Herbert Mehrtens

Herbert Mehrtens (5May 1946 – 27May 2021) was a German historian and historiographer of mathematics. Mehrtens is notable for his work in the history of lattice theory, and as an early researcher in the social history of mathematics, mathematical modernity, and the history of mathematics in Nazi Germany. Mehrtens' work in the history of mathematics "has had considerable, indeed field-changing impact."

Life

Mehrtens was born on 5 May 1946 in Bremen. After graduating from high school in 1966, he did his basic military service in Oldenburg from 1966-1968, and then studied mathematics, philosophy, and the history of science at the University of Hamburg graduating in 1974. He went on to take his doctorate under Christoph Scriba in 1977 with a dissertation on the history of lattice theory (published 1979). From 1977 to 1982 he was a research assistant at Technische Universität Berlin, subsequently holding several positions there, interspersed with "scientifically productive unemployment, fellowships", and the like. He received his habilitation from TU Berlin with the study Moderne—Sprache—Mathematik. Eine Geschichte des Streits um die Grundlagen der Disziplin und des Subjekts formaler Systeme (Modernity—Language—Mathematics. A history of the controversy over the foundations of the discipline and the subject of formal systems) which was published in 1990. In 1992 he became Professor of Modern History (History of Science and Technology) at TU Braunschweig, retiring in 2011.

After Mehrtens became chairperson of the (German Society for the History of Medicine, Natural Science and Technology) in 2001 he strived to achieve its reunification with the (Society for the History of Science [GWG]) which had split off in 1965. As a first step Mehrtens organised a joint meeting of DGGMNT and GWG at TU Braunschweig in 2006. The unification of the two societies under the umbrella Society for the History of Science, Medicine and Technology was achieved in 2017 after Mehrtens had retired.

In his later years Mehrtens shifted his focus away from the history of mathematics and worked collaboratively with his life partner the art historian in the area of science policy journalism. Mehrtens died on 27 May 2021 in Berlin.

The December 2022 issue of the journal Science in Context was entitled Modernism in science and mathematics: A second look at Herbert Mehrtens and contained eight papers discussing the work and legacy of Mehrtens. The introduction stated "the contributors revisit key aspects of Mehrtens' oeuvre to reflect on its influence on the course of the historiography of mathematics and science from the mid-1970s onwards, and to consider its connection to the present priorities of these disciplines."

Social history of mathematics

From his earliest writings on the history of mathematics Mehrtens emphasized the importance of the social aspect of the history of mathematics. and by Lukas M. Verburgt 3. Kronecker, Klein, Poincaré, Brouwer and the counter-modernity (intuitionist) approach; 4. The foundational crisis (Grundlagenkrise), computing machines, ideology of National Socialism (anti-modernity), and Bourbaki; 5. The social and institutional modernization of mathematics in Germany 1890–1914; 6. Discourse and language of mathematics: a linguistic and semiotic analysis of mathematics that provides the theoretical model intended as a guide to the historical interpretation of the previous chapters; 7. Applications of formalism in the modern era, e.g. artificial languages, Taylorism, art, biometrics, and mathematics and the language of the unconscious (Jacques Lacan).

In his introduction to the book Mehrtens described it in the following terms:

Mehrtens argued that the modernization of mathematics took place in terms of a dialectical process between the modernists and the counter-modernists. The two approaches were not to be understood in terms of simple opposition, but as a genuine ambiguity inherent in the very idea of modernity. Both sides offered views that were contradictory and complementary, needing each other to fully express themselves in a way that reflected the wider cultural and social changes of modernization itself.

A Cultural History of Mathematics in the Nineteenth Century (2025) summarised Mehrtens' approach to modernism: "At the close of the nineteenth century ... the recasting of mathematics into an autonomous self-defining discourse had become pervasive, and this led to an epistemic breach that separated those who resisted from those who welcomed it. Herbert Mehrtens retrospectively made this the very cornerstone of his interpretation of mathematical modernism. "The new language of mathematics does not need to be ascertained by an external reality, because it ascertains itself through its own constant work".

