Famous mathematicians with background in arts/humanities/law etc 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.
 A: 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.
A: Per Enflo is (sometimes) a concert pianist; see this section of his Wikipedia page, also his web page.
A: 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
A: Fermat was a lawyer.
A: Cayley was a lawyer by profession for 14 years.
A: Noam Elkies is a musician and composer.
A: 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.
A: 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."
A: Leibniz studied philosophy and law. He worked as a diplomat.
A: 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.
A: 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.
A: 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.
A: 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 
A: 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.
A: Hermann Grassmann studied theology, classical languages, philosophy, and literature in university, but not mathematics or physics.
A: Paul Halmos was a graduate student in philosophy, and decided that subject was too hard, and became a graduate student in mathematics.
A: 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. 
A: 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."
A: According to an interview of his, Kazuya Kato started off studying aerospace engineering (or something similar) before becoming interested in math. 
A: I once read that the higher-category theorist Eugenia Cheng is also occasionally a concert pianist (she accompains lieder singers if I remember well).
A: 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
