Intuition in algebra?

My algebra background: I've had 2 undergrad semesters of algebra, a reading course in Galois Theory, a graduate course in commutative algebra and one in algebraic geometry, and I've done (most of) MacLane and Birkhoff's Algebra on my own.

The problem is that I feel like I still don't have any idea how to do algebra. I do well in my classes and don't have any problem with most exercises in M&B I do. But my process consists pretty much entirely of fiddling around with symbols until I figure out how to apply theorems I know in a completely straightforward way. I'm able to do exercises from a book, but rarely able to prove theorems in the text on my own.

This is a complete contrast to how I feel in topology and analysis (not that I really know any topology or analysis), where I have a halfway decent intuition and really think I know why things are true (and moreover, why anyone should care). I'm able to prove theorems.

In topology and analysis, I am able to visualize things pretty directly in a way that I can get insight into how things work. For algebra, I have no picture. I've tried learning about Cayley graphs to visualize groups. I think these are neat, but I have yet to successfully apply any insight from them. I hoped learning about algebraic geometry would help me visualize rings. But the geometry in algebraic geometry is dictated entirely by the algebra. So how can you use geometry to help you with the algebra, when you have to do the algebra first to figure out what the geometry looks like? I don't get it.

So the question I'm trying to get at is: How do I develop some insight or intuition about algebra? I don't really know what form answers might take; maybe a reading suggestion, or just a general way to look at things. Maybe this isn't a good question, but I'm kind of at the end of my rope with this stuff.

A particular user on MathOverflow said he fell in love with algebra the first time he saw the axioms for a group. When I first saw the axioms for a group, I spent the next year trying to figure out why the heck anybody cared about groups (and frankly still only know this in a detached and academic way). So it's possible the only answer is: I'm barking up the wrong tree; algebra isn't for me and I should move on to something that comes more naturally.

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Have you seen terrytao.wordpress.com/2010/07/10/… ? – Qiaochu Yuan Dec 11 '10 at 22:50
Yes I have, and I think it's great, thanks. I've spent some time today revisiting Cayley graphs and trying to firm up my ideas here, and I think I'm actually making a little headway. – Mike Benfield Dec 12 '10 at 0:43

But the geometry in algebraic geometry is dictated entirely by the algebra. So how can you use geometry to help you with the algebra, when you have to do the algebra first to figure out what the geometry looks like? I don't get it.

Roughly speaking, geometric intuition suggests to you what should be true, and suggests a strategy for proving it; and algebra is how you carry out the proofs. For example, it's geometry that should inform your intuitions about things like intersections of varieties, but algebra that you use to actually rigorously define intersections and prove things about them.

Consider the "principle of continuity." If you intersect a circle and an ellipse, you'll generically get four points, although sometimes you might get two or one. If you intersect a circle and a parabola the same thing happens. In a perfect world, you might suspect that the intersection of two conic sections is always "four points," but this clearly isn't true for the usual definition of "four points." But if you broaden your definitions (count the points with multiplicity; count the complex points; count the points at infinity), you will eventually be led to complex projective varieties, where something like the "principle of continuity" holds: if you have two varieties that intersect in $m$ points and you nudge one of them or the other continuously, they will still intersect in $m$ points, if you count the points properly.

It's important to keep in mind that historically a major incentive for developing commutative algebra was to rigorize parts of algebraic geometry; some algebraic geometers (the Italian school) had begun relying too heavily on geometric intuition and had been ignoring special cases, etc. and commutative algebra was one way to fix their proofs. But the point is that they were doing algebraic geometry first! There is a long and interesting history here which I think it is very instructive to learn; you should try to find Dieudonne's History of Algebraic Geometry, as well as (for a personal perspective) Parikh's The Unreal Life of Oscar Zariski.

When I first saw the axioms for a group, I spent the next year trying to figure out why the heck anybody cared about groups (and frankly still only know this in a detached and academic way).

The group axioms are an abstraction of the notion of symmetry, and symmetry is a natural and beautiful idea, and symmetries are everywhere in mathematics. Perhaps you aren't acquainted with enough examples; it's hard for me to give more specific advice here without knowing what you find unsatisfying about groups. But you might be interested in Conway, Burgiel, and Goodman-Strauss's The Symmetries of Things, as well as in Mumford, Series, and Wright's Indra's Pearls: the Vision of Felix Klein.

