Very generally, the question I'm trying to ask is: Can someone be a good professional mathematician without being particularly good at (or even being – relatively – bad at) computations?
More specifically to me: I'm currently a freshman in college and I'm in a multivariable calculus/linear algebra class, and, not for the first time in my life, I am finding that my computational skills are lacking.
(Note that when I say "computations" I'm referring to more than just algebraic manipulations; I'm including stuff like finding [messy] integrals, doing [messy] delta-epsilon proofs, and even working with [messy] inequalities in general… I say "messy" because, as I'll elaborate on later, I understand the general forms really well, I think. To illustrate this, if I were given these two delta-epsilon proofs for the first time, I'd probably find this one easier than this one… perhaps that's not the best example, so don't over-analyze it, but it gives you a general idea of what I'm talking about.)
Part of this probably comes from the fact that way back in the day when I was first learning algebra and whatnot, I wasn't that into math, so I never drilled the computations the way some other kids did. And you might say I could go back and practice solving some tough algebraic equations, but I feel it's gotten beyond that point; I just don't think about math like that, and I can't see how I ever will… by the time I started getting really interested (read: obsessed) with math, I was in calculus, where computations were sort of secondary to concepts anyway. And I love the concepts. I read Wikipedia pages on mathematics that is way out of my league, but I look for those one or two English sentences that tell me stuff like
The tangent bundle of a differentiable manifold M is the disjoint union of the tangent spaces of M
or
A presheaf (of sets) on a topological space is a structure that
associates to each open set U of the space a set F(U) of sections on U, and to each open set V included in U a map F(U) → F(V) giving restrictions of sections over U to V
which I then sort of latch onto and think about. I particularly like reading about morphisms and category theory. I'm not claiming to understand much of anything about tangent bundles or sheaves. I'm just trying to give you an idea of what I like about mathematics… what I feel like justifies my desire to become a mathematician despite my relative weakness with exactly what people normally think of as "math." As a quick clarification, I don't feel like I can't do the computations I just feel like they take me so much longer than a lot of the kids around me (granted I'm doing the sort of geared-towards-math-majors class, so I'm with kids who have been taking college-level math for usually 1-2 years or more).
And this brings me to my main questions:
- Can I be a good mathematician doing something like algebraic geometry or category theory or Galois theory or whatever I end up finding I most enjoy while being actually sub-par at the more specific/concrete/technical type of math?
- If yes to (1), what branches of mathematics would be best for this? I've always sort of thought of abstract algebra as the "promised land" with respect to my strengths in math, but I'm becoming increasingly worried that this is only true if you already have all the concrete techniques down.
- Should I fight it? That is, should I continue to work on, and inevitably, agonize over the computations or should I shrug and say, "that's not my thing" and focus on the parts that I like? (The thing that I worry about w/r/t the latter path is that I might develop a sort of "fear" of certain types or flavors of math, and just sort avoid them at all cost.)
I also should probably say that, as of now, I'm sort of taking the "grin and bear it" (to use Spivak's words) path from (3), but the reason that I'm asking this question is to see whether I will, even as a mathematician, have to be fighting against this weakness or if it sort of "clears" up (at least in some fields) after a while. Because my feeling is that, as long as computations play a central role, I'm not going to be able to contribute in any particularly deep way (I'm not saying that I would definitely be able to contribute deep mathematics if I were working in a more abstract field, but rather that if I were to contribute something significant, it would probably be doing mathematics of essentially that type).
I know this was incredibly long (this is my last long, advice-type question for a while, as I've definitely asked more than my fair share recently). So I'm sorry for that, but this means a lot to me and it's something I've been obsessing over for a while now, and I don't really have anyone to ask about it at the moment. Even if you don't answer, I really appreciate that you took the time to read all this. And, of course, thanks so much if you do respond. I hope that this all didn't come across as just a self-centered plea for other people to "magically come up with a solution": I'm just looking for some insight from people who know what the road is like farther along… I also hope that others can benefit from all this, in whatever way they can.
