Too old for advanced mathematics?
Kind of an odd question, perhaps, so I apologize in advance if it is inappropriate for this forum. I've never taken a mathematics course since high school, and didn't complete college. However, several years ago I was affected by a serious illness and ended up temporarily disabled. I worked in the music business, and to help pass the time during my convalescence I picked up a book on musical acoustics.
That book reintroduced me to calculus with which I'd had a fleeting encounter with during high school, so to understand what I was reading I figured I needed to brush up, so I picked up a copy of Stewart's "Calculus". Eventually I spent more time working through that book than on the original text. I also got a copy of "Differential Equations" by Edwards and Penny after I had learned enough calculus to understand that. I've also been learning linear algebra - MIT's lectures and problem sets have really helped in this area. I'm fascinated with the mathematics of the Fourier transform, particularly its application to music in the form of the DFT and DSP - I've enjoyed the lectures that Stanford has available on the topic immensely. I just picked up a little book called "Introduction To Bessel Functions" by Frank Bowman that I'm looking forward to reading.
The difficulty is, I'm 30 years old, and I can tell that I'm a lot slower at this than I would have been if I had studied it at age 18. I'm also starting to feel that I'm getting into material that is going to be very difficult to learn without structure or some kind of instruction - like I've picked all the low-hanging fruit and that I'm at the point of diminishing returns. I am fortunate though, that after a lot of time and some great MDs my illness is mostly under control and I now have to decide what to do with "what comes after."
I feel a great deal of regret, though, that I didn't discover that I enjoyed this discipline until it was probably too late to make any difference. I am able, however, to return to college now if I so choose.
The questions I'd like opinions on are these: is returning to school at my age for science or mathematics possible? Is it worth it? I've had a lot of difficulty finding any examples of people who have gotten their first degrees in science or mathematics at my age. Do such people exist? Or is this avenue essentially forever closed beyond a certain point? If anyone is familiar with older first-time students in mathematics or science - how do they fare?
There is a question about mathematicians that learned mathematics at a late age. http://mathoverflow.net/questions/3591/mathematicians-who-were-late-learners
No, of course not, do what you like! As you can see at Kim Greene's link, there are certainly examples of people starting late. But this is really a person to person thing, and no one can predict how you (or an 18-year-old) will fare.
I know one mathematician who left math at the age of 20(with a B Sc), enrolled in grad school in his thirties and got Ph. D. in a very modern and hot topic, requiring lots of study and effort, by 38 or so. He is very good, in my view. So you shouldn't hesitate at all.
In my opinion the material collect so far on this subject seems sufficient to refelect a wide spectrum of opinions. I thus now vote to close.
This is indeed not a typical math overflow question, but never mind that.
Of course you can learn mathematics at the age of 30 after having stopped studying it at the age of 18! Examples are abundant -- in almost every math department I've ever been in, there are at least one or two older graduate students that took some years off (after high school, after college or both) and did quite well upon their return.
Being older than 18 may not be a bad thing. Many 18 year-olds are neither well-prepared nor well-motivated to study mathematics (or something else) at the university level: a lot of them are there because their parents want them to be, and most of them are there because their parents are paying.
It is true that essential skills get rusty after years of disuse -- when I teach "freshman calculus", older students often do not do very well, even if "older" means 21 or 22: they've forgotten too much precalculus mathematics. But you have been learning about calculus, differential equations and linear algebra on your own and enjoying it! You're looking forward to reading a book on Bessel functions!! You're well past the point where older, rusty students have trouble. You can do it, for sure, and it sounds like you want to, so you should.
By the way, 30 is not remotely old. I am a few years older and I think better and more quickly now than I did when I was your age.
But of course, according to Hardy, "If a man of mature age loses interest in and abandons mathematics, the loss is not likely to be very serious either for mathematics or for himself." This, of course, is about professional mathematicians producing interesting research.
We can safely file that quote under "Hardy has been known to talk like a pompous ass".
When you encounter Hardyesque opinions on older mathematicians, it's probably worth keeping this comic in mind: http://xkcd.com/447/
@fpqc: When I was your age, teenagers were more respectful of their elders. @SC: On the contrary, Hardy wrote his famous book after he himself stopped doing mathematical research. Many contemporaries and historians have written about his state of mind during the writing of his book; I believe the phrase "clinically depressed" would now be used. I think it has meaning only as a personal statement, and a dark one at that.
I just turned 20 yesterday! Anyway, I assume you've read the book, so how do you refute his argument? I mean, you've gotta at least come up with something if you want me to agree with you (Honestly, I probably will).
@fpqc: I think Hardy also said something like "Young men should be allowed to be arrogant; but they shouldn't be imbecile". I offer this merely as reported speech - I'm not going to commit myself to agreeing or disagreeing with Hardy's line.
Anyone who actually doubts the productivity of older mathematicians has only to look at Gelfand's output in his later years.
