Is Abstract Algebra a Necessary Foundation for Mathematical Maturity?

[Ulagatin]

Hi everyone,

I'm in Grade 11 this year (in Australia), currently studying from Apostol's first volume "Calculus". I have just recently started working on the theory of integration of trigonometric functions (just giving some information on my background). I am thinking that, perhaps once I've finished the calculus section of Apostol's text, I could move on to either;

(a) Multi-Variable Calculus
(b) Real Analysis
(c) Abstract + Linear Algebra.

I cannot really make a decision at this point in time, as I am unsure about abstract algebra (hence my post here). My interests are in physics, mathematics, computer science and philosophy, and so, I want to make a decision based on relevance to these interests.

A short time back, I e-mailed Professor Apostol himself (giving feedback on his textbook and asking for advice regarding his Mathematical Analysis and Calculus Volume 2 books). He suggested that his second volume isn't necessarily a pre-requisite for his more terse exposition in the Mathematical Analysis text. He also suggested, since I am seriously considering majoring in mathematics at university, that I take a look at his Intro to Analytic Number Theory text.

I would assume this step would come one after that of Real Analysis. But, may I ask, if I decide to pursue pure rather than applied mathematics, how useful would abstract algebra be to this field? What ability does a solid knowledge of abstract algebra present to you: i.e. what type of problems does it allow you to solve, and what is the topic's main idea/motivation?

How difficult is the topic (abstract algebra) in comparison with calculus: is it less visual and geometrically intuitive? My favourite aspect of mathematics is calculus at this stage, and also sequences/series, although I don't understand them as well as many of the integration/differentiation topics and theorems I have covered.

While I may not be the most sophisticated mathematically, I am interested in a rigorous but understandable presentation of the various topics. I'm not afraid to see proofs or try and prove things myself (although for the most part, I'm unsure of how to do that).

Would it be possible to pick up linear algebra as I go, if I decide to study abstract algebra? I'm aware that linear algebra has great importance in computer science and physics, but I've also heard that it's not a greatly interesting branch of mathematics. I'm aware that this is an entirely subjective question, but how does abstract algebra rate on this "interest scale"? Do people find it more fun than calculus?

My ultimate goal, after university, is to become a theoretical physicist, and I've heard that both calculus (analysis?) and algebra are very important to this field. Any information on the basics of abstract algebra, what it is, how it's used, and if possible, recommended texts on this field would be useful, and greatly appreciated.

So, based on all the information I have provided, what would you all suggest for me to move on to? Remember, analytic number theory is potentially an option too, but I fear it may be too advanced for me at this stage. Would abstract algebra be too advanced?

Any enlightenment would be grand. :tongue:

Cheers,
Davin

 

[sutupidmath]

In U.S, Abstract Algebra in most universities is considered and upper division course. It depends on the way it is presented in class, but it is totally proof based, and pure math.

In a first course in abs. alg, one usually coveres some of the following topics:

Group Theory:
1.Permutation Groups
2.Cyclic Groups
3.Quotient Groups
4. A little bit of Solvable Gourps
5.Additive Notation of Grops
6.Isomorphism
7.Homomorphisms
8.Congruences in Z
9.Rings
10.Isomorphisms of Rings
11.HOmomorphisms etc


There are other topics in between, but basically you can get an idea of what it is.

Group theory basically deals with the study of some set of something with some kind of operation defined in it. That is, it serves as a tool to study many things at the same time. i.e many mathematical objects, that satisfy a set of axioms.
Basically, in there one developes some theorems that apply to any mathematical object that satisfies the following axioms:

1. Closure,
2.Existence of Identity
3.Existence of Inverse
4.Associativity
5. and if they satisfy this property :Commutativity, they are called abelian groups.

And some other axiomos when it comes to rings.

I personally took this course this past semester( i will begin my second year this spring, university studying math) and i really enjoyed it a lot.
If you don't intend to study mathematics, i don't see of what use it might be for you.

As for being hard, i personally had to dedicate a lot of time to it (GOt an A at the end but...), because even the problems that we had in exams were all proving stuff. So, you will need some time to get used to it, i suppose, but of course it is subjective(you probbably are way smarter than me so it might work just fine).

