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

[Group_Complex]

After getting back a result in an Abstract Algebra exam (In which I only got 70%), a result just below the class average I am having extreme doubts about my ability to become a mathematician.
The real shock was that I believed I understood the material well enough to get at least 90%. I am aware of where I went wrong in the exam, and understand my mistakes, but there is a lingering doubt now (Due to how well my classmates did) that I have subpar mathematical ability, and should consider leaving the subject. This causes me a great deal of distress as there is nothing else I can consider doing, and It was and I think, still is my dream to become a mathematician. Furthermore this exam result will probably leave me with a B in the subject when I had hoped for an A+, which in turn has ruined my chances at a top grad school.

In hindsight I realize I did not put in enough time revising certain areas before the exam, but I do not believe any of my classmates put in a great deal more effort than I did, and yet they have received vastly better results with minimal effort. The ease at which some of them seem to pick up the concepts which I have to bang my head against a wall for is truly frustrating and I cannot believe this is anything other than an indication of a lack of ability in pure mathematics.
Is it possible to come back from this and contribute something to pure mathematics or should I start considering a different career?

 

[dustbin]

I do not have the experience to directly answer your question, but if you know what you did wrong and that your preparation for the exam was not adequate, perhaps if you correct these sorts of deficiencies you would be fine. You say you do not believe that your peers put in as much effort as you, despite your lack of preparation, but how do you know? Does it really matter what your classmates are doing? Perhaps they have a better foundation and you just have to play a little catch up (I do not know you, this is just a suggestion). Have you tried talking with your professor to fix the problems and see if you can turn your grade around? Perhaps some of your professors could better answer the question regarding your ability.

As well, I don't know what your other grades are like, but I doubt having one B is a for sure bar to gaining entrance into a great grad school... nor would having straight A's guarantee your admission.

 

[Group_Complex]

I do not have the experience to directly answer your question, but if you know what you did wrong and that your preparation for the exam was not adequate, perhaps if you correct these sorts of deficiencies you would be fine. You say you do not believe that your peers put in as much effort as you, despite your lack of preparation, but how do you know? Does it really matter what your classmates are doing? Perhaps they have a better foundation and you just have to play a little catch up (I do not know you, this is just a suggestion). Have you tried talking with your professor to fix the problems and see if you can turn your grade around? Perhaps some of your professors could better answer the question regarding your ability.

As well, I don't know what your other grades are like, but I doubt having one B is a for sure bar to gaining entrance into a great grad school... nor would having straight A's guarantee your admission.

I am sure if I spoke to my professor in regard to my ability/talent he would advise me not to become a mathematician.
I don't think it is just that I am a poor test taker, I think this may be the final blow in my quest to become a pure mathematician. Who has ever heard of a mathematician getting below average in a math exam?

 

[micromass]

After getting back a result in an Abstract Algebra exam (In which I only got 70%), a result just below the class average I am having extreme doubts about my ability to become a mathematician.
The real shock was that I believed I understood the material well enough to get at least 90%. I am aware of where I went wrong in the exam, and understand my mistakes, but there is a lingering doubt now (Due to how well my classmates did) that I have subpar mathematical ability, and should consider leaving the subject. This causes me a great deal of distress as there is nothing else I can consider doing, and It was and I think, still is my dream to become a mathematician. Furthermore this exam result will probably leave me with a B in the subject when I had hoped for an A+, which in turn has ruined my chances at a top grad school.

In hindsight I realize I did not put in enough time revising certain areas before the exam, but I do not believe any of my classmates put in a great deal more effort than I did, and yet they have received vastly better results with minimal effort. The ease at which some of them seem to pick up the concepts which I have to bang my head against a wall for is truly frustrating and I cannot believe this is anything other than an indication of a lack of ability in pure mathematics.
Is it possible to come back from this and contribute something to pure mathematics or should I start considering a different career?

You seem to think that success in mathematics or in grad school has something to do with natural ability. It's not really true. Way more important is being able to work hard and to love the subject. If the other students put in a minimal effort and don't work hard, then they won't succeed in grad school.
If you can get an A+ by putting in the work, then go do the work. Working hard and persisting is way more important in grad school than understanding everything immediately.

 

[Group_Complex]

You seem to think that success in mathematics or in grad school has something to do with natural ability. It's not really true. Way more important is being able to work hard and to love the subject. If the other students put in a minimal effort and don't work hard, then they won't succeed in grad school.
If you can get an A+ by putting in the work, then go do the work. Working hard and persisting is way more important in grad school than understanding everything immediately.

