Other Should I Become a Mathematician?

[mathwonk]

If you like Euclid, I think you would like pure math. It doesn't get any purer than that. I tend to divide pure math up into geometry/topology, algebra, and analysis. Many people like one of these areas best and not always the others, or feel an affinity for one of them more than others. So I would explore all of them and not be put off if one area does not appeal so much. Of course eventually one wants to try to gain some insights from all of them, and see how they enrich each other,

 

[stardust]

If you like Euclid, I think you would like pure math. It doesn't get any purer than that. In tend to divide pure math up into geometry/topology, algebra, and analysis. Manuy people like one of these areas best and not always the oters, or feel ana ffinity for one of them more than others. So I would explore all of them and not be oput off if one area does not appeal so much. Of course eventually one wants to try to gain some insights from all of them, and see how they enrich each other,

Thanks! Any thoughts on my original post? I was hoping to get some feedback from professional mathematicians.

 

[mathwonk]

short answer
; go for it. you are not too old, you don't need to be a genius, hard work is sufficient, and you will likely find a fellowship for phd. besides, you will be doing what you enjoy, what else is therte?

look at my mathoverflow post on age.

http://mathoverflow.net/questions/7120/too-old-for-advanced-mathematics/45644#45644

 

[waddlingnarwhal]

short answer
; go for it. you are not too old, uyou don't need to be a genius, hard work is sufficient, and you will likely find a felliowship for phd. besides, you will be doing what you enjoy, what else is therte?

look at my mathoverflow post on age.

http://mathoverflow.net/questions/7120/too-old-for-advanced-mathematics/45644#45644

This post and your mathoverflow post were very inspiring. Thank you for sharing.

I recently had an experience that illustrates why (for me) doing math provides a level of satisfaction that just reading about math does not. A few years ago my number theory class stated and proved Hensel's Lemma. At the time neither the lemma itself nor its proof were of any interest to me; I could not figure out why anyone would care about the result or how anyone had thought of the proof. Consequently, I did not pay attention and soon forgot about Hensel's Lemma. A few days ago, I became very interested in a small exercise that asked the reader to show that "x^2 ≡ -1 (5^n)" has a solution for every n. After calculating a few examples for small n, I figured out that you could consider (x + k*5^(n-1))^2, and that the key was in the term 2xk5^(n-1). Although this problem was very humble, I was extremely happy that I had figured it out. When I told my friend, they said, "Oh that's just Hensel's Lemma." I revisited Hensel's Lemma, and suddenly the statement and the proof were motivated, impressive, beautiful, and made sense to me.

Reflecting on this experience made me realize that your statement "besides, you will be doing what you enjoy, what else is there?" is something that is very true and meaningful to me. Although I have recently had to confront the realization that going through math graduate school will probably not lead to a job in academia, having the opportunity to spend 5 years doing math (!) for most of the day, to have the time to construct my own mental models of the math that is interesting to me, talk about math with brilliant people, and work on interesting problems, is something I will not pass up and will never regret. So again, thank you for sharing your interesting experiences and wisdom.

 

[mathwonk]

You are welcome. Of course one still has to earn a living, and for that exigency I was once told that even David Hilbert made sure he had a teaching certificate, just in case. In our day and age it would probably include experience with computers.

 

[FocusOnTruth]

Would it be viable to work in industry after completing undergrad and then trying to apply to a Ph.D program, or would that put me at a disadvantage?

 

[mathwonk]

I think that might be fine. It might even help. Of course it is possible you would need to refresh on the material which is tested on qualifying exams, but you might be better trained in using some of the material you had practiced in your job. You might also have acquired some useful intangible assets, like a renewed desire to benefit from classes.
Applied mathematician jobs can also teach you how to solve problems and perhaps also help you learn to find them. My friend who worked as an applied mathematician possibly before taking a PhD in topology said the fundamental principle of applied math was that the most likely conjecture for the solution of a given problem is something like: "the simplest statement consistent with the data". I always liked that.