Cheers!

 

[Ulagatin]

Hi there,

Thanks for your reply. Much appreciated.

First of all, I am glad to hear that you greatly enjoyed the subject. I would like to offer my congratulations for the A you scored in your course.

Group theory, which I gather is the main component of introductory abstract algebra, would appear to be an interesting, but certainly very challenging course, due to the proof-writing aspect. I feel that proofs appeal to me (from my coverage of calculus from Apostol's text so far), but that I would not be mathematically sophisticated enough at this point in time to tackle them directly.

So, in lieu of abstract algebra, what subject would you recommend for me to tackle after my completion of "One-Variable Calculus" by Apostol (the calculus segment)? Would analytic number theory be suitable? Perhaps I should tackle a real analysis text (based on single-variable calculus) before I tackle abstract algebra, so I have a better idea about proofs. If there is an error in my reasoning, please tell me. I could also naturally flow to multi-variable calculus, but a slight change in scenery would be grand, to have a better idea of a few fields of mathematics.

I find intelligence to be a slightly vague concept, difficult to define rigorously. I have spent many hours over a few months working on Apostol's text, and I've encountered difficulties. I've also encountered much more simple notions and ideas. This would have been defined by my strengths and weaknesses in mathematics, that may not mirror your own. I may be better with geometric intuition than yourself, whereas you may be better able to understand symbolic analytical arguments, for example. Perhaps a different kind of intelligence.

If I do decide to take up self-study of abstract algebra in spite of the level of difficulty you have cited here, may I ask a favour? I would like to ask that, perhaps, if you are agreeable, you tutor me, when I encounter difficulty? My first request would be regarding assistance with proof writing. This may simultaneously allow you to reinforce your own knowledge and understanding of abstract algebra, but I would also understand if you decline. I merely present you with this opportunity for a favour.

Thanks again for the speedy reply. :tongue:

Cheers,
Davin

 

[Edgardo]

If you want to know more about proofs have a look at the thread How to write Math Proofs.

 

[Ulagatin]

Thanks, Edgardo. I'll take a look. I think I've decided I'd be better just to move on to multi-variable calculus after completion of Apostol's "One-Variable Calculus". I intend to move on to his second book in the series. Abstract algebra appears too advanced for me at this stage.

Cheers,
Davin

 

[cup]

As far as I know (still a student), (Abstract) Algebra is very applicable. And not just to other kinds of mathematics.

Mathematicians, "pure" and "applied", "should" have some basic knowledge of algebraic structures and techniques.
I think even some engineers study abstract algebra.

It's the basis of many, many, different fields of mathematics that you never know when you'll run into.

I'll leave it to the other people on this forum to give more specific examples, but I know that group theory (that you'll study when you first study algebra) is extremely important in modern/theoretical/mathematical physics, for instance.

I don't know of any applications of algebra to philosophy, though.

Book tip:

https://www.amazon.com/dp/0201763907/?tag=pfamazon01-20

Very nice book for beginners.
Theoretically, anyone could start out with this book without prerequisites, but realistically you need to have acquired some "mathematical maturity".
I think you are ready, though.

That is not to say that you won't have to spend quite a bit of time with it.

Solve as many problems as you can. Ask for help here.

Good luck.

 

[Ulagatin]

Upon taking a look at this book, I think I really should go on to multi-variable calculus. Several reasons to do so: I won't get "burnt out" learning an entirely new branch of mathematics as some people have warned me (I recognise that there is so much to learn). I think my time is best spent learning calculus at this stage, as it is what I really need at this point in time.

I find I really do enjoy calculus also, so I would rather just "go with the flow" and continue study with my favourite branch of mathematics. I'm not a great risk-taker, and perhaps not being very adventurous, but it may be for the best.

Thanks for informing me about the applications of abstract algebra. I've even heard it's useful in chemistry... so I can imagine that some engineers would take the course.

