Is analysis more intuitive than algebra?

[tgt]

Of the limited number of people I've met, the ones who liked algebra disliked analysis. Does this apply to nearly everyone? They are very different fields.

Does the opposite apply?

What are your reasons?

 

[mathman]

What algebra are you referring to? There is the material leading up to calculus (college algebra), linear algebra, and modern algebra (groups, rings, fields, etc.) There are also more advanced topics such as Banach Algebra, topological groups, Lie groups and Lie Algebra.

 

[tgt]

What algebra are you referring to? There is the material leading up to calculus (college algebra), linear algebra, and modern algebra (groups, rings, fields, etc.) There are also more advanced topics such as Banach Algebra, topological groups, Lie groups and Lie Algebra.

Lets say abstract or modern algebra.

 

[redrzewski]

I am definitely more interested in analysis than modern algebra. But I don't "dislike" algebra. I just prefer spending my limited time primarily on analysis these days.

 

[tgt]

I am definitely more interested in analysis than modern algebra. But I don't "dislike" algebra. I just prefer spending my limited time primarily on analysis these days.

What's so attractive about analysis?

 

[phreak]

I love analysis, I love geometry and topology, but I hate algebra. It seems sometimes that the purpose of algebra is abstraction for sake of abstraction. I've never seen it applied to a large extent, so I've never really found motivation to enjoy it. I think a lot of people like the abstract nature of algebra, but it just doesn't work for me.

 

[redrzewski]

Why I like analysis? Some highlights off the top of my head.

Combines set theory with calculus, cool constructions like cantor sets, cantor functions. Characterizations of linear functionals as integrals. The relation between measures, functions, and derivatives (of functions) vs. derivatives of measures. Applications like applying the hahn-banach theorem from functional analysis to the poisson integral. Gives us insight into sets of functions on which Fourier transforms diverges.

etc

 

[MathematicalPhysicist]

I don't understand what there is to hate about them, really.
Both of them might be very hard, look at the monumental effort of people to select the free groups, does someone even have time to read all the books written on the selection of free groups?

 

[tgt]

I don't understand what there is to hate about them, really.
Both of them might be very hard, look at the monumental effort of people to select the free groups, does someone even have time to read all the books written on the selection of free groups?

I'd imagine most people start off liking both when first studying it but as one specialises in one, the other becomes more and more 'disgusting'.

What do you mean by selection of free groups?

 

[MathematicalPhysicist]

I meant simple groups, http://en.wikipedia.org/wiki/Simple_group

I haven't yet taken a course in groups, but I guess I can fully grasp Saunders Maclane and Birkhoff's Algebra book which I have at my disposal.

 

[MathematicalPhysicist]

But, your'e using algebra all the time in calculus and vice versa, analysis nowadys is being covered as algebraic tool, look at differential algebra.

 

[zhentil]

But, your'e using algebra all the time in calculus and vice versa, analysis nowadys is being covered as algebraic tool, look at differential algebra.

Not after a certain point. Nothing beyond the definition of an ideal is used in analysis, and you will never see a Fourier transform if you do research in algebra.

I enjoy analysis much more than I enjoy algebra. In algebra, they always seemed more interested in building the machinery than in solving problems.

 

[tgt]

Not after a certain point. Nothing beyond the definition of an ideal is used in analysis, and you will never see a Fourier transform if you do research in algebra.

I enjoy analysis much more than I enjoy algebra. In algebra, they always seemed more interested in building the machinery than in solving problems.

There was a passage in an algebra textbook that Artin advised his students when doing research in algebra to first discover a trick then find the right problem to use that trick on.

 

[shaggymoods]

I think most people are introduced to proof-based mathematics via analysis and Calculus. Thus people learn to utilize their geometric and physical intuition, whereas algebra is highly formal and more difficult to grasp with this type of intuition. I love analysis, and I find algebra to be very difficult, but still very rewarding and beautiful.

 

[confinement]

I used to say I liked analysis and disliked algebra, until I begin to study groups as differentiable manifolds, which for me contains the best of both worlds.

 

[tgt]

I used to say I liked analysis and disliked algebra, until I begin to study groups as differentiable manifolds, which for me contains the best of both worlds.

Why like analysis and dislike algebra?

Incidently I love groups and think manifolds are disgusting.

 

[Office_Shredder]

Algebra is just a subset of analysis; every equality can be written as two opposite inequalities :D

 

[confinement]

Why like analysis and dislike algebra?

Incidently I love groups and think manifolds are disgusting.

I like analysis because it always takes place in some kind of space, and usually the space is a continuum (uncountable and complete) wihch is the kind of structure that I find natural and intuitive to think about. It makes it easier to remember definitions or theorems and to prove results.

In contrast, in (discrete) algebra I often begin a proof by looking at several special cases, since I can't directly imagine the abstract general case. On the otherhand, the theorems of algebra feel more powerful to me because they are not so obvious, and it is this aspect of the subject that I have learned to like.

I am sorry that you find manifolds disgusting, I find them beautiful (at least, those which admit a differentiable structure). The only thing about them which I find disgusting is our relatively enormous lack of knowledge about them!