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I'm sorry if this question goes against the meta for posting questions - I attached all the "beware, this is a soft-question" tags I could.

This is a question I've been asking myself now for some time. In most areas, there's a "cut off age" to be good at something. For example, you're not going to make the NHL if you start playing hockey at 20. It just won't happen.

So my question then, how late is too late to start studying math and make a career out of it? By "start studying math" I mean, to really try to understand and comprehend the material (as opposed to just being able to do well in a formal, intuitional environment).

I don't mean this from a "do what you love, its not too late" motivational perspective. I mean this from a purely biological perspective; at approximately what age has your brain's capacity to learn effectively and be influenced by your learning stop? When does the biological clock for learning new math run out?

My reasoning for asking this question is (for those who care): I love math. Really I do. But , having spent the first 21 years of my life in sports/video games/obtaining a degree in a scientific field which I care nothing of/etc, despite all my best attempts at trying to learn math, am I just too late starting to ever actually be good enough at it to make it a career? I've almost completed my second degree (in Math), but find that in many cases, despite how I look at a problem, I lack the intuition to comprehend it. I'm going to single him out (sorry), only as an example, but Qiaochu Yuan is my age.

Note 1: If this question isn't a suitable post, I won't be offended at all if you vote to close - I know this question borders what's acceptable to ask.

Note 2: Thanks to everyone for reading and taking the time for the great responses. Really appreciate it!

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You might like reading some of the responses in these MO topics. Too old for advanced mathematics? and Mathematicians who were late learners... –  Nastassja Nov 14 '12 at 7:33
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@bizso09: Your facile LOL-cat generalization based on a Wiki article seems profoundly out-of-place here. "You obviously won't make it big?" Really? Compared to what? –  daniel Nov 14 '12 at 11:10
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A possible alternative is to have a disagreement over a girl and get shot in a duel –  wim Nov 14 '12 at 13:40
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I am almost 40, and wondering the same thing... 22 is nothing, try being near a decade away from the 1/2 millennium mark and then see about re-evaluating your career path... now that is a scary thing to do. Math from when you were in High School to when I was in High School to now, is an entirely different... you are too young to question can you learn something new. I am an old dog trying to learn new tricks... and a new career... good luck to you, but honestly you need to re-evaluate things if you think you are too old now. –  Matt Ridge Nov 14 '12 at 15:28
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You're 22 and you're asking this?!? It's not too late. It's probably never too late to do anything. When I started reading this I thought you were going to turn out to be 50 or something. As you get older you find out that it's possible to look up to people who are younger than you, and that's ok. You will always bring your own unique perspective to the table and that will always be valuable, no matter how old you are. You never have to know more than the next guy, just own your own space. –  Todd Wilcox Nov 14 '12 at 18:11

28 Answers 28

up vote 172 down vote accepted

Karl Weierstrass was in his 40's when he got his PHD. There are a dozen other counterexamples, a number fairly recent. A good set of examples can be found in the thread on MO here: http://mathoverflow.net/questions/3591/mathematicians-who-were-late-learners

This myth of "science is a game for the young" is one of the falsest and most destructive canards in modern society. Don't listen to it. You only get one life and when it's over,that's it. When you're dead a hundred million years,you'll be dead the tiniest most infinitesimal fraction of all the time you'll ever be dead. So stop listening to career advice from teenagers,grab a calculus book and get to work. That's my advice.

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@Matt Sigh.Ok,fixed. I can't win....... –  Mathemagician1234 Nov 14 '12 at 7:15
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Wierstrauss is very creative. Should it be pronounced "We're Strauss" ? –  Georges Elencwajg Nov 14 '12 at 10:35
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Weierstrass is actually a pretty bad example. He was already reading advanced mathematics when he was a teenager (he regularly read Crelle's journal...) and followed mathematics lectures next to his ordinary studies. He continued pursuing mathematics while he was a teacher and after more than a decade in scientific isolation (pursuing work suggested to him by Gudermann) he finally published and for this work he received a doctor honoris causa at 39. Two years later he was promoted to be a professor in Berlin. Biography –  commenter Nov 14 '12 at 23:01
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You should be more careful with the facts. He never spent "2 decades" in the German school system (that was 1841-1855). "He pursued higher mathematics as a personal calling after failing to secure a law degree" is a misleading way to put it. In 1840 (when he was 25) he already wrote stuff that impressed Gudermann deeply. He pursued advanced mathematics already when he was much younger, so your comment/example is really not convincing. –  commenter Nov 14 '12 at 23:54
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I have to agree with commenter. If the point of your example is to say that people can start out older and do well at mathematics, then giving an example of someone who started out with math quite young but just didn't get a PhD until later isn't a good example. The fact that he could do advanced math without formal training just shows how smart he was and thus says he is not a good comparison for most people. You could also list Ramanujan as someone who got a PhD around 40. What does that have to do with any of us? Are we on his level? –  Graphth Nov 15 '12 at 17:18

21 is not old at all. I personally know heaps of people my age (32) who started out at 18 as salesclarks/BA or BCom majors/lawyers/bookeepers etc and ended up having a PhD degree in some advanced math areas and landed a job in academia or industry.