I wouldn't expect there to be applications of algebra to philosophy! Only pointing out my interests! :tongue2:

Anyway, thanks for an interesting and informative post! I would not have imagined it would also appeal to applied mathematicians.

Actually, at this stage, I'm unsure about pure vs applied mathematics, but as I go further with my studies, I should find out what I honestly prefer.

Thanks for wishing me luck.

Cheers,
Davin

 

[cup]

Reading more advanced calculus is a fine way to become "mathematically mature".

Then, after reading advanced calculus (theorem-proof based), you are probably ready to read a book on Real Analysis.

The difference between "real analysis" and "advanced calculus" is fuzzy, so you will be well prepared for it.

Just know that abstract algebra can be read with virtually no prerequisites (especially if you use the right book, like Fraleigh).

So whenever you feel like changing subjects you should just go for it.

 

[mercurial]

Hi Davin,

It sounds like you've already made a decision, but I feel like jumping into the conversation anyway. I take you're about 17 years old? At that point in your career, it's impossible to tell which academic subjects will be relevant for interests you plan to pursue later -- and by that I mean "relevant" in the narrow sense that you seem to be reading it. The only safe assumption here: it's all relevant! Don't pass anything up just because it sounds to hard or weird at this point.

That being said, you really are best off to go with the flow, as you put it. If you like the calculus/analysis way of looking at mathematics, stick with it. You really do have all the time in the world to learn abstract algebra, should you choose to do so. The best way for you to find out how (or whether) something suits you intellectual tastes is to talk to someone more knowledgeable. If you happen to know someone who's already taken a course or two in abstract algebra at the university level, invite them for coffee and ask them what they find cool it. If you don't know anyone, you will by the time you're in your 1st or 2nd year at university: and then there's still plenty of time to decide with maths courses fit your schedule!

Jerry

 

[Ulagatin]

Hi Jerry,

Yes, I'll be 17 in May. Thanks for your advice.

At this stage, the only person I would know that has studied abstract algebra would be an associate professor of theoretical physics at my local university (very nice man). Although your suggestion is top-notch, it would be somewhat impractical to invite him for coffee/tea!

I am looking at eventually studying abstract algebra anyway: I was looking to do more advanced calculus, analysis and algebra courses when I get to university, and by that I mean the calculus sequence, real/complex analysis, abstract algebra and then some functional analysis at undergraduate level (dealing with Hilbert and Banach spaces etc). Of course, this will depend on my "intellectual taste" when I reach that stage - it may be that I change my mind down the track. I also believe it's entirely realistic to think that way.

I'm just going to go with the flow: multi-variable calculus sounds interesting (especially, when by the sounds of things, I'll get to look at things in "n-dimensions" - now that's appealing!).

Cheers,
Davin

P.S. Thanks for the reply, Cup, and pointing out the similarity between advanced calculus and real analysis. I suppose complex analysis wouldn't be much more difficult than real analysis? How much overlap is there?

 

[mercurial]

Hey Davin,

I'll bet you're still online, so I'll respond to your response before I go to bed. My suggestion of inviting someone to coffee to talk about math was meant exactly how you took it: someone closer to your own age will always give you the most honest advice. That's simply a social fact.

In any case I'm glad to read that you're already excited about learning more mathematics even before you've gotten to university. I didn't even think about mathematics as something I might like to study until I was finishing my sophomore year, that is, I was already 20. Incidentally I went on to get a Ph.D. in pure mathematics, which I neglected to mention in my last post so as not to lend the impression that that fact might possibly skew my advice. After finishing I wound up switching to something more applied and my current research focuses on computational physics. That's how you found me hanging out around here!

Whatever path you decide to follow, keep up the way you have been so far: let your passion and interest be your guide. You'll realize too that learning something completely new is just a matter of doing it -- even switching from pure maths to computational physics. Every one who's doing something scientific had to start from the very beginning at some point.

Yours,
Jerry

 

[Ulagatin]

Hi Jerry,

Congratulations on the Ph.D! I am interested to hear that you went from pure mathematics to computational physics - a fair leap I imagine (although I imagine the latter is probably a little less challenging mentally). What is it like adjusting to a different level of rigour?

