Other Should I Become a Mathematician?

[sunny79]

Lavinia! Appreciate the input. :)

 

[mathwonk]

Just want to thank Lavinia for the great post!

 

[mathwonk]

@narwhal: see if this flies:

if a simple group of order 168 had an element of order 21, then it would generate a cyclic subgroup of order 21 of course, and that subgroup would be abelian, so would be contained in the normalizer subgroup of every one of its elements. Thus the conjugacy class of each of its elements would have order a factor of 8. But G acts transitively on that conjugacy class, so the order must equal 8, or else we would get a homomorphism from G to a small permutation group, and the kernel would contradict simplicity.

So every non trivial element of the group of order 21is conjugate to exactly 8 elements. But look at an element of order 3 in there. It generates a Sylow 3 subgroup, of which there are either 1,4,7,10,... such subgroups. But there can't be < 7 or we get an action on a small permutation group by conjugation. And there can't be 10 or more since all such subgroups are conjugate, and our element is only conjugate to 8 elements.

I think it follows there cannot be 7 such subgroups, so there must be only 4 such subgroups, a contradiction. I.e. either our element is conjugate to its square or it isn't, hence the number of conjugates should equal either the number of Sylow 3 subgroups, or twice that number. Either way 8 doesn't work.I hope this makes sense. Even if this is right, I would not have solved that during the time given for most tests though, unless I was really in good form.

The moral though is again, it has to follow from Sylow and group actions, usually by conjugation.

 

[waddlingnarwhal]

Hi mathwonk. I've read your solution, and I think everything you said is correct. I will look at it again tomorrow to see if anything sticks out. Thanks for the reply! For now, I would like to offer two short, alternative solutions.

(1) This was how I eventually did it. The cyclic group of order 21 contains one of the Sylow-3 subgroups (call it P) of G and is contained in the normalizer of P. By Sylow's Theorem, n3 = 1, 4, 7, or 28. By orbit stabilizer, n3 = 168 / |N(P)|. Since N(P) contains an element of order 7, we have already divided the 7 out of 168, so n3 = 1 or 4. This contradicts that G is simple.
(2) This is a solution my T.A. proposed. I think it is very nice. G acts on the 8 Sylow-7 subgroups by conjugation. Since the group is simple, this gives an embedding of the group into S8, which contains no element of order 21.

lavina - Your post was great! I am so impressed by the level you've reached studying by yourself. It is very inspirational.

 

[mathwonk]

yes yours is much neater than mine, using one extra fact about n3 i ignored. your TA's solution uses a fact i did not know about S8. I tried a similar proof and got that G embeds in both A8 and A7, but did not know how to finish.

 

[waddlingnarwhal]

I believe that S8 has no element of order 21 because the order of a permutation is the least common multiple of the lengths of the disjoint cycles in the disjoint cycle decomposition.

If anyone is interested, here is another interesting problem I did this term. It is a probability problem.

You have a box of n toothpicks. Each time you pick a whole toothpick, you break it in half, put half back in the box, and throw the other half out. Each time you pick a half toothpick, you throw it out. Thus the box will be empty after 2n steps. At a given step, each whole toothpick has the same chance of being chosen as any other whole toothpick, and each half toothpick has the same chance of being chosen as any other half toothpick. Each whole toothpick is twice as likely to be chosen as each half toothpick. Let H be the number of half toothpicks remaining after the last whole toothpick is chosen. Give a closed-form expression (i.e. involving only factorials, fractions, exponents, binomial coefficients, or products) for P(H = k).

 

[mathwonk]

narwhal: thinking about your group theory problem again, I think your solution is the most natural one. And we really could have solved it quickly by applying the basic principles.

I.e. the fundamental fact that in a transitive group action, the order of the group always equals the product of the orders of the orbit and one isotropy subgroup, implies the most basic fact of all for conjugation actions: the order of the conjugacy class (orbit), equals the index of the normalizer (isotropy subgroup).

Since the most fundamental objects to let G act on are the sylow subgroups we have three cases, those for the primes 2,3, and 7.

Since the normalizers for these have orders which are multiples of 8, 21, 21, respectively, from what is given, the conjugacy classes have orders

dividing 21, 8, 8 (and greater than 1, since G is simple). But they also have (by sylow 3), orders congruent to 1, mod 2, mod 3, and mod 7. thus those orders are factors of 21, 4, 8.

This gives us actions on the conjugacy classes of those orders. But a transitive action on a set of (more than 1 and) less than 7 elements

implies G is not simple.This is your proof, but I see now it follows from pursuing the most basic facts: sylow theorems plus the mantra: "order of conjugacy class equals index of normalizer".

I'm a little out of practice after 4 years of retirement and maybe 20 since I taught the course. But I'm just trying to emphasize that these problems seem to usually follow from the same basic principles.

 

[neosoul]

I'm a physics major who was a math major at first. I LOVE pure mathematics but I switched to general physics because I wanted a challenge. However, I decided that I would take a few math electives. I loved them more than I like my physics class, and calculus III drew me in very quickly. I say that if someone loves mathematics, he or she should give it a try. Only try to double major if you love mathematics and WANT to learn. Doing something you want to do and love to do make things much easier.

 

[Nous]

Mathwonk, I think that the fact that you have been posting in this thread since it's inception over 8 years ago is fantastic! I found it while searching around for some "soft" information about what it's like to be a mathematician - the kind of information that is cultural and only really acquired by a member of said culture, in this case, the culture of mathematicians.

I read through the first several pages of the thread and intend on going back to read more. But in the mean time I wonder if you would entertain a couple of questions from an aspiring mathematician. Anyone else with relevant experience is welcome to respond as well :)

TL;DR - Do you suppose that a combined major is a liability when applying to grad school?
Is it unusual for students who have not completed an honors undergrad to get into grad school?
Are there any age-related stereotypes at work for professorships or research type jobs in mathematics that would bias a university to select a younger candidate who had an 8-year start on me?

