Should I Become a Mathematician?

[tgt]

Can't you just do a search yourself?

I've tried without success

 

[serkan]

hi mathwonk, this is my second year in university and i started to take mathematics as my second major. problem is, I'm not confident about my ability to prove. even simple proofs include some tricks that i think i won't easily come up with at the moment. what would you recommend? i think examining lots of simple proofs will help me at this point but any other advice will also be appreciated. so it would be great if you could point me some good resources where i can find such examples.

thanks

 

[Tac-Tics]

problem is, I'm not confident about my ability to prove. even simple proofs include some tricks that i think i won't easily come up with at the moment. what would you recommend?

Not all proofs are created equal. Some are so simple or obvious that it's not clear that they should in fact be called proofs. Things like proofs that (x + y)^2 = x^2 + 2xy + y^2. On the other end of the spectrum, there are proofs that seem to require infinite genius to have found. The proof that no rational number's square is two, for instance... the contradiction in the proof is just so damn subtle!

In a single topic in math, you'll notice that many proofs follow a similar pattern or employ a similar "trick". Many mathematicians make their careers off of becoming the first or best at exploiting some kind of mathematical trick. For example, in set theory, Cantor invented the "diagonalization" trick to show relationships between the sizes of sets. The same trick can be used over and over in different ways. You can use it to show that the reals outnumber the rationals. But you can also use it in contexts of computability and formal logic to show that the number of truths and functions outnumber provable truths and computable functions.

By studying proofs, you become more familiar with these tricks. If you study proofs in Point-Set Topology, you'll become much better at proofs in point-set topology. If you come across a theorem which you've never seen, but you recognize topological elements of the problem, you'll have a clue that you should begin looking for ways to reduce the problem to a statement about homeomorphisms, compactness, connectedness, and continuous functions.

As a student, most proofs you'll be expected to exhibit on a test are going to be fairly easy ones. The ones you're most likely to encounter are ones that are very closely related to a definition of some sort. At my school, linear algebra was the course used to introduce students to proofs. Proofs on the test were things like "Prove that the operation 'rotate a vector by 45 degrees' for R^2 is a linear operator" or "prove that a nullspace is a linear space." These kinds of proofs you should be able to do (in any subject) with a small bit of studying.

Sometimes, though, if a major theorem's proof is presented in class, your prof. may want you to reproduce it, or to prove a similar theorem. For example, I had a class where the prof. taught us the proof for the irrationality of the square root of 2, then on the test, asked for the proof for the irrationality of the square root of 3. But if you knew the first (and actually understood how it worked), the second is really easy.

When you're working on a proof which is neither obvious nor has been covered in your class, that's where you're doing real mathematics =-) There is no clear cut path how to solve a proof in general, but as you learn more, you'll pick up lots of useful techniques.

 

[Tom86]

serkan, from my experience (which is not much more than yours - I've only just done my BSc and am about to embark on a MSc) time and patience are your best friends when it comes to learning how to prove things. It wasn't until my third year courses in analysis and topology that I really began to appreciate the epsilon-delta definition of continuity (in metric spaces). To be honest, I was almost ready to drop out of my first year because I missed so many lectures at the beginning that I had absolutely no clue about abstract definitions and structures like groups, let alone how to prove things about them! Even with this poor performance in my first year, things eventually started to sink in and I graduated with one of the highest marks in my year.
As an aside - the only reason I didn't drop out is because I came across David Burton's book on Elementary Number Theory and it really helped me to appreciate the beauty of the subject (although it took me quite a lot longer to get to grips with analysis!).

My advice for you would be to keep going as you're going, but to take your time when studying proofs. If you need to, ask yourself questions like "why does proof by induction work?" or "why does proof by contradiction work?" and "why is one method of proof used in this circumstance and a different method used in another?" Spend a while contemplating what "necessary and sufficient" means. I would not, however, recommend spending hours agonising over a proof. If you get to a point where you are well and truly stuck, take a break or do some different work. Come back to the problem later and you might be able to see it from a different point of view.

