Is it worth studying mathematics?

[mathwonk]

[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.

 

[mathwonk]

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.

 

[jma2001]

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]

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.

 

[cragwolf]

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.

 

[mathwonk]

I think most mathematicians probably study and think about math because they simply enjoy it. It is an innocent activity, no one gets hurt, it stimulates the mind, and maybe just maybe, one will contribute something lasting to the intelelctual heritage of the human race.

This makes me reflect briefly however on the enjoyment aspect, as I happen to enjoy thinking about geometry or topology, more than about some aspects of analysis, which to me are very hard.

I tried and failed to get a PhD in several complex variables because I was always somewhat in pain while thinking abut the topic. It was too complicated and too hard to mentally envision the "infinities" required for analysis.

On the other hand I eventually managed to envision geometric objects having 12 or 15 complex dimensions, and even add something to their history. Then my exposure to several complex variables came in handy in algebraic geometry of higher dimensions.

Topology, which I liked, seemed almost too "easy' (you can always deform things so wildly to get whatever you want), so I landed somewhere in the middle, in algebraic geometry. It had enough geometry to be visualizable, but enough analysis to be somewhat unintuitive. So I wanted a subject that was hard enough to challenge me, but not so hard and unintuitive that I could not imagine how to proceed.

Ironically it now seems to me that the most powerful tools in algebraic geometry are borrowed or adapted from algebraic topology and several compelx variables, (cohomology, sheaves, charcateristic classes), and now quantum physics!

Fortunately after years and years of study, and the opportunity to teach courses in calculus and a few in analysis, and better to talk to brillaint friends in these subjects, I am beginning to enjoy that too.

I never liked combinatorics either, so what is the hottest area of examples in algebraic geometry for the last decade? "toric" varieties, with a combinatorial flavor.

We seem to enjoy what we understand, and not what we do not. So if you want your listeners to enjoy your talks or your courses, try to help them understand.

And eventually everything you ever learned or had a chance to elarn, may pop up as useful in your own specialty, so don't sdespise or neglect anything when its time comes around.

I took GH Hardy as something of a model as a young man, but that is hazardous. I liked his toast: "to pure mathematics; may she never be useful to anyone!"

this isolationist attitude is not healthy for a young person, as it can shut him off from the sources of inspiration available in physics and applied math. perhaps hardy only meant he opposed military destruction using science, but I took it as an excuse not to become well rounded.

 

[metrictensor]

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

I enjoy math problems for the same reason people like to put together jig-saw puzzles. But statements like "seeing the beauty of math" make no sense. It is like a painter marveling at their work forgetting that they are the ones who created it.

While they are certainly intelligent, mathematicians are actually somewhat intellectually lazy. Math is for those who want certainty and have trouble with ambiguity. The deep thinkers go into philosophy. Wittgenstein is an excellent example. This is evidenced by the fact that one can have an successful career as a mathematician without ever going into its foundations but the converse is not true in philosophy.

If you marvel at the beauty of math you must also marvel at the beauty of language. You may ask where does the truth of mathematics come from? This can be clarified by asking where does the truth of the sentence "The color of that house is green" come from. A statement the "color of the house is coarse" is mistaken because color does not have the property of being coarse. The truth of this comes from an appeal to the physical world. Certain things have certain relations to others and non-relations to others. The truth of these statements are definitely more than just mind created truths. They do have some empirical validity to them. But to marvel at the fact that color has the property of being green but not coarse is not what mathematicians are talking about when they make similar statements.

All mathematics says about the world is that there is a structure and that it can be captured by conceptual thought. There is no structure to mathematics that is revealed. No emergent properties that belong to some abstract space of "mathematical truth". Saying there is is like saying that the fact that in language the sentence "color of the house is coarse" makes no sense has some truth independent of the actual house and its properties.

 

[cragwolf]

While they are certainly intelligent, mathematicians are actually somewhat intellectually lazy. Math is for those who want certainty and have trouble with ambiguity. The deep thinkers go into philosophy.

One can, of course, argue the exact opposite. For example, mathematicians are so constrained by well-defined concepts and the strict logic that applies to them, that they can't get anywhere in the field without being intellectually disciplined. They are assailed every step of the way by the strictures of logical necessity. Philosophers, on the other hand, usually or often deal with poorly-defined concepts, and thus they have plenty of leeway to be lazy with their arguments, covering up their tracks with the fuzziness of their ideas.

Wittgenstein is an excellent example. This is evidenced by the fact that one can have an successful career as a mathematician without ever going into its foundations but the converse is not true in philosophy.

I don't see how this is evidence that philosophy attracts the deeper thinkers. It might be construed as evidence that philosophy hasn't yet reached first base, so to speak, and is still nowhere near settling its foundational problems.

But I'm talking out of my rear end, so don't take this too seriously.

 

[mathwonk]

when i was a young man looking for answers to life's problems, it seemed to me there was a sort of hierarchy of wisdom, in which philosophers seemed deeper than psychologists, and poets seemed deeper still. of course these categories also coincided with which was the less precise and more difficult to understand clearly.

but who would you rather read: freud, jung, or william blake?

or yogananda, ramakrishna, or ramlal, for that matter.

it may be true also that mathematicians are lazy in that the issues they choose to consider may not be the ones which are important to many people, but they often spend a great deal of energy on them.

 

[metrictensor]

when i was a young man looking for answers to life's problems, it seemed to me there was a sort of hierarchy of wisdom, in which philosophers seemed deeper than psychologists, and poets seemed deeper still. of course these categories also coincided with which was the less precise and more difficult to understand clearly.

but who would you rather read: freud, jung, or william blake?

or yogananda, ramakrishna, or ramlal, for that matter.

it may be true also that mathematicians are lazy in that the issues they choose to consider may not be the ones which are important to many people, but they often spend a great deal of energy on them.

I tried to make clear that (1) I wasn't saying that mathematicians were not intelligent and (2) that they do put a lot of work and thought into what they do. Our work on the inscribed sphere/cube was evidence for me. What I am saying is that from my own experience it is easier to seek security in problems that have a definite solution than those that don't offer such security. The trade off is that the questions answered by math are not as pertinent to the deeper questions of life posed by philosophy/poetry, etc.

I have a graduate degree in math and enjoy it but I no longer think that science/math can answer the questions I once thought they could.

 

[mathwonk]

i agree. math problems offer the entirely unreal security of actually being right or wrong. try getting that satisfaction in a discussion of the iraq war with someone, or even on the proper way to teach calculus!