That is what some people say, then there are others (Paul Halmos for instance in his biography) who would tell decent students but not brilliant ones not to go into mathematics.

If it were not natural talent, then the only other things I can think of possibly causing problems are:
1. Not taking notes in class (I just reread the textbook if I have to remember something)
2. Not doing extra problems. I have not really had the time to do this with many other subjects also requiring me to submit weekly homework.

 

[chill_factor]

I'm in physics, not math, but here's something that is relevant:

take notes no matter what.

its not for the formulas. its because the professor might solve an EXACT problem that's unsolvable in any other way except by already knowing the solution.

 

[Robert1986]

DUDE! Don't worry. I got like a 55 on my first undergrad algebra test. I don't remember the average, but it was much higher than that. I still remember the 20 point problem that really screwed me: If a^2=a for all a in G, prove G is abelian.

I'm now in grad school,. So don't sweat it.

 

[Arsenic&Lace]

Natural ability:
DNE! Anybody who's good has worked hard at some point. There are probably genetic variations in maximum ability, but I highly doubt a few tests truly represent ability to be good at math.

Think about it, if this were true, African Americans and women of any race would, on the average, have a lower natural ability than white men, which I doubt very, very strongly; they get less practice, encouragement, and might not grow up in an environment in which mathematics and science are interesting enough to warrant any attention.

Getting into a top grad school:
You can have perfect grades and not get into a top grad school, and you can have imperfect grades and get into a top grad school. Who cares about top grad schools anyways? Do they have magical powers to make you a better mathematician? If all you care about is doing mathematics, does it matter if you're doing it at Bob's Bible College or Harvard? Even if you do go to Harvard, if math is anything like physics, you'll probably still not become a professor, by virtue of the academic market.

Getting hit on the head is a good thing. It happens to everybody some time or another. Now you have to double down, and work twice as hard... I highly doubt a 70% will prevent you from getting a B+, A-, or A; the professor has designed an awful course if you're not allowed to mess up an exam (which is a frequent occurence) and still get a good grade in the course.

Also, this guy is always inspiring:
http://en.wikipedia.org/wiki/Stephen_Smale

 

[xaos]

it would take some arrogance to assume your are a brilliant mathematician by default (if you were getting excellent grades). you have indicated that you are not taking notes, having difficulty with the material, and perhaps also you are assuming understanding by passive reading. these are easily correctable: realize when you're behind in the material and push those extra hours needed to attainment. the office hours are there to be used by the student to ask questions.

 

[homeomorphic]

Who has ever heard of a mathematician getting below average in a math exam?

It's true that I smoked everything in all my theoretical math classes in my undergrad, but I got below average grades on many exams in lower level classes. Of course, I am not going to make it as a mathematician, but I think I will finish my PhD this year.

I feel like a complete idiot now, after having so much trouble writing my thesis and not always being on top of things in graduate level math classes. If it were only a matter of being an incompetent mathematician, maybe I would stupidly persist in it. But the real issue is that, on the research side, I am not motivated in the slightest by trying to compete in some kind of publishing game, doing "original" work for the sake of doing original work. I am only interested in somewhat selfishly pursuing my own understanding of math and physics, not what other people think is interesting and what are the "hot" topics. I always took it for granted that there would be a place for that in the mathematical community, but I was wrong. I am completely unable to guarantee any publications within a few years, if I am to pursue my true interests. The only possible way to succeed would be to pursue research that I am not motivated to do, and with no motivation, I can't overcome the difficulties that research presents.

who would tell decent students but not brilliant ones not to go into mathematics.

In the current state of affairs, I don't know that I would tell ANYONE to go into math, unless they have an insane compulsion to do it. The only people I would tell to go into math are the ones who are far enough ahead of the game they don't even need my advice on the matter, so I would say to anyone wishing to go into math, "don't say I didn't warn you." I'm not sure if "brilliance" is the right criteria, but I agree with the notion that it's not a good choice for most people, including most people who THINK it is a good choice for them (90% of math PhDs don't do any more original research after the PhD). I was pretty into math, but I am so burnt out now, it's not even funny. If you are into teaching, it's okay, but if you want to do it for research, I don't know if you have to be brilliant, but I think you at least have to be either way ahead of the game or you just have to have the right personality to fit into the mathematical research community, which means you are really good at cranking out papers as fast as possible, above all else. There is no time for the laid back philosophical musings and reflection on math that has been done already that people like me would like to indulge in, and I'm not alone in this. My friend who is about to defend his dissertation in PDEs said people just quote a lot of results that they don't understand and he felt pressured to rush through reading papers, instead of taking time to understand them. The ones who survive are not really the strongest, but the ones who are the best adapted to working in this kind of high pressure environment. Much like natural selection.