 

[FocusOnTruth]

I see. Also, is it difficult to find a fulfilling career as a mathematician without being educated at a prestigious institution? I have often heard that it's mostly those who graduate from top schools end up doing well. Excuse me if that is a stupid question,but I'm quite ignorant.

 

[mathwonk]

well i would say it is difficult for anyone, i.e. it requires very hard work for a long time and dedication. it is an advantage to graduate from a good school, at least at first, before ones own strengths or lack of them become evident, but eventually it is the ability and accomplishment of the individual that matters.

 

[Hornbein]

I went to math graduate school at age 38 and was not successful. Hard work didn't help. I think, as do most, that math is a skill that must be learned young. 16 years old is about as late as you can get. It's like sports, classical music, or learning a language. You have to learn it young so your brain can build special hardware.

Real mathematicians get started young, like ten years old, and have a sort of obsession with it. You can't compete with these people, who may have been at your present level when they were twelve years old.

Secretary of State Condoleeza Rice started out as a pianist. When she was 17 she went to Aspen, where she encountered 12 year olds who could sight read what had taken her all year to learn. She decided she had better get another way to earn a living.

There are very few top-level jobs. So even those people who DID start at age ten have a difficult time of it. Usually you can tell who is going to make it by age 18 or so. Things like the Putnam Exam tell you who is going to make it. You can have a look at older problems. I can't do any of them at all. One of my professors could do about a third of the problems. But a few students get a perfect score.

You have no hope with the Putnam. Take practice exams for the Math Achievement Test. I studied a lot but could never get better than 80% percentile or something like that. That's not very good. Looking back, I'm surprised I got even that high

There is a huge difference between a top mathematician and a pretty good one. In my graduate class was a top mathematician who got there by accident. All the other students were greatly outclassed. I've heard that there are only a few dozen people in the whole world making fundamental discoveries. I doubt that even he was good enough for that, but it's possible.

I was never able to understand the Riemann Hypothesis. I once picked up a graduate level book on it and wasn't able to make it past the first page.

I have been told that 9 of ten great math discoveries were made by those under age 25.

The best you can hope for is teaching math in a junior college or high school. Some enjoy that, but for the most part the students hate the subject and are there under compulsion. The teachers are highly overqualified. Not for me.

I had a boss who was quite good at math, much better than me, but gave up on it because it was too competitive. He went into Silicon Valley management. At age 60 he finally got his own company, making very small lenses. Challenging math there.

If you don't want to do such teaching, I'd look at electrical engineering, though even that math was not my style for some reason. Or computer programming, which is applied logic, not math at all, but a realistic possibility. A computer program is (or should be, though it seldom is) a big proof. But you'd have to get into writing programs from scratch, because maintaining existing code is usually like pumping out a septic tank.

 

[Hornbein]

Would it be viable to work in industry after completing undergrad and then trying to apply to a Ph.D program, or would that put me at a disadvantage?

It would help you in a few areas that require some math. As a friend of mine put it, "You've got a master's degree, so I trust you to add 2+2." But it wouldn't help you as much as getting a degree in the area you are applying for a job.

 

[mathwonk]

thank you hornbein for this story. this is not everyones experience but it is certainly valid. if you have read my posts you know I went back to grad school in my 30's and found it the most difficult experience of my life, but did eventually emerge with a degree.

 

[Zarem]

Real mathematicians get started young, like ten years old, and have a sort of obsession with it. You can't compete with these people, who may have been at your present level when they were twelve years old.

I can't do any of them at all. One of my professors could do about a third of the problems. But a few students get a perfect score.

You have no hope with the Putnam.

I was never able to understand the Riemann Hypothesis. I once picked up a graduate level book on it and wasn't able to make it past the first page

The best you can hope for is teaching math in a junior college or high school.

You seem quite confident about declaring what everyone else is or isn't capable of, which is interesting coming from someone who was not successful themselves.

 

[symbolipoint]

You seem quite confident about declaring what everyone else is or isn't capable of, which is interesting coming from someone who was not successful themselves.

