Is Pure Mathematics a Waste of Time?

[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.

 

[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.

 

[khemix]

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.

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.

 

[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.

 

[Steely Dan]

Kind of like jazz music - just musicians showing off (atleast I find anyway).

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.

 

[khemix]

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.

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.

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.

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.

 

[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".

 

[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.

 

[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.

 

[epenguin]

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.

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.

 

[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)
$$\oint_S \frac{q}{4\pi r^2} · d^2S = q$$
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.

 

[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.

 

[Dragonfall]

Real analysis is the opposite of jazz.

 

[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?

 

[symbolipoint]

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?

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!

 

[vanesch]

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.

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.

 

[lurflurf]

You can live very well without real analysis.

I would not really call that living.

 

[Gib Z]

I would not really call that living.

LOL hear hear.

Though he has a point =]

 

[vanesch]

I would not really call that living.

:rofl:

 

[deiki]

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.

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.

 

[Tac-Tics]

Furthermore, you can easily see where special cases refuse to generalize.

R^n, the n-tuples of real numbers, forms a metric space under the Euclidean metric. However, R^infinity, which can be interpreted as the infinite sequences of reals, does not. Why? Because the point (1, 1, 1, 1, ...) is infinitely far away from 0.

 

[uman]

Pure math is useless in the same sense literature is useless. If you don't like it or don't see the point, study something else.

 

[tim_lou]

Pure math is useless in the same sense literature is useless. If you don't like it or don't see the point, study something else.

Problem is, he's partly saying that pure math is useless in the context of Physics. Which is just absolutely wrong.

 

[Werg22]

Actually the real and only objective value of any piece of (pure) mathematics is the depth and breath of its applications! whether it simplifies things, or extends them.

 

[uman]

I respectively disagree: math is beautiful for its own sake, in addition to for its applications.

 

[CRGreathouse]

Actually the real and only objective value of any piece of (pure) mathematics is the depth and breath of its applications! whether it simplifies things, or extends them.

And in similar news, the Mona Lisa was finally put to good use, patching a leaky roof.

 

[khemix]

Ugh... I think I got my answer.

I didn't mean math is useless in a general sense. I meant in terms of applications. The only answers I've received were along the lines of "no way... math is very important". While I agree there is a certain beauty and elegence to it, it becomes more of an art at that point - where nitpicking is done for the sake of being done. This alone doesn't justify taking more courses in it. Summer reading? Sure, I like the mental exercise. But instead of taking say a course in inorganic chemistry or computer science? Why bother...

I mean I like poetry and music. But its completely useless. So I won't take courses in it. Math I thought was different... but again, its in the definition. Pure math is motivated for reasons other than application.And I am particularily criticizing modern math. I think its important to prove all theorems in calculus, but real anaysis is not needed for that.

 

[lurflurf]

Good thing you found out now in time to change your major to accounting. Also be sure not to take any more physics classes. Theoretical physics is motivated for reasons other than application.And I and paticularly criticizing modern physics. Electricity, heat, optics, sound, gravitation, mechanics and quantums will never be used for anything. I mean sure I like to sit in a warm room, listening to music from an electric device being run by nuclear power, and looking around with binoculars, while not floating away, but physics is not needed for that. I read physics book once, it was all equations and priciples and weird symbols. I don't want to know how a television works just where to plug it in and how to turn it on.

 

[Hurkyl]

Hah. I was just going to suggest that he try and do all his calculus problems by manipulating inequalities, rather than using more elegant and beautiful techniques like derivatives and integrals. But your reply is better.

 

[Doodle Bob]

Ugh... I think I got my answer.
... I think its important to prove all theorems in calculus, but real anaysis is not needed for that.

Actually, you apparently didn't get your answer, given the second sentence above.

Real analysis, at least at the entry-level, is the adult version of calculus. It is needed to prove all of the results that are typically used in calculus. In fact, that is the reason for its existence: to give calculus a logical foundation from first principles. It is necessarily unwieldy and often quite difficult the first time around.

