Is Pure Math Useless?

1. Jan 11, 2009

khemix

I'm a little dissapointed with the Real Analysis I recently learned. For one, I don't remember 90% of the results. Also, while it was a great mental exercise, I don't feel it enriched the way I view calculus in any way. It seemed like a technical exercise. Kind of like jazz music - just musicians showing off (atleast I find anyway). I know for a fact none of the originators of the theories or theorems discovered the results the way that is presented and proved. It seems like mathematicians have grabbed a hold of it and destroyed all intuition just so they can dazzle me with their algebraic tricks. Why are geometric results ofted discarded, and instead a 2 page proof is presented involing sophisticated sums and what not when a simple cartesian plot would suffice.

It has not helped me understand physics in any way. I no longer view integrals as sums of differentials, but as the unique number lying between two sums. Useless for application, as well as the way I now view chain rules and derivatives. I would like to give math another chance, but is there a point? My intention was to get insight into calculus and the real numbers. Instead I got a bunch of inequalities that I don't think I will ever comprehand. This is in contrast to the calculus I found in elementary books, whose results I remember clearly to this day. I mean is there anything out there that pure mathematics actually developed? Instead of taking someone elses pure idea and stampting their boot on it.

Last edited: Jan 11, 2009
2. Jan 11, 2009

Reedeegi

Keep in mind that it is the development of such ideas that lead to new ideas, such as topology, that have value in areas such as physics, economics, and even surveying. Pure mathematics is an abstract way of thinking that I believe will always have an application, no matter how abstract. Take, for example, category theory. This subject is often called "abstract nonsense" yet can be utilized in computer science.

While physical implications may not be readily apparent, applications will always be found.

3. Jan 11, 2009

khemix

Topology is way too technical and is quite useless itself. Saying it is used in physics is a stretch. String theory is one of the only fields that seriously applies it, but even the applicability of string theory is questionable. Even in other places it lingers like EM it isn't of fundamental importance, more like a specific case example.

The places where purely abstract math finds its uses are themselves minimally useful. I just don't see pure math as having done anything for the world. All it does is reforumlate results into more elegant form (and elegant turns out to be obfuscated more often than not). Maybe I'm wrong, and hopefull I am.

4. Jan 12, 2009

dvs

So, having just completed what sounds like a first-year analysis course, you think you are in a position to judge the entirety of pure math? And, in particular, you think you're qualified enough to say that topology is useless (when you probably don't even know what topology really is)? This is humorous.

5. Jan 12, 2009

Steely Dan

Sir, I insist that you have no idea what jazz is. Jazz music is not about musicians showing off. You probably saw some people take solos in the middle of songs and thought they were just being showoffy but that's not what it's about. People do that because the audiences like hearing it. I always enjoy hearing a great solo. If you don't enjoy hearing music, then you would think all music is bad, not just jazz. Also, jazz music is the toughest music there is - the most technically demanding, at any rate. Listen to some classical jazz, some Ellington or Basie or something - that's what jazz is really about.

Also, pure math isn't useless if its innovations can be applied in other fields.

6. Jan 12, 2009

khemix

Its only my opinion. Yes, I know I'm not expert in any of it. Maybe some day I will be. Maybe I wont. What I seek now is a reason to justify studying it (or atleast taking more courses in it). So far, with my introductory studies, I found none. Only aesthetics. This is in contrast to how useful calculus and geometry were.

Perhaps you can enlighten me. By the sounds of it, you should be more qualified.

I don't mean to bash jazz music, I just think its technicallity for the sake of being advanced and at the cost of melody. Neither am I bashing math, I just don't see it as being practical. But what innovations has pure math given? Based on my experiences (which are limited Ill admit) it seems pure math just crosses the t's of already known results.

7. Jan 12, 2009

snipez90

I think you are being unfairly critical in your first paragraph. You seem to have taken for granted the relative ease of access to certain ideas in mathematics that took many decades to develop. You say that intuition is often abandoned in the presentation of ideas (while unfairly generalizing this behavior to all mathematicians), but how hard have you tried to make the abstract proofs appeal to your intuition?

You seem to suggest that pioneering mathematicians constructed many geometric arguments to aid in their intuition, before presenting an abstract proof. Yet it is generally agreed upon that a diagram or picture does not constitute a proof. A rigorous proof ensures that our intuition is actually correct and reliable.

For instance, I'm not sure if you are familiar with the Three Hard Theorems from Spivak (IVT, A continuous function on a closed interval is bounded and also has a maximum/minimum value), but in each of these theorems, the geometric argument is quite clear, perhaps some would say undeniable for the IVT. My initial impression of the proofs (which are given in the chapter on least upper bounds) was dismissive. The mechanics of the proofs did not appeal to intuition and I thought they weren't worth the trouble.

