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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?

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closed as no longer relevant by Felipe Voloch, Gjergji Zaimi, quid, Bill Johnson, Alex Bartel Dec 24 '11 at 19:03

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There is a question about mathematicians that learned mathematics at a late age. mathoverflow.net/questions/3591/… –  Kim Greene Nov 29 '09 at 8:54
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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. –  Jonas Meyer Nov 29 '09 at 9:47
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Community wiki. –  Harry Gindi Nov 29 '09 at 19:18
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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. –  Regenbogen Mar 4 '10 at 16:32
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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. –  quid Dec 24 '11 at 17:45

37 Answers 37

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.

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We can safely file that quote under "Hardy has been known to talk like a pompous ass". –  S. Carnahan Nov 29 '09 at 21:54
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When you encounter Hardyesque opinions on older mathematicians, it's probably worth keeping this comic in mind: xkcd.com/447 –  Harrison Brown Nov 30 '09 at 2:09
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@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. –  Pete L. Clark Nov 30 '09 at 2:31
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@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. –  Yemon Choi Nov 30 '09 at 9:17
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Anyone who actually doubts the productivity of older mathematicians has only to look at Gelfand's output in his later years. –  Qiaochu Yuan Dec 1 '09 at 19:27

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.

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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. –  Jennifer Oct 14 '11 at 15:18
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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. –  roy smith Aug 6 '12 at 5:50

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!

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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!

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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.

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At Princeton, no less! Go, Cotton! –  Eric Zaslow Mar 8 '10 at 4:40

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.

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Too much information. –  Todd Trimble Dec 4 '11 at 21:15

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.

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youtube.com/watch?v=fdkVsnZk2MI $$ $$ –  Will Jagy Nov 11 '10 at 5:22
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Dear Roy, As I already told you, the privilege was ours. –  Pete L. Clark Nov 11 '10 at 5:32

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.

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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. –  Cory Knapp Nov 29 '09 at 21:29

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.

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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. –  Ryan Budney Dec 19 '09 at 20:27

My 2 cents (I'm too poor for 5 cents ;) )

I have a very problematic educational impediment, I am moderately to severely disabled, and it strongly effects my ability to learn mathematics. I'm not equating the problem of "being too old" and "being disabled", but just as a success story.

I will be completing my mathematics undergraduate degree in my 6th year at UC Berkeley (next year).

I would not let anything stand in your way if you want to learn mathematics. It is an amazing subject and I wish I could tell more people about why it's amazing. Your enthusiasm indicates to me that no impediment could stand in your way.

Pacing may be an issue since you are not working with an instructor... that's a different topic but I believe there is a post on here about pacing for learning new mathematics (for some reason I think I even posted it ><)

I have friends who are your age and older that have succeeded at UC Berkeley in mathematics, and, not to exaggerate, they made it through what is for most people a rather intense curriculum (from my vantage point, which is, of course, quite limited).

I never doubt the capabilities of the well motivated, so I believe you can do it! Keep with it! It only gets better :)

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One of the ablest mathematicians at my grad school went to Julliard and taught music until he was 30. He then spent 6 years at the U. Wisconson and got a PhD degree under Walter Rudin in function theory. He is a very successful professional mathematician.

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I agree with the consensus here: it is definitely not too late to be learning math. The fact that you have mastered several books worth of material on your own, without the benefit of an instructor, is proof enough. There is nothing to stop you from continuing in the same mode if you wish.

However, it sounds like you have heard the siren call of mathematics and would like to get into it in a bigger way, and more rapidly. As a mathematician I can only encourage you. Sure, there are some disadvantages to starting later in life, but there are compensating advantages as well.

My own testimony: I did complete a bachelor's degree in math straight out of high school, so I do not quite fit the profile you are looking for. But after working 23+ years in the software industry, I returned to grad school in 2002, completed my master's in 2004, and finished my PhD this last June (2009) at the ripe age of 51. I don't think my thinking is any slower than when I was young.

Whether you should return to school is a personal decision, and it no doubt depends on many factors. But that is really a separate discussion, a separate thread. The answer to your immediate question is that age should not be a barrier (especially for a young-un like you).

