# Why study math?

1. Feb 24, 2007

### oneofmany

Can you give me a nonselfish reason to study pure mathematics? How does one justify spending an entire life pushing around symbols and definitions which may have no connection to the physical world? I enjoy doing math, but I also enjoy playing chess and minesweeper, and even if someone were to pay me, I don't know that I could justify spending my entire life playing these games if I weren't one of the top players in the world.

2. Feb 24, 2007

### fourier jr

imho it would be very difficult to justify that. for example, group theory was developed to decide whether or not the general quintic polynomial was solvable by radicals. that alone was a major milestone, never mind that group theory has become indispensible to other parts of science. on a superficial level solving a problem about polynomials might not seem to have much use in the physical world but the ideas underlying the solution to that problem (symmetries, groups, permutations, etc) have great potential to be applied elsewhere.

"Though each mathematican must be free to pursue the research he prefers, he does have the responsibility to produce potentially applicable papers or papers that offer high aesthetic quality, novelty of method, freshness of outlook, or at least the suggestion of a fruitful direction of research. But far too many mathematicians take advantage of the facts that potential use is difficult to judge and aesthetic quality is a matter of taste. Hence, the good is swamped by the bad. Of course, as in the past, history will decide what is of lasting value. It is the deserved fate of inferiors to fall into oblivion."
Morris Kline (one of my favourite authors )

3. Feb 24, 2007

### Integral

Staff Emeritus
I am sure that there exists things that you are compelled do for reasons even you do not understand.

So why do you think someone else should have to justify doing what they are passionate about. As for the application to the "real" world, how can you judge that there is no application if the work has not been done? Both non-Euclidean geometry and matrix theory were developed with no physical application at hand. Only after the fact were physical applications found.

Personally I find small attraction to pure math theory, but am certainly glad that there are those who do.

4. Feb 24, 2007

### samh

I remember reading an interview in a newspaper with a mathematician who I think won the Fields Medal and finding one of his comments really interesting. He said that studying math is actually very ethical, as studying math doesn't pollute the environment, doesn't use natural resources, and doesn't hurt animals. I had never thought of it that way.

I also remember a cool comment one of my math professors made talking when about university budgets. He pointed out that the math department is not only important but uses the least amount of money! Chemists and so on need big expensive labs; mathematicians need only paper and a blackboard. You don't need to keep reconstructing math buildings and upgrading them with new equipment either because no fancy equipment is used.

5. Feb 24, 2007

### Gib Z

w00t! go the mathematicians!

As to studying pure math, saying its selfish can be quite insulting. I do it because I find it beautiful and interesting. So what if i get paid for it as well?

6. Feb 24, 2007

### trinitron

If we were all pursuing things to improve the physical well-being of humanity what would the purpose of humanity be? First you design an electric razor to shave, and then worry about having it be self-cleaning. Doing math, music, philosophy, etc. to me are in fact the purpose. Medicine, politics, etc. are just maintenance.

7. Feb 24, 2007

### complexPHILOSOPHY

Pure mathematics contains the elegance of logic and the beauty of reason, bridged by the familiar voice of intution and illustrated by formal symbolic language. It is art. It is visual. It is experience.

8. Feb 25, 2007

### arunma

Good question; I wish I could give you an answer!

I can certainly tell you why I personally started studying math. Back in 11th grade when I took my first algebra-based physics class, it raised a lot of questions about why certain formulas popped up. For example, I wondered why the moment of inertia of a cylinder is $$\frac{1}{2}mr^2$$, how the exact gravitational force between two non-spherical bodies could be computed, and why Maxwell's Equations were "forbidden territory" so as to not even appear on the section of my algebra-based book entitled "Maxwell's Equations.". That's why in college, I decided to take as many math courses as I could. It turned out that all the "big" questions that I had could be answered in a mere two years of calculus. To be honest, after taking several senior level courses in math (enough, actually, that I picked up a math degree as a physics undergrad), I figured out that I didn't really like math as much as I thought I did.

Don't get me wrong, I certainly don't hate math. And I can certainly understand why Complexphilosophy views math as an art. Indeed I can understand the appeal in elegantly constructing mathematical logic. But I certainly couldn't imagine doing math for a living, especially when I could be doing physics instead. Physics, and indeed all science, is directly applicable to the physical world, and it has predictive power. Math, at least in my mind, tends to be dry and boring. Alas, some people actually like doing it.

But from what I know, it turns out that many mathematicians end up working on problems in applied mathematics. So in a way, those guys turn into scientists too.

9. Feb 25, 2007

### complexPHILOSOPHY

Perhaps we should distinguish what is to be pure mathematics and what it is to be applied mathematics. We are using all using the same words but are we all alluding to the same thing?

10. Feb 25, 2007

### Gib Z

I completely agree with complexPHILOSOPHY and trinitron (isn't that the model name for some Sony Tvs?).

Arunmas response somewhat reflects me, I was interested in physics, and that got me interested in maths. I got to a point in physics where i need to learn calculus, so i did, and It just went on from there.

11. Feb 25, 2007

### arunma

The distinction I usually make is: numerical analysis, linear algebra, etc., are applied, and algebra, analysis, and topology are "pure."

12. Feb 25, 2007

### Gib Z

I would put number theory into pure math, unless im being ignorant and not including that into analysis

13. Feb 25, 2007

### complexPHILOSOPHY

I would agree with this distinction, however, topology, algebras and analysis had direct applications in theoretical physics. Do you not consider these pure maths to be important? It is also stated by some mathematicians, that topology is one of the great unifying ideas of mathematics.

Do you not consider these 'pure' maths to be important? Or perhaps my perception is distorted?

14. Feb 25, 2007

### Diophantus

There is a book: "A Mathematician's apology" by the number theorist G.H. Hardy. It is quite old (published about 1940) but it's main goal is to basically answer the very question you pose: "How does a pure mathematician justify his life?".

It's a rather short volume (it's more of an essay really) and can be read in a day but I have read it several times at different stages and it has really helped me to develop a satisfactory justification of my decision to become a pure mathematician.

One of his main themes is his arguament that pure mathematics is a creative art and that to have created something beautiful is justification enough. He also questions the value in arguing that pure maths is in some way useful to the good of humanity - while he accepts that maths can be so, he maintains that this sort of utility is likely (based on examples) to be somewhat inversely proportional to the beauty of the mathematics and how 'deep' it cuts.

15. Feb 25, 2007

### fourier jr

even number theory, if we date its beginning with the pythagoreans, was studied to better understand nature. i like hardy generally but i think he was wrong on this point. the most imporant problems come from the study of the real world, or their solutions will have useful applications in the real world.

16. Feb 25, 2007

### arunma

I wouldn't say that any math is unimportant. It's true that even pure math can be applied, at some level, to some sort of science or another. And even if it couldn't be, pure mathematics is useful for its philosophical value of making the discipline logically consistent. It's nice to know that when I integrate a function in three dimensions with a bunch of dirac deltas scattered around, what I'm doing can be logically justified somehow (even if I don't care to know the mechanics of it).

Of course, I'm not really speaking from an informed perspective here. I never took rigorous algebra or topology in college, so I don't really know all that much about the topic.

17. Feb 25, 2007

### fourier jr

lol that isn't a very good example but i know what you mean