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It is well-known that many great mathematicians were prodigies.

Were there any great mathematicians who started off later in life?

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    $\begingroup$ Am I the only one bothered by "well-known" and "great"? Unqualified by context, these are unreliable terms at best. $\endgroup$
    – Yemon Choi
    Commented Oct 31, 2009 at 21:09
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    $\begingroup$ So is "prodigy." But I think the intent of the question is clear. $\endgroup$
    – Qiaochu Yuan
    Commented Oct 31, 2009 at 21:12
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    $\begingroup$ My only response is a strong desire to go in and add <sup>[citation needed]</sup> to the first sentence. $\endgroup$
    – Theo Johnson-Freyd
    Commented Apr 25, 2010 at 2:23
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    $\begingroup$ I'm 26, currently in my last semester undergrad. studying Physics and Mathematics. You have no idea how encouraging this thread is. $\endgroup$
    – AmagicalFishy
    Commented Feb 26, 2015 at 20:46
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    $\begingroup$ This question helped me despite having been deemed "unlikely to help any future visitors." $\endgroup$
    – j0equ1nn
    Commented Oct 23, 2015 at 3:51

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Joan Birman went back to grad school in math in her forties, and is now one of the top researchers in knot theory.

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Karl Theodor Wilhelm Weierstrass (Weierstraß) Follow this link

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    $\begingroup$ BINGO,THE GREAT COUNTEREXAMPLE TO HARDY'S RIDICULOUS QUOTE. $\endgroup$
    – The Mathemagician
    Commented Mar 26, 2010 at 2:11
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    $\begingroup$ Andrew, all-caps is considered impolite on the internet; it's equivalent to yelling. $\endgroup$
    – Qiaochu Yuan
    Commented Apr 24, 2010 at 18:12
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    $\begingroup$ I disagree. All-caps is equivalent to speaking more loudly. Depending on the context, just like in personal conversation, this can range from yelling to genuine excitement (to a variety of other things). In this case, it's clearly an all-caps of excitement, which in personal conversation would not be construed as impolite. $\endgroup$
    – Cam McLeman
    Commented Apr 24, 2010 at 19:39
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    $\begingroup$ I consider them to be more gauche than impolite. I see it more of issue of something losing its original impact due to overuse during certain periods of internet development. It's the discussion forum analogue of the dancing baby animated gif. $\endgroup$
    – Ben Webster ♦
    Commented Apr 24, 2010 at 23:01
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    $\begingroup$ SIGH.I can't win in here......... $\endgroup$
    – The Mathemagician
    Commented May 11, 2010 at 5:37
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She didn't get started late, but I do know that Alice Roth wrote an important thesis in 1938, took 35 years off from research, and then did very beautiful and influential work in complex approximation starting at age 66.

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    $\begingroup$ WOW. A great example and a tragic reminder of the pathetic status of women in Western Culture for most of human history. $\endgroup$
    – The Mathemagician
    Commented Mar 26, 2010 at 2:19
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    $\begingroup$ @TheMathemagician Not only "western" culture. If you want to really be depressed, read about historical treatment of women in the East. $\endgroup$
    – JMJ
    Commented Dec 6, 2017 at 18:54
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Eugène Ehrhart (of Ehrhart polynomial fame) was born in 1906, taught in various French lycées (high schools), began his work on geometry in the 1950's, did his best work in the 1960's, and received a Ph.D. in 1966. See https://icps.u-strasbg.fr/%7Eclauss/Ehrhart.html

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    $\begingroup$ Awesome,Richard-thanks! And thanks for finding the time in your really busy schedule to post here. $\endgroup$
    – The Mathemagician
    Commented Mar 26, 2010 at 4:28
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According to this Notices article, Raoul Bott was undistinguished in high school, but displayed impressive talent once he reached graduate school (though his thesis was actually in electrical engineering, rather than mathematics).

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    $\begingroup$ I used to play hockey with (sometimes against) one of his former grad students -- who was a student of Bott's back when Bott was an electrical engineer. One game our teams got into a bench-clearing brawl. We skated up to each other and started talking about Morse functions on manifolds. The person I'm referring to is Dave Delchamps, at Cornell. $\endgroup$
    – Ryan Budney
    Commented Nov 12, 2009 at 1:53
  • $\begingroup$ An awesome story. Only in Canada. $\endgroup$
    – Todd Trimble
    Commented May 10 at 3:28
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Well, there's Witten. He got his degree in history, then attempted to be a political journalist and get a grad degree in econ before looking into physics and math, but got the Fields Medal.