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I always feel very encouraged when I come across threads like these. I'm in a similar position. I was a math major at Georgia Tech. I lost my way and ended up in pharmacy school. I'm now in my last year and plan on going back to Tech to finish my math degree as soon as I'm financially able (hopefully after one year of working as a pharmacist), after which I want to go to grad school. It's great to hear such positive stories from others.

If everything goes as planned, I'll be 30 when I enter grad school. So no, you have no reason to feel you're too old. I know of several mathematicians who obtained their PhDs in their 30s, and even a famous one who got hers at 41.

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You are NEVER too old if you love it. And this is coming from someone who is 47 and working on their PhD in probability.

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I just read that coffee table book full of mathematicians' biographies: "Mathematicians: An Outer View of the Inner World". The most striking part of it was the variety of backgrounds. While the majority of featured mathematicians were interested in math early, or had parents with analytic/technical backgrounds, a significant fraction of them only became interested in mathematics while in college, and some later than that.

BTW, you said you came to mathematics from music and acoustical applications, and that you're starting to feel overwhelmed by the math of those waveforms. There are actually many other connections between music and mathematics, and some of these other areas may be fruitful to you. For example, look at Combinatorics for the study of melody or harmony. But along the path you've described, of Fourier and DST and Bessel functions, take a look at this free online edition of "Music: a Mathematical Offering", by Dave Benson and Cambridge University Press: http://www.maths.abdn.ac.uk/~bensondj/html/maths-music.html . It looks terrific, and I'm going to download it myself now.

Also, it's very common to not understand some new mathematics the first time. Professional mathematicians will often plow through a difficult paper on the first read, skipping over equations to get the general idea of where it's going. Even Feynman recommended this technique. Don't let yourself get stuck trying to understand an equation, but keep going. Then you can come back and read it again, either immediately or later, with more understanding.

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As someone doing something similar, the biggest problem for me has been time as I recently got married. The coursework part of things isn't that bad but doing research on a topic takes focus and a lot of time just thinking and reading about a problem.

I don't always have as much time as some of the younger students who can essentially dedicate most of their waking hours to studying mathematics.

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I just turned 30 myself, and I'm in almost precisely the same boat. I read mathematics "religiously". I keep thinking of going back to school, but I've been too timid about it. I think the biggest obstacle I've felt learning on my own (or trying to) is the lack of a "click" to make sure you really understand the material. Rather than a test, exam, or problem sheet, you have to figure out how to verify your answers in more than one fashion. I'm sure that can be a great boon, but it also slows you down.

It's very good to know I'm not alone in this boat!

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I started at 32, almost 33. I've been worried that I've lost my best years but I'm getting through Analysis fairly well, teaching myself topology and preparing to take number theory & a real probability course in the spring.

In my second semester the topology (Munkres) and the Analysis (Rudin) were beyond me, but just a bit of work went a long way. Much of Concrete Math (Knuth) was beyond me last year as well, but I'm able to get what I want from it as I need to.

I'm slower in a lot of ways than I was when I was 18, but as some have said maturity does a lot for a person. There's no way I'd have done as well as I am then. Also, speed isn't what's important for mathematics. It's great if you have to take the Physics GRE and need to calculate on the fly, but I've found I tend to be able to grok proofs at a deeper level than I would have.

Also, professors love the work ethic us "older" people have.

The only suggestion I can offer is find a program that you fit well in, whether big or small, with strong guidance or with a lot of independent study and find a professor you enjoy working with if at all possible.

I couldn't be happier and my life couldn't be better were I to have chosen a different path. Give it a try, the worst that happens is you decide it's not for you.

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I am a PhD candidate in Comp Sc and will soon touch 30. And I don't think I've grown slower during the years. My intuition has probably never been "better" and I have nurtured the humility to understand that I am far from perfect, even in my opinions and thoughts. But the pursuit is to be perfect and I know that I can do a great job.

My question to you is: Are you sure that the "slowness" as you've got "older" is really true or is it something you've learnt from the popular culture? It is easy to get biased, very difficult to be true. This is, in particular true about the scientific theories of the effects of age on mathematical ability. Just take note of how many papers are published each year contradicting the earlier theories.

