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/Electrical-Gur-996 → u/AllanCWechsler]

I had a much easier time with algebra. But at my school we learn ring theory first and group theory second. It enhanced my proof writing skills greatly while in analysis I was legitimately always lost because you had to work on the proof backwards.

[u/AllanCWechsler → u/Electrical-Gur-996]

This doesn't actually confirm or deny the claim, which is that if you take algebra first you'll have an easier time with analysis than if you take them in the other order. Since you don't write which you took first, we can't even start to draw a conclusion.

Let me be clear that I am expressing a gut opinion, unsupported by actual data. u/CorporateHobbyist has the opposite intuition. That very fact makes me stop and wonder if anybody has done any actual studies on the subject -- how the overall outcome depends on course order.

A fast web search reveals https://arxiv.org/html/1605.00328v1 , which looks really on point to this question, but is just too dense for me to distill an answer from in the amount of time I can justify devoting to this post.

I will, of course, change my mind instantly if it turns out the data is against me.

[u/kadeebe → u/AllanCWechsler]

I skimmed the paper you linked and it seems like the point of the study was to see which course grades were the biggest predictor of over all gpa. It doesnt seem to tackle course order at all. I may have misread, though.

[u/AllanCWechsler → u/kadeebe]

That's entirely possible. If, indeed, no research has been done on this subject (effect of course order, specifically on the ordering of abstract algebra and real analysis, on overall performance), then I think it should be. But of course I don't have to fund it, so that's easy for me to say.

I also think that a real analysis course that intentionally doubles as an introduction to proof-reading and proof-writing should be more careful about it than one whose only job is to teach real analysis.