Zarem, what hornbein and mathwonk say makes sense. Really, individuals do not know for sure unless they try ; and often, keep on trying, hard, for a long time, and must be willing to work through things more than once. Individuals might be geniuses or some might not be geniuses; but regardless, hard, long work is usually necessary. If as mathwonk says, he earned his PhD through many years of hard work, I believe and trust what he said.

 

[mathwonk]

It took me a long time to PhD, but then I published over 30 research papers, and taught over 40 different college math courses. Then in retirement, I began a 3-4 year association lecturing and mentoring brilliant 8-10 year olds with "epsilon camp". That's me in the picture linked below, holding the chair for a youngster working on something related to the construction of the regular pentagon. Of course you and I won't be the kid in the chair, but we can be the one holding it for him.

http://www.epsiloncamp.org/

 

[Loststudent22]

The best you can hope for is teaching math in a junior college or high school. Some enjoy that, but for the most part the students hate the subject and are there under compulsion. The teachers are highly overqualified. Not for me.

Is this what a bitter grad student is like? This entire post was very negative.

 

[mathwonk]

well i would say hornbein's post is one of the more negative ones i have seen here. but it is his sincere feeling about his experience and we cannot discount it. i just think it is not universal. indeed i would say mr hornbein himself may find some good outcome if he modifies his aims realistically. my advice is just to aim at what you want most, try as hard as you can, stay the course for a long time, and accept what comes your way. as long as you are engaged in an activity you enjoy even along the way it will offer satisfaction. I myself did not become famous or win world class prizes but I did some good work and I did my best. I enjoyed as well my contact with top quality minds who were willing to talk to me. I also helped some more talented people to achieve their goals. And even here I try to give good advice for free. As the famous Cech nobel winning poet put it, hey it beats killing and murdering.

more precisely: to quote Jaroslav Seifert:

"Prague! Like a draft of wine her savor, Though she should lie in ruins round me, Though fate from hearth and home should hound me, And choke her soil with blood. Oh, never Will I forsake, though all forsake her! Here with the dead I'll wait, unbending, From early spring to winter's ending, Mute at the door till time awakes her. Though screech-owls call down death and mourning, Though God avert His eyes above, One tear upon His lashes burning Charms from our roofs the hovering curse. All my heart's burden, in this verse, I have brought and sung for you, my love! And Now Goodbye To all those million verses in the world I've added just a few. They probably were no wiser than a cricket's chirrup. I know. Forgive me. I'm coming to the end. They weren't even the first footmarks in the lunar dust. If at times they sparkled after all it was not their light. I loved this language. And that which forces silent lips to quiver will make young lovers kiss as they stroll through red-gilded fields under a sunset slower than in the tropics. Poetry is with us from the start. Like loving, like hunger, like the plague, like war. At times my verses were embarrassingly foolish. But I make no excuse. I believe that seeking beautiful words is better than killing and murdering."

 

[symbolipoint]

hornbein said

The best you can hope for is teaching math in a junior college or high school. Some enjoy that, but for the most part the students hate the subject and are there under compulsion. The teachers are highly overqualified. Not for me.

Is this what a bitter grad student is like? This entire post was very negative.

That is largely the truth. Not all students, but most of them. Expect that any grad student doing some teaching will be teaching students who do not like Mathematics.

 

[xaos]

i think the warnings are valid: 'you have to enjoy it' and 'you have to measure this against the opportunity cost of this path to other paths'. since my master's ten years ago, i have been at it almost continuously, finally reaching up to homological topology. but is only getting 5 pages of work done after twelve hours of semi distracted effort really worth it? shouldn't i be doing something with my life?

i really enjoy mathematics but i hate to program because it takes away time from studying mathematics. however i know i will never teach, so the only other option i see is to learn how to program. surprisingly there are jobs in programming but nobody cares about commutation diagrams.

you don't have to be a genius to understand mathematics, but you don't have to be a fool either. there is great reward in understanding things few others can barely grasp.