However, I made an effort to gain a better understanding of the proofs (admittedly, about 5 days before the exam). I read the end of the chapter on the Three Hard Theorems carefully. There, Spivak presented the motivation behind a rigorous proof of the Intermediate Value Theorem. At the same time, he explained the necessity of the existence number that would bound all the elements of a set to complete the proof. Soon it became clear to me why the LUB property is an axiom. Afterward, the proofs of the Three Hard Theorems were much more clear. In fact, I think the proofs improved my intuition (except for maybe the maximum value proof in which Spivak used a trick to reach a contradiction using the 2nd hard theorem) rather than hindering it. They presented arguments that were not as far away from the purely geometric arguments as I had imagined.

Now perhaps the biggest flaw with my example is that Spivak does everything for the student. Although I'm not taking real analysis until next year, I'm not blind to the fact that certain real analysis books, if not all, will not have nice diagrams and geometric arguments. However, I have faith that it is still possible to obtain intuition and motivation in the geometric sense or otherwise. I understand that this will take more work on my behalf to fill in what has not been presented. My rationale is that from doing the problems in Spivak, whether assigned or not, I am involved in the process of taking an intuitive idea and making it rigorous.

I think my second example will address some of the other questions you raise. I am currently working through Chapter 13 of Spivak, the chapter introducing integration. I believe what you're referring to as "useless for application" is known as Darboux Integration.

As a high school senior last year, I took Calc BC. We proved almost nothing in the class. My intuitive grasp of limits and derivatives was gained in precalc, and somewhat reinforced in Calc BC. I really cared about how integration would be presented, since my senior friends who took the course the previous year all claimed that differentiation was much more intuitive than integration. The presentation of the theory of integration lasted maybe 10 minutes, probably less. Essentially, the teacher drew some rectangles on the board, threw us a definition, and never referred to the definition again.

Soon the FTC was introduced and they were a bit easier to grasp through concrete, numerical examples, although no proofs were given. I quickly learned how to integrate and the rules and tricks came easily to me. Eventually, I held onto the notion of the sum of differentials to aid my intuition, even though this concept did not affect my ability to integrate at all. I retained the idea to comfort my intuition, but I couldn't really use it to do anything else.

Although I haven't proved many of the chapter theorems using the darboux integrability criterion, I can already see why the definition is useful. First of all, it made more sense than the other definitions I have encountered. I mentioned that my Calc BC class threw a definition that was not really used. I soon looked through Stewart's and was somewhat satisfied with his presentation involving sample points and taking a limit. By then, I figured the concept of integration was something like "we add more and more rectangles while making the dimensions of the rectangles smaller to approximate the area".

Spivak's presentation reinforced my intuition. The base of the rectangle was determined by partitioned subintervals, the height determined by sups and infs of f on the interval. I learned what it meant to "add more rectangles" by understanding why L(f,P) <= L(f,Q) where P is a partition with a lesser number of points than Q (similar idea for upper sums). Then the use of sup{L(f,P)} made sense (and also inf{U(f,P)}. It was easier to see how adding more and more rectangles that become smaller and smaller captured the true area from above and below. The application of the definition to many simple functions convinced me of the definition's usefulness.

So is this definition more useful to me than the notion of "summing the differentials"? Until I learn the rigorous formulation of the latter process, yes. I repeat that viewing integration as summing differentials was of no use to me when I think about it. Now I have a definition that makes intuitive sense to me, can be used in key proofs, can be used to prove certain problems by itself, and probably used to numerically approximate the area under the curve.

I don't expect you to agree with the last paragraph, and that's perfectly fine. But to me, there is a very strong rational basis for sticking with the darboux definition, partly because of its practical usefulness. I find that my intuition is improved with the definition and that the definition itself is applicable.

On a final note, I suggest you try taking a look at Understanding Analysis by Stephen Abbott. Although it might not be as comprehensive as other real analysis books (indeed, it is an "intro to analysis" book), I would place it more towards "analysis" than "theoretical calculus with rigorous proof".

Last edited: Jan 12, 2009
8. Jan 12, 2009

snipez90

Hmmm, my post turned out to be longer than I expected. My main focus is not to argue about the application to other disciplines, a topic I know little of as of right now. However, I am wondering if we are thinking about the same intuitive/geometric notion of summing the differentials. For me, this is something I can picture, but something I can't use to do much of anything because I haven't sought a rigorous treatment of the subject. I mean I can still integrate by mechanically doing the algebra ("symbol pushing"), perhaps making an appeal to the notion of differentials but it's not necessary.

My main point was that pure math (what little I know of it) need not obfuscate intuitive concepts. A more rigorous treatment of calculus has helped my intuition, and provided me with definitions that I can work with. From what I can infer, darboux integration, a component of such a treatment, is encountered in analysis as well.