Context: I am 28 years old, live in Canada, and have an undergraduate degree in education specializing in mathematics education. I actually started my post-secondary studies in mechanical engineering but botched that experience a little bit from being young, unguided, and unfocused. I transitioned into education in the interest of exploring University from a different perspective and mathematics became my chosen area of concentration out of convenience (I had the standard engineering preparation in calculus, differential equations, and elementary linear algebra).

When I left engineering and started taking pure math courses as part of my education degree I was blindsided by how much I loved it. (Engineering math basically skipped all the proofs and that instrumental approach reduced those courses to their computational aspects. I found that a bit dry.) I did a proof-heavy course in geometry, and introductory courses in abstract algebra and discrete mathematics.

Fast forwarding to today, I am heading back to university for a second undergrad in mathematics. Well, first I need to decide on my specialization and am divided somewhat between doing a combined major in mathematics and computing science vs. a single major in mathematics. I plan on continuing into a masters degree in mathematics after I am done but am concerned that doing a combined major makes me appear unfocused (whereas in reality I am simply interested in the intersection of those two fields). Do you suppose that a combined major is a liability when applying to grad school?

I would certainly opt for taking an honors stream of math if I could but that option is not open to second degree students at UBC where I will be studying. Is it unusual for students who have not completed an honors undergrad to get into grad school?

I'm not sure if combined majors are common in other schools but they are basically the same amount of coursework as a single major. Thus, there is less mathematics coursework in a combined major than in a single major.

Now I suppose that I will be at least 35 by the time I have my PhD and am ready to seek employment as a professor. I am concerned that I might be passed over for younger candidates, but I am not sure where this fear comes from. Are there any such age-related stereotypes at work for professorships or research type jobs in mathematics?

How competitive is the field currently for mathematics professor jobs and what trends do you see in the market for those who might enter it in several years?

Any other general remarks inspired by any of my post are certainly welcomed and desired. Thank you to those who read this far :)

 

[mathwonk]

Hello and welcome to the forum. I am retired going on 5 years now so my information is less first hand than it was, but may still be relevant. First of all there are not a lot of people wanting to be mathematicians and so there is not a strict system of excluding candidates due to lack of an honors degree or presence of a dual major. I.e. many programs are just glad to have applicants. The stipends are also not so large and the work is hard, so there is not a huge risk of accepting someone. They do want capable, smart, well trained ones, and hard working ones, but basically anyone who has impressed his undergraduate instructors that he can do a PhD is a reasonable candidate. I.e. rather than certain checklist of criteria we tend to just go with what the professors say about the student.

Beyond that we look at the course work and the reputation of his university. We may also offer pre graduate supplementary training in some cases to help candidates with inadequate background to firm it up. There are even funds set aside for this by congress in some cases (see VIGRE) to help American candidates compete with better trained foreign ones. (You have to be a little mindful of not getting into a program where you are exploited so much as cheap labor, that you may not have time to do your research work.) So anyone who has the ability to do a PhD has a good chance of getting that opportunity.

On the other hand getting a good job as a professor afterwards is very competitive. We have excellent people coming to the US for jobs at universities who are from all over the world. This is a prime place to live and work and attracts the best people from many places where opportunity is less, such as India, China, Russia, ...Some of them come here for graduate work then stay, and some are senior scientists who come here fully trained.

In my opinion, a dual major in math/cs is a wise idea, since the computer science side of things should lead to more attractive job options, either in academia or the business/high tech world. The latter will not be the high flying pure math research environment, but pays much better than university professorships.

As far as age goes, I received my PhD in my mid 30's, and we had some successful PhD candidates at UGA older than that. I may be naive, but in my experience, pure math is one of those areas where a person is judged pretty much on how strong he/she is, and not on age, gender, ethnicity, school attended,... If anything, membership in some non traditional group can be an advantage since the community now tries to increase participation in mathematics from "under represented" groups.

 

[Nous]

Thanks for such a thorough response! I appreciate it. From the reading I've done I also have the sense that training in computer science is an asset for employability.

 

[NathanaelNolk]

Hello guys. I've been wanting to post in this thread for quite some time now, and I finally decided to ask the questions I wanted to. I'm in my last year of High School (in Switzerland) and I am still hesitating between two career paths. Since I was a 10-year old kid, I've always wanted to become a physicist. I am curious and always liked the scientific explanations physics would give me. Thing is, as I started High School, I never really studied for physics. I mean, sure I loved physics, and I took supplementary courses, but not physics courses, only maths courses. For instance, I started reading about multivariable calculus, linear algebra, and differential geometry which really interested me and it became part of my free time to read and see online lectures about it. So I'm beginning to reconsider what I should study. Here's for the context, but now the real question : Do you think I could study mathematics and still end in a physics research department ? I'd really like to study nonlinear dynamics and plasma physics, but I'd also like to study about topology and other maths-related topics. What do you think would be best ? I know it's hard to answer given the little you know about me, but still I'd like to know what real mathematicians would say. I'm really lost right now and some advice would be great, so thank you if you could take a little time to help me.

 

[homeomorphic]

Here's for the context, but now the real question : Do you think I could study mathematics and still end in a physics research department ?

Possible, but very difficult, I would think.

I'd really like to study nonlinear dynamics and plasma physics, but I'd also like to study about topology and other maths-related topics. What do you think would be best ? I know it's hard to answer given the little you know about me, but still I'd like to know what real mathematicians would say. I'm really lost right now and some advice would be great, so thank you if you could take a little time to help me.

