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

  • #26
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No, I wouldn't feel left out of academia. I went to a lot of talks, and, frankly, they are usually very technical and boring. Most of what is being done is very uninteresting to me, maybe sometimes only because I don't understand it, but usually it just seems inherently uninteresting and ugly. Occasionally, there will be talks that I don't understand, but I can tell that the subject matter is interesting never the less. I did enjoy a lot of my advanced topology classes (topology talks on current research, much less so). As it is, I only understand a very narrow piece of academic mathematics. So small a chunk as to be insignificant, so there's nothing that I would miss because, for the most part, I don't understand any research level math, other than my own research.

Physics may be an empirical science, but it is a lot like math in that there is a lot of theoretical reasoning. It's like math where you can test things using experiments. Arnold said math is the part of physics where the experiments are cheap. I also am interested in the motivation for the axioms and definition. It's a lot easier to remember everything when it's not introduced by arbitrary decree. But the axioms and definitions are only the starting point. The theorems are interesting, too.

The math that people are working on today, by and large, is not foundational to other sciences. Sounds like what Gauss said, what you quoted. Math was a very different subject when Gauss was around. People were still developing the basics that are quite useful in physics and engineering. It no longer plays that foundational role for other sciences, for the most part. I hope to write a book one day about how useful recent research is and speculates on how useful current research will be. Most of the math that is really foundational to the rest of science has already been done, much of it over 100 years ago. There are some interesting applications of more recent subjects being developed, but it takes a lot of expertise to understand whether they are genuine applications or just the result of people trying to force-fit some math into things to sell the math. I used to be interested in fundamental physics, but I now find condensed matter much more compelling. I'm not sure a theory of everything with all the math it requires is that admirable or realistic of a goal. Condensed matter is a bit more practical.

My favorite areas of math are topology, complex analysis, and graph theory, for their visual appeal. Graph theory also appears to have very interesting applications.
 
  • #27
You see the reason I am so interested in math is for foundations. The way I see it, mathematics is the path to absolute truth, or at least as close as we humans may come to it. For some reason I have found Algebra less enjoyable than Real Analysis. For starters I feel that Real Analysis and Complex analysis come far more naturally to me, part of the appeal being the ability to visualise things in analysis. However, may favourite mathematical topic so far has been topology, and differential topology (the small amount I have read) seems like a perfect match for me (Is the field still active or has it been "mined out"?). If I do proceed to graduate studies in pure math, I feel the pressure to work in a field just as analytic number theory, with the goal of working on problems such as the riemann hypothesis, yet I only enjoy specific parts of elementary number theory, and am not very good at tricky olympiad/putnam number theory problems. Another field which interests me is logic, but even logic the foundation of all pure mathematics seems second rate compared to number theory and analysis. Is this just an ego or thing or is there something fundemental to these feilds that the others lack?

Algebra on the other hand has felt lest intuitive to me, and although I can visualise things, it does not "feel" the same as analysis. Furthermore everything seems less motivated, basic abstract algebra, such as cosets and normal groups annoyed me because I did not understand why we were making these definitions. Would it have hurt for the professor to have given some motivation as to why galois invented the concept of a normal group? Instead I spend most of my time chasing down motivation such as this, only to have exams sneak up on me and forcing me to cram problems and little techniques. Furthermore I have attempted to prove many of the things in my algebra course without reading the text, or going back over my professors proofs looking for better ones, and yet this has no benifit on my grade or even homework. It seems like no one else cares about the proofs, rather solving specific problems in the text or assignments.
Talking about this has made me realise I probably do want to stay in mathematics, even if I won't be the next Terry Tao. It saddens me that you want to leave the field, because I would love to have a professor like you. People like you are the reason I want to go into math, but the fact that you do not find it congenial is very troublesome.
 
  • #28
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The way I see it, mathematics is the path to absolute truth, or at least as close as we humans may come to it.
I think that is part of the attraction for many people that get into mathematics. However, in time, the luster on that jewel starts to fade. Bertrand Russell, who once suggested that mathematics was his salvation, remarked that the certrainty he had once hoped to find was as absent from mathematics as it was from anywhere else.

That's ok, though. It has a kind of truth, if not Truth, and as the man said, it does have an austere kind of beauty.


Talking about this has made me realise I probably do want to stay in mathematics, even if I won't be the next Terry Tao.
To cite Russell yet again, he once wrote that mathematics, too, needs its bricklayers and masons...or something to that effect.

I look at it like this: I have to make a living. If I'm fortunate, and work hard, then I might be able to make a living doing something that I enjoy. It doesn't matter if I'm famous for it, so long as I find it rewarding.

The funny thing is, I've spoken to many people in a wide variety of professions that voice the same complaint. "I don't make a difference," or the related sentiment, "I can only make a difference if I'm big enough, famous enough, important enough." Academics, lawyers, doctors, professors, teachers, businessmen--and yet, some in those professions are quite content. I think the difference is perspective: viewing fame, relative or otherwise, as a byproduct of success and not the goal or measure of it.
 