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    $\begingroup$ I am not sure that simply the fact that his first degrees were not in Physics or Mathematics is enough to deduce that he was a late learner. His father, Louis Witten, is a well-known relativist. Perhaps he was "home-schooled" :) $\endgroup$
    – José Figueroa-O'Farrill
    Commented Nov 1, 2009 at 8:40
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    $\begingroup$ Exactly.This thread,to me,is supposed to be about people who were at an age the rest of the world has given up on them and go on to have strong careers. $\endgroup$
    – The Mathemagician
    Commented Mar 26, 2010 at 2:18
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    $\begingroup$ economics is too hard. $\endgroup$
    – PatrickT
    Commented Dec 9, 2014 at 17:03
  • $\begingroup$ I think the claim that Witten ''got started late in life'' is questionable. He obtained his PhD in two years at around the age of 24 (earlier than the average age at which one would obtain the PhD). $\endgroup$
    – Hollis Williams
    Commented Aug 17, 2021 at 18:44
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I don't think that Stephen Smale really distinguished himself until after graduate school.

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    $\begingroup$ Somewhere there's a wonderful letter of recommendation written for Smale by one of his professors at Princeton. The letter basically says (in the first sentence) that Smale didn't seem very good until his final year, when he solved several open problems. The writer then suggests his improvement might be due to his having gotten married that year. The remainder of the letter is a digression about Smale's wife. (Does anyone know where this letter appears? I can't think where I might have seen it. My best guess was Stalling's webpage, but it's not there.) $\endgroup$
    – Dan Ramras
    Commented Apr 24, 2010 at 20:33
  • $\begingroup$ Smale basically corroborates Ben's answer in his interview for More Mathematical People. $\endgroup$
    – Todd Trimble
    Commented May 6, 2011 at 10:55
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    $\begingroup$ @dan: Smale's PhD is from Michigan -- perhaps you were thinking of the letter from Ray Wilder that appears at the bottom of the page here, and mostly on the top of the next page: books.google.co.uk/… The book is "Stephen Smale: the mathematician who broke the dimension barrier" vy Steve Batterson $\endgroup$
    – Paul Johnson
    Commented May 6, 2011 at 11:45
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    $\begingroup$ The letter from Wilder is also available on the 4th page of this pdf file from the Notices ams.org/notices/200311/comm-batterson.pdf $\endgroup$
    – Sam Lisi
    Commented Apr 9, 2013 at 10:58
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Here, Rob Kirby describes some of his experiences as an undergraduate at Chicago, and how he "snuck into graduate school".

As an undergraduate, I'd been far more interested in chess, poker, and almost any sport, than in the game of mathematics. I had little chance of getting into a good graduate school. However, I failed German and didn't get a B.S. in four years, so in my fifth year I took most of the graduate courses on which the Masters Exam (really a Ph.D. prelim) was based. With a B.S. I asked to be admitted to graduate school so as to take the Exam. They cautiously said yes if I got grades closer to B than C in the fall quarter. I got a B and a C (measure theory from Halmos and algebraic topology from Dyer) and a Pass, and no one told me to leave.

The Masters Exam could have four outcomes: you could pass with financial aid, pass without aid but with encouragement, pass with advice to pursue studies elsewhere, and fail. I got the third pass, but really liked Chicago and turned up the next year (1960) anyway.

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    $\begingroup$ Great story,Kevin. Don't know if Kirby qualifies in terms of age,but sure shows grades don't mean squat when determining the potential of someone to be a mathematician! $\endgroup$
    – The Mathemagician
    Commented Mar 26, 2010 at 2:15
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    $\begingroup$ So at the time the University of Chicago had a test you could take that guaranteed you admission to grad school? $\endgroup$
    – Daniel McLaury
    Commented Jul 27, 2013 at 17:42
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Preda Mihailescu is a good example https://en.wikipedia.org/wiki/Preda_Mihăilescu.

He received his PhD with 42, and proved the Catalan conjecture 5 years later.

The Catalan conjecture was open for 160 years.

He proposed in 2009 a proof of the Leopoldt conjecture, but I am not sure about the status of this.

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Sophus Lie didn't become interested in mathematics until after university, and before then didn't seem to have shown significant aptitude for it.