My personal case: I got a lousy BCom degree with little math at 22 and then worked in a primitive banking job. After a few years I realized I was growing stupid, so decided to do what I secretly always liked but never really had the balls to do: math and stats.

So I moved to another country, did a Masters degree in Computational Statistics, (2nd level honours) then started on PhD in Computer Science (mathematical modeling of AI). After 6.5 years of math I'm, like $n^n$ times smarter than I was at 22, I got a postdoc job in my area and got awarded a Doctoral degree this morning.

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Congratulations on your PhD. –  Potato Nov 14 '12 at 8:00
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Congratulations on your degree, and thanks for sharing! –  Thomas Nesbitt Nov 14 '12 at 12:40
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Ah... congrats on that PhD but much more and warmer congratulations on the drive and stamina to pursue your wishes!! –  DonAntonio Nov 14 '12 at 13:17
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For your sake I hope that $n>1$ ;). Seriously 'though, yours is a good inspiring story fitting for this question. And congrats! –  TheTerribleSwiftTomato Nov 14 '12 at 18:02
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Congratulations! Such a nice story. Instead, I am considering going to some investment bank after my phd, as it something that I always secretly loved but never really had the balls to do. So now I'm a bit scared of starting growing stupid if I go there –  Ilya Jan 10 '13 at 21:22

In Israel kids are expected to serve in the army when they are 18, and they serve for three years (men do, women serve two years). After this period it is common to find yourself questioning what you should do with yourself and not many people have answers. Therefore it is common to take another two years to work and travel the world before settling down and starting your academic education.

This means that most students in Israel begin their undergrad studies around the age of 23-24.

That been said there are considerably less mathematics students, and many of them start sooner. I started at 22, but I studied both with kids that didn't serve in the army and were 19-20 and people who took longer to settle for math and were 25.

So if you are just 21 and you want to start with mathematics, you're still younger than the average Israeli freshman.

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very interesting. thanks for the reply! –  Thomas Nesbitt Nov 14 '12 at 12:59
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@ThomasNesbitt to add to this, it's quite easy to find a great number of very accomplished Israelis in mathematics, and there are quite a few Israeli theoretical computer scientists as well (which depending on their age, might make them mathematicians as well). –  John Moeller Feb 20 '13 at 20:27
    
@AsafKaragila - Not to mention the tank battalion commander who left the post in his 30's and is now a prominent mathematics professor. –  nbubis Feb 21 '13 at 9:20
    
@nbubis: Who's what now? –  Asaf Karagila Feb 21 '13 at 9:26

Of course you can make a career out of it! When I started reading your question I though you were around 50, but 22 is not old at all to go after a career in anything but sports. This kind of time don't affect your brains ability to think at all. The only thing is, if you have great ambitions, you are probably not gonna be able to win the fields medal because it has an age limit, but apart from that there is nothing that can stop you from following a career in math and be great at it.

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Even if he WAS 50,so what?!? As long as he has no serious health issues-which would be the only real concern that should stop him-he should go for it and hope he lives to age 90. If not,he'll get as far as he can. Age is just a number. –  Mathemagician1234 Nov 14 '12 at 7:19
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My sentence really was ambiguous, I didn't mean that, I meant that 50 is the age where people can start to ask questions like that and have an answer like "Yes, its probably too late", but in a general career. What I really wanted to say is that at the age he is, any career except for some sports is a possibly sucessfull career, not only math. –  Ivan Lerner Nov 14 '12 at 7:30
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@Mathemagician1234 Do you seriously believe age is just a number? Even if there would be no effect on mathematical ability, it certainly influences performance in many areas. You might take a look at the olympics the next time it's on. –  Michael Greinecker Nov 14 '12 at 8:08
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@Micheal Obviously I didn't mean it LITERALLY-age clearly gradually takes a toll on us physically,although this is somewhat overrated. But this effect is really limited to pursuits where the maximal physical ability of prime youth is required and these are quite limited in number.It certainly doesn't apply in this case-unless of course you want to make the absurd claim that pursuing a scientific career is isomorphic to pursuing an Olympic competition dream.Assuming no physical illness or limitation,most people would be surprised to find what a human body or mind is capable of. –  Mathemagician1234 Nov 14 '12 at 8:53
    
"When I started reading your question I though you were around 50" <- This, except more like 40 to 45. –  Joe Z. Mar 12 '13 at 19:37

I'm not sure my personal experiences will be very helpful to a mere kid of 22, but here goes ...