If this post has been too general - or even condescending - I apologise. What I'm saying is you sound like you're doing just fine. If you get to your final year and you find yourself revising for exams and not knowing how to prove things, then you have reason to be concerned. :)

 

[mathwonk]

ok here is my secret: I decided to quit pretending I was smarter than others and to try to see how good I really was: i.e. I decided to see how good I could be by actually working as hard as possible.

The result? I was nowhere near as good as I fantasized, but much better than I had been.

best wishes to you. you all know what you should be doing. my advice is merely that if you start doing those things, they will work for you.

 

[serkan]

thanks everyone for advices, it's been helpful. well, i will try to do my best at this point, and working as hard as possible seems to be the way to go =).

 

[tgt]

Finally found that mathwonk got a gpa of 1.2 after first year. You were kicked out and worked as a meat slumber? How many years after did you get back into undergraduate again?

 

[arshavin]

I haven't studied any real analysis, except for basic stuff (open/connected sets, bolzano-weiestrass) but will do so soon.

I'm curious to know how it is different from advanced calculus?

 

[muppet]

ok here is my secret: I decided to quit pretending I was smarter than others and to try to see how good I really was: i.e. I decided to see how good I could be by aCTUALLY WORKING AS HARD AS POSSIBLE.

The result? I was nowhere near as good as I fantasized, but much better than I had been.

best wishes to you. you all know what you should be doing. my advice is merely that if you start doing those things, they will work for you.

Thanks for sharing all this mathwonk. It is encouraging.

 

[qspeechc]

Hi. I too am a second year student hoping to major in maths. I have a few questions, please bear with me.

How important is linear algebra to the mathematician? I have already taken a course in linear algebra, but I am thinking of studying it again over the break before 3rd year, since the course I took was not so good. Is it worth studying linear algebra properly, or should I focus on abstract algebra instead? Or both? I may not have time to revise both.

Should I study set theory and logic independently, or is it sufficient as it is given in the course of my undergrad years?

How good I have to be to get into grad school? Do I need 90% plus in my final year? Is that even acheivable?

Finally, should I do two majors or just maths? Would another major detract from my maths studies, or would two majors be a more 'rounded' degree?

Thanks.

 

[PowerIso]

Linear algebra is pretty important subject. The more you know from it, the better you'll be for it. If you feel your course in it was weak, then go study it independently. You'll find that many linear algebra concepts will be applicable to abstract algebra, so studying for linear algebra can help you study for abstract algebra.

I studied set theory a lot because it is rather important to what I study. However, it seems set theory and logic is something that you just kind of pick up as you go. At least, that's my experience.

Depends which graduate school you are applying for and if it is a masters or PhD.

You should do two majors if a second major interests you. I did mine major in mathematics and interior design. Don't ask why, but I did and I had fun, met my wife too, so it worked out pretty well. Sometimes it was hard to work through both majors but time management is key. If another field interest you, then go for it, if not, then you'll be pretty miserable.

 

[mathwonk]

i was out one year from undergrad. the bigger gap was from grad school. after teaching for a few years, i went back and finished the phd in my 30's. (that may sound old for a grad student, but i wouldn't mind being 50 again now!)

 

[mathwonk]

oh and linear algebra is crucial. in that vein, i offer my free book on my website, notes for math 4050.

 

[Tac-Tics]

I studied set theory a lot because it is rather important to what I study. However, it seems set theory and logic is something that you just kind of pick up as you go. At least, that's my experience.

Formal logic is really nice when you aren't quite sure if you cheated during a proof. And you get a really good understanding of how variables play together in an equation.

 

[morphism]

How important is linear algebra to the mathematician? I have already taken a course in linear algebra, but I am thinking of studying it again over the break before 3rd year, since the course I took was not so good. Is it worth studying linear algebra properly, or should I focus on abstract algebra instead? Or both?

You can hit two birds with one stone. First review the basic topics, such as vector spaces, dimension, linear maps, etc. Then look at more 'abstract' topics, such as, say, canonical forms of matrices, spectral theory, etc.

