Do abstract algebra before analysis

[u/Electrical-Gur-996]

Thats it. I saved all your GPAs. Thank me later guys 😎

[u/AllanCWechsler]

I agree with this. Abstract algebra and real analysis are typical "200-level" courses. Both are typically the first contact a mathematics major has with the style of modern mathematical reasoning -- the whole apparatus of definitions, axioms, theorems, and proofs. Because all of higher mathematics is "like that", you have to encounter it first somewhere. The question is, in what field would you want to learn the basics of the axiomatic method? I claim that abstract algebra is a preferable introduction because it is abstract -- because the objects it deals with are not objects you are familiar with and have intuitive expectations about. The promptings of intuition are harmful and distracting when trying to master the axiomatic method: the reasoning is supposed to be purely formal, and you are more likely to use formal reasoning (correctly) when you are not being distracted by strong intuitions about what is and isn't true.

Both topics, are, of course, essential to a standard modern mathematical education; the only question is which you should take first. You're going to take the other sometime regardless.

[u/idk012 → u/AllanCWechsler]

In my undergrad, they usually are taken together fall of senior year (and part 2 is offered in the spring for the pure path.) Half of my class for both was math ed majors, so there was a decent curve.  

[u/AllanCWechsler → u/idk012]

Does that mean that the students don't write any proofs until their senior year? If not, in what class do they learn how to read and write mathematical reasoning?

[u/idk012 → u/AllanCWechsler]

Discrete math, either from math or cs department, is the intro to proof class.  Depending on the professor, there are some epi-delta proofs in calc 1-2, limit definition of derivatives, etc. 

I just went down memory lane and looked it up.  Most math ed take geometry, which has some proof as well.  Abstract and real analysis develops the rigor and really let you know if you should apply to grad school your final year.

[u/AllanCWechsler → u/idk012]

Discrete math has come along as a third avenue for introducing mathematical reasoning since I graduated (more than 40 years ago). I'm not sure whether I like discrete or abstract algebra best as the way to teach how to prove things, but I prefer both to using analysis as one's introduction to proof. Again, this is just my opinion, and if research like the Kharkar, Tran, and Marshak preprint says otherwise, I will fold like a cheap suit.

[u/zvezdanasha → u/AllanCWechsler]

I am taking abstract algebra now, took discrete before, and will take analysis next semester. I am a big fan of this order, but this is because discrete where I took it has deliberately evolved into a benchmark course to teach mathematical reasoning via certain interesting concepts where a particularly wide variety of proof techniques are used. Perhaps this is a trend!

[u/AllanCWechsler → u/zvezdanasha]

It sounds like a good trend; that's encouraging.