I made a complete mess of being an undergraduate when I was 18 (until 21), and followed a career for some decades before I finally got round to seeing if I was actually capable of doing maths at a more advanced level.

I was almost 50 by the time I had published some research and gained a PhD in pure maths, so the effort could be considered a total waste of effort in terms of "my career", but it was hugely rewarding to me personally to have discovered that I was capable of finding out things in maths that professionals were interesting in reading - even if they were not in the same league as the millenium problems.

So - "go for it!" - you are never too old!

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youtube.com/watch?v=KHVYBiVKldU –  Will Jagy Nov 15 '12 at 0:34
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OK - I guess it really IS possible to be too old to "rock and roll" :) –  Old John Nov 15 '12 at 0:38

Oh my! $22$ years old?

You are still very young, and certainly not too old to pursue any passion, math or otherwise!

I've taken many (lengthy) breaks from math. But each time I've returned to serious and dedicated mathematical work (be it studying math, teaching math, or pursuing research), I've come at it from a fresh perspective.

Sure, when I've resumed mathematical work after taking "time off" from studying math, I've been a bit rusty at first, and have needed a bit of time to clear cobwebs from my brain. But that happens when taking time off from anything! So yes, "picking it up again" can feel discouraging at first, even perhaps overwhelming. But it's worth it; sometimes revisiting material you've encountered a few years back actually leads to a deeper and more comprehensive understanding of the material.

With respect to your impression that you "lack the intuition" to fully comprehend some of the problems you encounter, I doubt that intuition is "something you either have or don't have"; I'm convinced that mathematical intuition is something one can acquire/develop and honed over time, with practice and perseverance.

Take heart. Pursue your passions; with patience and persistence, the "cobwebs" will clear, and in no time at all, you'll feel very much at home in the mathematical world.

On a different note: I do think there is a myth floating around that you need to be a prodigy to excel in math, and there does seem to be a bias held by many academics that only the best and brightest among young students are worth investing in, in terms of mentoring, and/or admission into graduate programs. But perhaps the best way to defy such bias is by setting out to prove them wrong!


Added $(1)$: Your question reminded me of the article Is math a young man's game?. (Of course, I'd prefer it was entitled: *Is math a young person's game?* - but that's another matter altogether!)


Added $(2)$: On a light note, see xkcd for Too old for....

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If your career means Fields medal, you're probably too old. But 21 is not too old for studying math and be successful, in the sense that you still have a great chance to become a professor in a descent university after say 15 years, 36 is not old at all, I saw lots of people got their Ph.D around 30. But I am sure you will fell frustrated sometime but if you can hold on to it, you will make it.

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If I recall, Ed Witten didn't start math/physics until 21 ... and he still won a Fields medal! Of course, this is exceptionally uncommon, but even starting at 21 isn't always too late for a Fields medal. –  Matt Nov 14 '12 at 6:35
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WHO CARES ABOUT A FIELDS MEDAL?!? MY GOD, IT'S ONE THING TO BE AMBITIOUS,BUT YOU CAN'T MAKE A CAREER DECISION EVALUATING YOUR AGE BASED ON THIS! THIS IS RIDICULOUS. –  Mathemagician1234 Nov 14 '12 at 6:39
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To add yet another example, Szemerédi also started out quite late. He had attended a year of medical school and worked in a factory before he started into mathematics. –  EuYu Nov 14 '12 at 6:47
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This is perhaps off-topic, but I personally find "in the sense that you still have a great chance to become a professor in a descent university after say 15 years" quite unbelievable, as my impression has been that the job market is very competitive these days. Am I wrong? –  user27126 Nov 14 '12 at 7:49
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@Matt Getting the Fields medal is exceptionally uncommon, no matter what age one starts from. –  Michael Greinecker Nov 14 '12 at 8:04

Maths is life, when you start doing maths your age will be 0. Don't worry about age. Maths is life, live life as life whether its 10 days or 100 years.

All the best. I too started Maths very late :-), and after that only I rediscovered myself. As far as I remember, most of Euler's work done after he became blind (around 59 years ).