Reviewers described the book as "provocative", "daring ... a book that has been received with a mixture of enthusiasm and criticism", "remarkable and demanding" ... "[an] immensely rich work" ... "a major contribution to the history of modern science",

Specific criticisms of the book included:

The assignment of Klein into the counter-modern grouping, and Mehrtens' discussion of Klein's eugenic pronouncements and Klein's analysis of the rhetoric of Deutsche Mathematik in the 1930s. The assignment of Poincaré into the counter-modern grouping has been challenged by Jeremy Gray and Ivor Grattan-Guinness. The supposed opposition between the modernist and counter-modernist positions, with a marked partiality by Mehrtens towards the modernist side, when in practice this incompatibility almost never occurred.

Later commentators acknowledged the legacy of Mehrtens' work on mathematical modernity.

In 1996 Moritz Epple said: "Herbert Mehrtens opened the debate with his 1990 book Moderne—Sprache—Mathematik. Future contributions to the topic of modernity in mathematics and its possible end will have to be measured by this work, which confidently links a broad spectrum of mathematical historical research with themes from cultural-theoretical discourse from Foucault to Lacan." In 1996/97 David E. Rowe commented "By placing the mathematical issues that divided antagonists like Hilbert and L.E.J. Brouwer within a larger cultural and disciplinary context, Mehrtens offers a provocative new interpretation of the deeper issues at stake that galvanized the rival parties in the conflict." and "one of the most provocative and original contributions to the history of mathematics ... in recent decades." In 2008 Jeremy Gray published the book Plato's Ghost: the modernist transformation of mathematics in which he gave credit to Mehrtens for the first serious and detailed work in the field of mathematical modernity, and discussed at length the similarities and differences between his approach and Mehrtens'. Compared with Mehrtens, Gray's book has international coverage, expands the mathematical fields covered to geometry, algebra, and analysis (in addition to logic and foundations), examines the relationship of modernity with philosophy, and analyses mathematical language in the context of linguistics. A 2013 essay review of Plato's Ghost by David E. Rowe contained much material comparing and contrasting it with Moderne—Sprache—Mathematik. In 2011 Amir Alexander commented: "The insularity of the field has been a source of much soul-searching among scholars in the history of mathematics ... In recent years some historians of mathematics have worked to remedy the situation by emulating developments in the history of science and making use of social and cultural approaches. Herbert Mehrtens's Moderne—Sprache—Mathematik traced the role of mathematics in the modernist movement" In 2022 David E. Rowe commented: "The recently published collection of essays in The Richness of the History and Philosophy of Mathematics (Chemla et al. 2023) reveal how the larger issues raised in M-S-M have continued to spawn further reflection up to the present day."

History of mathematics and science in Nazi Germany

Mehrtens began his work in the field of the history of mathematics and science in Nazi Germany in 1979/80 with detailed literature reviews. Collaborative relations (Kollaborationsverhältnisse), or "the collaboration of the elites", referred to the complex, symbiotic working relationship between scientists, mathematicians, and the Nazi regime. Mehrtens argued that scientific work under the Nazis was not a battle between fanatical Nazi ideologues and oppressed, anti-Nazi scientists. Instead, it was defined by scientists and the state actively working together to fulfil mutual goals.

Reinhard Siegmund-Schultze summarised Mehrtens contribution: "The theoretically most informed approach to the history of mathematics in Nazi Germany (and, incidentally, to the history of science as a whole) is that by Herbert Mehrtens ... On the social level he considers, partly inspired by P. Bourdieu, mathematics as a system that exchanges knowledge products plus political loyalty in return for material resources plus social legitimacy."