From my personal experience I can say that when you enter the graduate school late, the expectations of the people around you are different -- most likely you'll have considerably more responsibilities like family, planning for the future, etc. So you surely need a stronger will power and time management skills than the under 25. This is an opportunity.

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I'm 29 (turn 30 in Spring) and I returned to finish my undergraduate degree in 2007 in mathematics after working in computers for 8 or so years (after dropping out of 1st college). I'm finishing up this Spring.. currently on my last required course for the major. I can tell you from direct experience that you're not too old. I initially returned to study biology (I was interested in bioinformatics) and starting taking math courses... taking these courses turned me onto math... at 27.

My first class back was an introductory Linear Algebra course; last math class was 7 years prior (a discrete math course) and took calc 2 8 years prior. I struggled for like 6 weeks and finally things started to click... the key was that I just worked at it; kept on trying. I was slower at doing the maths than my classmates, the majority of whom were 19 years old. While the beginning of the course was a struggle, the rest of the course was very clear to me after I put the time/effort in.

My second course was a vector calculus course and I found it easier than the first course I took upon my return. It was easier because I knew how I needed to approach the material better and I was more disciplined to follow that path. I compare this mindset to my mindset when I first was in college getting my CS degree; I was a space cadet... most people that age (18/19) tend to be. But now I know that I enjoy learning the different areas of math and the way to do that is to put in the time/effort to study and work on problems. I just didn't have the patience when I was younger. This was a very enlightening thing I learned by taking a 2nd math course.

From then on I have taken a variety of courses from ODEs, non-linear DEs, geometry, topology, more advanced linear algebra, and others... and I have greatly enjoyed my time learning the materials, even if I have lived int he library the past 2 years :-). One thing I've noticed is that being older, I tend to want to know the whole story. I am an applied math major and unlike my younger classmates, they take a lot of the theorems and methods for granted while I am interested in the why and how that got created. From what I've seen, this spans the pure and applied math realms at times.

Anyway, sorry to ramble on like this, but I just wanted to illustrate that it is possible and that you can do it!

Go for it! Good luck!

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Wow - this is totally encouraging. I just turned 50 in March and within the last year and a half have become obsessed with numbers. I am compelled daily to find the prime factors of zip codes, addresses, and RSA numbers. I saw the Christmas lights on my house in binary. I love the patterns and personalities of numbers.

Though I am not as wizardly as you all, I blew through an elementary algebra text in a weekend, find that geometry is returning easily, and I'm on my way to really enjoying trig and calculus, although I really want to explore discrete mathematics.

So reading through these posts has really boosted my confidence and enthusiasm. Thanks to all!!

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It has been done many times. Someone retires from his/her first career, then goes back to school. Occasionally even getting a Ph.D. in mathematics, and having a second career as a mathematician.

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Well, you can't be sure that you will do an academic career.

But you're definitely able to study mathematics.

Here are my 5¢:

  • You should not study mathematics unless you love the process of learning, problem solving, etc. I think the right motivation is the joy of studying, not a career. (But don't be upset if studying is hard for you — there is no royal road, it is hard for everyone.)

  • Also, it is not possible to get an insight into the real mathematics from applied topics and random books. As a first step, you should try to take some rigorous courses from working mathematicians and find a good advisor.

So give it a try.

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My experience is the older I get the quicker I am at learning new things, in mathematics or anything else. Maturity goes a long way at quieting the mind and a quit mind is much quicker at learning new things than the buzzing mind of the 18 year old that is still trying to figure out what is relevant and what is not. If anything your age is an asset so you should definitely go for it.