 

[FocusOnTruth]

Regarding my late start, I have certainly seen very mixed feedback. I'm not sure I've mentioned this already, but here's my brief background:

-Enjoyed mathematics intensely as a small child and managed to get several years ahead of the curve in the beginning. I could do some basic college level problems as a toddler.
- Environmental and emotional issues interfered with learning much further. My skills regressed, and I underachieved throughout school. (Something not math related but possibly useful in elaborating upon my situation: I had never even read a book until summer before my senior year of high school.)
- I took precalculus senior year and barely passed. I think I could have done better if it was my only focus,but I was terribly behind in every subject, so I was going through great struggles in virtually every facet of life. However, it may be pointless to speculate.
-Going into college, I didn't know how to factor polynomials or how to define a function.
- With persistence, I have improved greatly. I finished Calc I with a solid A and am doing similarly well in Calc II. I also study independently and am gradually making up for childhood and adolescence.

With all that I've heard, I honestly can't say whether or not I'll ever be able to catch up to those who started young and competed in IMO and Putnam. I certainly won't be on their level any time soon. Regardless, I love mathematics and think that having overcome environmental obstacles and personal issues to rekindle my joy in learning is also valuable and may benefit me in ways that years of practice may have not.

Hornbein certainly had a respectable view, but I can't say I agree. Perhaps it's my naivete,but I believe that by pushing myself to the edge of my ability and remaining curious, I can make worthy contributions. I also think I have more options in mathematics than high school and junior college teaching. Of course, time will tell whether or not I'm correct.On a somewhat related note: Since I've gotten a rather late start on academics as a whole, I am indecisive as to whether or not I should study non-math subjects such as humanities and biology. I certainly find all studying enriching, but I fear that by spreading myself too thin I may be incurring to great an opportunity cost. That is, I think it may be better to just focus, for now,on math and more strongly related subjects like physics, coming back to those other curiosities later when I have more time and a well-established career as a mathematician.

 

[Catria]

My grad-level instructors (especially in the courses I dropped) thought that I was one of the most mathematically-inclined students in years, if not their careers as physics professors, since most of the questions I ask in class is about mathematical assumptions and what to do when they aren't verified: non-commutating second-order partial derivatives, discontinuous Lagrangians/Hamiltonians (usually because of the potential term), inability to commute sum and derivative (or integral), inability to commute integration order, inability to commute derivative and integral, time-dependent masses (in discrete-body problems, rather than classical field theory, where time-dependent mass densities are common), and yet I feel that we can discover new physics partially by relaxing mathematical assumptions (ultraviolet catastrophe comes to mind). But they understand, by the same token, why I want to do theory on some level.

And also the instructor of the course I grade homework for has even accused me of caring too much about mathematical rigor... all of which lead me to mathematics (probably mathematical physics) as something I would do if I still want to do research after I cure the mental illness that caused me to consider dropping out of a physics PhD in the first place. Yet I once ruled out mathematics due to a poor experience with real analysis 2 in undergrad, knowing real analysis is a common topic for PhD math quals.

Now, the one roadblock I envision that would preclude success as a mathematician (applied or pure) would also preclude success in PhD programs in general. High-level coursework would frustrate me to no end, especially since I know there is an extant solution to a lot of coursework problems, whereas in research you're the one looking for a solution. And I feel coursework-induced frustration is IMO (IME?) poor preparation for research-induced frustration. For me how I handle frustration is highly source-dependent.

 

[dkotschessaa]

I think the "young man's game" fallacy arises from an error in reasoning, i.e. that mathematicians peak at a certain age because they are a certain age. I suspect it may have more to do with the age that one is indoctrinated into mathematics, which is *typically* around a standard age (mid 20s). Once the indoctrination is complete I think the ideas are less fresh.

I also think that the advantages of youth are largely physical. The illusion is that the young brain is somehow better but that is simply because the body is better. Physical health lends itself to mental acuity. I am finding that withstanding the rigors of grad school is actually more physical than mental. To "keep up" with the younger students I have to exercise a lot and eat well. I cannot study until 2:00 am or take a test on 5 hours of sleep, or skip a meal. Your brain gets old because your body gets old, but there's so many advantages to maturity. I have less innate skill than most of my peers but more balance and a lot of determination.