However, the beauty of it is that once those analytic foundations have been set, it opens up many generalizations of calculus from measure theory (which leads to probability theory which leads to current models of quantum physics) to differential geometry (which leads to Riemannian and semi-Riemannian geometry which leads to special and general relativity).

Real analysis also leads to complex analysis as well as functional analysis, both being mathematical disciplines that have been used extensively by physicists.

However, naming specific applications of mathematics in physics and other areas disguises the more important aspects of learning advanced mathematics, which is simply to understand mathematics from as deep a level as possible. With a greater understanding of how the mathematics actually works, you have a greater understanding of which models would be most effective and you have a greater ability to generalize concepts and make connections with seemingly disparate fields.

BTW a great deal of mathematicians are perfectly happy in declaring their subject "useless."

 

[Tac-Tics]

What use is a use?

In fact, what does 'useful' even mean?

People keep saying art and music are 'useless' in this thread. Maybe the use is in the joy they bring to people. Or the feeling of accomplishment they bring to the artist. Why be so literal about the meaning of the word? It's not like knowing chemistry is any more useful than knowing how to draw. How often in life do you perform quantitative experiments in your home or work? I'd say there is probably more use to a drawing class than a chemistry lab, because the former you can at least practice on your own.

Just playing a little devil's advocate.

The one aspect of pure math I like is its compressibility. Memorization is hard and time consuming. The more truth we can encode into something, the better. Pure mathematics is very, very good at this. The reason is that one of the unspoken principles in mathematics is simplicity. The axioms are few in number. General theorems trump special cases. Special exceptions to the rules must all be declared ahead of time. Even our books are terse to the point it's often hard to rebuild the main idea the author had when he wrote it. Personally, I find the absolute regularity appealing. It provides a confidence to your results that simply doesn't exist in the real world. (It also happens to remove the need for tedious experiments).

 

[phreak]

I think one point that is not understood here is that the rigorousness of mathematics is necessary for higher-level physics. Let us consider Hamiltonian mechanics, which is a necessary extension of classical mechanics. In Hamiltonian mechanics, symplectic manifolds are indispensable. A symplectic manifold is a 2nd countable Hausdorff topological space with a symplectic structure - that is, a closed, non-degenerate, differential 2-form. Sure, the results of a standard introductory real analysis class could be seen (and "proven") via purely geometric techniques, but how would you construct (or understand) any theorem concerning symplectic manifolds unless you already have a rigorous mathematical setup? The advantage of pure mathematics is precisely that it is so abstract, and so that we can work with and prove the very abstract constructs and theorems that exist in physics, and trust me, there is an abundance of them.

More than likely, you're not to the point in physics where you have noticed just how abstract it can become. In fact, almost all undergraduate classes are very 'physical' in nature - the mathematics behind the theory is almost never employed. However, if you were to become a physicist and take some graduate courses in physics, I think you would understand just how vital pure mathematics is to the state of physics.

Pure mathematics has always been heavily criticized, even after it has, time and time again, proven to be absolutely vital for the progression of physics. It was in the early 1900s when differential geometry was being criticized as a needlessly abstract and useless theory in pure mathematics, and then, Einstein revolutionized physics with his use of differential geometry in the creation of general relativity. As a mathematical physicist, I constantly have to remind myself to look at modern developments in pure mathematics for the answers. For example, even though abstract algebra is widely thought to be rather useless in physics, I wonder if some of the great unanswered questions in physics have solutions there.

 

[Gokul43201]

Topology is way too technical and is quite useless itself. Saying it is used in physics is a stretch.

You can't be serious - not after all that has happened with gauge theories, GR, soft condensed matter systems like liquid crystals, fractional quantum Hall physics, Aharonov-Bohm theory, (and probably dozens of other areas I know absolutely nothing about).

 

[deiki]

http://a.imagehost.org/0247/chern.jpg