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I've been motivated by this question about starting to study mathematics at an unusually advanced age. It would be nice to know examples of people who successfully switched from a very different field into mathematics.

Which well-known mathematicians, past or present, started out as law/art/humanities/business students, but later turned to mathematics? This excludes mathematicians who switched from the sciences or engineering to mathematics such as Raoul Bott.

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    $\begingroup$ If the question stays open, it should also be CW. Some kind of motivation for the question other than commendable curiosity would also be good $\endgroup$
    – Yemon Choi
    Commented Sep 28, 2011 at 0:17
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    $\begingroup$ Dear Francois, "At the time of Fermat and Cayley" encompasses more than two centuries, as well as two quite different countries. It's probably possible to make a more refined analysis. (Not that this would necessarily bear on the present question.) Best wishes, Matt $\endgroup$
    – Emerton
    Commented Sep 28, 2011 at 2:07
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    $\begingroup$ By the way, this question looks salvageable to me. Editing it to fix the problems (or simply voting it down) is better than permanently closing it. $\endgroup$
    – Douglas Zare
    Commented Sep 28, 2011 at 2:19
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    $\begingroup$ Douglas: Closing a question is never permanent! One of the main reasons for closing a question quickly is to prevent irrelevant answers from appearing before the question is edited to its more-or-less final form. $\endgroup$
    – François G. Dorais
    Commented Sep 28, 2011 at 3:10
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    $\begingroup$ The question is better but I still think is in danger of being "ahistorical", in that it presupposes a certain context of education and training which probably didn't apply to Famous People Who Will Be/Have Been Mentioned In The Answers (e.g. Fermat). Francois already made this point well $\endgroup$
    – Yemon Choi
    Commented Sep 28, 2011 at 19:41

21 Answers 21

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Persi Diaconis left home at 14 to work with Dai Vernon as a magician. Trying to protect himself from being cheated in dishonest casinos, he was led to Feller's textbook on probability theory, which he couldn't understand. He started studying calculus at the City College of New York at the age of 24. Some more details may be found in this article.

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I remember reading an interview with Vladimir Arnold where he tells the following anecdote about Hassler Whitney. I don't have the interview in front of me, so some of the details may be not quite correct, but if memory serves, the story goes on like this.

Whitney used to study music in America and at some point decided to spend a year in Germany. He arrived in G\"ottingen where it turned out that he had to take a course outside his main subject of study, which was music. He asked which one of the courses was the most difficult one. It turned out the most difficult subject was quantum mechanics. Whitney enrolled on that course. After the first lecture he came to see the professor and said

-- I was one of the best students in Yale in my year, Herr Professor; how come I didn't understand a single word of the lecture?"

--Well, you see, there are some prerequisites for this course. You have to know calculus and linear algebra and ....

-- Are there any books where I can read all this up?

It took Whitney a couple of weeks to work through the books the professor told him to read. In a month Whitney was able to follow the course, and he decided to switch to mathematics at the end of the semester.

Arnold tells this story to illustrate the dangers of early specialization.

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    $\begingroup$ I heard about Whitney's starting out in music from late Stanislaw Lojasiewicz (who also once contemplated a career in music), but could not find any confirmation in writing. $\endgroup$
    – Margaret Friedland
    Commented Sep 28, 2011 at 21:18
  • $\begingroup$ Margaret -- again if memory serves, Arnold heard the story from Whitney himself; he regretted he hadn't asked who the quantum mechanics lecturer was. $\endgroup$
    – algori
    Commented Sep 29, 2011 at 21:58
  • $\begingroup$ Apparently, $\L$ojasiewicz also heard it form Whitney- but I have only a recollection of him relating the story once at a lunch table. It is good that you were able to point to a more reliable reference; the story deserves to be known. I wish I had asked $\L$ojasiewicz about his own background in music. $\endgroup$
    – Margaret Friedland
    Commented Sep 30, 2011 at 13:46
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    $\begingroup$ Confirmation in writing, page 437 here: math.upenn.edu/Arnold/Arnold-interview1997.pdf $\endgroup$
    – Qiaochu Yuan
    Commented Oct 24, 2014 at 2:22
  • $\begingroup$ @QiaochuYuan that's a broken link, but ams.org/notices/199704/arnold.pdf appears to be the same interview. $\endgroup$
    – hobbs
    Commented Aug 29, 2016 at 8:33
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Edward Witten.(Fields Medalist) Witten attended the Park School of Baltimore (class of '68), and went on to receive his Bachelor of Arts with a major in history and minor in linguistics from Brandeis University in 1971. He planned to become a political journalist, and published articles in The New Republic and The Nation. In 1968 Witten published an article in The Nation arguing that the New Left had no strategy. He worked briefly for George McGovern, a Democratic presidential nominee in 1972. McGovern lost the election in a landslide to Richard Nixon. look at http://www.youtube.com/watch?v=f1BcyxQCnoE&feature=related