Hardy and people who think this way about age did not have a lot of information to go on. There are more of us now and so the world will have a chance to see what we can bring to the table.

-Dave K

 

[FocusOnTruth]

I have had some trouble finding out what exactly a day in the life of an industrial mathematician is like. Will I be able to spend most of my day solving mathematical problems, or is that only a small part of the job?

 

[mathwonk]

I do not know the answer to that, but I know the answer for an academic mathematician, in college. Namely most of your day will probably be spent with teaching classes, grading, counseling, office hours, meetings, reviewing grad exams, more meetings, preparing classes, office hours, applying for grants, writing reports, writing planning documents, evaluating staff, reviewing dossiers of potential students or potential hires, writing dossiers for people going up for promotion or awards,..., so you have to be strict about drawing off some reaearch time that is sacred, and shut your door and don't answer even a knock on it for that afternoon or that part of the day. I only had one brief period like this per week in my schedule. Most research got done during holidays, and some at home, sometimes late at night while not sleeping. It is very difficult for the average college teacher to find time to do math. So sometimes try to get leave at a research institution, or take in a summer meeting. If you can get a job at a place that offer sabbatical leave regularly as part of the conditions for employment,. that would be super. I never had that. UGA offered zero sabbatical leave, even after decades of service. Most laypersons seem to think sabbaticals are part of the academic lifestyle, but they are not everywhere. some places offer 6 months leave after 7 years, and recurringly. you should definitely prefer such a place.

 

[mathwonk]

some free math books from springer:

https://gist.github.com/bishboria/8326b17bbd652f34566a

 

[NathanaelNolk]

Hello everyone, I'm starting my undergrad in mathematics and I was wondering: Considering I'd like to end up doing research in mathematical physics, how much courses should I take in the physics department? Should I self-study physics instead? Are there some absolutely required courses in physics (apart from the obvious EM, mechanics, QM, etc) that I should take, like thermodynamics or something of the sort? I was of QFT/GR. Is it even possible for a mathematician to take such courses?
Thanks for anyone taking the time to reply, even though I know the question may be a bit too general.

 

[mathwonk]

I suggest you ask a physicist like zapperz this question, perhaps in his thread :can i get a phd in physics if my undergrad degree is in something else?"

 

[NathanaelNolk]

I suggest you ask a physicist like zapperz this question

Thanks for your help mathwonk, I'll do that.

 

[VCrakeV]

Hello fellow academics. I'm not interested in pursuing a career directly related to math, but I enjoy it to an extent, and so I'm thinking of doing a minor or secondary major in math (primary major is philosophy). Basically, I find myself loving some aspects, and hating others, so I want to know if I would enjoy a minor or major. I enjoy Algebra, derivative Calculus, integral Calculus, complex numbers, and concepts of infinity. But matrix Algebra and matrices in general bore me to tears. I also dislike 3D graphing, and 3D visuals in general. A major would also require some computer science, which I find quite a bore.

So, would I like either degree? What's math like in the higher courses?

 

[lavinia]

Hello fellow academics. I'm not interested in pursuing a career directly related to math, but I enjoy it to an extent, and so I'm thinking of doing a minor or secondary major in math (primary major is philosophy). Basically, I find myself loving some aspects, and hating others, so I want to know if I would enjoy a minor or major. I enjoy Algebra, derivative Calculus, integral Calculus, complex numbers, and concepts of infinity. But matrix Algebra and matrices in general bore me to tears. I also dislike 3D graphing, and 3D visuals in general. A major would also require some computer science, which I find quite a bore.

So, would I like either degree? What's math like in the higher courses?

I was also a philosophy major. I found mathematics answered many questions that stumped philosophers. Also mathematics presents a Platonic universe. Not a bad idea for a philosopher to know one first hand.