Most people should realize that they are likely to be destined for a teaching-oriented job, and therefore, interest in teaching is equally important to interest in math. For teaching-oriented people, a math PhD isn't quite as bad of a deal. All you have to do is make it through, then, you can go and focus on teaching.

 

[Group_Complex]

It's true that I smoked everything in all my theoretical math classes in my undergrad, but I got below average grades on many exams in lower level classes. Of course, I am not going to make it as a mathematician, but I think I will finish my PhD this year.

I feel like a complete idiot now, after having so much trouble writing my thesis and not always being on top of things in graduate level math classes. If it were only a matter of being an incompetent mathematician, maybe I would stupidly persist in it. But the real issue is that, on the research side, I am not motivated in the slightest by trying to compete in some kind of publishing game, doing "original" work for the sake of doing original work. I am only interested in somewhat selfishly pursuing my own understanding of math and physics, not what other people think is interesting and what are the "hot" topics. I always took it for granted that there would be a place for that in the mathematical community, but I was wrong. I am completely unable to guarantee any publications within a few years, if I am to pursue my true interests. The only possible way to succeed would be to pursue research that I am not motivated to do, and with no motivation, I can't overcome the difficulties that research presents.



In the current state of affairs, I don't know that I would tell ANYONE to go into math, unless they have an insane compulsion to do it. The only people I would tell to go into math are the ones who are far enough ahead of the game they don't even need my advice on the matter, so I would say to anyone wishing to go into math, "don't say I didn't warn you." I'm not sure if "brilliance" is the right criteria, but I agree with the notion that it's not a good choice for most people, including most people who THINK it is a good choice for them (90% of math PhDs don't do any more original research after the PhD). I was pretty into math, but I am so burnt out now, it's not even funny. If you are into teaching, it's okay, but if you want to do it for research, I don't know if you have to be brilliant, but I think you at least have to be either way ahead of the game or you just have to have the right personality to fit into the mathematical research community, which means you are really good at cranking out papers as fast as possible, above all else. There is no time for the laid back philosophical musings and reflection on math that has been done already that people like me would like to indulge in, and I'm not alone in this. My friend who is about to defend his dissertation in PDEs said people just quote a lot of results that they don't understand and he felt pressured to rush through reading papers, instead of taking time to understand them. The ones who survive are not really the strongest, but the ones who are the best adapted to working in this kind of high pressure environment. Much like natural selection.

Most people should realize that they are likely to be destined for a teaching-oriented job, and therefore, interest in teaching is equally important to interest in math. For teaching-oriented people, a math PhD isn't quite as bad of a deal. All you have to do is make it through, then, you can go and focus on teaching.

A quick question, how hard did you have to work to "smoke" the upper level math courses? Or was it for the most part just natural?
Also what do you plan to do after your Phd?

 

[homeomorphic]

A quick question, how hard did you have to work to "smoke" the upper level math courses? Or was it for the most part just natural?

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.

Also what do you plan to do after your Phd?

Basically, abandon the field I worked so hard to learn, except maybe in my spare time. Engineering, insurance, operations research, whatever I can find.

 

[Group_Complex]

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.





Basically, abandon the field I worked so hard to learn, except maybe in my spare time. Engineering, insurance, operations research, whatever I can find.

You had a professor exclaim at your brilliance? That makes me quite envious. Why would you give up on mathematics? Why not just work on some less interesting mathematics for a while to secure tenure and then win a fields medal or something?

Also how easily did you remember the proofs of certain theorems? I find myself unable to remember certain proofs especially the intricacies involved (Despite thinking about them for a long time in a conceptual manner). The ease at which some mathematicans can come up with original proofs whilst reading a text is startling to me and I am unable to replicate this.

 

[homeomorphic]

You had a professor exclaim at your brilliance? That makes me quite envious. Why would you give up on mathematics? Why not just work on some less interesting mathematics for a while to secure tenure and then win a fields medal or something?

