# Is it worth studying mathematics?

C0nfused
I am wondering if it is worth spending much of your time=>much of your life, in order to study mathematics. Why do we give mathematics so much importance? Of course, they are really convienient and make many things easier in our lives , but I don't think that mathematicians actually find this aspect of mathematics the most interesting. Studying for years number theory, or non-Eucleidian geometry shows that. So, besides being challenging for your mind, do they deserve to occupy much of your time and instead of enjoying other aspects of life, just sit in a desk for yours-days trying to understand a theorem, or trying to solve a difficult problem? Believe me, until now I have been really enthousiastic with maths. I have studied much but I haven't found a clear answer to the question "Why do maths appeal to me".

Thanks

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C0nfused said:
I am wondering if it is worth spending much of your time=>much of your life, in order to study mathematics. Why do we give mathematics so much importance? Of course, they are really convienient and make many things easier in our lives , but I don't think that mathematicians actually find this aspect of mathematics the most interesting. Studying for years number theory, or non-Eucleidian geometry shows that. So, besides being challenging for your mind, do they deserve to occupy much of your time and instead of enjoying other aspects of life, just sit in a desk for yours-days trying to understand a theorem, or trying to solve a difficult problem? Believe me, until now I have been really enthousiastic with maths. I have studied much but I haven't found a clear answer to the question "Why do maths appeal to me".

Thanks

Mathematicians don't study these subjects because they are important but because they are fascinated by them and challenged to contribute to the great tradition of productive mathematics. It is truly sad that so many people just can't see the beauty and excitement of mathematics.

Mentat
C0nfused said:
I am wondering if it is worth spending much of your time=>much of your life, in order to study mathematics. Why do we give mathematics so much importance? Of course, they are really convienient and make many things easier in our lives , but I don't think that mathematicians actually find this aspect of mathematics the most interesting. Studying for years number theory, or non-Eucleidian geometry shows that. So, besides being challenging for your mind, do they deserve to occupy much of your time and instead of enjoying other aspects of life, just sit in a desk for yours-days trying to understand a theorem, or trying to solve a difficult problem? Believe me, until now I have been really enthousiastic with maths. I have studied much but I haven't found a clear answer to the question "Why do maths appeal to me".

Thanks

Why does literature matter to people? Why art? Why sports? All such things have roles in society, but not necessarily the same kind of role as factory workers or paleontologists. Do what interests you, so long as money is no object...that's my two cents.

Neoma
There isn't one objective answer to the question "is it worth studying mathematics", it depends on your definition of value. If you aim at making much money or serving your country/the human race in the best possible way, I don't think mathematics is the way to go. However, if you're looking for a real intellectual challenge and if you're able to see mathematics' appealing beauty, it's definitely worth studying.

Working on mathematical problems is the utmost intellectual challenge. Although writing novels/poems, composing music or painting might beat mathematics in creativity (since there aren't really fixed rules one most hold, one can truly create a totally new and original world), mathematics requires the highest level of abstract thinking.

C0nfused
Mathematicians don't study these subjects because they are important but because they are fascinated by them and challenged to contribute to the great tradition of productive mathematics. It is truly sad that so many people just can't see the beauty and excitement of mathematics.

From this answer, and from the others of course, I got that four people(including myself) that love mathematics just can't explain why they do so, and they can't give a clear answer to the question. Ok, I got that most of you think that it's very mind-challenging, but maybe this also shows that we study mathematics in order to feel smarter,more intelligent than the ones that "just can't see the beauty of them" (mathematics have traditionally been considered as the most difficult subject,the one for the smart people...). I have felt great satisfaction by solving a difficult problem. But maybe -and there are also some facts about that- mathematicians, like every person, are fascinated by the fact that what they study can't be understood by most people, and especially seek recognition that will make them feel important. And if this is the case, I don't think that's a good reason to study maths in your entire life.

Just to make something clear: I am not a mathematician, I don't blame anyone and maybe I have drawn some false conclusions. But I am just trying to figure out what's so special about them.

Homework Helper
C0nfused said:
From this answer, and from the others of course, I got that four people(including myself) that love mathematics just can't explain why they do so, and they can't give a clear answer to the question.