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Somebody who probably fits the bill here is Albrecht Fröhlich who after fleeing Nazi Germany as a teenager, eventually attended university only when he was about 30. He later went on to jointly organize the Brighton conference which put class field theory on the mathematical map, essentially create a new branch of number theory and produce his most important work well into his fifties.

There's a biographical memoir here.

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Misha Cotlar, born in 1912 in Ukraine, emigrated to Uruguay in 1928. He never had a formal education. He got his PhD from Chicago University in 1953. He died in 2007. He is well known for his work in harmonic and functional analysis.

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  • $\begingroup$ Now HERE'S a good example.Thanks,Mike. $\endgroup$
    – The Mathemagician
    Commented Mar 26, 2010 at 2:17
  • $\begingroup$ It's true that he had no formal education. But he was interacting with mathematicians already in his early 20s. Looking at his published papers, his first publication was at 24 years old. So, he had a very unusual career, but he was not a "late learner". $\endgroup$
    – Martin Argerami
    Commented Dec 20, 2021 at 1:02
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    $\begingroup$ I agree with @MartinArgerami: not so much a late learner as someone who had to overcome serious obstacles and had a very indirect path to mainstream academia. (I think he worked as a tutor to university students starting when he was quite young?) $\endgroup$
    – H A Helfgott
    Commented Feb 19 at 0:21
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Persi Diaconis of Stanford had two careers -- the first as a violin prodigy studying at Juilliard, and then as a world famous magician who performed for the crowned heads of Europe. In his early twenties he decided that he wanted to learn enough math to understand Feller's two volume treatise, so enrolled at CCNY. His beginning was rocky (by his own admission), but he finished there well enough to be admitted to the Ph.D. program in Statistics at Harvard, and, the rest, shall we say, is history. He also worked as an advertising copywriter while he was attending CCNY.

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    $\begingroup$ So, two prodigy lives before math. Not sure this counts :) $\endgroup$
    – kcrisman
    Commented May 16, 2011 at 20:41
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    $\begingroup$ As a magician, he believed that knowing probability would be helpful (for example, in card tricks). But probability was so interesting that he gave up magic to study it full time. $\endgroup$
    – Gerald Edgar
    Commented Nov 22, 2023 at 14:44
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    $\begingroup$ Was it really worth bumping this ancient closed question to correct a spelling error? $\endgroup$
    – Andy Putman
    Commented Feb 19 at 0:19
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Dwork started out as an electrical engineer and was 31 when he received his PhD. The memorial article by Tate and Katz gives the interesting details.

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I've always heard that Hilbert was unexceptional (not bad, but not genius) as a student. He gained steam throughout his career, rather than bursting into prominence.

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Lefschetz didn't move to math until he lost both of his hands in an industrial accident at the age of 23.

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    $\begingroup$ Again,too young to really be on this list. $\endgroup$
    – The Mathemagician
    Commented Mar 26, 2010 at 2:13
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I've just been reading Peter Roquette's entertaining account of the remarkable career of Otto Grün. Grün was an amateur, never attended university but at the age of 44 sent some results around FLT to Helmut Hasse. There were considerable errors, but Hasse spotted enough originality to keep up a correspondence and helped guide Grün into becoming a highly respected group theorist with work fundamental enough to find it's way straight into group theory text books.

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How about Raymond Smullyan? According to his autobiography[1], he has published his first mathematical article at the age of 35, to which Marvin Minsky has reacted by saying Ray has decided to become a child prodigy at the age of 35. Does this count as starting off late in life?

[1] Raymond Smullyan, Emlékek, történetek, paradoxonok. TyopTeX, 2004, original title "Some Interesting Memories. A Paradoxical Life".

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One could perhaps also cite George Green, miller and mainly autodidact mathematician as an unconventional and relatively late bloomer. He entered University only at 40, seven years and seven months before his death. See for instance Green's Biography at MathTutor.

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    $\begingroup$ Green was an amazing man --- he left school at age 9 to work in his father's bakery, and no one really knows how he got his mathematical education. But his entering university at age 40 is a little misleading, as he had been publishing major papers since he was around 35. $\endgroup$
    – Nik Weaver
    Commented Jul 7, 2014 at 5:46
  • $\begingroup$ @NikWeaver Right - more of a Misha Cotlar type (or vice versa). $\endgroup$
    – H A Helfgott
    Commented Feb 19 at 0:23
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There's Thomas Kirkman of Kirkman's schoolgirl problem, who didn't start studying mathematics until he was into his forties. Aside from the problem that bears his name he went on to work publish papers in extremal set theory, finite geometries and the like, he was also one of the first to write about group theory in English.