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  • $\begingroup$ I don't want to downvote this; they are factually correct. But from his interview, Witten was a little dismissive of the narrative. He said that at Brandeis he was allowed to study whatever he wanted, and history was one that had fewer requirements, meaning probably that he also took some math and physics. He also told the story how he was exposed to calculus at the age of 11 (his father being a physics professor), but unfortunately didn't see much beyond this wonderful subject. His latest interview: grahamfarmelo.com/the-universe-speaks-in-numbers-interview-5 $\endgroup$
    – liuyao
    Commented Jun 12, 2019 at 21:55
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My colleague Tadashi Tokieda studied classics at university and switched to maths after being inspired by a book on the subject. I can't remember the exact details, but they are remarkable, as is he.

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  • $\begingroup$ I was fortunate enough to see him lecture this summer and I was stunned by his enthusiasm. Great example! $\endgroup$
    – Dedalus
    Commented Sep 29, 2011 at 7:37
  • $\begingroup$ The quanta magazine article has the details of his story: quantamagazine.org/… Not to spoil it, he not only studied classics (or philology, to be precise), but had a professorship before reading a book (a biography of a certain physicist, let me just say). $\endgroup$
    – liuyao
    Commented Jun 12, 2019 at 21:02
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Hermann Grassmann studied theology, classical languages, philosophy, and literature in university, but not mathematics or physics.

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    $\begingroup$ He had notable achievements in linguistics, too: he translated Rigveda (from Sanskrit to German) and observed some pattern in phonology that became known as Grassman's law. $\endgroup$
    – Margaret Friedland
    Commented Sep 28, 2011 at 18:59
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    $\begingroup$ Grassmann also compiled a dictionary, which people use up to this day to read Rigveda (or at least were using 15 or so years ago). $\endgroup$
    – algori
    Commented Sep 28, 2011 at 22:21
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    $\begingroup$ However, Grassman did his work on Indo-European studies and the Rigveda only after he got disappointed by the fact that his mathematics hadn’t been as accepted by the community as he hoped. $\endgroup$
    – k.stm
    Commented Mar 7, 2018 at 18:46
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Paul Halmos was a graduate student in philosophy, and decided that subject was too hard, and became a graduate student in mathematics.

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    $\begingroup$ Halmos did not decide anything in this respect, rather he failed his philosophy exams (and was properly mortified by the event). $\endgroup$
    – Did
    Commented Oct 26, 2011 at 8:51
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Karl Marx. So you didn't know he was a mathematician? A book of his collected mathematical papers is in our math library, which is more than most mathematicians can claim. (They are mostly attempts to understand the definition of a derivative if I recall correctly.) They were quite popular during the cultural revolution, Chinese mathematicians presumably figuring that the study of dialectical calculus was better than a one-way trip to one of Mao's holiday resorts.

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    $\begingroup$ From the current draft of the original question: "Which well-known mathematicians, past or present, started out as law/art/humanities/business students, but later turned to mathematics?" $\endgroup$
    – Yemon Choi
    Commented Sep 28, 2011 at 20:45
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    $\begingroup$ Let us not forget the famous mathematician Napoleon, having a theorem named after him. $\endgroup$
    – user9072
    Commented Sep 28, 2011 at 21:04
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    $\begingroup$ Another example in reverse: Edmund Husserl (he studied with Weierstrass and wrote about foundations of mathematics and logic). $\endgroup$
    – Margaret Friedland
    Commented Sep 28, 2011 at 21:15
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    $\begingroup$ Engels writing to Marx: "Yesterday I found the courage at last to study your mathematical manuscripts even without reference books, and was pleased to find that I did not need them. I complement you on your work. The thing is as clear as daylight, so that we cannot wonder enough at the way the mathematicians insist on mystifying it. But this comes with the one-sided way these gentlemen think. To put dy/dx = 0/0, firmly and point-blank, does not enter their skulls. . . You need not fear that any mathematician has preceded you here." $\endgroup$
    – Steven Landsburg
    Commented Sep 28, 2011 at 22:13
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    $\begingroup$ Here's a good summary: pballew.blogspot.com/2011/01/mathematics-of-karl-marx.html $\endgroup$
    – Alex R.
    Commented Sep 28, 2011 at 22:18
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Serge Lang started out as a graduate student of philosophy at Princeton, but he switched to math, because he had "finished it", it being philosophy.

Here's the relevant part from his biography here

"After returning to the United States, Lang went to Princeton University with the intention of studying for a doctorate in philosophy. After a year in the philosophy department, he changed to mathematics and Emil Artin became his thesis advisor."

