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It seems that 'mathematics' was within the category of 'philosophy', at least in universities, until 1799, when Gauss received a doctorate in 'philosophy' rather than a doctorate in 'mathematics'. So, here's the question. When was the first ‘mathematics department’ opened in a university?

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    $\begingroup$ This depends on the country. But in german-speaking lands, it was probably much later than 1799. In the early 1920s, Hans Hahn was a member of the philosophy department at the University of Vienna, along with all mathematicians and many physicists. Whereas in France one could say that the opening of the Ecole Polytechnique was the first break with philosophy more than 100 years earlier. So the first one is a good question, but I'm pointing out that the autonomy of mathematics took a long time to be established. Great question to find out when and where it all started $\endgroup$ Commented Jun 8 at 5:25
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    $\begingroup$ Even today we still receive Doctorates in Philosophy (Ph.D.) in the North American system. This too is regional. In German speaking universities one can encounter Dr. rer. nat. for doctorates in the natural sciences. Additionally to answer your question we'd have to agree what certain organizational structures across regions constitute "departments" is a "Fachgruppe/schaft" or "Seminar" or a "Institut" a department? Did the founding of the Lucasian Chair in Mathematics at Cambridge (1663) constitute the founding of a (one-chair) department in our modern sense? $\endgroup$
    – Georg Essl
    Commented Jun 8 at 8:27
  • $\begingroup$ Originally, all disciplines were within philosophy. Witness that as far as we can glean, the chief object of study for pre-Socratic philosophers appears to have been what we call natural philosophy in historical retrospective and science in modern practice. So mathematics definitely was. $\endgroup$
    – Mary
    Commented Jun 9 at 14:56

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The question is extremely broad, probably too broad to be answerable. But in lieu of an answer here are some thoughts and sources.

Studies that might shed some detail on the evolution of institutional organization of Mathematics in Germany. The classic reference is:

  • Lorey, W. (1916). Das Studium der Mathematik an den deutschen Universitäten seit Anfang des 19. Jahrhunderts. Abh. über den math. Unterricht in Deutschland, Teubner.

A modern update can be found in:

  • Scharlau, W. (Ed.). (1990). Mathematische Institute in Deutschland 1800-1945. Braunschweig: Vieweg. (see the chapter by Schubring, Gert. "Zur strukturellen Entwicklung der Mathematik an den deutschen Hochschulen 1800–1945." p264-278.)

The difficulty to even delienate the foundation of a department can be found in:

Nadis, S., & Yau, S. T. (2013). A history in sum: 150 Years of mathematics at Harvard (1825-1975). Harvard University Press.

One could say that the department officially began in 1727, when the first Hollis Professor of “Mathematics and Natural Philosophy,” Isaac Greenwood, was appointed. (Although the mathematics department consisted of just a single person at the time, earlier in the school’s history there were no departments at all. At the school’s very beginning, one instructor was responsible for teaching all subjects.)

This paragraph is perhaps the most cross-cultural and cross-temporal description of the dynamics here, in that a single appointment (chair, professorship) tended to seed topics and the formation of departments-like structure is a function of institutional shifts but also number of faculty.

Compare this historical blurb by the University of Vienna:

While there was originally one professor of mathematics, over the centuries the number of professors in mathematics at the University of Vienna grew gradually up to finally three at the end of the 19th century. In this time, more precisely, in the year 1876, the "Mathematisches Seminar" was founded, mainly at the instigation of Ludwig Boltzmann, then professor of mathematics.

From my personal experience it would be misleading to compare a mathematics department in Austria to one in the US naively, as the underlying structure is different. While in Austria there are still appointments by topic (Professor for Algebra for example), in the US professors are bundled into departments and the topical distribution is defined by departmental needs but not fixed topically. So though we might consider them the "same structure" their functioning and organization is different. In Austria if a chair is appointed there will very likely not be a second professorship on the same topic, maintaining the old topic=chair="institute/department" organization.

Translating these topics across different organizational principles (German, British, French, North American for example) is difficult and will require care and clear notions of comparison.

All that said, the Lucasian chair at Cambridge of 1663 (Knox & Noakes (2003). From Newton to Hawking: a history of Cambridge University's Lucasian professors of mathematics. Cambridge University Press.) may be a viable candidate - if we can accept the proposal of Nadis & Yau of a de facto one-person department.

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