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Many musicians believe that if one wants to become a professional pianist he/she should begin playing piano when he/she is a child.

Question 1: Is this true about mathematics? In other words, is it possible for a person without any academic math education at university level to become a professional mathematician after his mid-life (about 40 years old)? Is there any limitation in age for entering the realm of research level maths? What about legal academic limitations?

Question 2: If the answer of question 1 is negative and there is no limitation, what are good suggestions for such a person to make his/her entrance onto stage of professional mathematics easier and more successful? What are good examples of such persons?

Question 3: Is the first paragraph about professional pianists really true?! I heard this from one of my friends who is a conductor.

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  • $\begingroup$ Despite laws that forbid it, discrimination based on age remains a reality. $\endgroup$
    – André Nicolas
    Aug 21, 2014 at 4:22
  • $\begingroup$ Many of us do begin mathematics as children. Could you please clarify whether you are asking about people who have not had any education in mathematics at all or people who have not had any university level mathematics? $\endgroup$
    – J W
    Aug 21, 2014 at 4:53
  • $\begingroup$ @JW I mean university level education. I edited the post. $\endgroup$
    – user230
    Aug 21, 2014 at 11:33
  • $\begingroup$ There are quite a few posts over at academia.SE, here are a few: academia.stackexchange.com/questions/17430/… academia.stackexchange.com/questions/23613/… $\endgroup$
    – Chris C
    Aug 21, 2014 at 13:50
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    $\begingroup$ See also this old post on MO: mathoverflow.net/questions/7120/… $\endgroup$
    – J W
    Aug 21, 2014 at 16:34

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Question 1: No, there is no limitation. In some countries, such limitations would be illegal due to anti-discrimination laws, as maths are never safety-relevant. If it's possible, depends on the person itself and cannot be answered in general on a pure age basis.

Question 2: There is but one suggestion: Study math at the university of your choice and supplement it with private studies at home.

Question 3: No idea. That question doesn't belong here.

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Q2: Regardless of the age, the way to enter professional mathematics is to get a Ph.D. in mathematics.

Q1: Mathematics has been done as a second career. I know one case where the first career was in the Merchant Marine, and another case where the first career was in the U.S. Army.

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I am familiar with the situation in Russia and in the US, and your country may have different practices.

Your biggest impediment is that agility of one's brain, and one's ability to learn, really depends on age: it is easier to learn stuff when you are young; that's just biology of the brain. You know this, and other people know this. So while there may be no legal limitations on age, and most civilized countries would explicitly forbid age discrimination, the admission committees in most grad schools would likely screen you out and prefer younger candidates. Remember: the fame of the program depends on both what the faculty publish and on where the graduates are placed, and if they perceive that you will not be placed well five or seven years down the road when you are fifty (and everybody's thinking on the job market would be, "I have plenty of 50-year old professors with three dozen publications and 20 years of teaching experience in my department, why should we bother with a 50-year old Ph.D. graduate who will retire in 10 years?"), they may not take you upfront.

You have not defined what you mean by "professional mathematician"; there's a spectrum from teaching math in high school to working full time on math research at Courant Institute. You would probably be able to get up to Master's if you can pay for your education out of your pocket. But even getting admitted into a Ph.D. would probably require that the department knows you in and out, and knows your dedication to learning math -- which means you will be limited to the school where you can get your Bachelor's and/or Master's.

As a side note, in Russia, there is usually little way for you to get into a Ph.D. (or rather the 3-year research-only program called "candidate of science" in Prussian manner) unless it is right after your first college degree, especially in mathematics where agility of the brain is the primary issue. 90% of the post-graduate students would go to the candidate program back-to-back after their basic degrees (which were 5-6 year programs called "specialist", which are easily mapped to the Western Master's degree), and the remaining 10% were one or two years out of college, and would have done some work in a related area. For that candidate program, you'd have to take formal entrance exams in the subject (not unlike subject GRE), in English, and some sort of philosophy of science (unlike the American process based on application, standardized scores and recommendation letters). You will likely fail the subject exam unless you just had an intense math training in the immediately preceding years, and retained most of the math you've just learned.

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  • $\begingroup$ Note that there are many examples of mathematicians older than $50$ who are still doing brilliant research -- for example, Osher who just won the Gauss prize. So beyond $50$ one's brain can still be quite agile. $\endgroup$
    – littleO
    Aug 26, 2014 at 7:55
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    $\begingroup$ Yes, but only if you've been exercising it. There are plenty of people who can run a marathon under 3 hours at the age of 50. However if all you have done in the first 50 years of your life is walking to your mailbox, you will simply die if you really try to run a marathon. $\endgroup$
    – StasK
    Aug 26, 2014 at 18:57

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