If you are more interested in the workings of nature than math for math's sake, you should study physics. It may be possible to go towards that in a few math departments, but it's likely you'd end up like me and be dissatisfied with how much real physics you are learning. Unless you like the math for its own sake. At some point, you have to rein in your ambition to learn about a million different topics too soon or else you will spread yourself too thin. So, you have to pick what topics are most interesting to you and stick to those, and have some control over the impulses to learn every little topic that seems like it might be interesting. I think not being able to focus and control that may have been part of the reason for my downfall as a mathematician.

 

[NathanaelNolk]

Thanks for the reply. I think you may be right, I've always wanted to learn everything as soon as possible. But I think I'll study mathematics for its own sake since that's already what I'm doing in my free time. Besides, the more I think of pure mathematics, the more I see them as an art rather than a simple tool to apprehend the world. Anyway, thank you for taking your time to help, it's really appreciated.

 

[homeomorphic]

Besides, the more I think of pure mathematics, the more I see them as an art rather than a simple tool to apprehend the world.

There's some truth to that, even if you ultimately adopt a very applied mindset like I have because there's a certain playfulness that you need to have in math that goes beyond sitting down and just solving problems directly, so even if you just care about applications, you may sometimes be lead to think more about math internally and just be ready to grab something from that that you think might be useful to solve your problems. V.I. Arnold says "there are no applied sciences, only applications of sciences."

I still don't know what the hell he means by that, but I am guessing it could be something like what I am trying to get across now.

I would caution you against coming to conclusions too early on. At a topology conference, I met a grad student, finishing his PhD from Berkeley who said he thought he wanted to be a pure mathematician, but now he's not so sure--after all that time. Why? From my point of view, it's easy to relate to that. In practice, you might find that maybe a lot of pure math that is being done on the cutting edge isn't as "artistic" as we might hope. I'll link you to Baez's very astute post here that addresses this problem

https://golem.ph.utexas.edu/category/2007/04/why_mathematics_is_boring.html

That's part of the reason I quit pure math (the other reason was that I was really an applied mathematician trapped in a pure mathematician's body). I couldn't take it, personally. Baez loves his job as a math professor, even despite these issues, although it's clear that he is bothered by it. He says that math is one of the most exciting things in the world, yet people succeed, against all odds, in making it boring. I'm not sure all of math can be rescued from boring-land, but a lot of it could be.

Another thing is that a lot of what mathematicians seem to be concerned with these days is checking that things are true, rather than understanding why they are true. Take the 4-color theorem. I don't see any artistic value in asserting that "it's true because the computer said so". Doron Zeilberger, champion of computer-based mathematics, would probably call it a "beautiful" proof. I don't really see why he would say that, other than the fact that he loves computers so much, and perhaps it signifies that the theorem is so deep that it defies human comprehension. Can we say that the 4-color theorem is a beautiful theorem? I would actually say yes, but the problem is it's not really that the theorem is beautiful. It's more that the problem (figuring out if you only need 4 colors to color a map) is beautiful. The theorem doesn't add much to that by telling us that it's true, even though we don't know why. The issue goes beyond computers. Very technical proofs that no one understands are similar to computer-based proofs, as far as this goes.

So, I question your idea that it's all a pretty art form--maybe it could be more than it is now, though. Some mathematicians approach it much more like a sporting event where they set certain goals for themselves and the object is not so much to make beautiful things, but to pull off impressive stunts. If you have a more artistic bent, you might be put off by that side of things, and you might find it hard to avoid, if you aren't careful.

Maybe there's value to sporting achievements, though. Maybe they teach us more about to solve really hard problems. For example, the 4-color theorem is always one of my big examples I like to pull out when I talk about this stuff.

Another point is that I think a lot of the artistic value of math actually comes from the connection with applications, particularly physics (read any book by V.I. Arnold for proof of this), so it's not always the case that the art form is separate from apprehending the world. My big gripe about a lot of the math that I learned was the lack of motivation. It turns out that things like symplectic manifolds have a physical motivation. It's beautiful because there's an inspiration for it. If it's just some arbitrary definition that some mathematician pulled out of his butt, I don't find it beautiful. Some things have a purely mathematical motivation, but what annoys me is when the best motivation, coming from physics, is thrown out, in order to keep math more a of a "pure" science that's independent of the physical world and applications. The truth is that, psychologically speaking, the roots are not separate from the real world, even if it is possible to make it formally independent of it.

One thing I find somewhat objectionable about the "art form" point of view is how small an audience you may be talking about, the deeper you get into math. It's sort of like doing paintings that get locked away and only displayed to certain special people who have to work really, really, really, really hard in a sort of treasure hunt to be given their secret location (50 or at best, maybe a few thousand people if you prove a really accessible result). There's just something weird about it. But hey, whatever floats your boat. I'd be the first to say what's popular isn't always what's good, in a lot of ways, but still. This can be alleviated to some degree if we address some of the problems Baez was pointing out. On the other hand, you can always take the point of view that you're an explorer, so it's kind of cool that you are discovering things that no one else knows. Personally, I found it profoundly unsatisfying and anti-climatic when I finally managed to prove something no one else knew. I will admit in retrospect, it's slightly cool that I can look back on it and say that I did it, but it was unbelievably painful to carry out, so it's a fairly small consolation that I'm getting as my reward for all the blood, sweat, and tears that went into it. It doesn't always come cheaply. There was even an article in the AMS notices one time about the psychological dangers of being a mathematician that talked about poor little mathematicians breaking their backs to prove theorems that seem completely trivial in retrospect. So, it takes someone a little crazy or else unbelievably talented to think that the "art form" or the sport is so compelling as to justify the immense amount of effort required. I think Halmos or Hardy or maybe both of them talked about how you have to love math above all else, even your family and so on. Bertrand Russell has a quote that says something to the effect that you have to lose your humanity in order to make a great discovery or something like that. Mathematicians who happen to be more normal human beings can hope that maybe that's not true because it's a fairly hideous thought. If it is true, it casts the "art form" in an even stranger light. These things seem considerably less cold and sinister if the art form has practical consequences that can change the world for the better. All the madness seems worthwhile if it can help us figure out how proteins fold and create new drugs to treat horrible disease and save your grandmother. It's a double-edged sword of course, because maybe it helps the NSA to spy on you, make bigger bombs that blow up children, etc., but on the whole, it has so much potential if used responsibly.