I think I can dawg:

The non-linear brain evolved as a successful survival strategy in a nested, fractal, non-linear world. As a coping strategy, emerging from that brain came a nested, fractal geometry we call "mathematics" (if the world was linear, it would be linear, if the world was a circle, . . . you get the picture. Mathematics is a "reflection" of nature, you know, "when in New York, act like a New Yorker". To study math, is to study the reflections of nature.

Cheers,
Salty

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Gold Member
There is no science without math. Math is the purest of languages. It is the only known way to communicate observations and conclusions in a manner that is unambiguous and incontrovertable.

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Edgardo
I think mathematicians love the challenge. They see a problem
and they KNOW if they solve it, they will feel great satisfaction.
Look at Andrew Wiles who tried to prove Fermat's conjecture.
He spent 5 years on this problem. I think mathematicians are even
addicted to mathematics. Having proved a theorem dopamin is
produced (at least that's what I think).

From my personal experience I can tell you that if I'd written down
a proof, even if it was an easy one, I felt very happy.

C0nfused
Edgardo said:
Look at Andrew Wiles who tried to prove Fermat's conjecture.He spent 5 years on this problem.
He actually spent 8 years, 7 years before reaching the "first proof" and 1 year trying to correct the mistake it had. And the question that occurs: 8 years of life doing nothing else but thinking of a problem can be considered wasted, can't they?

Edgardo said:
From my personal experience I can tell you that if I'd written down a proof, even if it was an easy one, I felt very happy.
That was my point. You probably can find yourself a proof to a theorem you know without reading its initial proof, but won't be really happy because you haven't "written down" it yourself. So it won't be called Edgardo's theorem. That's what I mean by saying that many times mathematicians seek fame and recognition. But I guess this is inheret to humans after all. (I don't exclude myself from all these!)

Neoma
And the question that occurs: 8 years of life doing nothing else but thinking of a problem can be considered wasted, can't they?
When do you consider time being wasted?

As for me, I consider spending 8 years proving one of the most challenging mathematical problems ever not as much a waste of time as spending 8 years watching television.

Actually, I think the only one you can judge wether someone wasted his time or not, is the person himself. Since as long as we're not aware of any absolute purpose of life (personally I think there isn't any, but off course I can't be sure), the valueing (there has to be some better word...) fully depends on one's personal situation/preferences.

DanReit
look, I think that mathematicians learn that much because it is the only (maybe not...) thing that challenges them. I believe that a person should do whatever he wants, if someone like Geography (though I can't understand those pupils he or she should learn it and do it all their lives.
Well, I am a strange teenager but I like to learn interesting subjects (math and physics for example)

so, when people do something for all their lives it is mostly because they enjoy it (it refers to smart people)

Staff Emeritus
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DanReit said:
so, when people do something for all their lives it is mostly because they enjoy it (it refers to smart people)

Some fine mathematicians (I believe I am mainly thinking of Mark Kac here) have pointed to the thrill of their first independent proof, usually as a teenager, as motivating them to take up math as a life choice. Kac's was a different way to derive the solution for cubic equations than the standard one - he thought the assumption of two independent unknown quantities was inelegant. He worked and struggled and failed for months and then he got it. Subsequently it turned out that J.J. Silvester had published the same method in the nineteenth century, but that didn't matter because Kac had DONE IT ALL BY HIMSELF.

C0nfused
Neoma said:
When do you consider time being wasted?

As for me, I consider spending 8 years proving one of the most challenging mathematical problems ever not as much a waste of time as spending 8 years watching television.

Actually, I think the only one you can judge wether someone wasted his time or not, is the person himself. Since as long as we're not aware of any absolute purpose of life (personally I think there isn't any, but off course I can't be sure), the valueing (there has to be some better word...) fully depends on one's personal situation/preferences.
I guess this is after all true. It totally depends on somebody's opinions. There is probably not any "yes-no" answer to such things. I just mentioned that thing about wasted time because usually mathematicians study and work when they are young. So the best years of their lives are spent in mathematics. But on the other hand, it is indeed incredible to see what a human can achieve. I have seen the proof of Wiles and, although i didn't understand a word, I was so impressed because it is a 7-year work of 130 pages containing the proof to one of the most famous and difficult problems, done only by one man.

Mentat
C0nfused said:
So the best years of their lives are spent in mathematics.

And what else would you have them spend it on? This is the integral point, IMO.