EDIT:
Alright. Here are the reasons:
Video 15: http://www.simonsfoundation.org/science_lives_video/pierre-deligne/
“When one talked with M Hermite, he never evoked a sensuous image, and yet you soon perceived that the most abstract entities were for him like living beings. He did not see them, but he perceived that they are not an artificial assemblage, and that they have some principle of internal unity… Apropos of these reflections, a question comes up that I have not the time either to solve or even to enunciate with the developments it would admit of. Is there room for a new distinction, for distinguishing among the analysts those who above all use this pure intuition and those who are first of all preoccupied with formal logic?
M Hermite, for example, whom I have just cited, can not be classed among the geometers who make use of the sensible intuition; but neither is he a logician, properly so called. He does not conceal his aversion to purely deductive procedures which start from the general and end in the particular.” – Henri Poincaré (http://www-history.mcs.st-and.ac.uk/Extras/Poincare_Intuition.html)
“I have tried to avoid long numerical computations, thereby following Riemann's postulate that proofs should be given through ideas and not voluminous computations.” – David Hilbert
Page 19, in response to "Competition seems to be of widespread concern": http://web.mit.edu/uma/www/mmm/mmm0101.pdf (Artin, I think, is best read in a Tommy Lee Jones from No Country for Old Men voice)
Page 8, second answer in the first row: http://www.ams.org/notices/201009/rtx100901106p.pdf
“Since then I’ve had the chance in the world of mathematics that bid me welcome, to meet quite a number of people, both among my “elders” and among young people in my general age group who were more brilliant, much more ‘gifted’ than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle–while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured) things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates almost by sleight of hand, the most forbidding subjects.
In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective or thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone.” – Alexander Grothendieck
^"I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle"… that's EXACTLY how I feel.
You will, with good reason, say, "But these guys are geniuses… the gods… you cannot compare yourself to them as though they were just your average Joe giving advice," or perhaps, "of course, Deligne's definition of 'bad at computations is not the same as yours." And that is what I'm asking: Are these guys just being humble, or is there a place in mathematics that actually is not all that computational?
Last Edit:
I wanted to ask this one more time, not because I didn't receive good answers the first time, but simply because the topic is still relevant for me – even if I view it in a different light now – and the possible avenues for responses still intrigue me deeply, and I do not think it impossible that all of this would still be somewhat interesting to others. You'll probably think that I was and still am looking for a particular answer, which is not, of course, the best way to ask a question, but I don't think that's what it is. Hopefully my decision to make this edit doesn't come across as pure vanity, but if it does, then it is, in a sense, and I should be "moderated" accordingly. Many thanks again.
Best Answer
I have an analogy that I use with my students, and it is applicable here.
Mathematics is like literature. Things like elementary arithmetics (grade school $+$,$-$,$\cdot$, etc. on real numbers) are like the "abc"s. Things like algebraic manipulation (but not Algebra) (e.g. $log_2 (4 x)=9$, solve for $x$) are like words. (A lot of Americans reach only this stage and proclaim they hate math; it's as preposterous as saying you hate books when you can't even read the word
the). Things like elementary calculus and real analysis are like sentences, and by the time you have an undergraduate degree, you can probably read a board book or two (without help, gasp!).The whole rest of mathematics lies beyond.
The application is this: your "computations" are as fundamental to mathematics itself as the printed word is to literacy. It's great that you're excited about math, but I think it's extremely important to not just understand, but grok the basics before you try to build anything on it.
A lot of the stuff grade school kids learn is crap--like about not-dividing-by-zero, like about fractions, like about conics. And a lot of it is a dumbed-down-version or otherwise irrelevant once you get to higher math. But, a lot of it provides useful intuition. The analogy is: do you need to know what the letters' names are to read a book? No. Do you need to know that "?" is called "question mark"? No. As long as you understand that
ais different fromband that?is interrogative, then you're in good shape for understanding--but knowing those facts is useful too. If you know thataanderepresentvowels, which all words in English phonology require, for example, then you automatically know whysdfslkjhrwfbvis an ill-formed English word.The main points here are:
Under these circumstances, I recommend becoming competent. If you really do understand computations, this should be relatively straightforward. It will be well, well worth your increased understanding and increased ability to interact. You don't have to aim for complete fluency at first, but I'd still hit the grammar monograph before I tackle Tolstoy.