There is a lot of overlap between the ideas you see in linear algebra and certain ideas you see in abstract algebra. An example is the classification of finitely generated abelian groups and modules over PIDs -- this is pretty much a generalization of the notion of canonical forms of matrices.

Also, a lot of the topics you would see in an advanced analysis course will stem from linear algebra. Some people like to refer to functional analysis as "infinite-dimensional linear algebra," and with good reason. So if you have any interest in doing any advanced coursework in analysis, then you would definitely want to have a solid grounding in linear algebra.

 

[tgt]

i was out one year from undergrad. the bigger gap was from grad school. i went astray in 2nd or 3rd year, hung on until the fifth and took off for a 4 year job teaching.

then i went back and finished the phd in 3 more years, at 35. (does that sound old? it does sort of to me too for a grad student, but i wouldn't mind being 50 again now!)

What was your gpa at the end of your undergrad studies, just out of interest? So you got into grad school but decided to get out early to teach high school? If so, why did you decide to do that?

 

[qspeechc]

Thank-you everyone for answering my questions. Your answers have been very helpful to me!

 

[mathwonk]

qspeechc, you have learned a valuable lesson: namely, if you appreciate what you are given, you will receive more.as my former teacher said: "attention will get you teachers".

 

[CoCoA]

lets start a list of good free books.

"Algebraic Curves" by Fulton available free on the author's web site.

http://www.math.lsa.umich.edu/~wfulton/"

 

[mathwonk]

outstanding reference! grab this great intro to alg geom! it has been almost totally unavailable for years, and is just superb.

this teaches basic commutative algebra from scratch and uses it to prove the three fundamental results of curve theory: 1) bezout's theorem on degree of intersections of plane curves; 2) resolution of singularities of plane curves; 3) riemann roch for plane curves. i will give you a small impression of the atmosphere of the 60's by recalling that Bill Fulton taught the entire contents of this book in one week at brandeis, in about 1968.

 

[morphism]

I think that's going to be the text for the algebraic curves course I'm doing in the Winter. I'm glad to hear it's a good one!

 

[JG89]

What do you guys think about this linear algebra book:

ftp://joshua.smcvt.edu/pub/hefferon/book/book.pdf[/URL]

 

[Tom86]

Thanks for the link! I was actually going to post here asking for a good introductory text for getting into algebraic geometry. I've just started my postgrad to find out it's not being offered as a course this year. This is particularly annoying for me since one of the main reasons I chose to go elsewhere for my postgraduate studies was that algebraic geometry was offered (last year anyway!).

 

[thrill3rnit3]

mathwonk (or anyone)

Have you read this book called Geometry by Kiselev (Russian)? There's actually two books. My math teacher recommended them to me. Have you read that book, and if so, what do you think of it?

here's the link to the english translation version

http://www.sumizdat.org/

 

[zoner7]

Hey everyone. I've got a bit of a question.

I think it would be accurate to call myself a jack of all trades. My quantitative skills are verbal skills are quite similar when compared on an intelligence test; however, in comparison to most other students at my college, my verbal skills far exceed most others, simply because it seems like they have had a serious lack of education in that area. So far, at my liberal arts school, where study in all fields is necessary, I have been able to receive A's across the board.

I am currently debating whether or not I would like to pursue a mathematics or physics major. My passion lies in these two fields, and I also love to write. Unfortunately, I question whether or not I am talented enough to pursue a science or math major and still perform well. I thought Calc I and II were jokes last year. My intro physics class this year is quite intuitive for me. I am also enrolled in Calc III and a discrete mathematics course this year. The later is a joke while the former is definitely challenging for me, as is it for the rest of the class. This is quite discouraging for me; I'm used to quickly grasping concepts. If my limit for quick understanding lies at such a basic level of math, I question whether or not I am fit to continue.


Granted, my school has this fun thing called grad deflation, the opposite of what most schools have. As a result, homework problems and tests are absurdly difficult. While this is good for me in the long run, it sure makes things tough now. hmm... might also be important to note that multivariable calculus used to be taught in two semesters and is now squeezed into one, resulting in quite a challenging class. Perhaps my ability's appear dampened to me simply because of the rigor of the course.