You always have to ask yourself if one day you'll be bothered that you aren't doing something practical. Maybe one day it could hit you, like it hit me. "Hit" isn't really right because it was much more gradual. In light of all the things I've mentioned, this possibility might seem more real to a hopeful student who is in the honey moon phase of their relationship with math and doesn't see all the difficulties ahead.

I really wish someone could make a really strong and clear case, for the practical benefits that result as a spin-off of the art-form/sporting phenomenon that is pure math, so that more pure math students and maybe even profession mathematicians can sleep at night, without feeling so guilty about not contributing much to society. I've even toyed with the idea of writing a book that does just that one day, but I wonder if I'll ever be up to the task.

Finally, although I don't object to people getting their kicks in whatever manner they please, however strange (after all, we have a much bigger labor supply than we need, so it's useful to have some people just make a living by goofing off, so that some of us can get the jobs they would have been doing, instead of being unemployed), at some point, you have to convince people to pay you, which I don't think you can do if you can only say that it's an art form that only pathologically hardworking people are able to appreciate. So, keep in touch with the rest of us, the physicists, the engineers, the computer scientists. Don't distance yourself too much. It can be an art form, but it's got to have those useful spin-offs or it cannot survive. It's not only money but attention and interest from as wide a range of people as possible to help keep the subject alive and keep it from dying, forgotten in some obscure journal or even worse, not even fully written down, as was the case for so much of the intuition and folklore associated with the subject of foliations in the 1980s. Practically speaking, this may even make a difference in getting that grant money. In my branch of topology, if you said "quantum computer", that was sort of a magic phrase in some ways, even if it's only a hope of an application, rather than an actual one.

I caution people who think like me from getting into math because they may be unhappy there due to these problems, but maybe some of them should go, like I tried to for a while and be crazy enough to think they can change some of these things. I'm crazy enough to think maybe I can change some of these things as a hobby, while I find another way to pay the bills and contribute directly to things that affect people's lives in a positive way.

Sorry I have been so long-winded, but I hoped to give you a sense of the dangers involved with falling in love with math as an "art form". I don't mean to imply that it could not work out for you. Whatever makes you happy makes me happy, even if it's pure math. Just be careful. Think really hard about what you're getting yourself into first.

 

[mathwonk]

This just makes me want to remind you to search yourself for your own motivation. I just like thinking about and discussing and teaching math, i.e. understanding it and helping others understand it. I don't feel bad if my work does not cure cancer, or bring in huge sums of money. I don't agree that only people who work extremely hard can appreciate what i do either. I myself may have to work really hard to find a way to explain it to the average person but I enjoy that effort.

As to theorems that have long tedious proofs like the 4 color theorem, I have little interest in those computer proofs, but in teaching a young student I did have an enjoyable time thinking about an easier related result. He wanted a proof that no more than 4 plane regions could occur that all touch each other. This sounds like the 4 color problem, but is actually much easier, and he and I solved it together. This not only gave me the satisfaction of solving something but also helped me understand the difficulty of the 4 color problem. (Note that this problem is implied by the 4 color problem but not the other way round.)

My main focus in math is thinking about problems that interest me until they seem easy, and can be explained to anyone. This can take years. E.g. in differential geometry the concept of curvature is notoriously abstruse and complicated, involving tensors, connections and so on, but is actually, in its original conception by Riemann and Gauss, quite simple. Note that on a sphere a cap has more area compared to its circumference than does a disc in the plane. In the other direction, a disc in the hyperbolic plane has a larger circumference compared to its area, than in the plane. This simple visual fact lies at the basis of curvature; the sphere has positive curvature, and the hyperbolic plane has negative curvature.

Even excellent books that explain curvature do so in a complicated way that from my viewpoint leaves the understanding out. E.g. the beautiful little book by Singer and Thorpe starts from an abstract concept called a connection, then an abstract version of "parallel transport" defined in terms of a differential equation that takes the simplest formal expression, with no motivation from geometry, then defines a geodesic to be a curve where parallel transport coincides with the the tangent vector.

To me this is backwards, - the intuitive way to explain the concept starts with the idea of length, and curves of shortest length (geodesics), analogues of straight lines in the plane. For these parallel transport is simple, just move along the curve always keeping the same angle with the (tangent to) the curve. Once geodesics, i.e. "straightness" is understood, all the other concepts follow naturally and understandably. This approach is taken in an elementary book by David Henderson.
It is my opinion that all math can be made understandable if one takes the trouble to understand it oneself, and this is an activity I find pleasurable.

When I was in graduate school, choosing between several complex variables and topology or algebraic geometry, I decided that since it took so much time to do math, I had better choose the area that I actually found it pleasurable to think about, or else I was going to miserable for an awful lot of time. I.e. if most of your time is going to be spent thinking, you should probably choose a topic you enjoy thinking about. To me analysis was a bit painful, topology seemed too easy (of course it isn't), and algebraic geometry was both enjoyable and appropriately challenging, and I ended up in that. It also was important to find a very helpful teacher.

On the other hand, if you are a person who will mostly care about his salary, or his public reputation, or the political or practical impact of his work, or his scientific standing relative to others in his field, then those things will matter more to you. No one can decide this for you.

I will admit that there may come a time, after spending the day thinking enjoyably about your work, that you will have to pay some bills that are harder to pay than you think they should be, compared to the case of others who have chosen their professions differently. But it is possible to focus on the positives in ones choice of profession, the people one has helped, the natural beauty one has helped reveal, the scientific understanding one has gained and shared with giants of the past and present. If one is religious, one can try to work for the glory of the creator, as one compensation. Good luck!