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EXCELLENT POINT ABOUT EULER, USER. : ) –  Mathemagician1234 Nov 14 '12 at 19:53
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all caps is cruise-control for cool? –  Peter Sheldrick Nov 14 '12 at 20:15
    
@Peter No,it's for emphasis because I thought his/her point was that important. –  Mathemagician1234 Nov 14 '12 at 23:40
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@Mathemagician1234 You can use italics to emphasize certain parts of what you are saying. Caps lock is considered to be the same as yelling, and is generally discouraged. –  Eric Naslund Nov 15 '12 at 9:59
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He was yelling, so caps seem appropriate. –  bubba May 1 '13 at 13:57

OK, I understand nothing, but here's my opinion:

  1. To be a star you should have already started as a child, out of your own accord, and should have found yourself devoting all of your time to profession of choice - be that a star video game player, a star lawyer, a star anything.

  2. At the age of 21, after having completed a degree in a scientific field (for which you should have learned some math, regardless of caring for it or not) AND having almost completed your second degree in math, you are obviously qualified to make a career out of Math. Especially since you love it.

It might interest you to know that most (secular, non Arab) Israelis serve in the army till the age of 21-22, then usually take a one year vacation (backpacking around the world) and only then start their higher education. Didn't hurt us any.

As an obese person at the age of (OMG almost) 53 trying to survive Ninjutsu classes, I feel your pain :). I know I'll never be Sensei Hatsumi but I'm quite sure that had I decided to make a career out of it, I'd be able to reach instructor level.

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I am quite sure you could make a career of it too. That I think is the point. Given decent health and the will to succeed, there is really nothing holding one back other than the skepticism of others. –  daniel Nov 14 '12 at 11:14

I am a couple of months away from being your age and I, as well, just started doing math. I also wasted a good amount of time playing games, sports, etc. I was a psychology major but changed my major a year ago to math -- not that I hate psychology, I still love it. Of course I struggle understanding some of the concepts in math but that alone won't deter me from accomplishing the goals I want. To be honest, I really don't care if it takes me 50 years to completely understand math. My advice will be work as hard as you can and don't look back.

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It depends on you!

I can take myself as an example, I started studying college math when I was 15 years old, I was sick of math when I was 19, so many things to study, I didn't have the maturity to study such a serious subject, now I'm 29 and I'm doing my master's degree in math after two bachelor's degrees: psychology (duration 6 years) and math (duration 4 years). I feel like when I was 15, the same difficulties, in fact, I think I am better now than I was 15, really. And now I feel like I can study anything I want, my brain is really fresh and new.

However, I have friends who are old inside, what do I mean? I mean they don't like thinking, don't like to discover new things, they only live to work hard and take care of their family, very difficult to teach something new to them.

So forget research which says your brain power becomes less as you age. The tests are correct in some sense, but we have to ask ourselves why a person with older age makes less points in tests than younger people, this happens because when they become older, they stop to study to work hard, have children and so on.

So what you have to have in mind is not to lose your interest to study new things and don't be contaminated by people who have lost their will of life and curiosity. If you do so, you NEVER lose your capacity to study new things.

I really don't know why mathematicians have this kind of prejudice. I've never seen this in any other areas of science.

So remember, your problem is not biological, your problem is social, the social can influence our brains.

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Your story remind me of the soccer star Robin Van Persie with "a screaming little boy inside his body". –  Metta World Peace Feb 25 '13 at 23:47
    
@MettaWorldPeace I saw his interview, I feel the same :) –  user42912 Apr 6 '13 at 20:02

I am in graduate school now. Plenty of my classmates are aged over 30 and they performed very well. My mother went to college at 24 due to political reasons. She is a successful medical professor closing to retire now. I do not see any reason age should be an obstacle for your pursuit if you have passion, courage and following the right guidance.

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As I said in my post, I don't know why mathematicians has this prejudice with age. –  user42912 Nov 15 '12 at 16:56
    
@user42912 Part of it,I suspect,is based on an inferiority complex many mathematicans gain in thier student years at top universities,where teenage wunderkinds are commonplace and made them feel like old losers-as absurd as that is.Another factor is probably the fact that as one gets older,stamina and healing diminish steadily,making the rigors of intensive study that much more difficult.This part there's truth to.Lastly,there is SOME cognitive research that suggest creativity diminishes greatly after 30-but it's still highly controversial.I think it's still empirically an open question. –  Mathemagician1234 Mar 29 '13 at 21:00

We can be good at anything we set our minds to. If we make excuses, though, those become brick walls which hold us back from achieving greatness. In other words, if you approach math in the way that you'll be one of the best, and age won't matter at all. Plus, as someone who's researched mental decay, it tends to happen to people who let themselves mentally go, meaning that doing math or something mentally straining as you get older is great for your mind.

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> We can be good at anything we set our minds to. A heart-warming platitude, but clearly not true. In some cases, effort and determination can substitute for natural ability and physical characteristics, and in some cases they can't. It's a good idea to try to understand and accept our limitations. We all have them. –  bubba May 1 '13 at 14:05

Knowledge is provided for whoever want it. Like a Chinese quote,'Live and learn until you become old or even death'. Somebody believe in IQ or gifted or talented. But i believe the quality of hardworking and never give up is the true path to success. If you know something is right, just do it.

