Screwed up Abstract Algebra exam unsure if I have the ability to do math.

[homeomorphic]

Homeomorphic, what is your view on the existence of universal mathematicians these days. By this I mean a mathematician who has such deep knowledge that they may contribute to a wide range of mathematicial fields which may be totally disconnected. Poincare, Von Neumann, Hilbert are good examples of what I am talking about.
As an outsider it seems there is a trend to narrow specialization currently. However the sheer amount of mathematical resources and technology such as the internet seem to allow for a greater quantity of universal mathematicians than any other time in history.

It's too big. Even subfields are big enough that one person in that area can't understand another person's research without very serious effort. Even one paper takes a lot of time to understand thoroughly. So, a lot of times people don't understand them thoroughly, I think. They just take the minimum. Actually, a speaker I saw once told the story of how there was a theorem that the experts thought was in the literature somewhere, but they couldn't track it down when asked, so she had to reprove it. Often, a lot of the intuitive understandings aren't written down, so they are lost if the oral tradition of it breaks down. It's rare that people can contribute to many different fields these days. I guess maybe Terence Tao would probably qualify. But he doesn't do everything, like maybe someone like Hilbert was able to do. Some say Hilbert was the last universal mathematician. There's just too much to know.

Do you believe specialization or generalization is the best path to a deep understanding of mathematics?

It's probably good to have people who do both.

 

[Mariogs379]

@homeomorphic,

What're you going to do now? Sounds like you're not headed the academic route...

 

[homeomorphic]

Don't know what I will do yet. Finish up the thesis and try to go to industry. Engineering, computer programming, insurance, or operations research, something like that.

 

[Group_Complex]

Don't know what I will do yet. Finish up the thesis and try to go to industry. Engineering, computer programming, insurance, or operations research, something like that.

You really should just stay in mathematics, even if it is not pure mathematics, otherwise you are wasting your great ability for the abstract.

 

[homeomorphic]

You really should just stay in mathematics, even if it is not pure mathematics, otherwise you are wasting your great ability for the abstract.

I have no such great abilities. I have been put to the test and I am no good at research, especially no good as measured by whatever the yardsticks are that mathematicians get measured by (number and quality of publications, mainly, and I have a big fat ZERO publications, maybe 1-3 if I publish my thesis). The job market is competitive. I can't get a teaching recommendation as things stand for reasons I won't get into here. A teaching recommendation is a requirement for postdocs.

You have no idea how unhappy I am in grad school. It's like jail. Staying in academia just means more of that. Either I become predominantly a teacher, which was never what my interest really was, especially not with stupid traditional materials, textbooks, and lecture methods, or I become predominantly a researcher. Well, so far, I have hated research, not been particularly good at it, and to boot, I just have no chance to make it as a research mathematician.

Math is better kept as a hobby for me. Staying academia would probably be a much worse waste of my abilities than leaving it ever would. No one in academia really cares about my cute explanations of old math. They care about new math, and I don't care about new math, unless what I want to understand just happens, by sheer chance, not to have been worked out yet. That means I will have VERY few publications and thus not even a small chance to survive in academia. There's no room for such an attitude in academia. Really, the only thing I care about is making my expository materials. I have little to no interest in proving new theorems. Only in fixing what's wrong with the math we already have. There's no place for that in academia, except what ends up being just a hobby, anyway, not your main job. Either way, what I am really interested in will end up being relegated to "hobby" status. May as well do something useful as my day job, and something I actually believe in. I don't believe in teaching traditional classes, which is a requirement. And I don't believe in traditional research, at least not for me.

No, I am quitting for sure. Look for my expository stuff on the web when I get around to it, but I really have very little interest in publishing any papers in math journals.

 

[Group_Complex]

I have no such great abilities. I have been put to the test and I am no good at research, especially no good as measured by whatever the yardsticks are that mathematicians get measured by (number and quality of publications, mainly, and I have a big fat ZERO publications, maybe 1-3 if I publish my thesis). The job market is competitive. I can't get a teaching recommendation as things stand for reasons I won't get into here. A teaching recommendation is a requirement for postdocs.

You have no idea how unhappy I am in grad school. It's like jail. Staying in academia just means more of that. Either I become predominantly a teacher, which was never what my interest really was, especially not with stupid traditional materials, textbooks, and lecture methods, or I become predominantly a researcher. Well, so far, I have hated research, not been particularly good at it, and to boot, I just have no chance to make it as a research mathematician.

Math is better kept as a hobby for me. Staying academia would probably be a much worse waste of my abilities than leaving it ever would. No one in academia really cares about my cute explanations of old math. They care about new math, and I don't care about new math, unless what I want to understand just happens, by sheer chance, not to have been worked out yet. That means I will have VERY few publications and thus not even a small chance to survive in academia. There's no room for such an attitude in academia. Really, the only thing I care about is making my expository materials. I have little to no interest in proving new theorems. Only in fixing what's wrong with the math we already have. There's no place for that in academia, except what ends up being just a hobby, anyway, not your main job. Either way, what I am really interested in will end up being relegated to "hobby" status. May as well do something useful as my day job, and something I actually believe in. I don't believe in teaching traditional classes, which is a requirement. And I don't believe in traditional research, at least not for me.

No, I am quitting for sure. Look for my expository stuff on the web when I get around to it, but I really have very little interest in publishing any papers in math journals.

