Should I Become a Mathematician?

[mathwonk]

you ask an odd question. you say you have significant gaps in your training and then ask not whether you should fill them in, but whether you should avoid the goal you desire. I cannot identify with this approach. I suggest: Either do what is required for your goal or change goals. Is that your question?

 

[mathwonk]

here is a review of the whole course of differential calculus.

 

[20Tauri]

Hello! I was wondering if I could ask for some math advice. I am an undergraduate senior at a liberal arts college. I came to math a bit on the late side, but I really want to keep doing it. My options are basically to delay graduation a year and finish a math major (and maybe learn a little programming), or graduate this year as a creative writing major and go to grad school. (Strangely, yes, this is possible--I have sufficient math background to qualify as a pure math master's student.) I don't know what I want to do, ultimately, and I lack algebra/analysis experience and have only very minor programming abilities (I can write small functional programs in Mathematica). I have already been accepted to one school, and they assured me this shouldn't be a problem, but I am not sure what to do. As for what kinds of math I am interested in, I really enjoy number theory in both its purely theoretical and computational aspects. Do you have any thoughts?

 

[mathwonk]

if you are saying you like both creative writing and math and need to decide what to pursue now, I think math may be a good first choice. The reason is that creative writing ability may improve over time, while math ability fades. so you could probably go back to creative writing later, but I doubt you would ever return to math.

 

[20Tauri]

Oh, yes, I definitely agree. I was wondering more if it would be more practical to spend another year finishing a bachelor's in math (and possibly go to grad school for it later) or to go do a master's in math now, given that my background is a little weak. (One school has accepted me, but I don't know how many other options I'll have or how good they will be.)

 

[mathwonk]

those are rather strange alternatives. you say you have the option of either getting a bachelors in math or a masters in math? usually one precedes the other. but in general you may assume that if you are accepted into a program, you are thought to be qualified for it. In general one is benefited by moving ahead faster in terms of degrees, unless their is reason to expect failure.

Are you thinking that with a better bachelors in math you would get a better math grad school offer afterwards than you have now? That is possible. I would choose based on which program is better, with better teachers, courses, students, support, conditions.

I.e. I might stay at a great undergrad school ad opposed to entering a weak grad school.

 

[Mépris]

He or she might be from the UK, where they have so-called "undergraduate master's degrees" - the MMath or MSci (not MSc). Perhaps, they got accepted into a PhD program straight out of the BSc, which is awarded after completion of the 3rd year. The MMath is awarded after the 4th year.

At least, that would explain the above situation.

 

[20Tauri]

Yeah, it's a bit odd. I'm not from the UK--what's happening is that I have almost enough credits to finish a B.A. in math, which is why there are some places that will take me.

I think if I had a proper bachelors and some of the specific coursework they expect, I could get into better programs or get better funding, yes. I also wonder if it is more practical to have a bachelor's in math rather than a master's in pure math if I decided not to continue on for a PhD. It seems like a lot of math jobs want you to do things that are more applied, especially in terms of knowing how to program. Programming experience seems to be something I could pick up as an undergrad but not necessarily as a grad student, since the undergrad classes wouldn't count towards a master's. But I don't know if math-related jobs like that prefer advanced degrees or experience (or both).

 

[mathwonk]

here are the first couple days of my integral calc notes. It includes a second discussion of exp and log functions.

 

[mathwonk]

These are a little more advanced, and contain norm convergence of series o functions, and the proof of the series expansion of e^x, and a few other important functions, like sin, cos, ln, arctan. When they got this lecture, I think my first semester freshmen knew they were not in high school AP calc any longer.

 

[lecter]

The reason is that creative writing ability may improve over time, while math ability fades. so you could probably go back to creative writing later, but I doubt you would ever return to math.

Are you saying that I won't manage what I am doing now when I am at a older age? Also could you compile the important points from this thread into an article on what I could expect from a career in Mathematics, going through 185 Pages of this thread with low speed internet is really frustrating :P ... If you have already written an article on these lines please post a link here would appreciate it. Many of the posts by other people on the forum are very demotivating. Most of the posts go like ... long hours, low salaries, no recognition etc... but I am sure the field must be more rewarding than that, so basically would love to hear from some of the more positive aspects along with the negative aspects of a career in Mathematics. And what is the typical Work of a University Lecturer in Math, as in do you get time to do research or do you end up spending a lot of time on Administrative work, Correcting term papers etc?

