Proofs in Algebra vs Proofs in Analysis

[Fizicks1]

Both algebra and analysis are pretty much all about proofs...but a prof told me the proofs in algebra are very different from those of analysis. How are they different? Any input appreciated.

 

[pwsnafu]

Both algebra and analysis are pretty much all about proofs...but a prof told me the proofs in algebra are very different from those of analysis. How are they different? Any input appreciated.

The core of the disciplines is different. The vast majority of algebraic systems are of the form

• there is a set
• there is a finite number of operations
• the arity (number of variables) of each is always finite
• you only allow finite number of operations to be performed

Importantly, this means infinite sequences, infinite sums, and infinite products are not allowed*. In analysis you study things like convergence, limits, and continuity.
Proofs in analysis tends be more of the form "a $x$ approaches this number $y$ approaches..."
Proofs in algebra tend to be...static.
Does that make sense? You might want to actually do proofs in the two areas to see the difference.

*unless the "set" in question are these objects to begin with.

 

[Fizicks1]

The core of the disciplines is different. The vast majority of algebraic systems are of the form

• there is a set
• there is a finite number of operations
• the arity (number of variables) of each is always finite
• you only allow finite number of operations to be performed

Importantly, this means infinite sequences, infinite sums, and infinite products are not allowed*. In analysis you study things like convergence, limits, and continuity.
Proofs in analysis tends be more of the form "a $x$ approaches this number $y$ approaches..."
Proofs in algebra tend to be...static.
Does that make sense? You might want to actually do proofs in the two areas to see the difference.

*unless the "set" in question are these objects to begin with.

Thank you for the detailed response pwsnafu.

I have taken an introductory analysis course before and didn't do very well in it, and I found myself to be quite weak at proofs.

I want to take a course on algebra this coming fall semester, and due to my prior poor experience with proofs, am seriously considering whether or not I should take it.

From your description, would I be wrong to say that proofs in algebra seem to be more "mechanical" or "standardized" than in analysis? In my experience with analysis before, I get the impression that one often needs a sudden stroke of genius and creativity to form the proofs. Is that the case too for algebra?

 

[pwsnafu]

I have taken an introductory analysis course before and didn't do very well in it, and I found myself to be quite weak at proofs.

Proofs are a paradigm shift. Up to now, you have been taught methods and you are tested on how you can apply that method. A lot of students go through this time, and that included myself. In my case, it really took until group theory when I started to grok proofs, and even then I had problems with linear analysis. And yet my PhD is in real analysis.

You learn. Trust me. Just make sure you do lots of problems. Don't say "Oh I know how to do this" and skip. Do as many as you can. Especially the simple ones.

From your description, would I be wrong to say that proofs in algebra seem to be more "mechanical" or "standardized" than in analysis? In my experience with analysis before, I get the impression that one often needs a sudden stroke of genius and creativity to form the proofs. Is that the case too for algebra?

It varies. A lot.
An introductory course won't test you on creating new strokes of genius. You would tested on

• Definitions. Learn these exactly word by word. This is straight bookwork. You have no excuse if you get these wrong.
• Recalling the difficult proofs. You'll be taught these in class, so memorize the proof structure.
• Proof problems that would be on a similar difficulty to assignment problems.

The last usually are solvable if you understand the definitions/concepts, and not just as a bunch of symbols and words.

My experience with algebra proofs at undergraduate has a lot of 1) "fiddle around a bit" and then the answer pops out, and 2) you write down the definition of the questions and the answer is staring at you.

I definitely consider algebra proofs easier. Analysis has a lot of minute technicalities and the concepts are hard to understand if you had no experience. It doesn't help that the first proof subject students experience is real analysis. Its definitely a wake up call.

 

[lavinia]

All mathematical proofs except computations derive from an insight about a mathematical structure. Insights come from many sources and often mix algebraic, topological, analytical, and geometric ideas. For instance, the proof that every subgroup of a free group is free - an algebraic statement - is commonly proved using algebraic topology.
The representation theory of the fundamental groups of Riemann surfaces involves analysis and differential geometry.

While it is true that purely algebraic proofs tend to be computational - e.g. linear equation solving -while proofs in analysis use concept of limits and continuity, this cdoes not in my mind say everything about how ideas and proofs are arrived at in these fields.

 

[algebrat]

Outline of long winded answer: I talk about ability to visualize in analysis, and how to replace this technique in algebra.

I personally found calculus related proofs more comfortable than the algebra ones. I think this has to do with each person. While I am definitely good at the general symbolic reasoning compared to the general population, within the math population, I lean more towards the visual. But at some point, the subjects increasingly merge. As you get further in for instance manifolds, you start seeing algebraic manipulations like tensors and modules etc. In algebra, they definitely begin to treat questions that arise from continuous situatuations. Lie algebras for instance are a topic in both first year graduates' algebra and manifolds.

So when pwsnafu said "I definitely consider algebra proofs easier. Analysis has a lot of minute technicalities", I'm guessing their mind leans toward the ability to juggle many symbolic concepts in the mind buffer of their brain. I wouldn't doubt they also have many abilities to conceptualize and develop intuition to guide their steps, for instance if they don't just "see" the answer quickly, then you begin using everything you got.

If your mind buffer is not large enough, like mine, then you have to compensate with other skills, so for instance a problem I recently visited in algebra was show a nilpotent plus a unit gives a unipotent. I may not be remembering the problem correctly. But the problem came more or less down to a trick almost. And unless you could mentally run through your huge rolodex of math knowledge very quickly, you weren't likely to just see the answer. If I had remembered maybe some tricks from polynomials, I might have gotten it faster.

So, here's the technique I call upon in algebra, since you can't just draw yourself a picture like in analysis. I was pretty good at coming up with the proofs by visualizing in analysis, but this doesn't usually work in algebra, because most of the subject is about symbolic manipulation maybe. My new technique is, when stuck, come up with every example you can and try to figure out the relation. This is a last resort, most problems involve applying a definition, or figuring out how to manipulate the symbols correctly, by "seeing" it, or playing around with them for a little bit, as pwsnafu also said.

So for this problem, I had to come up with a manipulation that was harder to find. So I happened to know of a basic type of matrix which was nilpotent. I played around with that, trying different sizes of matrices, starting small, taking powers of them, and looking for some relation. I finally found it, and when done, while there were a number of terms in the relation, it was sort of simple in the end, resembled some polynomials tricks, too long to "see", short enough to write quickly, but hard to come up with. Knowing lots of examples can really help in algebra, not just to help undrstand the definitions, but as a testing ground when trying to investigate a statement.