Is it worth studying mathematics?

[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!