 

[homeomorphic]

This just makes me want to remind you to search yourself for your own motivation. I just like thinking about and discussing and teaching math, i.e. understanding it and helping others understand it.

Well, then you're a man after my heart! I did too, except that I was no good at teaching, and it was my expectation that no one would pay me to think about think about math that way. They expect publications, especially early in the career to be able to land that tenure track position. All a moot point since I am pretty sure I wouldn't have been able to get anything aside from an adjunct position, had I wanted to.

I don't agree that only people who work extremely hard can appreciate what i do either. I myself may have to work really hard to find a way to explain it to the average person but I enjoy that effort.

I sympathize, and there is truth to that, but as a general statement, it seems a bit optimistic to me. If I want to entertain the average person, I'd stick to more classical math, and even that can be difficult to explain. I like the old idea that a theory is not complete until you can explain it to the next person you meet on the street. I had a friend who is now a postdoc who studied functional analysis, and he said he didn't even try to explain what he did. A lot of mathematicians are like that. As for myself, I only try to give the flavor of what I was working on, and I could give you the short version or the long version. If someone was willing to sit down and sort of take a little mini-course from me, maybe I could explain something more substantial, but it's not every day that that happens. How many average people have that opportunity? Also, part of what I was saying is that it is that doesn't seem to be the way most of the mathematical culture is right now, and if you think along the lines you are saying, there's a lot out there to be disappointed by. I went to talks for audiences of professional mathematicians and I would be surprised if that many people understood much of it, unless it was close to their area. Some talks were okay, but they were more the exception than the rule. I had fun giving my talks because I went against this trend and tried to make things clear to people. Sometimes, I ended up being pretty successful at that, so maybe I'm not the worst teacher ever, after all. I still don't think I'm capable of doing it day in and day out and to all different types of students.

When I was in graduate school, choosing between several complex variables and topology or algebraic geometry, I decided that since it took so much time to do math, I had better choose the area that I actually found it pleasurable to think about, or else I was going to miserable for an awful lot of time.

As I found out, it's possible to be wrong about what you find pleasurable to think about.

It is my opinion that all math can be made understandable if one takes the trouble to understand it oneself, and this is an activity I find pleasurable.

Although I'm also an optimist to some extent about things that don't seem understandable initially being understandable with some effort, I don't know that it's all of math For example, the 4-color theorem, but maybe someone will find a better explanation some day. I do find it pleasurable to actually understand math, but it didn't seem like I was going to be able to do it all that much. I came across a quote a few weeks back from a mathematician who said "I've never been interested in research. I'm interesting in understanding, which is a very different thing."

And I wondered how he was able to make it as a mathematician, not being interested in research. Maybe with more competition, it's become harder now than it used to be.

 

[NathanaelNolk]

There's some truth to that, even if you ultimately adopt a very applied mindset like I have because there's a certain playfulness that you need to have in math that goes beyond sitting down and just solving problems directly, so even if you just care about applications, you may sometimes be lead to think more about math internally and just be ready to grab something from that that you think might be useful to solve your problems. V.I. Arnold says "there are no applied sciences, only applications of sciences."

I still don't know what the hell he means by that, but I am guessing it could be something like what I am trying to get across now.

I would caution you against coming to conclusions too early on. At a topology conference, I met a grad student, finishing his PhD from Berkeley who said he thought he wanted to be a pure mathematician, but now he's not so sure--after all that time. Why? From my point of view, it's easy to relate to that. In practice, you might find that maybe a lot of pure math that is being done on the cutting edge isn't as "artistic" as we might hope. I'll link you to Baez's very astute post here that addresses this problem

https://golem.ph.utexas.edu/category/2007/04/why_mathematics_is_boring.html

That's part of the reason I quit pure math (the other reason was that I was really an applied mathematician trapped in a pure mathematician's body). I couldn't take it, personally. Baez loves his job as a math professor, even despite these issues, although it's clear that he is bothered by it. He says that math is one of the most exciting things in the world, yet people succeed, against all odds, in making it boring. I'm not sure all of math can be rescued from boring-land, but a lot of it could be.

First of all, thanks for the long reply and the link to an interesting discussion. I'd like to say that, even though I said pure mathematics seemed like an art to me, I also think that stands true for its applications. After all, I wanted to be a physicist, so I really enjoy learning applied mathematics too. That being said, I find beauty in "elegant" proofs and equations. I don't really know how to explain it, I guess that's just my own personal feeling.

Another thing is that a lot of what mathematicians seem to be concerned with these days is checking that things are true, rather than understanding why they are true. Take the 4-color theorem. I don't see any artistic value in asserting that "it's true because the computer said so". Doron Zeilberger, champion of computer-based mathematics, would probably call it a "beautiful" proof. I don't really see why he would say that, other than the fact that he loves computers so much, and perhaps it signifies that the theorem is so deep that it defies human comprehension. Can we say that the 4-color theorem is a beautiful theorem? I would actually say yes, but the problem is it's not really that the theorem is beautiful. It's more that the problem (figuring out if you only need 4 colors to color a map) is beautiful. The theorem doesn't add much to that by telling us that it's true, even though we don't know why. The issue goes beyond computers. Very technical proofs that no one understands are similar to computer-based proofs, as far as this goes.

That's exactly my point in fact, I don't find this kind of proof elegant. But as I said earlier, it's really a matter of taste I guess. Some people will be completely astonished after seeing Wagner's Tristan und Isolde, others will just find it boring. Nevertheless, I think you have a good point.

So, I question your idea that it's all a pretty art form--maybe it could be more than it is now, though. Some mathematicians approach it much more like a sporting event where they set certain goals for themselves and the object is not so much to make beautiful things, but to pull off impressive stunts. If you have a more artistic bent, you might be put off by that side of things, and you might find it hard to avoid, if you aren't careful.