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I did a Mathematics Batchelors 7 years ago, but was always having difficulty trying to cram in theorems and proofs until they started to merge into one(!!!). Naturally, this didn't pan out too well as a revision method. But looking back on the notes now, that daunting information seems a lot more readable than it did before exams. (I've been reading up on Representations of Groups) So, in summary, No! I don't think you're too old. It would appear that the myths of being too rusty are bunk. Historically, Mathematicians have been known to have done their best work young. Then again, in the past a lot of them popped their clogs a lot more quickly than Mathematicians today. Besides, when if you were a lot younger, you might be just interested in just partying and all sorts of juvenile things at college thus being a little distracted from understanding things properly in the Mathematical World... Quite a lot of undergrads I knew just memorized proofs and exercise questions to get by in the exams, but didn't quite get an intuitive feel for it. The blank looks i used to get from friends with Firsts asking them if they can remember what a Stochastic Process or to get them to write the definition of continuity for a single variable...... :-D

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To encourage you there are two people in my Department who are about 40 years old and have just started their Ph.D. One of them was out of touch for nearly twenty years!

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I think the myth of being "too old" is a stereotype threat like others that tend to push people out of mathematics at various stages, at least in my experience in the USA. But it's not just about mathematical abilities, it's also about how you communicate, write or socialize; that affects whether you learn from others, whether they recognize your contributions to mathematical thought, and so on. Embarrassment and egos can get in the way of people who should be studying together and learning from each other. If you're studying math in college now, I hope you will reach out to some of your younger classmates and work with them.

A couple more words of encouragement. Sharpening and focusing one's mathematical thinking, as well as learning mathematical content, often feel like slow processes (or come in spurts), and repetition may be required; however the progress one makes in a year can look impossibly far before one makes it. I think that contributes to feelings of inferiority that people so often have when it comes to math. You should do more math in college, if you want to. But if you're keeping your job and have a real life and yet have room for math, you're in a great position, and the main obstacle might be figuring out how to contribute to the community. We don't just need published papers on new work; we also need time-saving sources, such as masters' theses, that clarify, organize, and modernize old work. Either way the trick would be getting math into circulation when you're not affiliated with a university. Contact with professors would help.

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I would not necessarily say that you're too old to learn mathematics. I'd even say it may be an advantage. Many mathematical problems you'll encounter during during your studies can be solved easily if you accept and understand how mathematicians proceed when they come across problems. This normally requires one thing: patience. It is normal to be stuck on a problem for days or even for weeks (my experience sometimes!). However, many (young) people tend to becoming very frustrated if a solution isn't obvious. Mostly, indeed, it isn't. These are then the interesting problems. There is a saying that the older you are the more patient you are. If you combine this with motivation and passion about mathematics and still some curiosity, you'll have no problem in succeeding (no guarantee, though, but it becomes more likely ;) ). My opinion is that you never become too old to do something academic. My dad for instance wanted too learn something about electrical engineering. So, he read some books about it and is able to solve the problems these books contain. And, he studied law and is 55 years old. I think this shows that with enought motivation you can do anything. Perhaps you'll have a decreased memory capacity compared to your fellow students, but only perhaps. I think, you never know, if you don't try. So just try it.

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It is important to consider that people have different intellectual strengths at different ages. In my teens and twenties computation was my strength and I would happily grind away on page after page of calculations eighteen hours a day, weeks at a time. In fifty five and that doesn't happen anymore. So I do feel I have lost a degree of mental nimbleness and speed. On the other hand I am a better communicator and educator now than when I was young. Someone wrote that age can bring a certain cunningness to solving problems.

Another pertinent issue is whether you have previously been maintaining an active intellectual and maybe even scientific live. You may have important skills, knowledge and perspective you can leverage and transplant into mathematics. I went back to college in math in my fifties and the fact that I had spent twenty five intellectually active years in the software development industry gave me important competitive skills that had a dramatic impact in how well I did in certain math classes. I took an advanced differential equations class that used Mathematica. My background as a programmer and twenty years experience with Mathematica allowed me to attack homework problems quickly with a large variety of methods so that after one hour's work I would have a ten or twenty page Mathematica notebook dissecting the problem I could share with my class. A broad background in relevant areas, attacking a problem in a number of radically different ways, and being interested in communicating and educating others are all strengths I developed with age.

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Depends on what you want. Too old to get the Field's medal? probably. Too old to study mathematics or have an academic career in mathematics? definitely not! I knew a few people who did their master's in math in their late 30's and went on to teach college math.

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