Next semester I am definitely taking linear algebra; however, in order to continue to take future math classes, I would need to take a course called principles of analysis, which is typically infamous for being the toughest course required of a math major. The kids who breeze through Calc III find it very difficulty. I question how I will fair.

While someone can always say I will just need to work a bit harder, I don't think this is too possible as this point. I have been blessed and cursed with a learning disability. Things take me a long time; however, I can complete many tasks others do not have the aptitude to complete. I already devote 30 hours or more to Calc III and week and see my professor multiple times as well. Because the college of the holy cross is a small school, we lack many of the resources of larger schools, meaning that tutors are scarce.

What do you guys think my options are? I love math. Should I sacrifice my perfectionist mentality and concede that I might not receive an A, or should I simply peruse something I enjoy slightly less - but still love - and perform well?

 

[Feldoh]

http://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx

Is this the same information you're going over in your Calc III class? If it is, use it Paul's Online Notes is a great resource. You also might want to try studying a different way, if your current method seems inefficient.

 

[zoner7]

http://tutorial.math.lamar.edu/Classes/CalcIII/CalcIII.aspx

Is this the same information you're going over in your Calc III class? If it is, use it Paul's Online Notes is a great resource. You also might want to try studying a different way, if your current method seems inefficient.

Hah, funny you should post that. I discovered that site just the other week and absolutely loved the guys teaching style. It really helped with the quadratic surfaces; it was assumed I understood these from high school, but since I was placed in all low level classes there, I had never seen any of them before. It made identifying 3-d surfaces quite difficult to say the least.

I am using Stewart's Calc III book and, quite unfortunately, despite the teacher of that website's incredible skill for explain complex concepts, it in no way covers the depth or breadth of my book and class. If someone was brilliant and could solve any problem simply through the application of concepts, that site would be great for him. It's a bit more difficult for the rest of us.

Thank you for the site, though. I am sure I am going to use it more in the future.

By the way, concerning the previous post... I think I should mention that I really don't intend to actually use what I am majoring in. I simply enjoy learning. I will likely do something with personal development in my future, self-employing myself. I will probably make a website.

 

[maze]

In math at some point everyone - I don't care who you are - hits a wall where your intuition/talent fails and you have to work hard.

If you are the sort of person who can look on this as a challenge and enjoy the fun of slowly figuring out the puzzle, then I would recommend math or physics for you.

On the other hand, if that sort of thing is not fun for you, then a lot of math and physics is just going to be a ton of pain so why put yourself through it?

 

[zoner7]

I do love the puzzles. I really do. I can spend hours and hours on one problem. I enjoy it. I'm just not sure if there will be enough time in the day for me to learn it all. I have had to work hard at school since a very young age, partly because I like to master material and partly because work simply takes me longer because of my learning difficulties. I've been doing 80 hour weeks of homework and classes combined since I've been at school, and it's only supposed to get harder. That's what I'm worried about. I don't want to get in over my head and then learn that I can't graduate on time, which would distinct possibility if I were to drop a class now or in the future.

My parents actually said that they would be okay if it took my longer to graduate. They know I work as hard as I possibly can. I simply don't know if I would feel okay making them pay 90k for the extra year, though. Plus, all my friends would be leaving. it would be tough.

serious ethical dilemma and case of over thinking here...

 

[maze]

edit: nevermind you already answered my question

 

[merjalaginven]

Hey everyone. I've got a bit of a question.

I think it would be accurate to call myself a jack of all trades. My quantitative skills are verbal skills are quite similar when compared on an intelligence test; however, in comparison to most other students at my college, my verbal skills far exceed most others, simply because it seems like they have had a serious lack of education in that area. So far, at my liberal arts school, where study in all fields is necessary, I have been able to receive A's across the board.

