Pure or Applied Math: Which Path for Deeper Understanding?

[Howers]

I would like the perspective of either a physicist or a mathemtician. Please indicate which you are before posting.

When learning mathematics, is it better to follow a strictly rigorous approach? By this I mean such things as being exposed to Calculus via Spivak, Complex variables by Ahlorfs and so on. Or rather that an intuitive approach (with some theory) be employed, with the likes of say Salas for Calc and Fisher for Complex then re-learned by the pure counterparts.

With the rigorous approach I see the benefit of being exposed to everything at once. This let's you take the math more seriously, and leaves a more lasting impression. You will not ask "why this is so" because everything is proved from the ground up. Not to mention all the extra practice you get with proofs and theory, which should prepare you for upper texts which follow that format. On the downside however, you lack much or all intuition. You may not even be able to apply your knowledge to a very simple problem. In a lot of cases, you may not be able to appretiate the theory and be put off by a dull approach. And as I've learned with Linear Algebra, the theorems don't stick at all!

Obviously with an applied approach you will learn more rapidly and get to fundamental results. But are you just learning bad habits?

The reason I ask is because I am debating whether to enrol in pure math courses, or in the applied ones. In the end I intend to learn the pure math, but is it better if I've had applied exposure? That way I can appretiate the theory more? Or should I start the right way from the start?

Spivak sure was fun when I already knew Calc 1... but would it have been fun without...

 

[mordechai9]

Mathematics is the ultimate interdisciplinary subject. It deals with the most generalized models and relationships, and it applies to almost everything. Hence, mathematics is applied mathematics by nature. So, you should always attempt to learn mathematics in the most applied way possible. But, if this means that you won't be seeing proofs or rigorous mathematics, than forget about it. Ideally, a mathematics course will have plenty of theorems, proofs, and rigor, but also many applications.

In other words, I recommend course which are both pure and applied in nature... figure out which ones those are...

 

[Howers]

When you speak of generality, I don't see applications. That sounds more pure to me. And what of math that cannot be applied?

 

[DavidWhitbeck]

Howers, applied math is not necessarily fast and sloppy. So let's just leave your generalizations to the side. You should enroll in both types of classes because now is the time to soak in a broad education in mathematics, you can worry about specialization when you're in grad school. Do you want to be computing or proving? Or do you enjoy both? Actually there is a role to play for both types in either pure or applied mathematics. You can work on existence theorems of pde's that arise in physical systems, applications of group theory to biology etc etc and that all would seem to be "pure" but is nonetheless applied.

 

[PowerIso]

Before I can even begin to see real meaning in a theory, I need to understand how the theorem works. That goes with pure or applied. If I don't have a good intuitive understanding of an idea, learning the theory won't do me much good because I won't be able to extend the theory. I can't even begin to ask the question, "why does it work." if I don't know how it works.

I think, like most things, it's about balance. Knowing your theorems are important but you should also know how to use them.

Also, I think people have funny views on what applied mathematics is. You'll prove a lot of things, and not everything in applied mathematics is plug and chug and read a theorem here and there.

Do you plan to learn mathematics so you can be a mathematician? Or learn mathematics so you can use it as a tool?

 

[Hurkyl]

Mathematician here.

"Rigorous" does not mean "lacking intuition". Similarly, "applied" does not mean "lacking rigor".

The usual way to gain intuition comes from doing lots of exercises. And when you understand a subject, you can usually make a direct translation from intuition to rigor. Rigor without intuition may be difficult, but correct. Intuition without rigor is quite dangerous: you run the risk of firmly implating wrong ideas into your head.

 

[Howers]

Howers, applied math is not necessarily fast and sloppy. So let's just leave your generalizations to the side. You should enroll in both types of classes because now is the time to soak in a broad education in mathematics, you can worry about specialization when you're in grad school. Do you want to be computing or proving? Or do you enjoy both? Actually there is a role to play for both types in either pure or applied mathematics. You can work on existence theorems of pde's that arise in physical systems, applications of group theory to biology etc etc and that all would seem to be "pure" but is nonetheless applied.

Therein lies the problem. I can't enrol in "both courses". There are either the computation approaches, which I consider applied; or there are the theoretical approaches, which is all proof. So I don't know whether I should go with the pure or applied approach.

Im not really referring "pure" applied math. Rather to things like ODE and calculus, which can be taught via theory or computation. What I'm asking is which approach is better.

Before I can even begin to see real meaning in a theory, I need to understand how the theorem works. That goes with pure or applied. If I don't have a good intuitive understanding of an idea, learning the theory won't do me much good because I won't be able to extend the theory. I can't even begin to ask the question, "why does it work." if I don't know how it works.

I think, like most things, it's about balance. Knowing your theorems are important but you should also know how to use them.

Also, I think people have funny views on what applied mathematics is. You'll prove a lot of things, and not everything in applied mathematics is plug and chug and read a theorem here and there.

Do you plan to learn mathematics so you can be a mathematician? Or learn mathematics so you can use it as a tool?

I'm planning on being a physicst. But I don't believe the view of many that mathematics is merely a tool... in fact, I don't know where this notion stems from ( I've only seen it in Griffiths EM book). I view math as a fundamental basis and a language that phyicists use to communicate ideas. So that's why I think having a pure background will give greater insight into the physics.

 

[PowerIso]

Hm, well I can't give you any insight then. I only know how to view math from the special viewing glass of mathematics.

But I can say this. English is a language used to communicate ideas. We know grammar, spelling, idioms, etc and we can communicate to each other with that. However, we don't need to know gerunds, imperative statements, infinitely long structure, post-modification to be great writers, but you do if you plan to be a linguist.

In the same way, I don't think you need to know mathematics exactly how a mathematician knows it to be able to communicate your physicists ideas. Will you get some great insight? I don't know, but it is possible, however, I don't think it is needed per se.