 

[homeomorphic]

Probably, math ability fades slightly with age, but I don't think it fades that quickly, plus, older mathematicians can make up for that with more breadth and so on. There seem to be plenty of examples of good work by older mathematicians.

People SHOULD be discouraged from pursuing math as a career, on the whole.

When you are growing up, it's easy to have this naive view of pursuing your dreams, but in many cases, it's just not realistic. My piano teacher just got his doctor of music and was looking for jobs. He says the number of positions available for piano professor in the country was something like 8. In the whole country. So, unless you started playing at age 4 or are at least are willing to put in 10 hours a day and have no life outside piano what so ever, you can probably forget about it.

Now, math isn't quite that bad, but there's a similar process of weeding out that goes on, leaving few survivors at the end. I just mentioned piano, just to shatter the naive childhood attitude of just saying "I want to be so and so when I grow up."

You can't always get what you want.

However, if you are really determined to do it, do it. Just don't say I didn't warn you if, at some point, you find it all a bit overwhelming and feel tempted to quit. Try to plan ahead. Start thinking about research as soon as you can. I didn't think about it enough until too late in my PhD. That doesn't necessarily mean you have to know what your thesis topic will be in undergrad already, but it helps to think about things like learning how to typeset in Latex, drawing mathematical illustrations, and so on. Practice typing up notes in Latex or something. Also, I think it might be helpful to think about what kind of skills you will need and how to learn things with research in mind, early on. I thought about it all a bit too late. Come up with exercises for yourself and try to invent things--don't just rely on books for that. That will give you some of the skills you will need.

If Atiyah was tempted to quit, anyone may be tempted to quit.

 

[Mépris]

Has anyone read this Lagrange book? Thoughts? [edit: after a search of this thread, I see you (mathwonk) have read it - any further thoughts, relating to this and Euler's book, i.e, any non-essential parts I could simply skim through instead of deeply studying? if I had it my way, I would read it all - trust me, I get very obsessive about this - but time is not on my side. no, I'm not dying but I need have other subjects to take care of]

Opinions on this text as well? I already have this book home and was wondering if the exercises in it would be suitable to supplement my reading of Euler. Viswuze, if you're reading this, note that I tried to get hold of the Allendoerfer/Oakley book but found no edition that ships to my country.

Lecter, I can see where you're coming from but there is simply too much information in this gem of a thread that it would require a herculean effort to compile it all in one article. In fact, I'd be willing to bet that there's enough valuable information/opinions here to make a few articles, directed to students in high school to those who already have doctoral degree!

Further, who can decide what information should "make the cut" and what should not? This is indeed mathwonk's thread and he is among the main contributors (or "guides"?) in it but there is, as I said, just too many good posts here for anyone to realistically put it all together in a concise way. It could be done but I think it should be a collective effort and even then, it will take a lot of time and one may accidentally omit one thing or another. At any rate, what I mean to say is, even if the people involved were to confine themselves to this thread alone (there's so much more information throughout the academic guidance section, and of course, in the whole website), it would be difficult to write "complete articles".

Anyway, this just my opinion and I could have missed something or could indeed, be completely wrong. :-) :-)

 

[sahmgeek]

Hello,

I have a question that may already be covered in this thread but I have not read all 185 pages. If my question has been addressed, could someone kindly direct me to the correct page(s)?

I am seeking advice on a receiving a math degree (e.g. Master's + Secondary Ed certification) however I have very little formal math training beyond high school. I'm one of those horribly misguided individuals with a social science/philosophy degree who thinks they can walk into the world of math/science with foolish confidence . Given that I would need to start from scratch, I wondered if taking the basics at a community college (Analytic Geo/Cal 1, 2, & 3, Linear & Abstract Algebra, Finite Math, and ODE) and, of course, doing very well in the courses, would provide enough preparation for applying to a graduate program. If not, are there other avenues one could take w/o getting another bachelor's degree (I already have an M.A. in another field)? I do know that most of the grad programs (I'm located in New York, NY and will not relocate) allow up to 4 non-matriculated courses so that could help. Also, I realize that these programs are very competitive and I would be up against applicants who already have a math degree. However, I also know that high school math teachers are scarce, especially in NYC.

Perhaps a little personal information about me would help you formulate your response. I'm 35, stay home with my 2 very young children and would most likely need to go back to school part-time. I might have some time during the day to work on math, but most of my free time would occur in the evening, 8pm and later. I am concerned that this isn't enough free time to really study this subject. I am not very concerned about my intellectual capabilities, but with my time constraints perhaps this is an unrealistic goal given the rigorous nature of math. I do, however, like the idea of studying math for it's own sake, even if the end result is purely for personal gain.