My undergrad profs were wrong. I never interpreted them as saying that I was destined to be a Fields medalist, just that I could have a successful career in math, if I worked hard. But they were wrong, even with that interpretation. I turned out to be terrible at math research. No good at all. Barely even capable of finishing my PhD, if that. Plus, I just don't think it's a very compelling or useful thing to do. I look at what engineers or physicists do, and I am jealous. I am proving some theorem that no one cares about. They are building cool robots, medical technology to improve people's lives, 3-D printing, all kinds of awesome things that have to do with real life. Genuine contributions to society. It's very easy to argue that math might be useful eventually because it's very hard to predict applications. However, you could argue that any kind of research COULD be useful. Like researching the best way to pick your nose. How can you predict that something useful won't come out of it? That may be an extreme example, but the point is that all research is not equal. I think things should be geared more towards applied stuff. Not that pure math shouldn't be studied, but there's way too much emphasis on it, and there ought to be more of a link to the applied stuff. Even "applied" math isn't really applied enough, a lot of times. Part of the problem is that the body of knowledge is now so large that it has become unmanageable and people are forced to specialize too much.

Math research is HARD. If you aren't motivated and don't really believe in what you are doing, it's just going to be completely miserable. I am already very burnt out doing that from grad school.

Also how easily did you remember the proofs of certain theorems? I find myself unable to remember certain proofs especially the intricacies involved (Despite thinking about them for a long time in a conceptual manner). The ease at which some mathematicans can come up with original proofs whilst reading a text is startling to me and I am unable to replicate this.

Retention is just a matter of reviewing and making things memorable. Review after one minute, one hour, one day, one week, one month, one year or some such thing. I used to review the contents of my electromag class in my mind every day, going through all the important points in the material up to the current material.

 

[DarrenM]

@OP: One poor test isn't indicative of anything except that you did poorly on one test. If you have a pattern of low scores, or significant gaps in comprehension, then you may want to consider reevaluating your goals.

It is very important that you get a handle on that kind of reaction. As long as you understand what you did wrong--and not in a general, abstract way, but in a very specific, detailed way--then you are moving in the right direction. The most important thing is not that you failed the test, but what you are going to do about it. Wring your hands and despair? Or dust yourself off and pummel those books until you can do that test in your sleep?

When you get a bad grade--and believe me, this won't be the last--give yourself a set amount of time to mope. 12-24 hours is good, so long as you have at least a night to sleep on it. Then take hold of yourself, tell the whiny, emotional, self-pitying part of you to get back in its box, and figure out how to improve so you don't make those mistakes again.

Plus, I just don't think it's a very compelling or useful thing to do. I look at what engineers or physicists do, and I am jealous. I am proving some theorem that no one cares about. They are building cool robots, medical technology to improve people's lives, 3-D printing, all kinds of awesome things that have to do with real life. Genuine contributions to society.

I don't know how to ask this without sounding accusatory, so I'll have to settle for assuring you that isn't my intention or tone. I'm just genuinely curious:

Wasn't that something you were aware of prior to going to grad school? I mean, for some people the fact that math isn't a "practical" field is part of the allure, or at least something they acknowledge before pursuing it as a study.

Was it something you were aware of beforehand, but only came to find objectionable later on?

 

[homeomorphic]

Wasn't that something you were aware of prior to going to grad school? I mean, for some people the fact that math isn't a "practical" field is part of the allure, or at least something they acknowledge before pursuing it as a study.

Was it something you were aware of beforehand, but only came to find objectionable later on?

I really wanted to do mathematical physics, which is not really practical, but is directly related to reality. But, the closest I could find as far as advisers to what I wanted was a topologist. He pulled me more in a pure math direction, but that was never really my intention. I wasn't really aware of how pure math it was going to be. I thought I could be more on the physics side than it turned out. Plus, back when I went to grad school, I still thought string theory was a promising approach, but it has been clearer now that it is probably not, though I don't want to make any definitive statements about it, since I don't know that much about it. The failure of string theory changes the situation quite a bit in this neck of the mathematical woods.

 

[Integral]

You seem to think that success in mathematics or in grad school has something to do with natural ability. It's not really true. Way more important is being able to work hard and to love the subject. If the other students put in a minimal effort and don't work hard, then they won't succeed in grad school.
If you can get an A+ by putting in the work, then go do the work. Working hard and persisting is way more important in grad school than understanding everything immediately.