Historiography of mathematics

Mehrtens was an early researcher on the applicability of Thomas Kuhn's theory of Paradigm shifts (scientific revolutions) in the field of mathematics; Mehrtens took issue with Crowe's use of the preposition in as tending to support mathematicians' own definition of what mathematics is and what it is not. As Mehrtens says "There is a danger for the historian of mathematics in this preposition in".

In 2022 wrote a paper summarizing Mehrtens' achievements in the history and historiography of mathematics and their implications for future research in the areas of mathematical modernity, technocratic modernity, and the history of mathematics and science in Nazi Germany. Ash concluded "In the broadest sense, this paper can and should be understood as a plea for a culturally relevant and politically engaged historiography of the sciences and humanities. That is surely what Herbert Mehrtens also sought to achieve." (published version of 1977 doctoral thesis) (reviews) ] 1980: Naturwissenschaft, Technik und NS-Ideologie : Beiträge zur Wissenschaftsgeschichte des Dritten Reichs (Natural Science, Technology and Nazi Ideology: Contributions to the History of Science in the Third Reich) (co-editor) (reviews) 1981: Social History of Nineteenth Century Mathematics (co-editor) (reviews) 1990: Moderne—Sprache—Mathematik. Eine Geschichte des Streits um die Grundlagen der Disziplin und des Subjekts formaler Systeme (Modernity—Language—Mathematics. A history of the controversy over the foundations of the discipline and the subject of formal systems) (published version of habilitation thesis) (reviews)

Papers

1976: T. S. Kuhn's theories and mathematics: A discussion paper on the "new historiography" of mathematics (reprinted with an appendix) (review) 1977: The interactions of mathematics and society in history some exploratory remarks (co-author with Henk J. M. Bos) 1979: Die Naturwissenschaften im Nationalsozialismus (The natural sciences in National Socialism) 1980: Das "Dritte Reich" in der Naturwissenschaftsgeschichte: Literaturbericht und Problemskizz (The "Third Reich" in the history of science: literature review and problem outline) 1980: Felix Hausdorff, ein Mathematiker seiner Zeit (Felix Hausdorff, a mathematician of his time) (reviews) 1981: Social History of Mathematics 1981: Mathematicians in Germany circa 1800 1982: Richard Dedekind - Der Mensch und die Zahlen (Richard Dedekind - the man and the numbers) 1984: Anschauungswelt versus Papierwelt: Zur historischen Interpretation der Grundlagenkrise der Mathematik (Intuitive world versus paper world: On the historical interpretation of the fundamental crisis of mathematics) 1985: Die "Gleichschaltung" der mathematischen Gesellschaften im nationalsozialistischen Deutschland (The "Gleichschaltung" of mathematical societies in National Socialist Germany) translated into English in 1989. (review) 1986: Angewandte Mathematik und Anwendungen der Mathematik im nationalsozialistischen Deutschland (Applied mathematics and applications of mathematics in National Socialist Germany) 1987: The Social System of Mathematics and National Socialism: A Survey (republished in 1993) 1987: Ludwig Bieberbach and "Deutsche Mathematik" (reviews) 1989: Mathematics in the Third Reich - resistance, adaptation and collaboration of a scientific discipline 1990: Der französische Stil und der deutsche Stil. Nationalismus, Nationalsozialismus und Mathematik, 1900-1940 (The French style and the German style. Nationalism, National Socialism and Mathematics, 1900-1940) 1992: Revolutions reconsidered 1993: Irresponsible purity: The political and moral structure of mathematical sciences in the National Socialist state 1994: Kollaborationsverhältnisse: Natur- und Technikwissenschaften im NS-Staat und ihre Historie (Collaboration relations: Natural and technical sciences in the Nazi state and their history) 1996: Mathematics and war: Germany, 1900-1945 1996: Modernism vs counter-modernism, nationalism vs internationalism: style and politics in mathematics 1900-1950 (reviews) 1998: Mathematik: Funktion – Sprache – Diskurs (Mathematics: Function – Language – Discourse) 2004: Mathematical Models