Thank you for any feedback relating to this post.

 

[homeomorphic]

I am seeking advice on a receiving a math degree (e.g. Master's + Secondary Ed certification) however I have very little formal math training beyond high school. I'm one of those horribly misguided individuals with a social science/philosophy degree who thinks they can walk into the world of math/science with foolish confidence . Given that I would need to start from scratch, I wondered if taking the basics at a community college (Analytic Geo/Cal 1, 2, & 3, Linear & Abstract Algebra, Finite Math, and ODE) and, of course, doing very well in the courses, would provide enough preparation for applying to a graduate program.

You would also need 2 semesters of analysis. Community college profs might not have that much credibility as far as recommendation letters go. Cornell or Columbia would be pretty hard to get into. Maybe there's a place in NYC that offers a masters in math that would more realistic. I don't know.

If not, are there other avenues one could take w/o getting another bachelor's degree (I already have an M.A. in another field)? I do know that most of the grad programs (I'm located in New York, NY and will not relocate) allow up to 4 non-matriculated courses so that could help. Also, I realize that these programs are very competitive and I would be up against applicants who already have a math degree. However, I also know that high school math teachers are scarce, especially in NYC.

I don't think you have to get the whole degree, although it helps. But you have to learn most of the same material.

Perhaps a little personal information about me would help you formulate your response. I'm 35, stay home with my 2 very young children and would most likely need to go back to school part-time. I might have some time during the day to work on math, but most of my free time would occur in the evening, 8pm and later. I am concerned that this isn't enough free time to really study this subject. I am not very concerned about my intellectual capabilities, but with my time constraints perhaps this is an unrealistic goal given the rigorous nature of math. I do, however, like the idea of studying math for it's own sake, even if the end result is purely for personal gain.

Sounds difficult. Taking classes would REQUIRE free time during the day, in most cases.

There are times when I do nothing but work, eat, sleep, and take a few breaks here and there for piano. Usually, at least one day a week, I take it easy (only work a little bit, maybe a couple hours). I suppose a lot of this work is self-imposed, due to the fact that I feel the need to drastically reformulate most of the math I come across in order to make it as intuitive and well-motivated as possible.

Poincare is said to have worked on math research for just 4 hours each day, but it will probably take a bit more work than that for several years to get to an appropriate level.

 

[sahmgeek]

Thanks homeomorphic. Yes, I would need to take time out during the day for classes. That is almost certainly true. I suppose I just need to begin with some basic community college classes and go from there. No sense in trying to plan ahead at this point. It does seem to me that it would be very challenging given my background and familial responsibilities. I can assure you that I am not, nor will I ever be, like Poincare. But that's not the goal...

 

[homeomorphic]

I can assure you that I am not, nor will I ever be, like Poincare. But that's not the goal...

I just mentioned him to suggest that 4 well-spent hours a day is probably sufficient, eventually, if you ever plan to do research.

 

[mathwonk]

people here are giving good advice on what mathematical background you might well need, but since your goal is to obtain a degree, it may be more efficient, to choose the school where you would like to get your degree, and ask them exactly what will be required to obtain an MA.

 

[Mépris]

http://www.alljapaneseallthetime.com/blog/potheads-planners-and-players

Something I found a website that MissSilvy referred me to. It's about learning Japanese, as the name would suggest, but I think some of us might benefit from this post. I know I did. (math, physics...schooling related, in general)

 

[gnarly]

http://www.alljapaneseallthetime.com/blog/potheads-planners-and-players

Something I found a website that MissSilvy referred me to. It's about learning Japanese, as the name would suggest, but I think some of us might benefit from this post. I know I did. (math, physics...schooling related, in general)

So we should smoke pot?

 

[ironman1478]

hello physicsforums.com,

i just have a question about proofs in general, but i didnt think it warranted a thread and i think this is the right place to put it.

if i am doing a proof and i get to the end, how do i know i am right? i am doing extra problems from my linear algebra book and from "Elementary Geometry from and Advanced Standpoint" and whenever i do a proof, i have no way of knowing that i have done it correctly since there is no solution given in both of these books. its not like finding a solution to an equation or a physics question because usually i can just plug my solution back into an equation and confirm my results, but with proofs its a bit different.

sry if this is a stupid question, but i am hesitant to continue doing problems from the books because i feel like i might finish the book, but i would have learned nothing since i did the problems incorrectly.