Well,does the fact that Karl Weierstrass-one of the founders of rigorous analysis-got his doctorate with a treatise published when he was 39 after many years working as a teacher and studying on his own and continued to publish until his death in 1897-count as a good counterexample, fpqc? Keep in mind-this was in an age when people lived much shorter lives and in much poorer general health. In fact,he died immobile of pnuemonia and continued to work when he could until his death. And we haven't even mentioned his remarkable career as a doctoral mentor to such names as Georg Cantor.
Dear bitrex: your enthusiasm is heart-warming!
I have had students much older than you and they have always been a joy to teach: their maturity more than compensated for their potential knowledge-gaps and they fared very well on their exams.
The nicest success story is a professional cellist who didn't even have the "baccalauréat", a French diploma for the end of secondary school usually taken at age 18. He started learning math because he had married a math teacher (!) and became an excellent student. He passed his D.E.A. (a sort of undergraduate thesis) brilliantly and unfortunately couldn't accept my suggestion to do a Ph.D. because of his professional activity ( I sometimes hear him at concerts...).
So my advice is to go on with your mathematics: I can't predict the future but my feeling is that your age is not very relevant. Good luck!
With all this unanimous enthusiasm, I can't help but add a cautionary note. I will say, however, that what I'm about to say applies to anyone of any age trying to get a Ph.D. and pursue a career as an academic mathematician.
If you think you want a Ph.D. in mathematics, you should first try your best to talk yourself out of it. It's a little like aspiring to be a pro athlete. Even under the best of circumstances, the chances are too high that you'll end up in a not-very-well-paying job in a not-very-attractive geographic location. Or, if you insist on living in, say, New York, you may end up teaching as an adjunct at several different places.
Someone with your mathematical talents and skills can often find much more rewarding careers elsewhere.
You should pursue the Ph.D. only if you love learning, doing, and teaching mathematics so much that you can't bear the thought of doing anything else, so you're willing to live with the consequences of trying to make a living with one. Or you have an exit strategy should things not work out.
Having said all that, I have a story. When I was at Rice in the mid 80's, a guy in his 40's or 50's came to the math department and told us he really wanted to become a college math teacher. He had always loved math but went into sales(!) and had a very successful career. With enough money stashed away, he wanted to switch to a career in math. To put it mildly, we were really skeptical, mostly because he had the overly cheery outgoing personality of a salesman and therefore was completely unlike anyone else in the math department. It was unthinkable that someone like that could be serious about math. Anyway, we warned him that his goal was probably unrealistic but he was welcome to try taking our undergraduate math curriculum to prepare. Not surprisingly, he found this rather painful, but eventually to our amazement he started to do well in our courses, including all the proofs in analysis. By the end, we told the guy that we thought he really had a shot at getting a Ph.D. and have a modest career as a college math teacher. He thanked us but told us that he had changed his mind. As much as he loved doing the math, it was a solitary struggle and took too much of his time away from his family and friends. In the end, he chose them over a career in math (which of course was a rather shocking choice to us).
So if you really want to do math and can afford to live with the consequences, by all means go for it.
FWIW, I think this is an excellent answer, esp. in context of all the other answers' unbridled enthusiasm. In all subjects, it's common for students who love the subject to consider "getting a phd" simply the next logical step. In reality, there are many factors, including lifestyle and career outlooks, that will weigh in along with the pure love for a subject in determining one's happiness or success. What I get from this answer is "Just because you love the subject does not mean a phd is right for you"--something I think a lot of current/former phd students wish they'd thought about.
... and this caution may be especially relevant at an older age, when you're more likely to, say, have to ignore an existing family, vs. putting off creating a family. I think @Deane is giving the very wise advice that, even if age does not preclude the mental ability to succeed, it may have an effect of the perceived magnitude of the sacrifices required to succeed in the path.
"You should pursue the Ph.D. only if you love learning, doing, and teaching mathematics so much that you can't bear the thought of doing anything else, so you're willing to live with the consequences of trying to make a living with one..." This advice is directed at someone who works in the music business, is it not? I'm sure it is familiar to bitrex if he knows any aspiring professional musicians, because the same advice applies to them, much more strongly.
When I interviewed at my school, I met another faculty member whose job it turned out I took. A few years later he returned for a visit, having moved to private industry, and was then earning triple my salary. Guess what, I don;'t envy him! I enjoyed doing what I got to do. Go figure.
Here is an example that no one has mentioned yet. Raymond Smullyan is a well-known mathematician who has had quite a career, publishing many books and papers on recreational mathematics and logic. He got his undergrad degree at around age 35 and his PhD at age 40. It didn't seem to hurt him much. He is now 91 and is still publishing(!). So yes, these people exist!
When I was 27 I was a lugger in a meat market. I got my PhD at 35, and spent the next 33 years on the faculty at UGA. To be honest, getting a PhD was hard and not especially lucrative, and certainly not everyone in my graduate class has earned a living doing math at the college level. But I would not have wanted to do anything else. I have met many brilliant and generous people and been privileged to discuss math with them all these years. If you pursue your goal but don't find a suitable college teaching job, and you like discussing math with interested kids, I suggest looking at private high schools. Some of my most rewarding teaching occurred when I volunteered one year at a good local high school.