Precisely what I was going to post. Well said Micromass!

 

[DarrenM]

I really wanted to do mathematical physics, which is not really practical, but is directly related to reality. But, the closest I could find as far as advisers to what I wanted was a topologist. He pulled me more in a pure math direction, but that was never really my intention. I wasn't really aware of how pure math it was going to be. I thought I could be more on the physics side than it turned out. Plus, back when I went to grad school, I still thought string theory was a promising approach, but it has been clearer now that it is probably not, though I don't want to make any definitive statements about it, since I don't know that much about it. The failure of string theory changes the situation quite a bit in this neck of the mathematical woods.

That is very unfortunate. I'm sorry that your studies were not what you had hoped. Thanks for indulging my curiosity.

 

[Group_Complex]

I have seriously been considering leaving mathematics over the last few days.

Things I have been considering:
1. I do not seem to enjoy "doing" mathematics as much as I should, rather I enjoy thinking about what it would be like to be a pure mathematician of the caliber of say Gauss or Euler, I seem to have romantic notions of mathematics.
2. I have no idea what else to do with myself, engineering does not really appeal to me, and neither does going into say academic physics rather than mathematics. The class I am enjoying most at the moment is a philosophy course, but I don't intend to become a useless philosopher. I seem to enjoy asking deep questions, but most of the mathematics I see and or do does not really have me doing this.

Are these signs I should just find something else to do?

 

[homeomorphic]

I have seriously been considering leaving mathematics over the last few days.

Knowing me, I would probably support such a decision, but it shouldn't be based on one test score. A single test score is irrelevant. If it is the catalyst to make you realize you don't want to do it for other reasons, that's a different story. In my case, my thesis went too slowly and there were several mishaps, which made me doubt my abilities. However, maybe I just got intimidated too much by grad school. And maybe it's just hard and takes a lot of work. So, it isn't so much that I doubt my abilities. The real issue is I realized this isn't what I want to be doing. Temporary lack of success was just the catalyst that made me realize it.

Also, I think math can be a great second major if you don't mind spending the extra time or money for that.

Things I have been considering:
1. I do not seem to enjoy "doing" mathematics as much as I should, rather I enjoy thinking about what it would be like to be a pure mathematician of the caliber of say Gauss or Euler, I seem to have romantic notions of mathematics.
2. I have no idea what else to do with myself, engineering does not really appeal to me, and neither does going into say academic physics rather than mathematics. The class I am enjoying most at the moment is a philosophy course, but I don't intend to become a useless philosopher. I seem to enjoy asking deep questions, but most of the mathematics I see and or do does not really have me doing this.

Are these signs I should just find something else to do?

It may be that you are actively looking for reasons to do something else, so you are biasing your decision. But that in itself is a sign that maybe it isn't what you want to do. Or it may just be temporary discouragement. In my case, I really don't care if my decision is biased--I just want out, no matter what. If I could have been an academic mathematician, that has been ruined, now. But actually, I am happy about not being an academic mathematician.

Deep questions. Hmm. There may be various different standards for what "deep" is. "Deep" to mathematicians might mean "difficult" or something requiring lots of background knowledge to understand. It might not mean something that anyone cares about, outside the math bubble. Group theory is kind of deep to me because it's secretly not really about a gadget that has a binary operation on it with associativity, identity, and inverse. To me, it's really about symmetry. All kinds of symmetry. Of course, there is the more nitty-gritty technical side of it, where you do things like classify all the groups of order 85 ad so on. A lot of things don't seem deep because the mathematical community has huge issues with providing the motivation for stuff right now. To the people who would tell me you just have to do the problems, the issue is that that they didn't put it in the problems. If you want to motivate it by problems, that's certainly an option, but I don't see it. I just see problems that teach you how to use it, not where it comes from.

I don't know that there's a profession that deals solely with "deep" stuff, though. There's always the nitty-gritty technical side, too.

 

[Group_Complex]

Knowing me, I would probably support such a decision, but it shouldn't be based on one test score. A single test score is irrelevant. If it is the catalyst to make you realize you don't want to do it for other reasons, that's a different story. In my case, my thesis went too slowly and there were several mishaps, which made me doubt my abilities. However, maybe I just got intimidated too much by grad school. And maybe it's just hard and takes a lot of work. So, it isn't so much that I doubt my abilities. The real issue is I realized this isn't what I want to be doing. Temporary lack of success was just the catalyst that made me realize it.

