What are the differences in mental skills required to master abstract algebra and analysis??

Question

I had took undergraduate-level abstract algebra and analysis courses before. And I find I can do proofs in analysis faster than in abstract algebra. However some other students is opposite to me. I find this phenomenon is interesting.

To me when I trying to do proofs in analysis there is some kind of visualization coming first in my mind, and that aids me a lot to write down the proofs. In algebra I can barely have visualization in mind when doing proofs, so I easily get stuck in practicing this subject.

So I wonder and would like to ask your opinions: What are the differences in mental skills required to master abstract algebra and analysis?

Answer 1

There is a large body of psychological evidence that mathematical skill falls into two general categories:

• formal and abstract
• visual and geometric

Most mathematicians excel in one of these domains (e.g., Ramanujan, formal; Desargues, geometry), but a few in both (Cox).

To speak from my own experience: I am certainly a visual/geometer and can visualize curves and forms in space, count faces and vertexes of complex geometric solids in my mind, and so on, but have greater difficulty with "non-geometric" fields such as number theory.

However, when I studied Abstract algebra in the Mathematics Department at MIT (which was taught in a formal, abstract way), I started out no better than average in the class. But then I came across some books that showed how to view groups, rings, and proofs geometrically (with Cayley graphs, among other constructs). Everything became so much easier. I could visualize the proofs, understood core concepts such as right-cosets, inner automorphisms and such visually. Once I understood things visually, I could then "fill in" the rigorous formal proofs and calculations--and I did much better in the class.

I came across a book years later, Nathan Carter's Visual group theory, which was a revelation. I felt as if this book and my cognitive style were perfectly matched.

So my humble recommendation to you is to know your cognitive style and try to cast your mathematical field or domain as much as possible into that style. You'll learn faster, remember more, and enjoy it more.

Now... not all formal math can be "geometrized" easily or naturally, nor geometric math be "algebraicized" easily or naturally. But do what you can!

Answer 2

(Some of this has already been pointed out by David G.Stork)

I think a problem with how abstract algebra is taught many times is we as students lack a lot of motivation or background as to where groups come from. So we have very little idea of what the people who developed and axiomatized the field had in their mind at the the time. This can cause us to have a very limited view as to what the key ideas behind the theory are, since we know very few interesting examples of groups this can sometimes lead to a lack of inspiration for proofs.

On the other hand in real analysis we have been taught since grade school how to visualize the number line and $\mathbb R^2,\mathbb R^3$. In my case it happens fairly often that trying to find a "visual" justification for a certain phenomenon can actually develop into a proof for analysis.

To wrap things up I think that it is not that thinkers can be "visual" or "combinatorial", but simply it is easier to work with objects you are used to and for which you have a certain amount of familiarity than working with objects for which you may only know some rudimentary examples or maybe just some definitions which were given out without sufficient motivations.

Of course some students are better than others at proving theorems just from knowing the axioms, but this is probably because they have previously learned a subject the same way (just trying to deduce theorems from axioms without having proper prior motivation).