Philocrat
C0nfused said:
I am wondering if it is worth spending much of your time=>much of your life, in order to study mathematics. Why do we give mathematics so much importance? Of course, they are really convienient and make many things easier in our lives , but I don't think that mathematicians actually find this aspect of mathematics the most interesting. Studying for years number theory, or non-Eucleidian geometry shows that. So, besides being challenging for your mind, do they deserve to occupy much of your time and instead of enjoying other aspects of life, just sit in a desk for yours-days trying to understand a theorem, or trying to solve a difficult problem? Believe me, until now I have been really enthousiastic with maths. I have studied much but I haven't found a clear answer to the question "Why do maths appeal to me".

Thanks

Our Natural Langauge (NL) already substantially captured the QUANTITATIVE CONTENTS of the Causal and Relational structure of the world. If the study of mathematics is to make this process much sharper and clearer, then by all means this might be well worth our while. But if it is for the sole purpose of servicing fantasy, elegance and ellitism, then we may be heading intellectually into oblovion!

NOTE: Since the structure of the world and the structure of quantitative and logical space are formally equivalent, our sole purpose of studying Formal logic and Mathematics should be to simplify NL to its finest detail and be re-engineered and re-programmed into the native speakers of it. Or is it not?

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C0nfused
Mentat said:
And what else would you have them spend it on? This is the integral point, IMO.
It depends on what you enjoy most or what you think are the most important things in your life.

Telos
Well, if it's true that mathematics can describe the entire universe, isn't it in one's best interest to know that math, since it is in one's best interest to know about the environment in which one lives?

If math can describe the entire universe, shouldn't everyone be a mathematician? Because it is such powerful and useful knowledge?

I think the value of math is integrally related to how useful it is in describing the universe. If it's only useful within a limited range (macro, micro, chemical, biological, financial, etc.) then it's only useful with respect to the range you're working in... and not overwhelmingly useful in principle.

You pays your money and takes your chances. I like spicy food, my wife does not. I like science fiction and thrillers, my sons read non-fiction. I get goosebumps when I hear Maria Callas sing most anything. The first opera I ever heard was Meistersinger; I thought it so beautiful I cried, as I have over Puccini, and Strauss's Rosenkavalier, or Charlie (Bird) Parker playing Embracable You or Parker's Mood, or ... (I can't stand heavy metal or punk music.)

When I was a kid, long ago, astronomer's pictures blew my mind. I wanted to know. I became a professional physicist for many reasons, definitely because of the challenge, the joy of mastery, the beauty and elegance I encountered in math and theoretical physics, the balancing act of combining reason and intuition, like many of my peers, I was, frankly, obsessed, loved being in an elite priesthood. As Duke Ellington remarked, "It don't mean a thing if it ain't got that swing".

No longer a physicist, officially at least, I've spent years in the market research business, where we try, among other things, to understand taste, perceptions, motivations for behavior,... At its best market research is practical or applied neuroscience. Market researchers, neuroscientists, psychiatrists, psychologists all try to answer that question, "Why does ... behave/like ...that way?" The plain fact is, nobody really knows, but we are getting better at learning how to attack such seemingly ephemeral questions. I will note that I have three grown sons -- they all were discernably different at birth, and have remained so as adults-- none are particularly enamoured of math.

Just to be clear, I love math and physics and neuroscience and I love to play jazz music, and I'm proud of it.

Regards,
Reilly Atkinson

joeboo
Mathematicians study mathematics for the same reason that a good orator can keep you on your toes, and a bad one can put you to sleep. What is the difference between those two?
Grace, Elegance, Succinctness.
That is the beauty of Mathematics.
I remember the day I first learned what a Topology was. It seemed so esoteric, so simple, so benign. I was left wondering, "What is the point?"
I also remember the day I learned the $\epsilon - \delta$ definition of continuity of a real function. I remember thinking, "This is very important"
But then, I remember the day I learned the topological definition of continuous.
Topology, which at one time seemed so weak, so pointless, had, in one graceful, elegant and succinct definition, done what Analysis could never do: Define the notion of continuity in an arbitrary topological space. And in a manner so appealing, I proved it was equivalent to the Analytical Definition, and then put my $\epsilon$'s and $\delta$'s to bed forever.
That is the beauty of Mathematics.