Also, I think math can be a great second major if you don't mind spending the extra time or money for that.




It may be that you are actively looking for reasons to do something else, so you are biasing your decision. But that in itself is a sign that maybe it isn't what you want to do. Or it may just be temporary discouragement. In my case, I really don't care if my decision is biased--I just want out, no matter what. If I could have been an academic mathematician, that has been ruined, now. But actually, I am happy about not being an academic mathematician.

Deep questions. Hmm. There may be various different standards for what "deep" is. "Deep" to mathematicians might mean "difficult" or something requiring lots of background knowledge to understand. It might not mean something that anyone cares about, outside the math bubble. Group theory is kind of deep to me because it's secretly not really about a gadget that has a binary operation on it with associativity, identity, and inverse. To me, it's really about symmetry. All kinds of symmetry. Of course, there is the more nitty-gritty technical side of it, where you do things like classify all the groups of order 85 ad so on. A lot of things don't seem deep because the mathematical community has huge issues with providing the motivation for stuff right now. To the people who would tell me you just have to do the problems, the issue is that that they didn't put it in the problems. If you want to motivate it by problems, that's certainly an option, but I don't see it. I just see problems that teach you how to use it, not where it comes from.

I don't know that there's a profession that deals solely with "deep" stuff, though. There's always the nitty-gritty technical side, too.

Homeomorphic, how much retention do you have for previous math courses? I was looking over some of my previous analysis proofs and realized I could not do most of the proofs without reading them in a textbook. How often should a mathematician have to go back over proofs he/she has already learned or is this a sign that they have not been well understood?

Also, how do you go about balancing your "conceptual" understanding with the need to get stuck into difficult and intricate problems. I seem to enjoy the conceptual side of mathematics (proving the fundamental theorems of the subject) but often feel unmotivated to do every difficult exercise in my textbooks.

 

[homeomorphic]

Homeomorphic, how much retention do you have for previous math courses?

Well, grad school just overwhelmed me so much, I wasn't able to review properly and a lot of stuff just slipped away from my memory. Part of it was the volume of material, part of it was the lack of a proper conceptual framework to tie it together. I don't even know what I remember or what I don't remember anymore. I remember a lot from my undergrad, still, where I could review more thoroughly. I also remember certain things from grad school that I focused on or things that I cared about a lot.

I was looking over some of my previous analysis proofs and realized I could not do most of the proofs without reading them in a textbook. How often should a mathematician have to go back over proofs he/she has already learned or is this a sign that they have not been well understood?

I don't know how often you have to review. Atiyah said that you don't need to have a good memory to be a mathematician because you can just remember how to derive it. Feynman also said in his lectures on physics that he didn't remember any of it from freshman physics, but just remembered the statements and invented explanations for them. I don't know that there is an easy answer to the question of what exactly you should try to remember. Apart from the question of whether I "should" remember things, I always derived a certain pleasure from going over everything in my mind and having a sort of database in my mind where everything was neatly stored and I remembered exactly where each thing was. I don't know that there's a right way or a wrong way to do it. However, you don't want to just put all that work into and then end up forgetting it all, including how to derive it. Once you get to grad school and see the volume of things that you have to know, the problem becomes quite different in nature. In undergrad, I never had an issue of having to manage trying to remember so many different things. One thing that helps is when you can see connections between things, so you build this tightly interconnected web of facts that are all associated to each other and all mutually reinforcing. When you go through these boot camp classes in grad school, there's hardly any time to even think about whether you are building effective data structures in your mind. And of course, part of the trick is to make things memorable. If I understand something, my brain feels happy. If I don't understand it, my brain feels unhappy. I keep thinking about it until my brain is happy, until those little voices in the back of my head telling me I don't understand it shut up. Often, by the time the voices shut up, I have come up with a fairly different version of the subject than the one that was presented to me in textbooks or lectures.

Also, how do you go about balancing your "conceptual" understanding with the need to get stuck into difficult and intricate problems. I seem to enjoy the conceptual side of mathematics (proving the fundamental theorems of the subject) but often feel unmotivated to do every difficult exercise in my textbooks.

I don't know. The exercises help reinforce things. You don't have to do every exercise. Usually, I didn't mind doing exercises. What really sucks is when you write a thesis, which is one big exercise that's so damn long you can't even see that you are making any progress relative to the whole thing. I guess it's a pretty different psychological battle you have to fight from just doing homework, which is much more short term, and therefore easier for human minds to deal with and motivate themselves to do.