 

[chiro]

hello physicsforums.com,

i just have a question about proofs in general, but i didnt think it warranted a thread and i think this is the right place to put it.

if i am doing a proof and i get to the end, how do i know i am right? i am doing extra problems from my linear algebra book and from "Elementary Geometry from and Advanced Standpoint" and whenever i do a proof, i have no way of knowing that i have done it correctly since there is no solution given in both of these books. its not like finding a solution to an equation or a physics question because usually i can just plug my solution back into an equation and confirm my results, but with proofs its a bit different.

sry if this is a stupid question, but i am hesitant to continue doing problems from the books because i feel like i might finish the book, but i would have learned nothing since i did the problems incorrectly.

Hey ironman1478 and welcome to the forums.

This is not a stupid question.

If you don't have access to someone else like a professor, instructor, lecturer, TA or even one of your peers then I strongly make the suggestion to post your query on here in the relevant mathematics forum.

If you provide all the steps then I gaurantee someone will take a look and critique it.

 

[Nano-Passion]

hello physicsforums.com,

i just have a question about proofs in general, but i didnt think it warranted a thread and i think this is the right place to put it.

if i am doing a proof and i get to the end, how do i know i am right? i am doing extra problems from my linear algebra book and from "Elementary Geometry from and Advanced Standpoint" and whenever i do a proof, i have no way of knowing that i have done it correctly since there is no solution given in both of these books. its not like finding a solution to an equation or a physics question because usually i can just plug my solution back into an equation and confirm my results, but with proofs its a bit different.

sry if this is a stupid question, but i am hesitant to continue doing problems from the books because i feel like i might finish the book, but i would have learned nothing since i did the problems incorrectly.

Do the odd problems so you can check if you got the correct answers in the back.
Check if your book is on cramster.com, they have step-by-step solution to virtually every problem.

 

[homeomorphic]

i just have a question about proofs in general, but i didnt think it warranted a thread and i think this is the right place to put it.

if i am doing a proof and i get to the end, how do i know i am right? i am doing extra problems from my linear algebra book and from "Elementary Geometry from and Advanced Standpoint" and whenever i do a proof, i have no way of knowing that i have done it correctly since there is no solution given in both of these books. its not like finding a solution to an equation or a physics question because usually i can just plug my solution back into an equation and confirm my results, but with proofs its a bit different.

sry if this is a stupid question, but i am hesitant to continue doing problems from the books because i feel like i might finish the book, but i would have learned nothing since i did the problems incorrectly.

You have to try to figure out for yourself whether it's right or not. What good is knowing math anyway, if you always need someone to tell you whether you did it right? In the context of a job, the person who told you whether you were right may as well just do it themselves. So, you should aspire to be one of the people who knows what is right, rather than one of those who has to be told when they are right.

Just check all the steps and see if each step follows logically from the previous ones.

 

[jbunniii]

You have to try to figure out for yourself whether it's right or not. What good is knowing math anyway, if you always need someone to tell you whether you did it right? In the context of a job, the person who told you whether you were right may as well just do it themselves. So, you should aspire to be one of the people who knows what is right, rather than one of those who has to be told when they are right.

Just check all the steps and see if each step follows logically from the previous ones.

Easier said than done if you're working outside your comfort range, which is how you learn anything new. This is why peer review exists. Logical errors can be subtle and hard to spot, especially one's own logical errors.

 

[homeomorphic]

Easier said than done if you're working outside your comfort range, which is how you learn anything new. This is why peer review exists. Logical errors can be subtle and hard to spot, especially one's own logical errors.

You don't have to be perfect in order to learn something. You don't have to eliminate all mistakes.

I learn boatloads of new stuff that is outside my comfort range all the time and I never need anyone to tell me if I'm doing it right. It doesn't matter that much if I get something wrong because misunderstandings are almost always temporary if you keep learning in a rigorous and questioning manner.

Peer review is there, but it's only the last stage. If you can't tell right from wrong by yourself with reasonable reliability, you will never get to the peer review stage.