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It's easy to find any excuse to give up, but it's hard to slog it out and learn something in depth.

I'm almost 50, have never done physics before, but I'm slowly but surely getting my way through Special relativity, which is meant to be easy. After that I'll go through General relativity.

Years ago at uni, we had a guy who was 8o years old in his first year, he went on to do his masters and Phd in the German language, by that time he was 91.

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+1 for this great post. But it would help your case if you identified the university so that people here could verify it. Otherwise,it's easy for the opponents of the view to dismiss it as fantasy. –  Mathemagician1234 Nov 15 '12 at 8:41

The bigger problem for mathematicians is not age, but running out of ideas. There is a historic trend that the best work has been done my mathematicians while they were relatively young, and after that they just kind of floundered through the rest of their careers, riding on their past glory.

If you start late, you may still have a chance to go through the same thing, at a later age. The excitement of exploring something unfamiliar may be there for you, in contrast to others who have long burned out already.

This window of creativity may have nothing to do with age; it may just have to do with an individual's capacity, which is "emptied out" like a vessel, earlier or later.

Sometimes, the more you know, the more excuses you have to reject some idea or not to pursue some avenue of exploration. It's the young academics who make the advancements because it's a time for them where they don't have the breadth to get in the way of digging deeply in some direction.

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Could you give some examples for your first paragraph? –  Rudy the Reindeer Nov 15 '12 at 8:07
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a historic trend... Wow! Nothing less? Makes one curious to see the data supporting this claim. –  Did Nov 15 '12 at 20:59

Pierre de Fermat was a French lawyer and amateur mathematician. I think he only started doing math in his late 20s. He is most famous for Fermat's Last Theorem, conjectured in 1637 and solved in 1994 by Andrew Wiles.

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When I started my undergrad degree, the head of the physics department was a guest speaker in another class I was taking. He told us that physicists don't come up with any ideas that are groundbreaking after age 40. So, even if a physicist were to publish a ground breaking paper at age 60, it is most likely based on ideas they first came up with when they were younger. I think this probably is true for the most part.

However, there are many things here to consider.

  1. This statement isn't absolute. There are counterexamples. And, it is based on his observations and not on a statistical experiment that is well controlled and well documented.

  2. You are still much younger than 40 and even after you get a PhD, I think you still will be, though I'm not sure about your age.

  3. This is probably the most important one. Most PhDs in any field don't do anything that is groundbreaking at any age.

So, what is your goal? Do you want to be a world famous mathematician? If so, if you are a genius then it might not matter when you start (i.e., even if you do start late, your genius can make up for it). And, if you are not a genius, then it might not matter when you start (i.e., even if you started young you still probably won't do anything world famous).

Otherwise, if your goal is to be a professor and do research that is fun to you (but not groundbreaking or world famous) and teach other students, then you're definitely not too old for that. Where I go to grad school, half the students are 30+ when they graduate. And, most of the people I know in pure math in my department go on to jobs where they are required to do no research or are required to do limited research. This is something you could achieve. Is this what you want to do?

If you want to do more research, you could do it on your own, in your own free time. If you want your job to be doing research, then it's possible but not likely no matter when you start. Most of the professors where I am getting my PhD got theirs at schools like Berkeley, Penn State, Yale, Harvard, Illinois Urbana Champagne, Wisconsin-Madison, etc. They now work here, a research university, and a large portion of their job is research. Can you get a PhD from a top school? That can't be answered here. We don't know enough about you.

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I would like to add that you can do research if you work in industry. People from my grad school do that, though I don't know a ton about it. That's probably a different type of research considerably but that's another option. If you want to be a math professor whose main duty is research, you have to be a top student, and again, that's hard for most people irrespective of age. –  Graphth Nov 17 '12 at 0:27
  1. You're never too old to learn anything.
  2. You're only 22 -- These are the years you're supposed to be deciding what to do with your life, so if you're interested in mathematics, do it!

This is a similar situation to my own. I'm going for it, you should too!

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Apart from age, this makes me think about this basic fear that if you're not enormously brilliant at doing logic and math puzzles, you're not going to make anything in maths, hard sciences or even applied sciences.

It's true that if you are not sure of being a crack, you are probably not one of them. But I see many people doing well in applied maths that are really not "cracks" (and I'm sure there are also some in very scientific fields).

Also, I usually hear about the major "super brains" in maths that they used help from fellow researchers on writing down theories, etc. (I was thinking about A. Grothendieck, sure you can find some examples).

