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Anything that can help make this subject more approachable is alright in my book. In my experience, the hardest part of Abstract Algebra isn't the abstract part

Anything that can help make this subject more approachable is alright in my book. In my experience, the hardest part of Abstract Algebra isn't the abstract part of it. It's the lack of examples and clarity around how different named concepts behave. The proofs aren't terrible even except for the fact that they require a familiarity with number theory that many (at least I myself) don't have. I love the subject though. Really interesting.

gh02t 11y ago
For me, it has always been the terminology. The basic theorems are relatively easy to understand... once you can remember the vast number of algebra-specific definitions. The field also has a proclivity for umm "colorful" choices of names. It's not unlike say Haskell, which IMO is pretty straightforward once you have a feel for all the terminology.
- goalieca 11y ago
  Like this from wikipedia on Monoid? Suppose that S is a set and • is some binary operation S × S → S, then S with • is a monoid if it satisfies the following two axioms: Associativity For all a, b and c in S, the equation (a • b) • c = a • (b • c) holds. Identity element There exists an element e in S such that for every element a in S, the equations e • a = a • e = a hold. In other words, a monoid is a semigroup with an identity element. It can also be thought of as a magma with associativity and identity. The identity element of a monoid is unique.[1] A monoid in which each element has an inverse is a group.
  - cmrx64 11y ago
    It seems to me that the definition presented there is relatively free of jargon, except the "in other words" section which links it to other (linked) structures.
    - gh02t 11y ago
      > except the "in other words" section which links it to other (linked) structures. That's what I was getting at. I took a few classes in AA and its sibling topology, including at graduate level. Most of the time was spent defining things, because there's so many terms with very specific definitions. Only after maybe 2/3rds of a course do you really start getting into deeper material. In my brain, reading definitions was always a process like: read through, substitute definitions for jargon terms, read through, substitute... iterate until you hit a fixed point. That's standard for math, but AA is particularly well known for its colorful choice of names. Some people enjoy it, but I was always more attracted to the analysis-family areas of math. I've always found the other name for category theory amusing though: https://en.wikipedia.org/wiki/Abstract_nonsense https://en.wikipedia.org/wiki/Abstract_nonsense
      - cmrx64 11y ago
        Fair enough! My biggest struggle taking AA was also remembering the definitions of everything. There are so many things all tied together.
  - kmill 11y ago
    I think I only finally remembered what a monoid was when I realized it was a category with one object ("mono-"). Or that a groupoid is a category where every arrow is an isomorphism, so then a category which is both a monoid and a groupoid is a "group."
throwaway999888 11y ago
> Anything that can help make this subject more approachable is alright in my book. Alright? Yeah, why wouldn't it be alright?
cottonseed 11y ago
Lack of examples? What? Algebra is literally overflowing with examples. Algebraic structures generalize concrete structures you've probably already seen elsewhere in mathematics: from arithmetic, addition and multiplication, as the video explained, from discrete math, you can write down lots of interesting, finite examples, symmetry groups of geometric figures generate lots of interesting finite and infinite (Lie) groups, 1- and 2-dimensional wallpaper patterns are classic examples, etc, etc. In comparison, for example, it's quite non-trivial to describe a single interesting 4-manifold.
- xyzzyz 11y ago
  I think the issue is that people try to jump in abstract algebra too early, before they acquaint themselves with a variety of concrete examples. It's even worse with category theory -- how one is supposed to understand why functors are useful before encountering them in a natural context is beyond me.
peterlk 11y ago
I like to generate examples by trying to add random things together, and then adding constraints. Forget, for a moment, words like "group" and "ring". At the risk of fatally oversimplifying things, I'll try to help. What happens if I add water and... oil? Well, first of all, let's specify that if I add water to water, I still get water; and if I add oil to oil, I still get oil. Now, I could have water on top of the oil, oil on top of the water, water in the oil, and oil in the water. Let's collapse this so that now we have a collection of combinations that are water, oil, water and oil mixed, water on top of oil, and oil on top of water. If we can agree that we can "add" things in this collection together and get other things in the collection, then we have created an abstract algebra. The math is just a formalization that gives us extra power to reason about this collection of things. In a more computer sciencey case, what might be the outcome of "adding" two Twitter users? Does one follow the other? Do they both follow each other? Is there something else that happens? Monad was a buzzword for a while because it made questions like this solvable with some goodies like easy parallelization. What if 200 million accounts suddenly needed to be added together? It sure would be nice to have an idempotent, parallelizable way to do so. Monads were one way of doing that, and abstract algebra gives us mathematical methods to rigorously approach problems like this (and many that aren't like this one). It might be worth it to check out algebird

gh02t 11y ago

For me, it has always been the terminology. The basic theorems are relatively easy to understand... once you can remember the vast number of algebra-specific definitions. The field also has a proclivity for umm "colorful" choices of names. It's not unlike say Haskell, which IMO is pretty straightforward once you have a feel for all the terminology.