 

[Group_Complex]

In regards to your issues with Grad school, is it possible that you simply did not go to the right Grad school or have the right supervisor? I mean if you were in a top 10 grad school as it seems you had the talent for, the work expected of you may have increased compared to your undergraduate university. On the other hand if you went to a less well known grad school, maybe you did not have the quality of instructors interested in your conceptual version of mathematics. I mean you seem to have issues with the entire pure mathematics system when it may be quite possible it is just the certain university or people you have been around.

Also in regards to time to review things. I am taking four upper division math and theoretical physics courses, as well as a logic/philosophy course, is this a suitable load to allow conceptual thinking or should I try and focus on specific key parts of mathematics, i.e. analysis, algebra, rather than trying to branch out too early in my undergraduate career?

 

[homeomorphic]

I don't have issues with the ENTIRE pure math system. I have issues with big parts of it. There are profs/textbooks that do a great job presenting topic A, but totally butcher topic B.

These issues can be found even at the top universities. The program I am in is not top 10, but it's been ranked in the top 20 recently. It's gone up since I came here.

My adviser is famous in his area. I don't have issues with the way he is doing things mathematically, as far as the stuff I am working on. He is way more mathematically talented than I am, plus has a lot more knowledge/experience, having been thinking constantly about math all the time for the past 30 years or more. He might not be the best at bringing out my talent. Somehow, I just don't care that much about what I am doing. It just seems like an obstacle that is in the way of pursuing my real interests.

I read textbooks that come from all over the place and that is a big part of my objection. I also think many papers are not very well written.

I don't really have that much talent. You have to realize that most of my graduate education is not even relevant to what I am doing in my thesis. So it's a non-issue, really. The issue is not that the mathematical community is screwed up, which it is, but rather that I suck at research and have no motivation to stop sucking at it because I feel like it isn't important or relevant to the reality. The motivation is just to get the darn thing finished, just because I started it, and maybe to practice writing math.

I don't know how many classes you can take. Depends on how hard you are willing to work. I think there's a trade-off between trying to study a lot of different stuff and just trying to learn a few things well. Some people like to just focus on a few things and learn them well, some like to focus on the big picture. It's a matter of style. But you don't want to go too fast or too slow. I think grad school did me some good in making me study things faster. I skip things when I get stuck, now. Sometimes, that is a good idea. If you want to fit into the system, though, I think the best thing is to specialize early on. But that's part of what I don't like about the system. It rewards specialization too much. If you cut out anything that is not relevant to writing a thesis on X topic, that is the most effective way to write a thesis. As a side effect, you would end up with a fairly narrow knowledge of mathematics, but in the end, you will look better than someone like me who tried to learn a million things not related to his research.

 

[Group_Complex]

Would you not feel "left out" by leaving academia, I mean say there are some breakthroughs in the fields which interest you, you probably won't have the time or resources to keep up to date with the tools and progress in the field.

The reason I prefer pure math over say theoretical physics is that you can justify everything logically in math. You can ask why is this so? and rather than having to accept experiment, you can see the logical path which justifies things from first principles. In my view mathematics is the queen of the sciences and all our objective knowledge essentially is derived from it (A mathematician said this once, but I can't recall who). I mean if physicists ever come up with a unified theory which models all physical forces, it will be a mathematical one, justified via mathematical reasoning. To me that is why the study of mathematics is the most important and also why I am hesitant to leave it behind, for other fields do not have such a claim to studying the foundations of "reality". I guess that is what I meant by a deep question, rather than deriving things from axioms, I would rather ask why the axioms are formulated in the way they are. I understand proving results contribute to this understanding, but it seems as though that is not the goal of the professor or the textbook. It seems more like puzzles for the sake of puzzles. Furthermore none of the questions in exams or assignments ask us to reprove a theorem from class, which makes me feel like I am wasting my time decomposing and thinking about the larger theorems in the subject.

May I also ask what areas you find most interesting in mathematics?

 

[homeomorphic]

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.

 

[Group_Complex]

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 fundamental to these fields 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 realize 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.

 

[DarrenM]

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 realize 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.

 

[homeomorphic]

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 fundamental to these fields 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 realize 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.

 

[Group_Complex]

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?

 

[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.