So I think there is a place for "less genius brains" in science and maths.
And if you're thinking about applied maths, there is place to make money with maths today more than ever.
If you love maths, go for it (there are too few people like you!).

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The comment on "crack talent" caught my eye. I was reminded of the half of the Dunning-Kruger effect that says some competent people exhibit low self-confidence because of their mistrust in their competence. –  rschwieb Nov 14 '12 at 18:13
    
@rschwieb Thanks for the link to Dunning-Kruger effect. One summarizing quote, referenced there, caught my eye: "the miscalibration of the incompetent stems from an error about the self, whereas the miscalibration of the highly competent stems from an error about others". Errors about self meaning overestimating the competence of self, errors about others meaning overestimating the competence of others! –  amWhy Nov 14 '12 at 18:44

I have looked through the answers so far, and don't see one from anyone really old. I am 82 and still learning maths, and lots of other stuff, as part of a hobby that includes electronics and amateur radio. I help with on-line tutoring on a UK course for those taking their exam for a Full amateur radio licence, and I have no trouble sorting out the answers to the problems students have. So age is not a hindrance.

From boredom and ill health, I just scraped though a Pass degree in Maths and Physics when I was 21. After ten years as an engineer I studied for a Teaching Certificate, and at the same time became interested (1964) in computing. I got a Master's degree in Technology (computer Science) at age 43, and thereafter taught computing at a Polytechnic that became a University (and a very good one) a few years before I retired.

My advice is to attempt what you fancy, age mostly brings maturity that makes it easier to study.

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Congrats on being 82 years old. –  Joe Z. Mar 12 '13 at 19:38
    
As opposed to the alternative,of course.......lol –  Mathemagician1234 Mar 29 '13 at 20:52

I don't think that age is an issue with you. This lack of "epiphany" and "intuition" you are having is not because of age. It is because of inexperience. You just have been out of touch with math for too long. Or maybe you were never exposed to enough mathematical thinking to begin with. Thinking logically is hard. And it can take a lifetime to get there.

So its not that your brain is incapable of being good at math. Its just that you need to spend some time in math mode. I have been buried in math consistently and constantly for the past 9 years and still at this point even if I just step away for one summer I start losing some of it. For me its exposure. If I see a problem, usually something in the past that I have seen helps me. Rarely I think of something truly original. So what I do is just expose myself as much as possible so my arsenal keeps getting bigger and bigger.

Very few people (like Gauss and Mozart) are truly gifted and rock the world as soon as they are born. The rest of us just learned and refused to give up. I think the only thing which you have no control over is creativity. Good GOOD mathematicians always had a lot of creativity and that is something you can't get with age or experience. That's the only thing you need to be born with.

Just keep at it and you will get better, guaranteed. Humans (including our brains) are remarkable at adapting.

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If, on the other hand, if this being really good requires a tenure stream academic career, I believe you need to think about how most hiring committees would view someone starting out at the age you have completed your Phd.

Mine was planned for age 50 but delayed due to revisions for three years and I discovered that 53 was too late. Onetime I actually found out the committee decided they would prefer the younger candidate (no I did not sue) and the one position I was hired for, it was pointed out that I starting rather late for academe. It was a very good Department, but I chose to leave after the first year rather than relocate my kids just before they finished high school.

You should be done well before that, I wish someone had pointed this out to me.

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Age 50 and age 27 are very different for starting a new career. Half the people where I go to grad school took a few years off and are in their 30s by the time they graduate. –  Graphth Nov 15 '12 at 17:11

I started working on my bachelor's in Computer Science at 24 at UCSD, and I came in with almost no knowledge of programming, and the highest level of math I had accomplished at that point was Pre-Calculus. So in addition to taking the entire CS curriculum, I had to take all 5 engineering calculus courses as well. I just tackled everything I needed to do, and with the help of good people around me, I finished with pretty good grades.

So, no. You're not too old.

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This popped as a newsletter question, and it seems appropriate to share personal experience.

It is true that most significant math research seems to be done before 30. I remember Hirzebruch from my Alma Mater who had published 50% of his research by 25 - or at least that was what people said. At the same school, my Algebra professor, Jens Franke, was a year or two younger than me (see below), having already spent time at Princeton and Moscow before this. One of my teachers in grad school (economics) had 4 seminal papers in 4 different areas by 25 (and more or less retired after).

In my first semester of college, I was 25. In my first semester of math, I was 26. Being surrounded by people 7 to 8 years younger than me wasn't very encouraging - in particular the regrettable attitude of many math students to (falsely) claim that they solved an exercise you agonized over with no problem at all, and to spend 1/2 their time shortening and obscuring their arguments, probably because of a misunderstood understanding of what is elegant and cool. My first homework buddy told me what I never forgot, and have found quite true:

"If someone has a better solution than you, just assume he copied it from a better book than you had access to."