What is wrong with focusing on teaching? I mean you have just said you have no interest in proving new theorems, but you would rather rethink old theories, that sounds perfect for teaching mathematics at a top university. You sound like you would be the perfect math professor for a student trying to learn the subject. You would also have a better chance to fix what is wrong with mathematics as you see it if you are a well known mathematics educator/professor rather than as an engineer.
Why not do enough research to get by, but focus on the teaching side of things?

 

[homeomorphic]

The bulk of the teaching load is calculus, diff eq, or lower level stuff. That's not interesting to me. I am not interested in dealing with mathematically crippled people and helping them to learn the basics. There's no real market for the in between level. No one just teaches grad students and math majors and doesn't do research.

I don't need to be prof to fix math. I just need to write my stuff and make it available online.

And as I said, I don't believe in the traditional way of doing things. That's what is expected of you. You have to be a cog in the machine. I refuse.

Besides, I can't even do enough research to get by. It's torture to do research, unless you are really, really interested in what you are doing. You may as well suggest that I put myself on a torture rack, just to make my life interesting.

I am also most definitely NOT the perfect professor. When I have taught, the students don't really like me. True, those are lower-level students, but, as I said, that is the bulk of the teaching load. Far from being good at teaching, I'm liable to be kicked out because the students are complaining about me. I am not particularly good at lecturing. Tutoring, I can do. I am pretty successful as a tutor. Lectures, not so much. I have to work like a mule on my lectures, just to keep the students from complaining about me, and even then, they aren't even that happy with me. It doesn't come naturally to me. Some people say teaching is easy. They don't struggle with it like I do. I have a lot of talents, so why should I being stuck doing one of the few things that I struggle with?

 

[R.P.F.]

To the OP:

I once got a test score below average but I wasn't very upset because I knew exactly what went wrong: the exam was too time pressured and I couldn't work well in such an environment. I have gotten a lot better at taking time-pressured exams now.

I whined about exams too. But I never thought about quitting. I love math enough that I never thought about quitting. Do you really like math?

To homeomorphic:
It sounds to be that you'd be better off if you had gone to a department whose strength is more aligned with your research interest. You might end up with a more positive view of your research that way.

 

[homeomorphic]

It sounds to be that you'd be better off if you had gone to a department whose strength is more aligned with your research interest. You might end up with a more positive view of your research that way.

I'm not sure that would help. I've been to conferences and seen speakers from other departments. I think the mathematicians that are aligned with my interests are extremely rare, if they exist at all. Yes, I probably didn't have the best possible environment for my interests, but that was pretty much unavoidable. I only got accepted to one place out of the 5 I applied to, so not much choice. It's possible I should have gone into physics instead, but even now, I'm not sure if that would have really worked very well for me.

I was interested in my research at first, and, actually, it's not even completely uninteresting to me now, but I wish I had time to study it from a physics point of view, not a pure math point of view. That's what leaves me feeling really unfulfilled by it. That's why it feels like an intellectual prison. I don't have time to find out about the physics, and without that, I find it pretty empty. Mathematical physicists are just not that common, so it was a pretty narrow target as far as that goes. As far as something like topology and many other pure math areas goes, physics is the main thing that can redeem them in my eyes.

Maybe I should have gone for some sort of applied graph theory. That might have worked out better. At any rate, it's an inherent peril of mathematics that it's hard to get an idea of what a field is really like until you are already in it.

Still, I think the general climate of academic mathematics would be very objectionable to me. Many of the things I don't like are independent of that.

 

[sshai45]

Well hard work vs natural is a very prevalent false dichotomy. I would say it was pretty natural, but a lot of that had to do with smart work, rather than hard work. A professor at my undergrad said I was brilliant at my graduation, but secretly, I knew I was CONSCIOUSLY DOING (as opposed to my brain doing them for me, as many uninformed people would presuppose) a lot of things that no one else would have even thought to do, such as carefully spaced review, and constantly mulling things over in my mind in just the right way, so that I always had every theorem and even its proof at my finger-tips. In grad school, the pace was too fast and my old methods failed to keep up. It was not possible to be so thorough. So, I didn't stand out in grad school, anymore. I made it to become a doctoral candidate, but not in the most impressive way. I may have some genetic gifts, but it's always been my experience in everything I am good at that it's not just how hard you work or how good your genes are--a lot of it is how smart you work. That's sort of the moral of my life, I think, and it annoys me to no end that most people seem completely oblivious to the whole idea. You can play chess all day and improve only a little bit. But if you don't use such a stupid strategy and actually try to learn what to look for, not just blindly practice, then you improve. Two people with equal ability can put in the same work and get drastically different results because one knows that tricks of the trade, and the tricks of learning in general, and the other is blindly try to practice it in any old random way, hoping to get better.

So do you think this thing that "most people" are oblivious to this is a *problem* (since it *annoys* you), then? And if so, how can a "most people" person learn how to "do it smart"? How did *you* learn that?

 

[homeomorphic]

So do you think this thing that "most people" are oblivious to this is a *problem* (since it *annoys* you), then? And if so, how can a "most people" person learn how to "do it smart"? How did *you* learn that?

Of course, it's a problem.

I learned it from books and websites, some of which I can't even recall the names of. Also, I learned it by thinking about how to learn for myself. "Most" people can learn it the same way I did. I am sure "most" people think about it a little bit, but they *seem* to be unaware of the full possibilities, in my experience. I do believe their is such a thing as innate talent, as well as an advantage from starting at a younger age, but I am agnostic, in the absence of conclusive proof either way, as to how important it is and whether people without it can overcome it by working smart and hard.