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I really, really recommend that you read the book about Zariski, by the way. It's too easy to think of definitions of algebraic objects as handed down by an inscrutable god and to forget that they were forged by human mathematicians to make sense of problems. Even if the historical route to a concept isn't the most efficient, it's still valuable to know and gives it a human face. – Qiaochu Yuan Dec 12 '10 at 1:53
Great post! Let me second Indra's Pearls. It is impossible not to like it and close to impossible to fail to appreciate, what a natural geometric concept a group is. If you are reasonably efficient at coding, you will enjoy it even more, but even if you aren't, you should read it! – Alex B. Dec 12 '10 at 2:17
Dear Qiaochu, I think one should be careful in arguing that commutative algebra is the way to attain rigour in algebraic geometry. It is one way, but there were plenty of correct (and correctly proved) theorems in algebraic geometry prior to Zariski and Weil and Chevalley's (and others') algebraization of the theory, just as there are plenty of modern theorems which are proved by analytic methods. – Matt E Dec 12 '10 at 2:47
@Matt: I only meant to indicate what I know of the historical development of ideas. I'll edit the wording. – Qiaochu Yuan Dec 12 '10 at 2:50
Dear Qiaochu, Thanks for your reply. I'm sorry if I sounded overly critical; this happens to be a touchy point with me. While I personally love the algebra associated with algebraic geometry, I like to remember that it is a branch of geometry, not algebra! Incidentally, I strongly agree with your first comment above. (And I agree with your assessment of The unreal life --- a great book about one of the truly great mathematicians.) – Matt E Dec 12 '10 at 3:15

A lot of intuition about abstract algebra and commutative algebra can be developed by visualizing by means of commutative diagrams.

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One suggestion is that you learn some algebraic geometry from a different perspective, and, having done so, return to commutative algebra and see if you have more insight by virtue of the intuition in algebraic geometry that you (hopefully) will have developed.

You wrote in your question that "the geometry in algebraic geometry is dictated entirely by the algebra", but this need not be true (and certainly is not true historically). If you find it is this way because of the particular books you've been reading, I suggest that you try some other books.

Assuming that you are a graduate student (based on your posts), you might try Mumford's book on complex projective varieties. (I forget the precise title, but it won't be hard to find.) It is quite geometric. You could also try looking at Mumford's Red Book, which tries to unify the geometry and algebra. There are other books, too: Griffith's and Harris is one, although it requires a reasonably strong background in differential and algebraic topology.

As I've tried to emphasize in some other posts of mine here and on MO, algebraic geometry needn't be done via commutative algebra at all; some of the most important recent work in algebraic geometry has relied almost entirely on analytic methods. There are also topological approaches, via Morse theory and its generalizations and variations. (Lefschetz, who was one of the founders of algebraic topology, was motivated by the applications to algebraic geometry.)

Of course, you needn't focus your study on algebraic geometry at all: I am mentioning it mainly because it came up in your post, and it is an area of mathematics that is very central, and very multifaceted, so that there are lots of ways to approach it, which in turn makes it a good way to build up intuition in lots of different areas of mathematics. (In your case I am suggesting it as a way to build up intuition in algebra; for myself, algebra was always one of my stronger suits, but through studying algebraic geometry I was able to strengthen my geometric and topological intuition.)

Depending on what other answers appear here, I might write some more later which connects with some of other aspects of your post.

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It seems you have a kind of geometrical intuition. If that's the case, why don't you construct geometrical representations of the algebraic entities you encounter? Have you had any representation theory till now? If not, that might help you figure out why algebra is "important" or why anyone, even outside the field of algebra, cares.

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You could try looking at axiomatic planar geometry for a new perspective on rings and fields. Given enough axioms of abstract planar geometry, one can construct a model of that geometry as $R^2$ for some ring $R$ (it's the same thing as happens when we construct vectors out of the affine plane). It turns out that various geometric theorems are then equivalent to algebraic statements about the ring $R$, like that it be commutative, or finite, or a field, etc. For example, if I remember correctly, then Desargue's theorem holds iff the ring $R$ is a division ring. I don't have references for this at hand, but I can look some up at work next week.

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I have one small insight just for finite groups which works for my own personal intuition. Since you seem to feel like topology and analysis make sense and are well motivated, then why not think about the elements of a finite group as points on a continuous manifold, in other words, represent the elements of a finite group using $N\times N$ matricies.

Lie groups are continuous manifolds with a group structure, and they are manifestly topological. Furthermore, a finite subgroup of $GL(\mathbb{R}^N)$ is, by definition, a representation of some finite group. So the elements of a finite group are represented by points on a manifold associated with some subgroup of $GL(\mathbb{R}^N)$, and these elements are connected to each other by continuous paths.

By the way, if anyone is looking for a good introduction to linear groups and representation theory, I highly recommend B. Hall.

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This may be a bit fundamental, but i found that Exploring Abtract Alegebra Package for Mathematica had some nice visualizations that helped me develop an intuitive understanding.

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