I am currently debating whether or not I would like to pursue a mathematics or physics major. My passion lies in these two fields, and I also love to write. Unfortunately, I question whether or not I am talented enough to pursue a science or math major and still perform well. I thought Calc I and II were jokes last year. My intro physics class this year is quite intuitive for me. I am also enrolled in Calc III and a discrete mathematics course this year. The later is a joke while the former is definitely challenging for me, as is it for the rest of the class. This is quite discouraging for me; I'm used to quickly grasping concepts. If my limit for quick understanding lies at such a basic level of math, I question whether or not I am fit to continue.


Granted, my school has this fun thing called grad deflation, the opposite of what most schools have. As a result, homework problems and tests are absurdly difficult. While this is good for me in the long run, it sure makes things tough now. hmm... might also be important to note that multivariable calculus used to be taught in two semesters and is now squeezed into one, resulting in quite a challenging class. Perhaps my ability's appear dampened to me simply because of the rigor of the course.

Next semester I am definitely taking linear algebra; however, in order to continue to take future math classes, I would need to take a course called principles of analysis, which is typically infamous for being the toughest course required of a math major. The kids who breeze through Calc III find it very difficulty. I question how I will fair.

While someone can always say I will just need to work a bit harder, I don't think this is too possible as this point. I have been blessed and cursed with a learning disability. Things take me a long time; however, I can complete many tasks others do not have the aptitude to complete. I already devote 30 hours or more to Calc III and week and see my professor multiple times as well. Because the college of the holy cross is a small school, we lack many of the resources of larger schools, meaning that tutors are scarce.

What do you guys think my options are? I love math. Should I sacrifice my perfectionist mentality and concede that I might not receive an A, or should I simply peruse something I enjoy slightly less - but still love - and perform well?

Do what makes you happy- you only get older and life gets shorter-
I am a new mom with no time at all on my hands yet I manage.
If your parents money is the issue then apply for a student loan-
i am in debt bc of mine yet my world is still in equilibrium and everything is ok!
Linear algebra was fun when you think about it and not just memorize.
Calc 4 is the same way- and then you enter what you are talking about- advanced calc analysis in one or several variables- topology- abstract abgebra (my fav!)
These classes are MEANT to be challenging. Sometimes I would spend ten hours (while entertaining the little one lol) trying to figure out the puzzle of the proof- how to prove a sequence converges monotonically to----- lol whatever else-
and I too- have limited resources- our campus tutors are not qualified and I do not have a sitter to attend any extra study sessions- but-
I love it- so I pursue it-
Please do the same- do not be discouraged!
'Perfectionist mentality' - do you know what great minds in the past were farrrrr from perfect- they were DIFFERENT and PASSIONATE!
You will find words only get you so far- do what CHALLENGES your mind not what comes easy to it- good luck!

 

[merjalaginven]

i would say this definition is only a small step in a long chain of work going back to the greeks who showed the area of a circle was a number that could be neither less than nor greater than pi R^2 essentially by showing that is was a limit of quantities that differed from pi R^2 by less than any given amount (any epsilon).

so many many people for hundreds and thousands of years gave arguments essentially equivalent to what we have as the epsilon delta definition of limit. i.e. limits were well understood by the masters for a long time before they were stated in the form we have now, and their use of them is roughly equivalent to ours.

i would say the discovery of the method limits by the greeks stand far above the much later precise statement of that method. the statement came from analyzing the method, not the other way round.

Yes I concur :)
I hate the outlined epsilon - N notation- genius, yes- still I am amazed at things like the rhind papyrus (obviously not applying to limits)- so old, so simple (now--maybe :) )- yet so important- see even then they thought math was all the 'mysteries' and 'secrets' of life... ;)

 

[thrill3rnit3]

mathwonk (or anyone)

Have you read this book called Geometry by Kiselev (Russian)? There's actually two books. My math teacher recommended them to me. Have you read that book, and if so, what do you think of it?

here's the link to the english translation version

http://www.sumizdat.org/

nobody has read Kiselev's Geometry in here??

 

[mathwonk]

you can be first!

 

[thrill3rnit3]

yeah...I guess

i ordered both books from Amazon. The book also had pretty good reviews. I guess I'll give them a shot.