 

[DavidWhitbeck]

Therein lies the problem. I can't enrol in "both courses".

Why? I don't believe you. It would be hard for me to imagine a math degree where one did not take one single class that revolved around proving theorems. In fact I can not imagine a math degree where you did not have to take analysis and algebra. It would be like a physics degree without mechanics and e&m!

Since you are a physics major I'll tell you what classes I found useful, and what I should have but didn't:

Complex Analysis: the calculus of residues are the bread and butter of the Green's functions you find in quantum field theory, very useful.

Partial Differential Equations: if only to learn the ins and outs of Fourier analysis, it appears in more physics classes than I can count.

Numerical Analysis: the toy problems you solve in lower division math are not what you encounter in research or even advanced grad level classes. Understanding and gaining experience in numerical methods is extremely useful. Numerical methods are as clever and sophisticated as an elegant proof, and is worth the time studying.

Abstract Algebra: Algebra appears all over the place in physics, so the undergrad class should be useful but it's not! Physicists need to learn group representation theory and Lie algebras since those topics occur over and over in quantum mechanics and field theory. And guess what? You almost never see this taught in an undergrad math class. Since both physics and chemistry majors need to understand these topics you think math departments could try to teach algebra using more than just Z_n as examples. But apparently they can't be bothered, so go figure...

Differential Geometry: This is obviously extremely useful if you want to learn General Relativity. Sadly enough I'm also going to say pass on this, because the undergrad classes many times just teach it as advanced vector calculus, and then it's nearly useless. It's another case where the real class is at the grad level.

 

[Howers]

What this thread is about is whether to take a theoretical approach to math or a computational. I can't take say Complex Variables and then Complex Analysis, because they are equivalents but differ in approach. Analysis being geared for the pure mathemetician.

If I were to enrol in a pure counterpart of every math course I'd do, it would be like retaking ever math class and I'd be in undergrad for 6 years.

 

[MathematicalPhysicist]

in this case take the pure courses, obviously you'll have there examples and exercises (along computing integrals, a great feature jk).

I don't know of a maths/physics/logic course which doesn't have some exercises in it, on what the exam would be, only proving known theorems?!

 

[zhentil]

The answer seems pretty clear to me: take the applied course if you can't handle the abstraction/level of the pure course. If you want to understand why a theorem is true, read the proof. Courses that teach "why" a theorem is true are usually designed for people who can't understand why a theorem is true.

Taking your example, I learned complex analysis with Ahflors (fantastic, btw). There were a lot of exercises on contour integration, conformal mapping, etc. The difference between Ahflors and an applied complex textbook is this: in Ahflors, there are five exercises on contour integration, all of which require a different method that the reader needs to develop himself. In a lower-level text, there are 50 exercises that apply one of the methods the author already showed (and in many cases, the reader is given a hint as to which method to apply).

I chose complex as my example because it's so clear-cut: you will learn everything in a pure class that you would learn in an applied course.

A different, more complicated question is PDE. In a pure PDE course, taught from, say, Evans, you will learn existence and uniqueness proofs. You will learn the general theory of Green's functions. However, if you're a physicist, you will not learn Green's functions in 8 domains, nor will you learn spherical harmonics or using Fourier series to solve the Laplace in a cylinder. However, I would still recommend the pure course, if you can handle it, for the simple reason that you will be prepared to develop whatever techniques you will need on your own.

 

[zhentil]

Just one other comment (I'm a mathematician, btw). You said that people who learn math the rigorous way often can't apply their knowledge to a very simple problem. I'm not sure where you got this idea, but it's simply not true, unless they're just memorizing the statements of theorems.

 

[DavidWhitbeck]

However, if you're a physicist, you will not learn Green's functions in 8 domains, nor will you learn spherical harmonics or using Fourier series to solve the Laplace in a cylinder.

Well you have me on the 8 domains thing but...

(1) Spherical harmonics--> wavefunction of the Hydrogen atom
(2) $$\nabla^2 \phi = 4\pi \rho$$ in cylinder--> electric potential inside a charged cylinder
(3) Green's functions in field theory involve complex integration over several variables

I'm especially baffled as to why you say physicists don't learn spherical harmonics when they show up in the most popular quantum mechanics solution that all physicists and chemists learn!

 

[MathematicalPhysicist]

well, you know how they learn it, not through rigorous acount as mathematicians.

If you ask me, if you have a chance to learn maths rigorously, then do it.

only then you can fully grasp the maths being used in physics.

 

[Howers]

Just one other comment (I'm a mathematician, btw). You said that people who learn math the rigorous way often can't apply their knowledge to a very simple problem. I'm not sure where you got this idea, but it's simply not true, unless they're just memorizing the statements of theorems.

I am basing that on books like Spivak Calclus & Calculus on Manifolds, which if I'm not mistaken serve as pure books for Calc I,II,III and vector analysis. I've noticed they miss out on a lot of application when compared to some lower level books. Having originally learned the material from lower counterparts, this wasn't a problem for me and enriched the experience. But I don't know how confident I would have been with calculus if I used Spivak orginally.

 

[zhentil]

Well you have me on the 8 domains thing but...

(1) Spherical harmonics--> wavefunction of the Hydrogen atom
(2) $$\nabla^2 \phi = 4\pi \rho$$ in cylinder--> electric potential inside a charged cylinder
(3) Green's functions in field theory involve complex integration over several variables

I'm especially baffled as to why you say physicists don't learn spherical harmonics when they show up in the most popular quantum mechanics solution that all physicists and chemists learn!

I stated that awkwardly. I meant that physicists want to know about Green's functions in different domains, because of their association with physics. You won't learn that in a pure PDE class.