It is truly sad that so many people just can't see the beauty and excitement of mathematics.
I used to work in my university's "Math Help Room", as an undergraduate. A Graduate student there noted that my hours were some of the most popular for the other students to attend. When I asked him why he thought that was, he said, "Alan, because you show them that it can be fun".

Thoughts II --The study of math through at least secondary school is very worthwhile. At the minimum it helps students to learn how to use their minds, gives an often useful window into the world, and, ideally, helps folks understand percentages. Past that it seems to me: if you like doing math do it; if you don't, don't. Not to worry. And if you want to worry a lot about your choice, take up philosophy.
Regards,
Reilly Atkinson

C0nfused
Thanks for your posts reilly and everybody .It was nice to hear your opinions.Although there is of course no certain answer to the initial question, your comments were really helpful. As for worrying about my choice, the problem is that i now "have to" take this choice. However i don't know if it's really wise to study philosophy at the age of 18. I sometimes think that I make things look complicated although they are really simple. Anyway, i guess you just have to enjoy what you are doing otherwise don't do it.
Thanks again

cbb2c
It seems to me that the nature tends to behave in ways that are describable in mathematical terms--the laws of physics are written in the language of mathematics. Often, mathematics and physics develop simultaneously, as with string (M) theory. So, we can imagine the progression of our understanding of the universe as a person walking on two legs, one at a time--mathematics and physics.

Also, I personally find it amazingly beautiful that nature behaves in ways that are described mathematically, while mathematics is based simply on logic--i.e., nature behaves logically, by defined logical rules. Doesn't anyone else see the beauty in a "logical nature"?

Homework Helper
But does Nature actually behave logically? Parts of it can be modeled to a reasonable accuracy with mathematics, but it is a huge leap to say therefore it "beahves logically".

Telos
C0nfused said:
I sometimes think that I make things look complicated although they are really simple.

I know the following opinion will not carry much weight around here, due to my youth and social science education, but I have a hunch that all mathematics is actually very simple. There really is only one rule to mathematics and that is "follow the rules." You go where they take you and no where else. That's it.

Change the rules all you want. Go ahead and dissolve the parallel postulate and see where you end up. But you cannot break a rule and keep it. It's as simple as that.

Since the natural world arguably has rules that cannot be broken, and we are all necessarily subject to them at all times, what could be more natural and instinctive than to learn mathematics? Why is it so difficult?

(And don't tell me mathematics is by nature difficult - I am assuming it isn't in a reductio ad absurdum endeavor).

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Gold Member
Telos said:
I know the following opinion will not carry much weight around here, due to my youth and social science education, but I have a hunch that all mathematics is actually very simple. There really is only one rule to mathematics and that is "follow the rules." You go where they take you and no where else. That's it.

Change the rules all you want. Go ahead and dissolve the parallel postulate and see where you end up. But you cannot break a rule and keep it. It's as simple as that.
I think those are nice observations, but how simple is it to follow complex rules? How will you know you are actually following the rules? How do you come up with the rules in the first place? Etc., etc.
Since the natural world arguably has rules that cannot be broken, and we are all necessarily subject to them at all times, what could be more natural and instinctive than to learn mathematics? Why is it so difficult?
It doesn't follow- we don't need to know the physical rules to be subject to them. If people did need to know the physical rules, well, we'd know them. ;) (Unless you believe in forgotten innate knowledge.)
I think some people find math difficult because they aren't motivated to learn it, for whatever reason. Abstractness would be the other main difficulty. I love math and don't find it difficult (unless I jump ahead when I shouldn't ), but I still find it much easier to deal with concrete objects than abstract ones. In fact, I picked up some linguistics books the other day after spending a long strech reading logic book after logic book, and I couldn't get over how much easier the linguistics books were. Of course, every field has its share of abstractness, but others hardly compare to math. Math reminds me of the poet's "airy nothing".

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Telos said:
I know the following opinion will not carry much weight around here, due to my youth and social science education, but I have a hunch that all mathematics is actually very simple. There really is only one rule to mathematics and that is "follow the rules." You go where they take you and no where else. That's it.

Change the rules all you want. Go ahead and dissolve the parallel postulate and see where you end up. But you cannot break a rule and keep it. It's as simple as that.