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  • $\begingroup$ VERY good and little-known example,Physics. $\endgroup$
    – The Mathemagician
    Commented May 11, 2010 at 5:36
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Alberto Calderón (of Calderón-Zygmund Theory/operators - one of the great analysts of the 20th century by any account). He studied Electrical Engineering in Buenos Aires, graduating at age 27. Zygmund met him during a visit to Buenos Aires and was very impressed with his mathematical originality, so he invited him to pursue a PhD in Chicago, which Calderón completed when he was 30.

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You could read the autobiography of Paul Halmos (RIP- he died just a few years ago) "I want to be a mathematician". He started mathematics much later in life, first he did chemical engineering then philosophy then mathematics. Halmos wasn't quite the genius in mathematics (as he has described it) but later in life he got into it and succeeded. John von Neumann (who was Halmos' countryman) also started from chemical engineering before going to mathematics.

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    $\begingroup$ He was a prodigy in the other fields, though, so he had a Ph.D. at 22. Not that much of a late learner. $\endgroup$
    – Harrison Brown
    Commented Nov 1, 2009 at 9:11
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    $\begingroup$ John von neumann is the antithesis of a late learner. He had a private tutor in math when he was in his early teens $\endgroup$
    – ngc1300
    Commented Apr 20, 2021 at 20:11
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    $\begingroup$ John von Neumann was absolutely not a late learner. He only took a chemical engineering course formally to keep his father quiet who kept complaining that pure math was useless, and said it cost him no effort at all to get a diploma in chemical engineering on the side whilst working on pure mathematics. $\endgroup$
    – Hollis Williams
    Commented Aug 17, 2021 at 18:46
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According to an interview of Arnold in Notices (p437), both Whitney and Kolmogorov switched subject at university after a couple years and chose mathematics (Whitney was studying violin, Kolmogorov was into history). So they discovered math after high school, but the interview makes it clear both were very smart (not late bloomers).

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    $\begingroup$ Kolmogorov, from my memories of his autobiography, first attended the history department, even though since the high school he was actively learning math from a popular science encyclopedia. $\endgroup$
    – Ilya Nikokoshev
    Commented Nov 1, 2009 at 9:54
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    $\begingroup$ Likewise, Andrei Okounkov transferred to the Mathematics Department (from economics?). $\endgroup$
    – Victor Protsak
    Commented Jun 8, 2010 at 0:53
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    $\begingroup$ Kolmogorov is not for sure, since for instance at 19 he discovered the first example of Fourier series that diverges almost everywhere. $\endgroup$
    – timur
    Commented Oct 24, 2010 at 3:49
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R. H. Bing taught high school for several years before entering graduate school.

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There's Serge Lang. Apparently, he finished his undergraduate degree in physics at CalTech, before a short tour of duty in Europe. When he returned for graduate studies, he was initially enrolled in Princeton's philosophy department. According to the biography, he switched to mathematics after his first year, and worked with Emil Artin.

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    $\begingroup$ Uh,wasn't he still in his early to mid 20's at that point,Steven?I don't thing Lang really qualifies unless you think anyone that doesn't get thier doctorate by 22 is done. $\endgroup$
    – The Mathemagician
    Commented Mar 26, 2010 at 2:13
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    $\begingroup$ Another great story about Lang is that he nearly flunked graduate school. $\endgroup$
    – Victor Protsak
    Commented Jun 8, 2010 at 0:54
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Here's a great example:Makus Fisz. "Who?!?" An expert in probability and statistics who was born in 1910 and grew up in war-torn Poland-and as a result,his career kept getting interrupted. He finally got his doctorate at the age of 40 and published a number of well known papers as well as an acclaimed text on the subject that was translated into a half a dozen languages and became very popular in Europe.He was finally appointed full professor of mathematics at New York University after many visiting positions. Tragically,he died of a heart attack at the age of 54. A great story and career with a very sad ending.