Maybe there's value to sporting achievements, though. Maybe they teach us more about to solve really hard problems. For example, the 4-color theorem is always one of my big examples I like to pull out when I talk about this stuff.

I know that this is the mindset of many mathematicians out there, but that really isn't my motivation for studying mathematics. Does it stop me from wanting to do research ? Hopefully not. I think that one can be a great mathematician without wanting to impress people.

Another point is that I think a lot of the artistic value of math actually comes from the connection with applications, particularly physics (read any book by V.I. Arnold for proof of this), so it's not always the case that the art form is separate from apprehending the world. My big gripe about a lot of the math that I learned was the lack of motivation. It turns out that things like symplectic manifolds have a physical motivation. It's beautiful because there's an inspiration for it. If it's just some arbitrary definition that some mathematician pulled out of his butt, I don't find it beautiful. Some things have a purely mathematical motivation, but what annoys me is when the best motivation, coming from physics, is thrown out, in order to keep math more a of a "pure" science that's independent of the physical world and applications. The truth is that, psychologically speaking, the roots are not separate from the real world, even if it is possible to make it formally independent of it.

What I was trying to say is that mathematics isn't just a tool, but more of an art form that can and must be related to nature and how we apprehend the world. That's why I think that physicist are wrong when they seem math as means to an end.

One thing I find somewhat objectionable about the "art form" point of view is how small an audience you may be talking about, the deeper you get into math. It's sort of like doing paintings that get locked away and only displayed to certain special people who have to work really, really, really, really hard in a sort of treasure hunt to be given their secret location (50 or at best, maybe a few thousand people if you prove a really accessible result). There's just something weird about it. But hey, whatever floats your boat. I'd be the first to say what's popular isn't always what's good, in a lot of ways, but still. This can be alleviated to some degree if we address some of the problems Baez was pointing out. On the other hand, you can always take the point of view that you're an explorer, so it's kind of cool that you are discovering things that no one else knows. Personally, I found it profoundly unsatisfying and anti-climatic when I finally managed to prove something no one else knew. I will admit in retrospect, it's slightly cool that I can look back on it and say that I did it, but it was unbelievably painful to carry out, so it's a fairly small consolation that I'm getting as my reward for all the blood, sweat, and tears that went into it. It doesn't always come cheaply. There was even an article in the AMS notices one time about the psychological dangers of being a mathematician that talked about poor little mathematicians breaking their backs to prove theorems that seem completely trivial in retrospect. So, it takes someone a little crazy or else unbelievably talented to think that the "art form" or the sport is so compelling as to justify the immense amount of effort required. I think Halmos or Hardy or maybe both of them talked about how you have to love math above all else, even your family and so on. Bertrand Russell has a quote that says something to the effect that you have to lose your humanity in order to make a great discovery or something like that. Mathematicians who happen to be more normal human beings can hope that maybe that's not true because it's a fairly hideous thought. If it is true, it casts the "art form" in an even stranger light. These things seem considerably less cold and sinister if the art form has practical consequences that can change the world for the better. All the madness seems worthwhile if it can help us figure out how proteins fold and create new drugs to treat horrible disease and save your grandmother. It's a double-edged sword of course, because maybe it helps the NSA to spy on you, make bigger bombs that blow up children, etc., but on the whole, it has so much potential if used responsibly.

This is a really interesting comment and made me think a lot about why I want to be a mathematician. Of course, only a few will understand what you want to prove and even fewer people will understand how you're trying to prove it. But still, I think as research more of a personal "quest" to understand something that is really important to you. Nevertheless, you're right about the danger of being a mathematician, but I think the same holds true for a writer, a musician or an artist. If you get obsessed by what you're trying to do and are not careful, you might just get burned.

You always have to ask yourself if one day you'll be bothered that you aren't doing something practical. Maybe one day it could hit you, like it hit me. "Hit" isn't really right because it was much more gradual. In light of all the things I've mentioned, this possibility might seem more real to a hopeful student who is in the honey moon phase of their relationship with math and doesn't see all the difficulties ahead.

I really think it depends on the type of person you are.

I really wish someone could make a really strong and clear case, for the practical benefits that result as a spin-off of the art-form/sporting phenomenon that is pure math, so that more pure math students and maybe even profession mathematicians can sleep at night, without feeling so guilty about not contributing much to society. I've even toyed with the idea of writing a book that does just that one day, but I wonder if I'll ever be up to the task.

If you give it a try, be sure to tell me where I can buy it, I'd be really interested in reading it. But that's where your opinion diverges from mine, in my point of view, you're being useful because you're making humanity progress further in our quest of seeking the truth. Besides, what was once considered as pure mathematics might turn out as being really useful in other sciences, e.g. riemannian geometry for general relativity.

Finally, although I don't object to people getting their kicks in whatever manner they please, however strange (after all, we have a much bigger labor supply than we need, so it's useful to have some people just make a living by goofing off, so that some of us can get the jobs they would have been doing, instead of being unemployed), at some point, you have to convince people to pay you, which I don't think you can do if you can only say that it's an art form that only pathologically hardworking people are able to appreciate. So, keep in touch with the rest of us, the physicists, the engineers, the computer scientists. Don't distance yourself too much. It can be an art form, but it's got to have those useful spin-offs or it cannot survive. It's not only money but attention and interest from as wide a range of people as possible to help keep the subject alive and keep it from dying, forgotten in some obscure journal or even worse, not even fully written down, as was the case for so much of the intuition and folklore associated with the subject of foliations in the 1980s. Practically speaking, this may even make a difference in getting that grant money. In my branch of topology, if you said "quantum computer", that was sort of a magic phrase in some ways, even if it's only a hope of an application, rather than an actual one.