I'm curious about how you view chain rule and derivatives as well. From my experience, the calc book used in BC gave a terrible explanation of the chain rule (some analogy to gears) with no proof. I forget Spivak's intuitive explanation of the chain rule, but nevertheless I found the proof more involved than proofs for the other differentiation rules, but not much more.

In general, the concept of the derivative presented in Spivak was more or less the one I was presented in high school. Short of going through a numerical example, Spivak mentions the physical meaning of the derivative. I think Feynman's treatment of the derivative in Volume 1 Chapter 8 is very good. Probably the best result of relearning differentiation in a more theoretical setting is determining whether certain not so simple functions are differentiable and where.

Needless to say, my view of limits has changed considerably, for the better I think.

Last edited: Jan 12, 2009
9. Jan 12, 2009

Gib Z

To the OP - it seems you are learning maths as a tool to physics, which is your main interest. If that is the truth, then I would have expected analysis is not your cup of tea, and don't expect you to bear too much interest in it. My advice would be to go to another physicist who is senior to you and ask them what particular results in analysis (the ones connected to topology I will expect) that you will need for your future applications or whatever it may be, and keep those ones in mind. To some extent, you will have to bear with it, but I'm sure you will come to appreciate taking the course later.

As an aside, I love maths and don't necessarily need an application to find beauty in a result. Whether or not it is useless is sometimes answered by "no, because we will find applications for it later" but really that is not the best answer. Hardy discussed this in detail (he was a pure mathematician) and you could read something up on him if you desire. Basically, Ill attempt to sum up both his and my views on the matter. Some people do painting, some write music, some play a sport. Do they have real world applications that necessarily solve problems? Not quite so, but we still do them because we enjoy doing them, and don't need an application to justify ourselves doing it. In very much the same way, I regard Pure Mathematics.

10. Jan 12, 2009

epenguin

With respect I disagree with this. Now I see Gib Z has state some of my reasons.

The OP was not presumptuous to be slammed down, and to say keep at what you are not seeing the point of (though he seems managing to do it) it and after five years brainwashing you will see the light is not an answer really. After all, even if he is wrong surely there is many a Professor of physics, engineering, chemistry or economics who thinks the same? Doh I see, you're right after all , but still it raises a question about syllabuses and teaching.

I wonder if a lot of the problems that students bring here aren't results of them taking a narrow-eyed concentration on symbols and formalisms than a wide-eyed overall intuitive vision which should be encouraged not discouraged, though I am expressing myself badly as it is a vague idea.

11. Jan 12, 2009

tim_lou

Without analysis, how do you understand Stokes' theorem rigorously? How would you use Stokes' theorem in terms of differential forms on manifolds?
Without topology, how do you understand contour integrations?

How would you understand (0 is inside the sphere S)
$$\oint_S \frac{q}{4\pi r^2} ยท d^2S = q$$
when standard calculations give
$$\nabla^2 \frac{q}{4\pi r} =0$$

What about manifolds? Lie groups? How would you understand what a representation is and how to calculate them? How would you gain an understanding of what a tangent space is when living on the manifold?

Physicists often talk about analytical continuation, poles, singularities, symmetry groups and many many more (because I don't know enough physics yet!). Even though sometimes they can get away with non-rigorous arguments, occasionally, you get some tricky singular cases that require more care. Without rigorous tools, it may be hard sometimes to show what the right answer is. Furthermore, if analysis doesn't give you a better sense of what those epsilon expansions that is done almost a million times in physics are, I don't know what else does. And don't tell me that you understand the uniqueness of differential equations, inverse function theorem that is used (another million times in physics without justification) without analysis.

Last edited: Jan 12, 2009
12. Jan 12, 2009

Tac-Tics

Physics-grade calculus is knowing how to drive a car. Mathematics-grade analysis is knowing how to build one.

Physicists work on a very small playing field: analytic functions defined on three-dimensional Euclidean space plus a few "band-aids" when the mathematical model they know doesn't quite fit. It's probably true that to do most of physics, it takes little to no understanding of analysis.

Analysis is for people who are interested in finding a solid theory of calculus. Think of it this way. Let x be a real number and f be a real-valued function of one real variable. We know that f(x) (that is, applying the number x as the parameter for f) is also a real number. But what is "dx"? It's intuitively described as a "very small value". But what is it really? It is a weird entity: neither a number nor a function. To a physicist, it's a mental cue to remind him how to take the integral. To a mathematician, it's a historical wart.

Analysis is a good starting point to understand mathematics. One of the most important lessons mathematics has to offer, in my opinion, is that as humans, we often fail to see the subtle complexities of topics we have an intuition over.

