What does “working mathematician” mean?

Question

What does "working mathematician" mean? Is this term derogatory? What properties of a "working mathematician" are considered undesirable, and what attitude contrasts them?

Answer 1

The way I understand it the working mathematician is a mathematician which works, researches, publishes.

I recall seeing this mainly in the context of "Theory X for the working mathematician" which means a mathematician which is not particularly interested in X, but knowing about it could help, and would help. For example surely most mathematicians won't care about large cardinals, but if there was a method based on large cardinals which would have solved a lot of problems it would sure be nice if one knows about it even if they are not set theoretically inclined. In such case a book called "Large cardinals for the working mathematician" would be a reasonable titled book.

If I were ever called a working mathematician I'd be insulted, but then again I believe that lack of work is much harder than work. Mathematics just writes its own! :-)

Answer 2

The term "working mathematician" has been occasionally used to distinguish mathematicians from metamathematicians (those working in foundations, logic, philosophy, etc). This term predates Mac Lane's book on category theory. For example, it was used by Bourbaki in his 1948 ASL address Foundations of Mathematics for the Working Mathematician as well as in his Elements series. Below are some pertinent excerpts from Chapter 7 of Leo Corry's book.

It is not unusual to come across pronouncements of Bourbaki members, who insistently characterize Bourbaki’s approach as that of the “working mathematician” whose professional interest focuses variously on problem solving, research and exposition of theorems and theories, and which has no direct interest in philosophical or foundational issues. Thus Bourbaki formulated no explicit philosophy of mathematics and in retrospect individual members of the group even denied any interest whatsoever in philosophy or even in foundational research of any kind. (Jean Dieudonné (1982, 619) once summarized Bourbaki’s avowed position regarding these kinds of questions “as total indifference. What Bourbaki considers important is communication between mathematicians. Personal philosophical conceptions are irrelevant for him.”)

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Thus, in spite of declarations to the contrary elsewhere, Bourbaki here implicitly admitted (concealing this confession, as it were, in a footnote) that the link between the formal apparatus introduced in Theory of Sets and the activities of the “working mathematician” (Bourbaki’s declared main addressee) is tenuous, and, at best, of purely heuristic value.

See also these excerpts from Kneeebone's Mathematical Logic and the Foundations of Mathematics, an introductory survey.

Hilbert's outlook was accordingly that of a working mathematician, and the opinions that he expressed at the Paris congress may thus be taken as direct testimony from a mathematician as to the nature of his activity. A comparison at once suggests itself with certain utterances of Nicolas Bourbaki, who has a clear right to speak for the mathematicians of a more recent generation, and we find indeed that Bourbaki confirms much of what Hilbert has said or implied. Bourbaki has remained very much a working mathematician, and he has not made any systematic study of the foundations of mathematics such as that which Hilbert undertook in his later work.

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Russell's Principia Mathematica and Hilbert's metamathematical investigations, and equally the various intuitionist and constructivist undertakings that we have been examining, were concerned less with helping the working mathematician to attain the rigour that he seeks in the actual presentation of mathematical theories than with answering fundamental questions that arise in the realm of philosophy of mathematics. In recent years, however, mathematical and metamathematical inquiries have been found to converge, and now at last the working mathematician and the metamathematician or logician are able to re-establish contact.

Answer 3

To me, it means a mathematician that is active, i.e. publishing research.

Answer 4

I was trained as a mathematician but work in industry on algorithms, usually of a probabilistic inference or machine learning flavor. I rarely publish, but for what it's worth, I still think of myself as a "working mathematician".