I caution people who think like me from getting into math because they may be unhappy there due to these problems, but maybe some of them should go, like I tried to for a while and be crazy enough to think they can change some of these things. I'm crazy enough to think maybe I can change some of these things as a hobby, while I find another way to pay the bills and contribute directly to things that affect people's lives in a positive way.

Sorry I have been so long-winded, but I hoped to give you a sense of the dangers involved with falling in love with math as an "art form". I don't mean to imply that it could not work out for you. Whatever makes you happy makes me happy, even if it's pure math. Just be careful. Think really hard about what you're getting yourself into first.

Oh but don't misunderstand me, I still love the applications of mathematics and other areas than pure maths. Physics remains my first love, even if I won't get into physics research. Nevertheless, you make good points and I will keep that in mind. Thanks for the insights of a mathematician, this is really appreciated.

 

[homeomorphic]

But that's where your opinion diverges from mine, in my point of view, you're being useful because you're making humanity progress further in our quest of seeking the truth.

I've heard the "extending the frontiers of knowledge" rationalization from a lot of people, but I think it's just using vague language to hide the lack of a real sense of purpose. We don't really need any more theorems to just be able to enjoy math. You could spend your life just trying to understand what's already been proven. If you just seek truth for its own sake and don't do eventually do anything with it, I don't see the point. It doesn't have to be "practical", but at least it should have some kind of philosophical significance. A subject like cosmology in physics is interesting to me because it is telling us something about the nature of the universe and reality, even if it's not practical. Not every theorem needs to be useful, but rather than using vague and empty language like "extending the frontiers of knowledge" to justify it, I would take a different approach. The way I see it, it's like shooting at a target. Not all your shots hit the target. In the same way, not all the theorems are useful. Where people might disagree with me is that they should actually aiming at any sort of target. And that's a genuine disagreement. I do think the target is applications, even though not everything has to hit that target. I guess it is the sort of target that you can sometimes hit when you're not aiming for it, which is why it can be okay not to try to hard to hit the target. Sometimes, maybe you learn things by shooting at other targets that can help you hit the target. I don't want to strain the analogy too far. At some point someone does have to try to put the theory into practice, though, or else it won't happen.

Here's a nice clip from someone who came from a pure math background:

Besides, what was once considered as pure mathematics might turn out as being really useful in other sciences, e.g. riemannian geometry for general relativity.

That's what I'm talking about when I say there are useful spin-offs.

 

[mathwonk]

general advice department:

my son is amazed at his success at tutoring subjects he has never taught. Reminds me: what is the difference between a professor and a student? (drumroll): ...the professor reads the book the night before the class.

 

[FocusOnTruth]

Hello, I intend to become a mathematician. I am about to start my second of five years of undergrad and am going into Calculus 2. I am quite a bit behind where I would have liked to be,but I suppose I have somewhat extenuating circumstances. I entered college with little mathematical knowledge-- really without having copied homework assignments from more responsible students and begging for leeway with teachers throughout my compulsory schooling, I would have not graduated high school. As a freshman in college I was learning how to factor and what a function was, among a variety of other basics. I was not completely sure what I wanted to do my first year, so I wasn't entirely devoted to math, though I did study diligently, managing to begin closing the gap with a high A in Calculus I (unlike my inflated grades in high school, this was actually deserved). I realize that I am far from having the skill that I would like to have, and am willing to work as hard as necessary(and even harder) to become a capable mathematician.

My plans for my sophomore/second freshman year:
Fall Semester:
Take Calc II
Text used: Anton 10th edition
Self-Study: Various YouTube channels(Professor Leonard),PatrickJMT. Complete courses on integralcalc.com. Use Khan Academy to review basics and gain more proficiency. I put a lot of emphasis on learning basic math, as I did not really learn anything more than I needed to pass along in my compulsory schooling. I also use Stewart's Calculus (I needed it at my previous school) to do extra problems.
Extra: I may try to get ahead and test out of Calc 3, but I will see how much I improve.

Winter Break: Work through as much as possible of Mathematical Proofs: A Transition to Advanced Mathematics by Chartrand and How to Read and Do Proofs by Solow.

Spring Semester:
Take Sets and Proofs Class
Take Calc III if I haven't tested out
Possibly take intro level stats

Self-Study: I plan on continue with Khan Academy and watching Youtube videos for basics, and I want to further strengthen my foundation with more rigorous material. To do this I plan on working through The Art of Problem Solving series. If I feel that I've caught up enough, I will try to start with Spivak for my more rigorous introduction to calculus.

This is my short term tentative plan, and any feedback is much appreciated. I hope as I become more educated and mathematically literate, I may contribute to the PF community.

 

[mathwonk]

welcome to the community. just keep working and trying to enjoy the journey, as it is a long one, and there are many related destination, so maybe try not to put too much presure on yourself to attain any particular one in any particulr time frame. Just try to keep working near the edge of what you feel capable of. And always try to understand whatever you are doing.

 

[FocusOnTruth]

First of all,thank you for the warm welcome and advice. Secondly,I have been wondering for sometime now what role a physics education plays in my role as a mathematician. I certainly find physics interesting but am more drawn to math. Would it be advantageous to devote some of my time to learning physics, or am I better of sticking to only math? I assume physics could be particularly helpful in becoming a flexible thinker with a broader perspective, but I am not sure as to whether or not it is worth the opportunity cost. Furthermore, I've heard that among pure mathematicians, it is a fairly common sentiment that physical intuition adulterates pure mathematical thinking. I am not quite sure what to make of all this.

 

[mathwonk]

I myself don't know much physics but I consider that a hindrance to understanding math. Physical intuition is very valuable at generating hypotheses as well as giving background for and examples of many mathematical constructions. Not long ago a physicist, Ed Witten, actually won the most prestigious prize in mathematics the Fields medal. Great mathematicians like Riemann and Newton were also very adept at physics. Most people I know believe it a big loss to modern mathematics that the two subjects have become more separated. They have much to offer each other in my opinion.