Since the natural world arguably has rules that cannot be broken, and we are all necessarily subject to them at all times, what could be more natural and instinctive than to learn mathematics? Why is it so difficult?

(And don't tell me mathematics is by nature difficult - I am assuming it isn't in a reductio ad absurdum endeavor).

Well, mathematics is simply the application of logic to a set of axioms and hypotheses to see what may be deduced. However, it is far from easy.

First it requires a great deal of hard work to even know or understand the rules. Then some natural talent to see how to apply them and what it is one wishes to prove.

After all, getting from A to B is simply a matter of following a map, but what if you don't know what B is?

However, with a certain amount of application most people could do it, tosome degree.

Painting is easy: it's just putting paint on a surface, hence I am picasso...

motai
matt grime said:
However, with a certain amount of application most people could do it, tosome degree.

Painting is easy: it's just putting paint on a surface, hence I am picasso...

The art that I like the most are the ones that operate off of highly mathematical constructs. For instance, the abstract nature of the recursive elements in Pollock's works appeal to me more than say a Van Gogh piece.

cbb2c said:
Doesn't anyone else see the beauty in a "logical nature"?

Yes, and it does work for many aspects of the observable universe, but there are still some parts that are counterintuitive (such as the differences between QT and GTR).

...Though I do smile whenever I see Newton's Law of Universal gravitation next to Coulomb's law.

matt grime said:
Well, mathematics is simply the application of logic to a set of axioms and hypotheses to see what may be deduced. However, it is far from easy.

First it requires a great deal of hard work to even know or understand the rules. Then some natural talent to see how to apply them and what it is one wishes to prove.

After all, getting from A to B is simply a matter of following a map, but what if you don't know what B is?

However, with a certain amount of application most people could do it, tosome degree.

Painting is easy: it's just putting paint on a surface, hence I am picasso...

You shouldn't be looking down at picasso like that, or any kind of art for that manner. (sorry if i am misunderstanding)

Using the paintbrush to paint is merely using the tool. That is the same as using a integral or etc as a tool in mathematics. It's *HOW* you use such tools (or invent them) that shows true measure of Art. (Art not just meaning the visual painting one)

Picasso draws somewhat weird, but i bet if you asked him to draw something realistically, he could *draw*, i mean DRAW to make your jaws drop.

Homework Helper
Hmm, you really did misunderstand my point. In fact, your post was exactly what my point was. I was pointing out that saying mathematics was simple (as in easy to do) since it can be reduced to the maxim "follow the rules" is as non-sensical as the final sentence in my post. Otherwise why is obtaining a good mathematics (graduate) degree so hard?

matt grime said:
Hmm, you really did misunderstand my point. In fact, your post was exactly what my point was. I was pointing out that saying mathematics was simple (as in easy to do) since it can be reduced to the maxim "follow the rules" is as non-sensical as the final sentence in my post. Otherwise why is obtaining a good mathematics (graduate) degree so hard?

Ah..damn. I apologize...

Homework Helper
[if you put your epsilon and delta to bed after learning the nice clean open set definition of continuity, how do you prove that x^2 is continuous?
i.e. abstract definitions are ok for talking about math, but calculations are required for doing examples.]

I was a beef lugger in the boston meat market in southie. I really enjoyed the honest labor, and hanging out with the other guys, and being extremely strong. I liked sticking a flower behind my ear, taking a stroll, and when some construction workers hollered "hey give that hippie a big kiss", being able to say back, "come on down, you might catch a surprize." and see the puzzled looks on their faces.

But the meat was starting to get heavy, and all the old guys, say 50 or so, looked really worn out. And there was some dishonesty, and bribery, in the butchering business there, and I felt compromised by them. Also I noticed that every year another one of us got murdered. Since there were only about 20 of us, I did not like these odds. Even my friend "Bigman" who easily won the bar fight he was in one night after work, got sent to prison for manslaughter, and I had to drive out to Walpole just to see him.

Then one day I read a newspaper article about the 25th anniversary of the atomic bomb, and the physics sounded interesting, and I realized my brain was atrophying there sitting playing whist and drinking by the railroad tracks between jobs.

So I decided to get a PhD in algebraic geometry instead. It has been way more fun and intellectually exciting, and the life expectancy is much greater.

As to beauty and elegance, can you imagine a 165 pound man swinging a 360 pound forequarter to his shoulders and walking out of a freight car with it? I hoisted slightly a 300 lb hindquarter myself once but did not walk anywhere with it.