Given that my high school math education wasn't very good, for faults of my own and my teacher's, and I averaged I'd say a B+ (below my general average), with a glorious F in my final oral exam (my only F ever), I felt both inferior, and did objectively struggle at the beginning. Add to this that in informal rankings (there are no official ones in Germany) my school, Bonn, was considered one of the two best for math, and so attracted many former math Olympics participants (the most accomplished one had participated in grades 10-13, winning 2 silver and 2 gold medals). Before meeting them, I didn't even know there were math Olympics...

I worked very hard, and I think for most this is simply necessary in math. As a result, within about a year, in a crowd of about 200, I started being considered pretty good - certainly behind the math Olympics guys, but probably around there. I went on being admitted to a top 3 grad school in the U.S. - not for math as I still lacked the confidence, but with a goal to study the most theoretical part of economics (decision, game theory and such), which boils down to applying algebraic topology and other fairly hardcore math disciplines. My course work consisted mostly of taking the theoretical stats grad level classes of the first two years (in stats, my school was - then - traditionally considered as competing with Berkeley to be the top-ranked stats school), scoring consistently A's with the rare A-, and I did very well (ranked 'outstanding' and 'excellent' by my school the first two years). Being a rather troubled human being, I proceeded to boycott my own future as an academic (which is a different story not belonging here), but still my main thesis paper was 3rd rounded by Mathematics of Operations Research (with feedback "Clearly worthy of publication", asking to tie up details), before I stopped pursuing it, having gone to industry, not academia.

I'm now in my 40s, and am just done auditing algebraic topology at Columbia. 22 is certainly in itself not too old. While you say that you don't want a "do what you like!" pat on the back, I can't help giving you one. Just expect it to be tough originally, to have to work very hard (making it your life for a while), and to pretty much keep feeling that you should see things faster than you do. That's the doing math experience: every new problem is a true challenge.

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A fairly interesting story! A question: Do economists spend a lot of time "obscuring their arguments" too? –  Metta World Peace Feb 22 '13 at 14:50
    
@MettaWorldPeace: As wanna-be mathematicians, for sure! :) –  gnometorule Feb 22 '13 at 18:54

While I think you are never too old to start anything, I really think Math sports needs to be learned and cultivated from a young age to be truly effective.

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When I was 21 I entered college not even knowing how to solve a quadratic equation, as I was a high-school dropout (I eventually got a GED). Ten years later I had a Ph.D in math.

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By now, you, Thomas Nesbitt, see this answer, you should be somewhere over 22, based on what you said and the time that has passed by. Several of the answers mention the Fields medal as an example of an award which has an age limit. So does the Nevanlinna prize. Assuming you had started to study mathematics now, add another 12 years, and because the Fields medal or the Nevanlinna prize are awarded every fourth year, the worst case scenario is that you would have only two years to come up with something extremely important in pure mathematics (http://en.wikipedia.org/wiki/Fields_medal) or information sciences (computer sciences, scientific computing and so on http://en.wikipedia.org/wiki/Nevanlinna_Prize). Certainly among pure mathematicians the Fields medal, say, is the most prestigious medal, to the extent of my knowledge. Having said that, if you care about landmarks for your career, there is no shortage of awards, both in the sense of having no age limit; the Abel prize, for example (http://www.abelprize.no/c53676/artikkel/vis.html?tid=53705&strukt_tid=53676 ) or in the sense of applied mathematics, like in the Gauss prize (http://www.mathunion.org/general/prizes/gauss/details/). Some awards are won, like in a competition, and some others are simply given. They reach you. I won't detail any of the awards I got, because that is totally beside the point, which is that one shouldn't be worried about awards, except that when you get and if you get one, it is only a landmark of work you have been doing anyway.

As for the that period of yours where you found a "lack of intuition", I most strongly recommend the book and the short course by Keith Devlin: "Introduction to Mathematical Thinking". The course can be found online, and you can sign in through Coursera (https://www.coursera.org/). I took it last year, and it really changed me in the way of understanding and doing mathematical proofs.

Now, my personal experience, which is not the most outstanding but hopefully useful for the issue that we got here at hand: Probably all my life have been inclined to do mathematics, but without a formal guide. As a child I found this book of Martin Gardner "Mathematical Magic Show" (1977; in the Spanish edition it has an apple and four matches making a kind of shovel, with the idea of taking out the apple from the shovel, moving only two matches, and reconstructing the shovel). I mention this, because at the time I couldn't do long division. It took me quite some personal effort, but I wanted to learn how to do it. The algebraic concept of "unknown quantity" was at the beginning so puzzling for me, that I took the letters at face value, so I used to count their position in the alphabet, and use it for computing the final value.