  • #29
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You see the reason I am so interested in math is for foundations. The way I see it, mathematics is the path to absolute truth, or at least as close as we humans may come to it.
What good is truth, though, if it isn't used for something or at least tells us something of philosophical significance? There's beauty in it at times, which is an end in itself, I guess. Not a compelling enough end for me, though, since it can only be appreciated by very few people. Also, it seems to me most mathematicians are creating ugliness, rather than beauty, in the pursuit of very technical questions. Let's face it, if you want to solve a problem, no matter what, you just want the answer, you can't really be bothered if it is ugly. You just have to solve it anyway. But I am not like that. If it's not beautiful, I am not interested and will move on to a different things. There are times when I might put up with temporary ugliness, though, in hope of a better solution later on.


For some reason I have found Algebra less enjoyable than Real Analysis. For starters I feel that Real Analysis and Complex analysis come far more naturally to me, part of the appeal being the ability to visualise things in analysis. However, may favourite mathematical topic so far has been topology, and differential topology (the small amount I have read) seems like a perfect match for me (Is the field still active or has it been "mined out"?).
Yeah, differential topology is the study of smooth manifolds, which is pretty active, still. Geometric topology is the study of manifolds, so differential topology is a subfield of it.


If I do proceed to graduate studies in pure math, I feel the pressure to work in a field just as analytic number theory, with the goal of working on problems such as the riemann hypothesis, yet I only enjoy specific parts of elementary number theory, and am not very good at tricky olympiad/putnam number theory problems. Another field which interests me is logic, but even logic the foundation of all pure mathematics seems second rate compared to number theory and analysis. Is this just an ego or thing or is there something fundemental to these feilds that the others lack?
From what I have seen, the basics of logic, like computability and Godel's theorem is vastly more interesting than current research, but that's just an outsider's point of view, having been to a talk or two, and knowing some grad students who thought logic would be cool, but changed their minds what they saw what it actually involved these days. As far as the Riemann hypothesis, yes, that's probably an ego thing. I don't see it as a particularly compelling problem, but then, the only thing keeping me from thinking number theory is totally lame is a sort of duty to think that it's interesting. Baez had a fun post about this.

http://math.ucr.edu/home/baez/week201.html

So, my point of view is the same as that of Baez before he came to appreciate number theory, except that I am aware, as he says, that there might be some interesting theory lurking behind it. So, I sort of take it as an article of faith that it's interesting. "If you say so, number theorists..."

I do like analysis, but I don't have a very good idea what research is going on in that area.


Algebra on the other hand has felt lest intuitive to me, and although I can visualise things, it does not "feel" the same as analysis. Furthermore everything seems less motivated, basic abstract algebra, such as cosets and normal groups annoyed me because I did not understand why we were making these definitions. Would it have hurt for the professor to have given some motivation as to why galois invented the concept of a normal group? Instead I spend most of my time chasing down motivation such as this, only to have exams sneak up on me and forcing me to cram problems and little techniques. Furthermore I have attempted to prove many of the things in my algebra course without reading the text, or going back over my professors proofs looking for better ones, and yet this has no benifit on my grade or even homework. It seems like no one else cares about the proofs, rather solving specific problems in the text or assignments.
Yes, that's just how I was. If you want to spend time doing that, I would recommend not taking a very heavy course load. My courseload in undergrad was a little light after I changed majors, simply because I didn't need that many more classes to graduate, but I had an extra year to play with because I was a fifth year senior after changing majors my senior year. You get in trouble, though, if you care about that kind of stuff. The most interesting part of math just doesn't seem to be valued very highly. I don't know how math should be done, anymore, though. I just know what I find interesting. The subject just seems so big and unweildy as to be completely unmanagable. Even successful research mathematicians feel like quitting sometimes because it's so hard and they feel like they are out of ideas.


Talking about this has made me realise I probably do want to stay in mathematics, even if I won't be the next Terry Tao. It saddens me that you want to leave the field, because I would love to have a professor like you. People like you are the reason I want to go into math, but the fact that you do not find it congenial is very troublesome.

It should seem troublesome. If you care about rethinking old subjects, you're less likely to get far with research. The people who just want to pounce on the homework and don't care about anything else are probably the most likely to be successful. Those who just take a lot of theorems on faith and get as much work done as possible. That's what is rewarded.
 
  • #30
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.

Do you believe specialization or generalization is the best path to a deep understanding of mathematics?
 
  • #31
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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.
 
  • #32
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@homeomorphic,

What're you gonna do now? Sounds like you're not headed the academic route...
 
  • #33
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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.
 
  • #34
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.
 
  • #35
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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.
 
  • #36
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?
 
  • #37
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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?
 
  • #38
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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.
 
  • #39
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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.
 
  • #40
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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?
 
  • #41
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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.
 

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