This slightly tongue in cheek but factual account, is just to remind you philosophers that real life also plays a role in what mathematicians do, as with other humans.

It was very embarrassing for one thing to learn that in some cases academicians are judged less objectively than are meat luggers. I.e. luggers who could lug were welcome in boston. On my first attempt at college teaching, I was released because I had no PhD, even though the other professors said they considered me the most knowledgeable among them. Of course I was also denied work at a redneck meat packing plant in the same small western town because I had long hair, ability and experience not being relevant at that place either.

I am glad I was forced to get a PhD though, because indeed there is a genuine exhilaration associated with seeing how to prove a theorem, especially something no one has been able to prove for years. It helps if you have worked a long time on it too, Sometimes I have solved problems instantly that others have been stumped by for weeks or more, but that did not mean as much to me, since it seemed so easy.

So sometimes the thrill is from pride of accomplishment, sometimes just the beauty of the insight. I admit too sometimes it has taken me years just to appreciate what someone else has done long before, even when it was staring at me the whole time. So I love the feeling of appreciating the depth of others' work as well, but if it is on a topic I too have studied deeply, I may simultaneously feel very foolish.

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Homework Helper
Right now I am reading Riemann's collected works. I have only read 12 pages of his inaugural paper, and already my ego is vaporized. He dispatches entire topics from scratch in every paragraph. So far in 12 pages he has introduced the geometric point of view in complex analysis, the cauchy riemann equations complete with motivation, the nature of branch points for a complex mapping, and the associated permutation of sheets, the Green's theorem, the Cauchy integral theorem, and the homology of surfaces, including bordered ones. And he proved everything, convincingly if not rigourously.

This is unreal. He just writes down the results of a calculation, often without even doing it, and then explains the intrinsic meaning in a few words. His insight is amazing. He goes right to the heart of every topic. And then get this, after proving a certain thing satisfies A leq B and B leq A, he spends three lines to explain why A = B. Give me a break Bernhard.

Sometimes I have heard people criticize his lack of rigour, but it is obvious even after reading this much, that people were probably delighted when his proofs needed compeltion, as that gave them something to do. Essentially everything he said was gospel, he just did not give all the details. We have been poring over it for the last 150 years, and people are still working on the Riemann hypothesis, which occurs in a paper about 12 pages long.

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Gold Member
If you want to know why someone would spend a lifetime studying mathematics, read _A Mathematician's Apology_ by G. H. Hardy. It is one of the great books of the twentieth century and it answers all of the questions raised in this thread, and more. I'll try to write more about it in the book review section, when I'm feeling up to the task.

C0nfused
jma2001 said:
If you want to know why someone would spend a lifetime studying mathematics, read _A Mathematician's Apology_ by G. H. Hardy. It is one of the great books of the twentieth century and it answers all of the questions raised in this thread, and more. I'll try to write more about it in the book review section, when I'm feeling up to the task.
I read it about a week ago and I found it really good. He had a very interesting point of view and he seemed confident and happy about his decision throughout all his life. However, I don't think that most mathematicians have the life of Hardy. He was wealthy, he did,as he states, almost no teaching at all and so he had much time for research with the help of two great people. Anyway, I have come to think that mathematics,for some strange reasons, attract many people who decide to dedicate their life in studying maths. Some of them may come to answer the question of this thread, while others will just state that maths was an honest way of making money in this life

It is indeed interesting however to see that some people think that mathematics exist even if people stop studying them and that mathematical truths are not related to humans but are independent. I have come across a similar opinion while reading a book. It states that Godel had expressed such thoughts. And from what I have heard and read, Godel is considered the most important logician of the 20th century. And more interesting is the fact that his obsessions killed him.

So mathematics definitely has some weird aspects that maybe only weird minds can understand. And it has a beauty that no other subject has. But as to whether it is worth studying it, this, I have now come to think, is a question with no objective(with its mathematical meaning) answer.

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cragwolf
C0nfused said:
But as to whether it is worth studying it, this, I have now come to think, is a question with no objective(with its mathematical meaning) answer.

It has no "objective answer" when it is applied to any human activity or vocation. So the question is utterly useless in that respect. Although it's interesting to read people's answers.

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