‘An inalienable prerogative of a liberated spirit’: postulating American mathematics

Abstract

In the early twentieth century, researchers in the United States engaged with foundational studies in mathematics by building and evaluating postulate systems. At the same time, their contemporaries were evaluating the meaning and politics of knowledge more broadly. This article argues that the study of postulates in the United States was tied to important Progressive Era questions about the nature of knowledge, the status of the knower, and the development of American Pragmatism. While most investigations of postulate studies have considered their implications within mathematical research and education, this article looks instead to the role of postulate studies in the professionalization of mathematics in the United States and to its cultural status more broadly.

Acknowledgements

I would like to thank my advisor, Suman Seth, and doctoral committee members Ron Kline and Aaron Sachs.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Notes

1 Mathematicians like Moore often considered ‘modern’ research in terms of topics and results that had redefined their field since the early nineteenth century. More generally, scientists used the terms abstract, practical, pure, and applied for different purposes in the late nineteenth and early twentieth centuries. See Kline 1995.

2 Hilbert’s approach to mathematics was never straightforwardly formalist. See Corry 2006; Rowe 1994.

3 On the prioritization of abstract mathematics in the early-twentieth-century United States, see Duren 1989; Feffer 1997; Parshall 2000; Servos 1986; Siegmund-Schultze 2003. On the relationship between pure and applied mathematics in Germany, see Pyenson 1983; Schubring 1981.

4 Examples include Finkel 1902; Halsted 1902a, 1902c, 1902d; Hedrick 1902; Sommer 1900; Veblen 1903.

5 On Moore’s seminar and the department at Chicago, see Chapter 9 of Parshall and Rowe 1994.

6 Robert Lee Moore would later become known for his axiom-based, competition-driven ‘Moore Method’ of teaching. He would also become known for his blatant sexism and racism, designating his women and Jewish students as inferior and refusing to engage with Black students and mathematicians altogether. See Parker 2005.

7 Some of these references included papers that Huntington had only read reviews of, including one read at the 1900 International Congress of Mathematicians and reported in the Bulletin of the American Mathematical Society by Dr. Charlotte Angas Scott.

8 In his address, Moore called for ‘evolution’ not ‘revolution’ in making changes to mathematics and education. Many of Moore’s contemporaries, including Dewey, called for solutions to problems like class conflict in the same terms. American social reformers around the turn of the twentieth century were keenly aware of the violence of events like the Chicago Pullman strike, not to mention the Civil War and Reconstruction. Evolutionary change was not only how Darwin explained the development of species, it also seemed like the safest and best way to further the development of society.

9 Huntington referred readers to a further explanation of the symbology by Christine Ladd Franklin. See Huntington 1904, 292.

10 That Huntington’s postulates in this paper ‘do seem more usable’ is discussed in (Hollings 2017, 444).

11 While Keyser was away, he received mail from President Butler back at Columbia. ‘Rumors of your progress have been coming in steadily,’ Butler wrote, ‘and they are all of the most complimentary and flattering character. I am sure that you have stimulated the whole country … ’. Butler to Keyser, March 1908, Columbia University Rare Book and Manuscript Collection, Cassius Jackson Keyser papers, Box 5.

12 James Pierpont had similarly referred to the ‘golden age’ of mathematics in St. Louis in 1904 (Pierpont 1904).

13 In many ways, Keyser’s student, E T Bell, would do the same, with a particular focus on algebra (Hollings 2016).

14 For references to postulationalism, see Carmichael 1930; Wiener 1964. Postulationalism was similar in many ways to what Corry calls the ‘axiomatic image’ of the French Bourbaki collective. See Corry 1997.

Additional information

Funding

Research for this article was funded in part by National Science Foundation Doctoral Dissertation Research Improvement [grant number 1753998].