Quick answer: No, you are not too old. Yes, such people do exist. It sounds like you're off to a good start. Don't let your age worry you.
I dropped out of college when I was 21 to work as a software engineer. Admittedly my work was technical, but my background in abstract mathematics was basically nonexistent. Two+ years ago, when I was 34, I decided to get a PhD in math. I returned to college to study math, basically from scratch. Like you, I'd learned some math on my own, but I feel I made faster progress in a more structured environment working with people who were also trying to learn math. It was a great decision. I'm now a first year graduate student, well on my way to a second career in math.
I turned 40 late this past year after a long and unhappy life taking care of sick family and then enduring my own illnesses that have slowed my progress considerably. But I'm nearly finished with my Master's in mathematics after getting a subpar bachelor's in chemistry and I'm planning for my PHD. My health is the number one concern here as far as being able to do it. I know when I'm on my A game,I'm as good as anybody. The problem is that happens less and less these days. Am I scared? SURE. Especially hanging around with 19 year olds who can run rings around me because they don't need to sleep........ But I can't give up now. I've suffered and lost just too much. And anyone walking the same path shouldn't have any other attitude except that. PERIOD. I've got a Master's degree almost, some terrific grades in some very hard graduate mathematics courses and some not so great grades and incompletes.I've learned some awesome subjects and had some fantastic teachers.I've been reviewing textbooks for the Mathematical Society of America pro bouno for almost 2 years and I LOVE the job.I have a blog on mathematics I don't write in often enough,but I intend for that to change.I have the same philosophy on mathematics that the late great science fiction writer Fredric Brown had on writing:His wife said after his death he hated to write,but LOVED having written. I love having written mathematics...... I don't even care about grades anymore. I was obsessed by it once,but you know what? After watching half your family die slowly in agony of cancer and seeing entire families live out of thier cars after falling behind on thier mortgages,it really puts a perspective on things.I doubt the fact I got a C+ in advanced ordinary differential equations because I got sick the last 3 weeks will matter if I publish 6 significant papers on additive number theory over the next 2 years,do you guys?
Yeah,I know,in academia,what I just said is heresy. But you have to keep it real. I'll make it or I won't. I'll get a PHD in mathematics and spend a few years making contributions and teaching or I'll die of a massive heart attack trying, It's as simple as that.
First of all, bitrex, of course you're not too old, and it's probably your imagination that you're slower now than you were at 18. What's more likely is that you now have a better awareness of what you're missing, so it seems slower, even though I'll bet that you're actually learning better. "Cognitive decline" (or at least any decline which would affect doing mathematics) doesn't set in until your 60s, and even then it's highly variable.
I think it's sad that anyone feels this question even needs to be asked. One of the things I find most frustrating about mathematical culture is how impressed everyone is when good mathematics is done by younger people, as though your age appeared next to your work, like in a grade-school art contest. I really hope the Chern Medal supplants the Fields as the premier prize--I think providing something aspirational to over-40 mathematicians will have a very salutary effect on the field as a whole, to say nothing of the salutary effect it will have on a lot of individual over-40 psyches.
But here's something odd that I've noticed: the mythology about "older mathematicians" seems to cut women more slack. I guess sexism can cut both ways.
I thought the justification for the "under 40" requirement for the Fields medal was to encourage future research. This is still a bit of a cop out, but it's worth noting.
The mythology I've heard is that men might be brilliant at math at a young age and then decline, whereas women might be pretty good for years (but never brilliant, is the implication). If that's what you're referring to, that addition always reeked of a post-hoc justification of the same old sexism to me.
I started college in Jan 2007 when I was just about to turn 27. At the time I knew nothing: I didn't really remember trigonometry. I didn't know what it meant to raise a number to a negative power. I'm graduating this December (at 29; I was motivated enough by feeling like I was years behind schedule that I was able to finish this degree in 3 years), in the process of applying to graduate schools, in 3 graduate classes now.
Yes, I do regret that I didn't do all this sooner. And honestly it still feels weird. Sometimes I go to class and it just seems so strange that I'm actually going to college. Still, overall, this is one of the best things I've ever done. I guess you have a couple years on me, but it doesn't matter.
The unfortunate thing is that, at least at my school, you don't get to do much mathematics until your junior year, so you may be in for a long slog of general education classes you are not particularly interested in. (I actually think this is a big problem... but that's a whole different issue I guess.)
Also, you say "I'm getting into material that is going to be very difficult to learn without structure or some kind of instruction." Maybe. But it's also possible you just haven't yet developed the skills/habits to learn that kind of thing.
Re "the unfortunate thing" -- this is a common story at many places. I think much of the reason why I entered mathematics and not any other subject was that the mathematics program at the University of Alberta allowed students to jump right in and learn mathematics in a very "aggressive" mode from the beginning. I found the intro courses in most subjects to be devoid of any real expectations on the student. I was in university for a challenge. Challenge me!! That's how I felt.