Difficulty of abstract algebra in relation to calculus

[robertjford80]

How difficult is abstract algebra or group theory, plus complex analysis in relation to calculus?

 

[micromass]

If your quote from another thread is accurate:

but proofs are just too hard for me, they're written in a highly opaque language so now I mostly skip over them but a few proofs I understand.

then you're going to have a bad time.
I suggest you start learning proofs first. Good books are Velleman and "How to think like a mathematician" from Houston.

 

[robertjford80]

could you go into a bit more detail. why are understanding proofs essential to abstract algebra. can't you just learn the algorithms and do the problems?

 

[micromass]

What algorithms??
Abstract algebra are proofs. The exercises are proofs.

If you eliminate the proofs from an abstract algebra text, then the only things that's left is the table of contents.

 

[Number Nine]

could you go into a bit more detail. why are understanding proofs essential to abstract algebra. can't you just learn the algorithms and do the problems?

No. Absolutely not. A course in abstract algebra will consist entirely of proofs. You're entering big boy mathematics now; gone are the days when problems can be solved by plugging in some numbers and applying an algorithm (it's extremely likely that you won't deal with numbers at all during the course, except in the most abstract possible sense).

 

[robertjford80]

what about complex analysis? Also I thought group theory was used for Quantum Field Theory. Certainly they use number in QFT

 

[Number Nine]

what about complex analysis? Also I thought group theory was used for Quantum Field Theory. Certainly they use number in QFT

Complex analysis will be the same; almost everything stated will be proved, and every question on every assignment will require a proof.

Yes, group theory is used extensively in QFT. Group theory is the study of symmetry; symmetry arises very frequently in physics.

 

[Undecided Guy]

How difficult is abstract algebra or group theory, plus complex analysis in relation to calculus?

I found it easier. Calculus professors have a tendency to wave their hands at justifications that are at the heart of most conceptual misunderstandings. If you do not understand a single line in a proof, you should not move on until you understand it completely. To do otherwise is cheating.

 

[micromass]

I suppose you can learn complex analysis without the proofs. But then you won't really know complex analysis. You'll be more like a monkey repeating steps that he learned.
Furthermore, I don't know any book that would teach complex analysis without relying heavy on proofs.
Perhaps Boas will be good. It contains some complex variables and it's not heavy on proofs.

 

[Number Nine]

I found it easier. Calculus professors have a tendency to wave their hands at justifications that are at the heart of most conceptual misunderstandings. If you do not understand a single line in a proof, you should not move on until you understand it completely. To do otherwise is cheating.

I found it far easier as well, but for a different reason -- calculus (particularly integral calculus) requires long and involved computations that offer many opportunities for errors to slip in unnoticed, even if you've mastered a technique. Abstract algebra only requires that you fully and completely understand the subject matter; if you do, then you can usually tell whether or not you're on the right track with a solution.

 

[WannabeNewton]

I think you are underestimating the level of mathematical rigor in QFT textbooks. Sure they aren't the typical proofs found in mathematics textbooks but, for example, some of the very first problems in Srednicki's QFT text are proof type problems. The whole plugging something in and getting a number answer will barely ever come up.

 

[robertjford80]

I found it far easier as well, but for a different reason -- calculus (particularly integral calculus) requires long and involved computations that offer many opportunities for errors to slip in unnoticed, even if you've mastered a technique.

This is so true

 

[robertjford80]

I think you are underestimating the level of mathematical rigor in QFT textbooks. Sure they aren't the typical proofs found in mathematics textbooks but, for example, some of the very first problems in Srednicki's QFT text are proof type problems. The whole plugging something in and getting a number answer isn't frequent at all.

So you mean QFT is more about how the objects relate to each other rather than what the quantity of the objects are? Feel free to go into more details.

 

[Number Nine]

So you mean QFT is more about how the objects relate to each other rather than what the quantity of the objects are? Feel free to go into more details.

You'll be dealing with plenty of quantities in QFT; in your upcoming courses, unfortunately, you won't. Pick up a book like How to Prove It and start working through it before the semester starts, because you're going to be neck deep in proofs for the durations of the year. Every problem on every assignment and every test will require you to supply a proof.

 

[mathwonk]

these are good bits of advice. don't be afraid. when i was in second grade i was afraid to go to third grade because they had to swim across the pool. but when i got there i also was ready. this is not a diss. the point is that every year you get stronger.

 

[Sankaku]

How difficult is abstract algebra or group theory, plus complex analysis in relation to calculus?

More than tuesdays but less than a rabbit.





(i.e. They are not part of an ordered set - the elements are not comparable).

 

[algebrat]

More than tuesdays but less than a rabbit.

That was awesome, did you make that up?

Robertjford80, many students who try out college math get to the level where proofs begin to creep slowly in, and greatly resent this uncomfortable step in their education. But look at it this way, the alternative is writing 20 page papers for a humanities class, or long technical reports for other science classes. Not that there is anything unusual in this discomfort, but in any of the disciplines, including math, it is of benefit to you to learn how to perform and communicate your ideas at a professional level.

 

[robertjford80]

That was awesome, did you make that up?

Robertjford80, many students who try out college math get to the level where proofs begin to creep slowly in, and greatly resent this uncomfortable step in their education. But look at it this way, the alternative is writing 20 page papers for a humanities class, or long technical reports for other science classes. Not that there is anything unusual in this discomfort, but in any of the disciplines, including math, it is of benefit to you to learn how to perform and communicate your ideas at a professional level.

Well, I actually liked proofs in logic. I had a feeling that I was getting to the heart of what true reasoning was all about. I was hoping it would be the same for math, but too often mathematicians use esoteric notation, skip over steps, and express their thoughts in a sort of opaque fog. However, I'm really determined to get to a high level, so I'm predicting success.

BTW, I actually belong more to the humanities than the sciences. I used to debate a lot in forums and I found out that I really didn't have the knowledge I needed to debate with my opponents. So I began to study philosophy and to my horror I discovered how ignorant I was. It then became clear that philosophical debates were really about how best to interpret scientific facts. That's when I decided to remedy my atrocious lack of scientific knowledge.

 

[Sankaku]

If you like proofs in logic but have these criticisms about mathematical proofs, perhaps you should ask yourself if the fault lies with mathematicians or with your level of preparation.

That was awesome, did you make that up?

 

[robertjford80]

logic proofs are rather simple, there are only 18 operations acting on essentially two numbers, True and false, math employs I don't know how many operations acting on an infinite set of numbers. But don't get me wrong, logic proofs can be quite difficult, but they deal with situations that would never arise in an actual argument.

 

[Number Nine]

logic proofs are rather simple, there are only 18 operations acting on essentially two numbers, True and false, math employs I don't know how many operations acting on an infinite set of numbers. But don't get me wrong, logic proofs can be quite difficult, but they deal with situations that would never arise in an actual argument.

Everything you've studied in elementary logic will arise in mathematics, in one form or another.

 

[diligence]

If you do not understand a single line in a proof, you should not move on until you understand it completely. To do otherwise is cheating.

No offense, but I think this is probably the worst advice you could possibly give to someone who wants to learn proofs and hopefully move on to high-level mathematics. Cheating? That's beyond ridiculous!

It's about doing whatever it takes to understand the material, beyond that is personal preference. This whole "math is only for the genius" mentality is juvenile at best, and disgusting at worst.