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How do mathematicians think about abstract algebra? 
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#1
Mar911, 07:35 PM

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Hi Folks.
I was hoping to pick the brains of some of the mathematicians and mathematically inclined on this site. I'm very interested in how mathematicians think about abstract objects that don't seem to be grounded in anything concrete. In particular, how do mathematicians think to themselves about abstract algebraic objects like groups, rings, fields, and such stuff. For vector spaces, there is a nice geometric and intuitive picture of collections of arrows. What about more general "sets with binary operations"? Do mathematician's have tricks to visualize such sets and structures, or is it more like learning rules to manipulate symbols on a page? I'm teaching myself some very basic group theory and Cayley diagrams are neat ways of thinking about the structure of groups, but is this really how mathematician's think or is it just a crutch and eventual hindrance to further learning? I vaguely remember an interview with Paul Halmos (I think) talking about some group theorist he knew. When the group theorist was asked what he thought of when thinking about a group, he responded by saying "the letter 'G' ". Is this more than just a funny anecdote; do algebraists really not know how they do their magic? Are they just some kind of mental demigods? I mean, in geometry or linear algebra, sometimes you are told something that is so obviously true because you can "see" it, even before you prove it. Is there a way to develop such an instinct and intuition for abstract algebra, or is it more like you have in your mind a catalogue of "if...then..." statements and the best you can do is get really good with using these statements. In other words, does being good at abstract algebra just come down to being good with logic and symbolic manipulation, or is there such a thing as algebraic intuition, analogous to geometric intuition? I ask because I'm teaching myself some basic group theory now and am enjoying it. But, it's exactly because of the intuitive aids like Cayley diagrams that allow me to "see" groups, that make me like it. I hated high school algebra because I foud it to be just manipulating "x,y,z"s on a page until they are configured in the desired way. If abstract algebra is like that too then I don't think I'll continue to enjoy learning it. I apologize if some things in my post are vague and unclear; I'm still just a newbie so I don't have many concrete examples to draw on. 


#2
Mar911, 08:27 PM

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#3
Mar911, 08:47 PM

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One thing with group theory is that in accordance with the axioms, every operation is "undoable" or has an inverse.
Think about things like say solving a rubix cube or playing chess. A rubix cube that ends up at a state in the future as a result of scrambling with its states has a path back to its original state (ie the solved cube). Its the same with chess. You can do the same sort of thing with the cube but in the reverse. In the cube example you know the end state (the scrambled state) and you want to go back to the solved state but in chess you might have an idea of what the end will be like and instead you want to go from the checkmate state back to the ordered state (the start of the game). In chess you are making logical guesses at what you want your end state to be, but in the cube you already know the scrambled state. There are plenty of systems like the chess and rubix cube example and it means that if something is a group, then we know that there is a path to go from "final" to "initial" in at least one way. 


#4
Mar911, 09:47 PM

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How do mathematicians think about abstract algebra?
I can only speak for myself: a group is just a set and some axioms. That's more or less what Halmos said, I think.
On the other hand, knowing something about a wide range of examples of groups of different types (finite, infinite, abelian, cyclic, permutation groups, etc,) is a big aid to thought  so long as you don't forget that "a group" is not equal to "the collection of examples of groups that you happen to know about". 


#5
Mar911, 10:58 PM

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Algebraic objects like groups, rings, fields, and such stuff are concrete. It is valuable to consider collections of similar things to organize a body of knowledge, to avoid duplicating effort, or to increase understanding. The same thing is done in most if not all subjects. Biology is a good example, but no one asks biologist how do you thing about mammals? A giraffe and a zebra are so different how can they both be mammals? It is obvious that in some collections members will have some some differences and some things in common, and it makes sense to talk aout such collections. I think a problem some people have with algebra is they either cannot think of many examples, or the examples are too much alike.



#6
Mar1011, 01:36 AM

P: 294

I like lurflurf's comparison to giraffes and zebras as specific occurrences of the more general and abstract concept of 'mammal'.
Giraffes and Zebras are very evocative things to most people, the integers mod n are rather dead and lifeless by comparison, at first! But I have found that the more I study group theory the more these various 'manifestations' of groups, matrix groups, dihedral groups etc. have become lively and colorful. All the connections between them, the deep structure and symmetry. I would look at a mathematical object one day completely blind to all its group structure, all the homomorphisms and isomorphisms running in and out of it and within it, and now I see. And so it has become embossed and enriched with all this extra texture and meaning. Your right that its harder to reason by geometric intuition in abstract algebra than in some other areas of mathematics, but all that structure and symmetry lends itself very well to other types of reasoning, such as that by example, analogy and metaphor. I have not studied any algebraic topology yet, but I have read an author describe group homomorphisms from these higherdimensional manifolds as our lanterns which allow us to investigate surfaces where our geometric intuition has at least partially begun to fail. I'm looking forward to learning about it personally, maybe you would be too. 


#7
Mar1011, 05:09 AM

P: 124

both joke and truth When mathematicians think about abstract structures they think about the properties those structures have. For groups they think about the order, whether it is abelian, etc. For vectorspaces they think about the dimension (in fact, two vectorspaces over the field F are isomorphic if and only if they have the same dimension). 


#8
Mar1211, 06:51 PM

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i myself and i , think about these guys by examples. a field is a generalization of the rationals, reals, or rational or meromorphic functions. a group is a generalized version of the automorphisms of some object. a ring is a general version of the integers or the polynomial rings over various rings. non commutative rings are something like the rings of linear automorphisms of a vector space.
the letter G is just a joke. that has no value whatsoever. 


#9
Mar1811, 07:02 AM

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i can't speak for anyone else, but i envision groups as quotients of free groups.
that is, a group element is a sequence of "letters" (a word), which might be a sequence of actions, a code, a composite function, gee, i dunno. and every particular group comes with its own alphabet, and its own set of rules. i no longer think of integers, for example, as numbers, but rather as elements of the free group on one letter (it is somewhat ironic that this way of looking at them probably comes close to the tally system that preceded ancient systems of arithmetic). the idea for me, is to reduce the patterns that occur in a group, to smaller patterns. if i can replace ABA* with B, it's all good. my feeling is, is that the reason people are more comfortable with linear algebra, say, than group theory, is that the free vector space over a finite set S of cardinality n, is isomorphic to the space F^n, free vector spaces are so much more constrained by the axioms, that they almost HAVE to behave in predictable ways. whereas groups can behave more unpredictably, because they have that much more freedom. i tend to apply the same approach to other abstract objects. if at all possible, i want to know what object has "just enough" structure to qualify, without any other conditions. if i can get a feel for how it behaves, then "restricting" it (preferrably via a type of preferred subobject) becomes a matter of filtering out information that is uninteresting. 


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