We all know sqrt(2) is irrational. That is, there are no integer a and b such that (a/b)^2 = 2. But what about other irrational numbers? What is the nature of an irrational number? Rationals are simple. The integers are taken as fundamental, and a rational number is a pair of integers satisfying certain conditions. The rules for how to add, multiply, and compare rationals is straightforward. But what about the reals? How do you "construct" a real number? If I have two irrational numbers, how do I add them? Or multiply them? If I add two irrationals together, can I get a rational number back? What if I take the diagonal of a square with irrational-measuring sides? Will it be rational? Irrational? Or maybe something more exotic like a super-irrational number? You don't know until you come up with a concrete definition for the real numbers. And that's chapter 1 of your analysis text.

Continuity is a central concept of analysis. We are taught in high school (or maybe even junion high), that a continuous function is one whose graph can be drawn without lifting the pencil. It *seems* simple, but it's not. The standard epsilon-delta definition works pretty well, but when you're working in the rational numbers instead of the reals, you suddenly admit pathological functions. For example, f:Q->Q where f(q) = {0 if q^2 < 2, 1 if q^2 >2}. Under the epsilon-delta definition of continuity, f is continuous, even though it takes on no value between 0 and 1. Uncool.

So what analysis tries to do here is pin down what we *really* mean by continuous, without appealing to our intuition. It's a game. Just as physicists try to pin down what happens to a particle in terms of high-level calculus, the mathematicians try to pin down what happens in high-level calculus with set theory.

It's true that doing epsilon-delta proofs all day long will net you little. I've been doing math as a hobby for a number of years now, and I still have to scratch my head to get the exact formulation down. But it's really the idea that's important, not the details. Proofs are a superb way to exercise the idea, while at the same time showing the near-infallability of math.

The central concept of analysis is the limit. The idea is simple: if you want accurate output, you need to provide sufficiently good output. This idea grows like a fractal all over math! The limit of a sequence is a number which the sequence can get arbitrary close to. The real numbers are just limits of rational in disguise. Limits of functions are used to create derivatives, integrals, and a hundred variations on both. We can talk about series and suddenly we have things like infinite sums and analytic functions and functions defined in terms of their harmonics.

The intuition behind mathematics is, in my opinion, the most important part. Studying it as a hobby, I have the luxury of doing only as many problems as I feel like. I can imagine how taking it at a university might affect your opinion. For sure, when studying math, you can reach esoterica quite quickly. But the fundamentals of set theory and basic analysis are useful and healthy.

13. Jan 12, 2009

Dragonfall

Real analysis is the opposite of jazz.

14. Jan 12, 2009

LightbulbSun

There are plenty of applications for pure math. There's another way to look at it too. Math teaches you to be more logical in your thinking. Unless you hate logic too?

15. Jan 12, 2009

symbolipoint

That is not well satisfactory toward the original complaint from khemix. He wants to know how Real Analysis or Pure Mathematics directly helps understanding & problem solving in the real world. Recheck that he just finished a Real Analysis course and does not find the topics relavent to realistic problems. Would students with more experience and mathematicians give an answer which do not rely only on learning logical thinking, finding Math as beautiful, or how Math relates to art? The answers to khemix's question very likely exist, but what are they? I suspect that intricate sequences & series may be one element from the set of possible answers since they can give ways to calculate values for functions - but my own Math background is fairly limited. Experienced students and Mathematicians, tell us more!

16. Jan 13, 2009

vanesch

Staff Emeritus
It also depends what kind of physics you're interested in. If it is "engineering physics" (that is, down to earth physics with which you can do some "simple" calculations to see how a certain design will work, and eventually how it can be improved, and how to do the numerical work that goes with it etc...) then probably you are right.

You could say that mathematics is to physics what grammar is to language. You can talk and write mostly without knowing a lot of grammar. However, if you want to study, say, ancient greek or something, then you will be totally lost without grammar - simply because you cannot count on your intuition anymore. With math and physics, it is the same.
We have, through our visual sensory system, a very good intuition of 3-dim euclidean geometry, and that's most of what's needed for "down to earth physics". So no need to go into the formal and the abstract. And for many applications, that's good enough. But there are things where you can't count on your intuition anymore, and then a more solid training is the only way out.

That said, if you don't see the point, and if what you want to do in your life won't depend on it, then don't study it. You can live very well without real analysis :-) 99% of the world population does.

17. Jan 13, 2009

lurflurf

I would not really call that living.

18. Jan 13, 2009

Gib Z

LOL hear hear.

Though he has a point =]

19. Jan 13, 2009

vanesch

Staff Emeritus
:rofl:

20. Jan 13, 2009

deiki

Geometric intuition can lead you to algebraic conclusions as well as algebraic calculations can lead you to geometric conclusions. The problem is that many geometric results are just special cases of algebraic computations, and you will often need algebraic results which can't be described geometrically.

Also, a good mathematical foundation will allow you to draw generalizations which will save you from having to learn tediously tons of special cases from scratch.