goalieca 11y ago
Like this from wikipedia on Monoid? Suppose that S is a set and • is some binary operation S × S → S, then S with • is a monoid if it satisfies the following two axioms: Associativity For all a, b and c in S, the equation (a • b) • c = a • (b • c) holds. Identity element There exists an element e in S such that for every element a in S, the equations e • a = a • e = a hold. In other words, a monoid is a semigroup with an identity element. It can also be thought of as a magma with associativity and identity. The identity element of a monoid is unique.[1] A monoid in which each element has an inverse is a group.
- cmrx64 11y ago
  It seems to me that the definition presented there is relatively free of jargon, except the "in other words" section which links it to other (linked) structures.
  - gh02t 11y ago
    > except the "in other words" section which links it to other (linked) structures. That's what I was getting at. I took a few classes in AA and its sibling topology, including at graduate level. Most of the time was spent defining things, because there's so many terms with very specific definitions. Only after maybe 2/3rds of a course do you really start getting into deeper material. In my brain, reading definitions was always a process like: read through, substitute definitions for jargon terms, read through, substitute... iterate until you hit a fixed point. That's standard for math, but AA is particularly well known for its colorful choice of names. Some people enjoy it, but I was always more attracted to the analysis-family areas of math. I've always found the other name for category theory amusing though: https://en.wikipedia.org/wiki/Abstract_nonsense https://en.wikipedia.org/wiki/Abstract_nonsense
    - cmrx64 11y ago
      Fair enough! My biggest struggle taking AA was also remembering the definitions of everything. There are so many things all tied together.
- kmill 11y ago
  I think I only finally remembered what a monoid was when I realized it was a category with one object ("mono-"). Or that a groupoid is a category where every arrow is an isomorphism, so then a category which is both a monoid and a groupoid is a "group."

throwaway999888 11y ago

> Anything that can help make this subject more approachable is alright in my book. Alright? Yeah, why wouldn't it be alright?

cottonseed 11y ago

Lack of examples? What? Algebra is literally overflowing with examples. Algebraic structures generalize concrete structures you've probably already seen elsewhere in mathematics: from arithmetic, addition and multiplication, as the video explained, from discrete math, you can write down lots of interesting, finite examples, symmetry groups of geometric figures generate lots of interesting finite and infinite (Lie) groups, 1- and 2-dimensional wallpaper patterns are classic examples, etc, etc. In comparison, for example, it's quite non-trivial to describe a single interesting 4-manifold.

xyzzyz 11y ago
I think the issue is that people try to jump in abstract algebra too early, before they acquaint themselves with a variety of concrete examples. It's even worse with category theory -- how one is supposed to understand why functors are useful before encountering them in a natural context is beyond me.

peterlk 11y ago

I like to generate examples by trying to add random things together, and then adding constraints. Forget, for a moment, words like "group" and "ring". At the risk of fatally oversimplifying things, I'll try to help. What happens if I add water and... oil? Well, first of all, let's specify that if I add water to water, I still get water; and if I add oil to oil, I still get oil. Now, I could have water on top of the oil, oil on top of the water, water in the oil, and oil in the water. Let's collapse this so that now we have a collection of combinations that are water, oil, water and oil mixed, water on top of oil, and oil on top of water. If we can agree that we can "add" things in this collection together and get other things in the collection, then we have created an abstract algebra. The math is just a formalization that gives us extra power to reason about this collection of things. In a more computer sciencey case, what might be the outcome of "adding" two Twitter users? Does one follow the other? Do they both follow each other? Is there something else that happens? Monad was a buzzword for a while because it made questions like this solvable with some goodies like easy parallelization. What if 200 million accounts suddenly needed to be added together? It sure would be nice to have an idempotent, parallelizable way to do so. Monads were one way of doing that, and abstract algebra gives us mathematical methods to rigorously approach problems like this (and many that aren't like this one). It might be worth it to check out algebird