My interest in mathematics is only fairly recent. Early last year (until about midway through), I was struggling with mathematics at high school (I was doing an extended mathematics course: year 11 pure mathematics). I knew that, since I wanted to be a physicist I had to pick up so I could pass the course and study pre-tertiary pure mathematics to get into a physics degree (I was quite concerned).

I then dedicated myself to getting higher marks in mathematics, and it was around this time that I read one of Ian Stewart's books (Taming the Infinite) which gives a brief overview of the history of mathematics and various fields such as algebra, analysis, topology and geometry (it's a brilliant book). Topology and (complex) analysis were the subjects that grabbed me most while I was reading the book.

I went from C's and C+'s to A's and B+'s at the end of the year, and towards the end of the year, I started with Apostol's calculus text. My mathematical ability has developed significantly over the year - in fact, almost immeasurably.

I also found that, once I started achieving at a good level in the subject, I started to appreciate it more deeply. And now it's grown on me! My mum knows how passionate I am about physics, but thinks I'll end up a mathematician! Quite a significant turnaround!

In one respect, the amount of mathematics I have yet to learn is daunting, but in another it is encouraging, as long as I retain a strong interest and passion for mathematics. I'm well aware it is a long journey, but what good is a journey if it's not enjoyable? Sometimes, as I see it, it can be just as important as the end result - the destination.

My dream would be to become a professor of theoretical/mathematical physics, doing research in high energy particle theory and cosmology. My ambition would be to do research in string theory and quantum field theory. Of course, I may never achieve this, but to pursue this dream, I have to have a very solid mathematical grounding, and so I am willing to take the initiative and work away towards that dream.

If I don't achieve it, then the mathematical ability and understanding I'll develop over the journey will not have gone to waste - obviously it will be very useful in many fields. I may also find an interest in other fields of physics, like computational physics, or solid state physics. I may go off on a tangent and do further training for engineering, or perhaps I could become a pure/applied mathematician. My passion will dictate, but to some extent, so will the economy.

Anyway, thanks again for your post and your advice, Dr Jerry! :tongue2:

Cheers,
Davin

 

[wsalem]

Hello,
I hope I'm not late in the discussion, please allow me to add a few thoughts, which are merely what I believe would have done if I were you,

First, (real line) analysis and Algebra are very different, in terms of proof techniques and ideas. I think you at this stage would benefit greatly from being exposed to the idea of an algebraic structure, techniques of obtaining new structures such as products, quotients, substructures, and of course the kind of mappings between such structures, and what it tell us about such structures.

Even if you're not going further in Algebra anytime soon, a solid knowledge of linear algebra should make other undergraduate analytic topics such as multi-variable calculus, functional analysis much easier to understand. It's hard to imagine doing quantum mechanics without linear algebra. You should devote a considerable time to be comfortable with matrices and the basics of linear algebra in disregard of the route you choose! I think working with Apostol should have given you some background but I have found Blyth's Basic Linear algebra, which can be covered quickly and also motivates abstract algebra, and there's Friedberg and Insel' Linear Algebra, which is more complete but still very accessible and rigorous.

In addition, you can't survive differential geometry and topology without abstract algebra, and even some notions in point set topology (a central subject for real analysis that is not algebraic) becomes easier to understand when you examine say the Zariski topology.

And it goes without saying that any "higher" study that is of geometric or topological nature requires a great deal of algebra (you always resort to Algebra when the situation gets difficult, in expense of losing (some) intuition!).


Perhaps the fastest route for applications in physics (and analysis) is the one that emphasizes linear algebra, group theory and the idea of symmetry, with the classical groups, (some) representation theory, linear and multilinear algebra being treated extensively, although it is highly geometric this approach requires a dedication to linear algebra and mathematical sophistication (which is not merely constructing sound arguments), an example would be the canonical Artin's Algebra, but also Knapp's Basic Algebra(perhaps with Stillwell's Naive Lie theory for some Calculus tossed in for the classical groups at an elementary level, and Armstrong's Groups And Symmetry for an easier geometric introduction). However if you're concerned about rings or commutative algebra you'll have to wait until later in the course.