At high school I was lured by genetics, so I got into a high school with medical preparation, including that subject. The school included basic calculus, and I struggled so much with a problem of limits (I think something that direct substitution would give infinity over infinity), that it triggered the first Lucid Dreaming I ever had, solving the homework, at the end. However I later learn that I could enter the Mathematical Olympiads, and prepared myself for it, with the couching of the teacher who explained all about algebra, including how to understand the "unknown quantities". I didn't get too far in the competition, but the preparation I got was so good, that when I entered the engineering school I got over all of calculus, vector calculus, statistics and so on very, very easily.

At University I made my choice of an Engineering with specialized in Biotechnology applied to Agriculture (where I did good grades in all the mathematical subjects). Still lured by genetics, molecular engineering and so on. I wanted to study, mathematically, the relationships (if there were any) of chaos, fractals, entropy, quantum mechanics and genetics, with the (extremely) ambitious idea of pondering how could a genetic problem be reversed, with the person alive and already grown up. With this idea in mind, I went to Cambridge University in the UK to ask who was working on the subject. Professor Burgess told me that no one was working directly in the mixture that I asked for, and also told me that I could be admitted to do Part III Mathematical Tripos, even coming from an Engineering degree, if I got good grades.

Once in Cambridge, I interviewed several professors of what I was concerned, and I managed to talk briefly with professor Geoffrey Grimmett. Upon hearing the list of subjects I wanted to combine, he told me "I think what you want is to find out Why do we die, isn't it?. I am not good at the big things, I work small ones [...]". But I was already in Cambridge! what to do? at the examinations of Part III, I did horrible, but still I wanted to do something. So, I found David J. Wales, and I did a C.P.G.S. dissertation "Chaos in Inert Gas Clusters" (about the Kolmogorov Sinai Entropy). Cambridge finished, I lectured Linear Algebra for one semester, and then I managed to obtain a scholarship to do a Ph.d. in Plasma Physics (controlled fusion and/or neutron generation), at the Czech Technical University in Prague. I finished it with the dissertation "Self-Organized Structures in Z-Pinch Plasmas". A teacher over there (Jiri Gregor) told me "you should do research, and not just teaching; otherwise you will become a potato seller".
Back in my country I found no job for me on Plasma Physics, but I was offered to work for the Mexican Petroleum Institute. I found myself since then doing mathematical modeling of flow and transport in porous media, and exploring the application or applicability of concepts like fractals, quasi-hyperbolic differential equations (transport equations), inverse theory, percolation, Poisson processes, symbolic mathematics, and many others. I am not claiming that I am any closer to an expert in any of those subjects; could I be replaced by a younger, more able person? sure! but the colleagues that I have would feel a kind of "gap", if wasn't here, and those more brilliant, younger people, are working somewhere else, and in other subjects. There is so much work to do in applied or pure mathematics, that, yes, there are The Problems for outstanding mathematicians, and there are also the day-to-day nagging-odd problems that need to be solved by someone, maybe a poor sap like me. At the moment I can tell you that I don't see myself attempting a proof Information Theory, Algebraic Geometry or one of The unsolved problems in Analytic Number Theory; and then again, who knows? I keep track on a few of them, and maybe one day I might just gather some bits and pieces, feel very, very, very inspired, and try to get my luck in one of them. Maybe a little, partial result could come out of it. Maybe never, but the point is that I have already a lot of work to do, a lot that will keep coming, some of my little, but own ideas, and if that is doing a "career" in the sense that you referred to, then there is in fact no age limit, and possibly no single path towards becoming a full-fledged mathematician; even today I wouldn't dare to call myself "mathematician", without feeling that I could be challenged by, or frown upon, by someone who did study mathematics since he or she was five years old. However some time ago, I got a colleague to whom I had a conversation about my intention to do another Ph.D., this time in mathematics, and he advised me not to waste my time, and just another paper to hang in my wall; I told him that there are people -among pure mathematicians- who do not like that I hold not a Ph.D. in mathematics neither my bizarre trajectory in studies and he answered me "I don't like your shoes". His point being that I should concentrate on the work that I did have, and not being worried about opinions of people outside our labor.

Summarizing, I tried to show you what you asked for: not just a pat on your back, telling you "everything is possible". Well, at least for me the Fields medal I am sure is impossible, because I am well over forty. But with some effort there are many other things in the course of a career that are possible, that are not unattainable, provided that you focus, relax, and keep working at it, but above all, the most important thing is that you find a subject that passions you really, so much that you will forget, at least for some moments, the difficulties of getting over an unknown subject, or a puzzling concept.

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