scientist, should go about learning higher level mathematics?

Would it be optimal to learn some interesting topics in mathematics that they think they may need ahead of time (in isolation to their field) or would it be better to just learn their field (theoretical CS for example), and dive into learning the relevant math only when encountering some topic that needs it.

I think that learning mathematics "just in time" would be the way to go, since the practicality of the subject is known beforehand and the concepts can be applied immediately, thus reinforcing the learned material. For example, if one takes 2 semesters of advanced linear algebra without practicing it much after that, a year (months even!) later most of the material is forgotten. However, when one only learns linear algebra thoroughly when he discovers that he needs it to complete a project in machine learning, he might learn the mathematics more optimally.

The question arose because I'm considering if I should just take a lot of advanced math courses this semester or concentrate on CS subjects (for example cryptography), and learn the required math individually (for example number theory) when the relevant concepts are encountered.

Thank you.

PS. Mind you, the question really isn't about learning just enough mathematics to understand the CS concepts in hand. It is more about how a computer scientist should go about learning mathematics, when he ALSO aims to become a decent mathematician.

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closed as primarily opinion-based by Clayton, anomaly, user147263, Eric Wofsey, Em. Feb 10 '16 at 4:42

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ If you want to learn math, learn math. If you don't want to take a class, don't. After a point, you (and I'm talking here about an abstract person, not you in particular) are expected to take charge of your own education. If you can't bothered to study something without a direct application spurring you onward, math probably isn't for you. I don't even know how you would learn something "more optimally" in this case; linear algebra is a basic, fundamental area of math and one with a wide swath of applications, not anything esoteric or difficult. $\endgroup$ – anomaly Feb 9 '16 at 21:47
  • $\begingroup$ Thank you for your response. I think that the optimality comes from the fact that when you need to learn some area of mathematics for some other incentive than just for the sake of mastering it, you are in some sense more exposed to it (you can put the knowledge into practice right away). In my opinion, the situation is comparable to someone learning a new language. You could in some sense rigorously learn it with a dictionary and some textbooks, or you could just start reading a lot of books you are interested in anyway in the foreign language and translate the words that you don't know. $\endgroup$ – stensootla Feb 9 '16 at 22:40
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In my experience, learning "just in time" has several advantages:

  • The leaner has the incentive to learn, seeing the near need
  • Having applications at hand, specifically applications of immediate interest to the learner, makes understanding easier
  • Learning at the hand of concrete examples avoids being led astray, concentrating on the correct way to apply the learned material
  • Learning "ahead of time" might mean that, by the time the use comes around, the material is already forgotten
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  • $\begingroup$ Thank you, I wanted to know what an experienced mathematician thinks of this approach, and you have answered my question completely :) $\endgroup$ – stensootla Feb 9 '16 at 22:41
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I expect that there are quite a few people on this site (myself included) who have earned degrees in mathematics and computer science at separate times and have experience both with the "just in time" learning and covering things far in advance of applying them.

Even if you do a complete undergraduate math degree before getting seriously into CS, I believe you would still find that you'll be learning a lot of the math you need for CS on a "just in time" basis. That was my experience, at any rate. Many years afterward I still find myself learning applicable mathematics from time to time.

On the other hand, if you're going to need much in the way of math to do your CS work, it's highly desirable to be able to teach yourself what you need to know "just in time." (Otherwise, you'll be hard pressed to learn it in time at all.) If you pick up a graduate-level math textbook, or even one for upper-level undergraduates, how difficult will it be to navigate your way through it and glean a sufficient understanding of the things you need to know? Just painstakingly difficult, or apparently-insurmountably difficult?

When you're self-teaching, there's a lot to be said for already being comfortable with the way higher-level math is presented and with the kinds of thought processes that are involved. You may pick up these things differently than I do, but I personally find that needing something in a hurry to use in a particular application is not a conducive situation for picking up on the deeper levels of thought in a mathematical paper or book.

So at some point (not necessarily this semester) it might be wise to take some math courses where you have to work out some relatively abstract material by your own effort. Exactly what courses those would be is something you'd have to discuss with an adviser at your university who knows the field and knows the nature of the courses offered there. Linear algebra, for example, could be the perfect course to take or completely the wrong course.

Besides, if you actually like any kind of math, and can find the right courses at your university, the math should be interesting for its own sake, even if you don't see an immediate payback in your CS work. (I think it's likely there would be some immediate payback, though it might not be a direct and obvious application of the math you learned.)

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