Another interesting route is the historical route, via the study of (the insolubility by radicals of) equations and constructions by ruler and compass, i.e Galois theory.
This approach usually emphasizes Fields (and extensions), polynomials, groups of permutations. It could be more intuitive in that it (somehow) resembles school algebra, you can learn a great deal from reading the history of subject matter, and there's a major "bang" in the end, and with very little work, you'll find applications to linear differential equations. I can't think of a canonical text, but some texts that requires little prerequisites are Chambert-Loir's A field guide to Algebra, Stewart's Galois theory, Howie's Fields and Galois Theory, Shallow's Exploratory Galois theory, or perhaps Fraleigh's A first course in abstract algebra (with proper selection of chapters).

The third approach, which I would have taken if I were you at this stage, is the arithmetical route where Rings takes a central stage, first you'd be given a dose of elementary number theory (e.g factorization, congruence, euclidean algorithm) and some properties of polynomials, this would allow you to work on your proving skills while you get to meet quite familiar objects/ideas, the common properties of the integers are abstracted to form a Ring, you see how the classical results of elementary number theory such as factorization works in this structure, which results works in Rings with extra hypothesis, others in quite general rings, ... how about obtaining new structures (products, quotients, substructures, ..). You'll get to meet Groups when you study units and quotients. The major "bangs" here is seeing if there's an analogue results between the integers and the polynomials over a Ring (or more restrictive rings say Fields) such as factorization, the several variable ones should lead you to algebraic geometry--and algebraic curves (with complex analysis, you'd get a tour de france at the undergraduate level), and of course Algebraic number theory.

This approach should keep you interested to study further algebra and it will pave the way for cryptography and coding theory(since you're interested in computer science), algebraic number theory (if you're interested!) and algebraic geometry as mentioned earlier.

Some texts include Cameron's Introduction to Algebra (a well written undergraduate text that also covers Modules, Finite groups and Galois theory), Childs' A Concrete Introduction To Higher Algebra(Cryptography is a major theme), Chatter's An Introductory Course In Commutative Algebra, Irving's Integers, Polynomials, and Rings (the most elementary).


Surely you'll meet the major structures (except perhaps modules in some books) in these routes, and these routes will eventually overlap with each other. Anyhow, I admit that I have omitted many great books (and different routes), but my post is merely a personal reflection on the standard "flavorless" approach that merely introduce the major objects, prove a few theorems, provide a lot of disconnected examples and ends, of which there are many, one ends with no satisfaction of achievement and worse one haven't studied enough of the structure to be prepared for mature treatments. I would have likes being introduced to algebra with a goal in mind, and later move to texts where the emphasize is on the structure itself, say Mac Lane and Birkhoff Algebra, the detailed Hungerford's Algebra or the adorable and sexy Grillet's Abstract Algebra.


To summarize, I believe for someone like you who's self studying, you should choose a route that has a goal for you so that you won't get lost in abstraction, and I think the third approach is more manageable, and if you find it too easy, you can switch to other routes, or go ahead with a graduate text like the three I mentioned in the earlier paragraph.

 

[Ulagatin]

Hi Wsalem,

Early wakeup for me, but since I saw your post, I thought I'd take the time to reply.

Perhaps the best route for me, once I finish Apostol's calculus text (volume one) would be to do perhaps both - I could interchange between Apostol's second volume for multi-variable calculus and an abstract algebra text such as the ones you recommend (I doubt I would find it so easy that I would require a graduate text though!). I'm not THAT talented.

As long as I have a goal in mind, I think I should be OK though. Maybe I have been convinced to do some abstract algebra at this stage, or at least give it a chance!

Algebraic geometry would probably appeal to me. As you can probably tell, I like geometric ideas and intuition, which calculus brings to the table in spades. No doubt algebraic geometry would too.

If the third approach is the least abstract, then I agree, it would be the best to pursue. But for now, I think I've decided I will do some abstract algebra once I complete Apostol's first volume. This will probably allow me to develop my understanding of calculus further, and no doubt allow me to better understand multi-variable calculus (as one reader pointed out earlier, there is a fuzziness between advanced calculus and real analysis).