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In May 2006, the AMS Notices printed a remembrance article for Serge Lang. Dorian Goldfield was one of the contributors, and as an undergraduate, he described himself as follows:

Of the many people who had serious interactions with Serge, I am one of those who came away with fierce admiration and loyalty. In the mid-1960s, I was an undergraduate in the Columbia engineering school on academic probation with a C–average. In my senior year I had an idea for a theorem which combined ergodic theory and number theory in a new way, and I approached Serge and showed him what I was doing. Although I was only a C–level student in his undergraduate analysis class he took an immediate interest in my work and asked Lorch if he thought there was anything in it. When Lorch came back with a positive response, Lang immediately invited me to join the graduate program at Columbia the next year, September 1967.

Then again, Goldfield was not a "late learner" as he was 20 when he finished college and 22 when he earned his PhD. But...

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    $\begingroup$ Having a lousy academic record at a top-flight school and having someone give you a chance anyway is NOT what this thread is about,Bman.And Goldfield was a genius who was either undisciplined or just did lousy on tests. $\endgroup$
    – The Mathemagician
    Commented Jun 7, 2010 at 21:57
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    $\begingroup$ @AndrewL: What is this thread about then? The OP stated that many great mathematicians were prodigies & asked if there were ones who started off later in life. I mentioned Goldfield as an example who had no "genius" attached to him until much later. You term Goldfield a "genius" now, but think about it before he became a bigshot in mathematics, he didn't distinguish himself until later. $\endgroup$
    – Bman
    Commented Jun 7, 2010 at 22:51
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    $\begingroup$ (contd) I also took into account such objections at the end of the post with my "Then again..." comment. Anyways, what is your point? When did you--in your abrasive and rude manner--determine the the permissible contributions to a thread? In the future should I numbly submit my posts to the shrine of Andrew L before I dare post them on this site? $\endgroup$
    – Bman
    Commented Jun 7, 2010 at 22:52
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    $\begingroup$ It's Goldfeld, not Goldfield--Dorian Morris Goldfeld. He was very good at weiqi, addicted, played at 1 dan level. In 1974, at the Princeton Institute, most everything was still ahead of him. He seemed to decide to go to and work with Bombieri then but somehow his wiki bio does not mention Bombieri at all. $\endgroup$
    – Włodzimierz Holsztyński
    Commented May 3, 2013 at 19:28
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Ludolph van Ceulen (1540-1610) was 60 when he became professor of mathematics. Prior to that, he was a fencing instructor. He is known for calculating pi to 35 digits (not easy without calculus!).

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A similar question was discussed some time ago over at http://quomodocumque.wordpress.com/2009/01/21/claimed-proof-of-the-abc-conjecture/#comment-3179. There it was pointed out that William H. Young (of, say, Hausdorff-Young fame) didn't publish much before 40. After 40 he had a successful career, with a prolific publication record. This said, it is reported that he was a very talented student.

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Dennis Sullivan who won the 2010 Wolf Prize comments on this in his own words in the magazine published by the New York Academy of Sciences:

http://www.nyas.org/Publications/Detail.aspx?cid=14047af0-9f26-481c-9b50-9434130c89db

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    $\begingroup$ WTF are you talking about,Joseph-Dr.Sullivan was 23 YEARS OLD when he began serious mathematical work! I have many friends who are students of Dr.Sullivan and I plan to study with him next semester.He's certainly one of the greats,but HARDLY call him a late bloomer.If he's a late bloomer in his mind,I'm sure it's because he studied at Princeton with teenagers publishing in major journals before they were old enough to drive.Hardly a common perspective. $\endgroup$
    – The Mathemagician
    Commented Apr 29, 2010 at 20:26
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    $\begingroup$ Here is what Dennis said: "I was a late bloomer academically in the sense that I didn't have any pressure to study when I was growing up. In college I got back into academics again and made a fresh start. I was able to attend Rice University in Houston, which at the time was like a scaled-down Caltech. I rediscovered my academic self there after being a quasi juvenile delinquent, running around working on hotrods!" He has some additional remarks not pasted in here. $\endgroup$
    – Joseph Malkevitch
    Commented Apr 29, 2010 at 23:38
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    $\begingroup$ I read his words and I've heard him say them in person,Joseph.He was 23 years old at the time. I don't understand how that makes him a late bloomer.And I'll ask him that when I see him. $\endgroup$
    – The Mathemagician
    Commented Apr 30, 2010 at 4:26
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    $\begingroup$ The link in the answer no longer works, but based on the quote in the comment, it might have been this article: How Math is Like a Ladder to the Moon (Wayback Machine). $\endgroup$
    – Martin Sleziak
    Commented Feb 20 at 10:04
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