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  • $\begingroup$ Halmos also reports starting out in philosophy, but he doesn't say he "finished it". The way he tells it (in his automathography), it was more like it finished him, after a disappointing qualifying exam, and he switched to mathematics. (Much to the benefit of the latter.) (Oh, and oops, I just noticed the answer by Michael Hardy and Did's comment which also gives this information.) $\endgroup$
    – Todd Trimble
    Commented Apr 12, 2015 at 6:09
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I love this one: Harald Bohr was such a good soccer player that he was member of the Danish national team at the 1908 Olympiads. Two years later he got his PhD (apparently there was a large crowd at the event, a quite unusual occurrence for the math department) and went on to become a famous mathematician.

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    $\begingroup$ Where is there any switch in fields. He was studying maths since 1904, starting aged 17/18. In addition, he had a hobby at which he was really good. $\endgroup$
    – user9072
    Commented Sep 28, 2011 at 21:09
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    $\begingroup$ I agree this does not qualify as an answer if not marginally. On the other hand, I do not understand your persistence in proving that. Several other answers are even worse! (what about Fermat or Leibnitz? shall we talk about Plato?) The main reason I posted this is that I love this fact snippet, and in particular the scene of a huge hooligan crowd gathering at the discussion of a PhD thesis in math :) $\endgroup$
    – Piero D'Ancona
    Commented Sep 29, 2011 at 9:00
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    $\begingroup$ Yes, it is a fun fact; and in some sense more interesting than Leibniz, Fermat, Euler,... in my opinion also more than the music-semi-examples, which are not all that surprising or unique. And both Yemon and I 'complained' in general about the too old examples (in my present this is only implict but I head a more explict one that I deleted not to clutter the general comment thread to much). So, I/we are not singularly 'complaining' about your answer. Perhaps we can agree that yours is off-topic in an fun/interesting/original way. $\endgroup$
    – user9072
    Commented Sep 29, 2011 at 16:24
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    $\begingroup$ Just to echo quid: for all the good it does, which is little, I have downvoted Richard Borcherds's egregiously off-topic answer. But hey, Fields Medallist! Upvote! Fun story! Upvote! [sigh] $\endgroup$
    – Yemon Choi
    Commented Sep 29, 2011 at 20:29
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    $\begingroup$ Ok. Pointless discussion I guess, the original question was not too serious to begin with $\endgroup$
    – Piero D'Ancona
    Commented Sep 29, 2011 at 22:08
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Fermat was a lawyer.

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Per Enflo is (sometimes) a concert pianist; see this section of his Wikipedia page, also his web page.

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  • $\begingroup$ As re Eugenia Cheng above: does this count as someone with a background in "non-maths" turning to maths? $\endgroup$
    – Yemon Choi
    Commented Sep 29, 2011 at 21:14
  • $\begingroup$ @Yemon: I think so, but I'm no expert on music so I could be wrong. In the autobiography on his web site he discusses his teenage years, and it seems to me that he was considered a prodigy and accomplished at this time (e.g. had played as soloist with the Royal Opera Orchestra of Sweden and many other concerts, and studied with masters and so on). He also mentions that "I did little or no systematic study of mathematics in these years." He also suggests there that he's never really given piano away, but there is a clear point where mathematics (later) enters the picture. $\endgroup$
    – Philip Brooker
    Commented Sep 30, 2011 at 4:32
  • $\begingroup$ In any case he started to study math at the university right after high-school at age 18 (like 'everybody else'). $\endgroup$
    – user9072
    Commented Sep 30, 2011 at 9:36
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Cayley was a lawyer by profession for 14 years.

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Christiaan Huygens (1629–1695). A musical prodigy who by age 10 read fluently in four clefs and played the organ, two years before his first study of mathematical sciences (though admittedly 12 is not an "unusually advanced age" for that either...). This according to a display case at Leiden University which I saw at the ANTS-IV conference in 2000, and which reproduced some of his harmony exercises!

Huygens kept up his interest in music, later in life publishing a treatise on a tuning of 31 equal notes to the octave, an idea that apparently still has some currency in the Dutch music scene. According to Huygens' Wikipedia entry, the 20th volume of his 22-volume Collected works is titled Musique et mathématique. Musique. Mathématiques de 1666 à 1695.

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Marcel-Paul "Marco" Schützenberger studied medicine before obtained his second doctorate, in mathematics. He also work in formal linguistics with Noam Chomsky and Stephen Cole Kleene.

For a short biography, see

http://www.gap-system.org/~history/Biographies/Schutzenberger.html

or

http://en.wikipedia.org/wiki/Marcel‐Paul_Schützenberger

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Leibniz studied philosophy and law. He worked as a diplomat.