 

[PrinceRhaegar]

Hey guys. So I'm in my second semester of college as a mechanical engineering major, but I'm thinking about switching to math. The reasons are simple; recently I've found that I'm better at math than any other subject (especially physics, which is likely what I'll be spending most of my time doing for the next few years considering my current major), and I just think math is cooler than any other subject I've seen so far. The reason I'm really hesitant to do so is because firstly, I have no idea what I'd do with my degree after I graduate, and secondly, and this may seem a bit shallow, I know that I'll likely be making more money as an engineer than as a mathematician, especially right after college. So I guess my question to you guys is what are some of the more lucrative career options for someone with a math PhD (I know that I'll be going to grad school regardless of my major), and what would I likely see myself doing for those first few years after I graduate? Thanks for all the help, sincerely.

EDIT: I should probably add a few more points. In a perfect world I'd major in math and get a job as an engineer (or at least in an engineering company). This is because I love math and I feel like I'd get a TON of satisfaction out of doing useful stuff for the world while also doing what I love. So I guess I should rephrase my question; how easy is it for someone with a math degree to work in an engineering firm? And I know that this will likely vary greatly from person to person, but, mathematicians of the board, how much satisfaction do YOU personally get from doing the more "normal" things that a mathematician does (research, possibly teaching, etc.).

 

[alanlu]

You can count the money you make when you apply it to *.

Try using a Lebesgue integral to count your money.

Or if you want you can be an enlightened hobo.

 

[20Tauri]

ironman1478, that is one of the tricky things about studying on your own. If you knew how to do the proofs correctly already, you wouldn't be studying, so it can be hard if you don't have access to the answers. I'd suggest getting a friend, prof, or a forum group to take a look at your answers. Sometimes you can also find proof solutions by Googling if it's a relatively common problem type, or you could check Proof Wiki. The homework section of Physics Forum also is good for this stuff, as I think others have mentioned.

PrinceRhaegar, I have heard the more lucrative math careers are in finance. You can make quite a lot of money as an actuary, although I don't think it's something you would do if you had a PhD.

 

[Locrian]

I personally wouldn’t call actuarial work lucrative, but if you can get a job and some experience it has been very stable historically. It certainly pays better than most office jobs.

There are people with math PhD’s that get jobs as actuaries, but they’re rare. Actuarial mathematics is very specific and, if you’re in the US, you’ll learn it from the exams anyways. So why spend the extra years of poverty? The fantasy people have entering grad school wears off long before a math PhD is earned, so a Masters in math is much more common in this line of work.

To PrinceRhagar, getting a PhD in math with hopes of working at an engineering firm sounds like a recipe for disappointment to me. Don’t get me wrong, with enough craft and luck I’m sure it’s possible. It’s just not probable. Still, you’ll have lots of other options, too, so maybe it’s worth a try.

 

[00Donut]

Ironman, I sometimes struggle with the same thing. I do a proof, one which I feel is especially hard for me at the time, and in the end, like as soon as I finish it, I'm sitting there wondering whether or not whatever I did was correct. Usually what I do in these situations is examine every single step in my proof as much as I can, like, I will review the exact form of any theorem I may have used, critically examine and "poke at" any kind of things I may have "constructed" to aid in my proof, and so on. Also, another thing which is, in my opinion, extremely helpful is to walk away from your finished proof for like 2 days, then come back to it and read it over. Many times, you will not be able to see a mistake you may have made in your proof if you examine it immediately after you've finished it. Walking away gives your brain time to let other ideas and stuff in, like you stop thinking about math. There have been times where I do a proof, and I examine it immediately after and find no mistake in it. But then, days later, I do the same thing, and I find this HUGE mistake in it, and it's because when I checked immediately after finishing it, I was walking through the same path I went through when I made the mistake, and so it doesn't seem like a mistake, if that makes any sense at all... So yeah, my advice is that you walk away for a couple days, and then re read your proof. I feel like gaps in your understanding are much easier to find when you do this.

 

[abelian]

What's a good resource to learn about simple closed curves and intersection numbers (geometric and algebraic)? I don't know if this is obvious but I'm looking at this from a surface topological perspective.

Thanks.

 

[mathwonk]

milnor's topology from the differentiable viewpoint, differential topology by guillemin and pollack, and algebraic curves by william fulton.

 

[Advent]

I've only read half topic but it has an insane amount of advice, references, and enjoyable stuff. Thank you all, seriously.

 

[Bearded Man]

I have a good PreCalc book to recommend people. It starts with logic and set theory then moves to the field axioms. It covers a wide variety of topics from there, including the fundamental theorem of Algebra, logs, one-to-one functions and their inverses, trig, imaginary numbers...

https://www.amazon.com/dp/B000H5ESKG/?tag=pfamazon01-20

Though I have yet to read Spivak, I imagine this would be wonderful preparation for it.