Topology appeals to me from the outset (provided it's not all set theory as it seems to be). Working from Apostol, set theory was perhaps one of my least favourite topics, but if it brings some geometric intuition to the table and paves the way for topology, then, fine.

Thanks again for a great post.

Cheers,
Davin

 

[wsalem]

Thank you,

What makes a set (or a topology) appear as a trivial and lifeless object is its lack of (a rich) structure, but once you impose some restrictions on it, it become richer, full of life, perhaps you'll have to wait till you meet it in analysis and algebra.

I think I have forgot to mention an important point: investing your time in Calculus was the smartest thing to do, the same goes to Linear algebra (it need not be a full fledged abstract algebraic treatment) as both are "backbones" (in terms of contents and techniques) for any serious study in physics or mathematics, pure or applied.

So, if doing Abstract Algebra will consume a lot of your time so as to hold you back from achieving a mastery of Calculus and Linear Algebra, then postpone it later.
If not, then, aside from Abstract Algebra, and after (single variable) Calculus, try Analysis on R, something stimulating like Abbott's Understanding Analysis, aside from getting better at sequences, series, uniform convergence and appreciate topology, you'll actually get to meet a lot of freaks out there, this alone should convince you why mathematical rigor is important and intuition is less so!
Ok, I need to shut up and let you enjoy Apostol's Calculus :D

Wish you best of luck.

 

[Ulagatin]

Thanks Wsalem.

Yes, I have Abbott's book - looks good! A friend of mine studying honours applied mathematics (4th year) recommended it to me.

Also, thanks for informing me about topology. I'll have to wait until I do analysis/algebra to come across it.

Time to start work on Apostol for the day!

Cheers,
Davin

 

[CMoore]

Hi Davin,

In my opinion, you could do no better in your choice of calculus texts. I frequently reference Apostol's calculus texts to remind myself of things I have forgotten. When you have worked your way thoroughly through them, you will be well-prepared to takle more much more advanced analysis texts such as Munkres Analysis of Manifolds and Spivak's Calculus on manifolds. After this, you will be well-prepared to begin differential geometry, tensors and the mathematics of general relativity (which, since you want to be a theoretical physicist, I assume you want to explore)

With respect to abstract algebra, I personally don't think there's anything more conceptually difficult there than in analysis. It is, I guess, a question of "mathematical maturity" as to whether you perceive it as difficult, but most introductory algebra texts don't assume a lot of prior knowlege. If you want a nice, easy - and fun - introduction to abstract algebra, I would suggest taking a look at Joyner's "Adventures in Group Theory". Joyner introduces and develops enough group theory to solve Rubik's Cube and other mathematically-oriented puzzles. But, keep in mind, it's a real math book - despite the fact that I was actually able to purchase it at Borders!

 

[Ulagatin]

Exciting stuff, CMoore! I greatly look forward to the advanced mathematics I am yet to learn such as differential geometry.

I will take a look at that book. I want a text that is fun - sure I like learning seriously (Apostol-type text) but I also like to have some fun of course.

Cheers,
Davin

 

[wsalem]

Davin, take a look at the MAA review of the book just mention, at "http://www.maa.org/reviews/joynergroups.html"

If you've had a couple of semesters of algebra, you'll probably find the book entertaining, mathematically speaking. If you haven't, then you won't get that much out. While the algebra is (mostly) self-contained, it is presented quickly, and the author gives just enough to do what he wants. So don't expect it to read like an algebra textbook.

...The audience seems to be mathematicians with a fairly good knowledge of algebra and an interest in permutation puzzles. If you fall into that category, this book is for you.

A manuscript of the second edition is provided at the author's web-site, at http://sage.math.washington.edu/home/wdj/rubik/agt2_0b.pdf

Also, the purpose of a first course in Algebra is to provide a sufficient training for handling algebraic structures in general, which will not be attained with Analysis(or mathematical maturity!) alone, even if(for the sake of the argument) Algebra was easier that Analysis!