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Noam Elkies is a musician and composer.

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    $\begingroup$ True (and I'd say that "composer" implies "musician"), but $-$ though I have no direct recollection $-$ I have it on good authority that this is not an example of "starting out" in arts and then "turning to science", because here fascination with numbers came first, albeit by only a few months. $\endgroup$
    – Noam D. Elkies
    Commented Sep 28, 2011 at 0:53
  • $\begingroup$ Sorry, when I answered, I only paid attention to the title of this thread, and you certainly count as a mathematician with "background in" the arts. $\endgroup$
    – David Corwin
    Commented Sep 5, 2012 at 18:15
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    $\begingroup$ @DavidCorwin: and a member of MO! $\endgroup$
    – José Hdz. Stgo.
    Commented Feb 18, 2014 at 2:15
  • $\begingroup$ And a world-class composer of chess problems (also, at least at one point, a chess Master), and from the lore that I've heard, quite a respectable weight lifter as well! Life isn't fair... $\endgroup$
    – Todd Trimble
    Commented Apr 12, 2015 at 6:24
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Daniel Bernoulli. I am not sure if he counts, because he knew he wanted to study mathematics. Yet his formal education was in business and medicine. According to Wikipedia,

"Around schooling age, his father, Johann Bernoulli, encouraged him to study business, there being poor rewards awaiting a mathematician. However, Daniel refused, because he wanted to study mathematics. He later gave in to his father's wish and studied business. His father then asked him to study in medicine, and Daniel agreed under the condition that his father would teach him mathematics privately, which they continued for some time."

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Henri Poincaré was a mining engineer. His first job was at the Corps des Mines as an inspector of mines. He participated in the rescue of miners trapped after an explosion, himself descending the shaft into the mine to investigate the cause of the explosion! Check this link for details.

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    $\begingroup$ Engineering is excluded. $\endgroup$
    – user9072
    Commented Sep 28, 2011 at 21:05
  • $\begingroup$ Sorry, didn't realize that. $\endgroup$
    – Slobodan Simić
    Commented Sep 28, 2011 at 21:38
  • $\begingroup$ I've never thought of mining engineering being related to maths. Possibly the world's most famous mining engineer, Major-General Sir Richard Hannay, KCB, OBE, DSO, Legion of Honour, never struck me as the maths-ey type... $\endgroup$
    – ADL
    Commented Sep 30, 2011 at 10:32
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    $\begingroup$ @Alan, are you informed about the French higher eductations system (at Poinarés time)? It is not overly surpising that he was in the Corps des Mines. And most high-school math teachers are not research math types either. Shall we include all French math that attended the/a ENS because clearly they originally all wanted to become high-school teachers. And those at the Ecole polytchnique are actually soldiers. $\endgroup$
    – user9072
    Commented Sep 30, 2011 at 11:18
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According to an interview of his, Kazuya Kato started off studying aerospace engineering (or something similar) before becoming interested in math.

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    $\begingroup$ "This excludes mathematicians who switched from the sciences or engineering to mathematics such as Raoul Bott." If not for this provision, one could name even more people, e.g. Solomon Lefschetz. $\endgroup$
    – Margaret Friedland
    Commented Sep 28, 2011 at 20:14
  • $\begingroup$ Could you give a link to the interview- an interview with K. Kato sounds quite interesting $\endgroup$
    – sisn
    Commented Oct 12, 2011 at 7:45
  • $\begingroup$ he studied astronomy, if i recall correctly. i think he liked thinking about different stars. $\endgroup$
    – user676464327
    Commented Jul 9, 2021 at 8:48
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I once read that the higher-category theorist Eugenia Cheng is also occasionally a concert pianist (she accompains lieder singers if I remember well).

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    $\begingroup$ Well, the information is correct, but it didn't involve change of field, I'm pretty certain. $\endgroup$
    – Todd Trimble
    Commented Sep 29, 2011 at 18:55
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    $\begingroup$ Nope, no change of field - she does both. She told me once that if she changed careers she'd become a pastry chef - she's darn good at that too. $\endgroup$
    – John Baez
    Commented Sep 30, 2011 at 7:20
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Frank Ryan isn't really a famous mathematician, but he was at one point famous and he did manage to get a Ph.D. in Mathematics. See Sports Illustrated article on Frank Ryan

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    $\begingroup$ That's a lovely story, but in the interests of consistency I guess I must downvote $\endgroup$
    – Yemon Choi
    Commented Sep 30, 2011 at 3:28
  • $\begingroup$ Perhaps back then (1965) attitudes in mainstream US culture towards science were different... $\endgroup$
    – Yemon Choi
    Commented Sep 30, 2011 at 3:30

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