 

[BubblesAreUs]

I myself don't know much physics but I consider that a hindrance to understanding math. Phyusical intuition is very valuable at generating hypotheses as well as giving background for and examples of many mathematical constructions. Not long ago a physicist, Ed Witten, actually won the most prestigious prize in mathematics the Fields medal. Great mathematicians like Riemann and Newton were also very adept at physics. Most people I know believe it a big loss to modern mathematics that the to subjects have become more separated. They have much to offer each other in my opinion.

I am not sure if that's possible anymore. The areas of maths and physics are so much larger now than in the past. It's true that there's far greater integration now than ever before, but I can't see one picking up all that.

For instance those who are more into abstract algebra and a more "rigorous" ( i.e. thinking in variables) may find it easier to incorporate computer science than physics. I'm also incorporating a lot more probability theory ( and statistics) into my journey seeing that both borrow a lot from Set Theory. Statistics is actually pretty rigorous and my upcoming computational theory feed ought to suffice as well. :)

Mind you, I'm not belittling physics. Theory of Partial Differential Equations ( highly rigorous) is heavily driven by physicists. While, I'm considering taking the honours/ grad-level offering, I doubt I could ever do intermediate ( electromag, fluids, thermo, waves etc) or higher level physics ( i.e. condensed matter, atomic, theoretical physics, etc). I do do look into the areas as a layman, but I generally like to stay within my league...

 

[MostlyHarmless]

I myself don't know much physics but I consider that a hindrance to understanding math. Physical intuition is very valuable at generating hypotheses as well as giving background for and examples of many mathematical constructions. Not long ago a physicist, Ed Witten, actually won the most prestigious prize in mathematics the Fields medal. Great mathematicians like Riemann and Newton were also very adept at physics. Most people I know believe it a big loss to modern mathematics that the two subjects have become more separated. They have much to offer each other in my opinion.

I disagree, it's not that modern mathematics has diverged from physics. It's just that there are now more fields that are interesting mathematics. PDE's are still heavily physics driven. Even other disciplines that's were thought to be strictly pure mathematics are having application is quantum physics. Topology for example.

 

[stardust]

So I have some questions I am curious to ask.

To give some context: I have taken Calculus I, II, III, Statistics, Linear Algebra, and Differential Equations. I have also taken General Chemistry I and II, Physics I and II, and General Biology I and II.

So, I love science and math immensely and find a lot of ideas/concepts fascinating and interesting. I entered school at 26 (having dropped out of high school at 18), having to take some amount of remedial math courses. My intention was to enter into engineering for job outlook as opposed to physics (which had always been my original desire). However, I found engineering not suited to my way of thinking, and decided I would pursue physics as I had always wanted to. I have found that while I enjoyed physics/chemistry, their methodology also does not appeal to my perspective. I enjoy proofs immensely, and absolutely abhor assumptions/lack of rigor. I think what I've always really been searching for is elegant beauty that is mathematics.

I find myself plagued by doubts. I have done fairly well in my math classes (all A's), and understand the course material pretty well. I have also completed honors versions of Calc II/III and Diff EQ. However, I come by this road with difficulties. It is not always immediately obvious to me how to progress in a proof, and I find myself having to look at ideas online to complete them. The reason I mention this, is that it makes me question my ability to be a mathematician. I believe I have some amount of intelligence, but I am by no means possessed of a powerful mathematical intuition/skill. Also, I am late to the game, so to speak. I am now 29, and just wanting to enter into mathematics. My teachers have expressed confidence in me, but I sometimes wonder if that confidence is at times misplaced.

So after that long winded diatribe, here are my questions:

Is 29 too late of an age to start a path to earn a PhD in mathematics (I am in my junior year) and pursue it as a field (with all considerations, including bias against my age)?

Is it possible for me to cut it in mathematics (can hard work carry me through a lack of genius)?

Will I often find myself at a disadvantage in regards to other candidates (jobs, grad school, etc.)?

If I do pursue a graduate degree in mathematics, is it likely to find a graduate program which will pay for my education, while providing a stipend for living costs (living modestly of course)?

Thanks for any help in this matter.

 

[IGU]

...I have taken Calculus I, II, III, Statistics, Linear Algebra, and Differential Equations... I enjoy proofs immensely, and absolutely abhor assumptions/lack of rigor. I think what I've always really been searching for is elegant beauty that is mathematics.

I'm wondering if you've taken any classes meant for pure mathematicians, like number theory, or analysis. Unless you've done so, I'm not sure you can know whether you will like classes devoted to rigor and proofs. You might want to do that before committing yourself in a particular direction.

 

[stardust]

I'm wondering if you've taken any classes meant for pure mathematicians, like number theory, or analysis. Unless you've done so, I'm not sure you can know whether you will like classes devoted to rigor and proofs. You might want to do that before committing yourself in a particular direction.

Well, I mastered the epsilon-delta proof of limits from my calculus book (was not covered by the teacher). My teacher indicated that was usually tackled in a real analysis course. Besides that, I've taught myself some amount of non-euclidean geometry (hyperbolic and elliptical), and have progressed about halfway (so far) through Euclid's Elements.

What would be a good book to help me get a notion of how well I would like pure math as a field, based on my current mathematical training?

 

[IGU]

What would be a good book to help me get a notion of how well I would like pure math as a field, based on my current mathematical training?

Hard to say. I think the right book depends heavily on the person who's doing the learning. But if you want to build on what you know you might want to look at Terry Tao's Analysis I (https://terrytao.wordpress.com/books/analysis-i). Actually I would just use his original notes that are free online rather than buying the book, at least at the point where you are. If you need help working out the problems, ask on the appropriate forum here or elsewhere online; lots of people are happy to help.

Of course you might do better to do something related to the various forms of discrete math, maybe logic or set theory or combinatorics... With luck, others will chime in.