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."
Work
History of lattice theory
Mehrtens was awarded his doctorate in 1977 for the thesis Die Entstehung der Verbandstheorie (The emergence of lattice theory) which was published in 1979 and was generally well received. The book is divided into four chapters: 1. The development of the algebra of logic up to the concept of a lattice by Ernst Schröder, 2. Richard Dedekind's theory of "dual groups" (lattices) in the 1890s, 3. the rediscovery and reformulation of lattice theory around 1930, and 4. the establishment and growth of lattice theory in the 1930s.
Garrett Birkhoff, one of the pioneers of lattice theory in the 1930s, reviewed the book for Historia Mathematica and described it as "a thoughtfully written and thought-provoking account of the formation of lattice theory as a mathematical discipline". Birkhoff praised Mehrtens research methods, including the detailed questionnaires which he had sent to living mathematicians who had participated in the development of lattice theory. Birkhoff concluded "All considered, by its faithfulness to truth, recognition of the importance of personality, and imaginative use of original sources, Dr. Mehrtens' thorough yet humane book sets a style which could be followed to great advantage ... by other historians of mathematics trying to describe progress in other 20th-century mathematics."
Ivor Grattan-Guinness in Isis commented "In this excellent history Mehrtens shows himself to be a critical appraiser of lattice theory rather than its paladin ... Mehrtens's account of the nineteenth century progress of lattice theory is a valuable contribution at the interface of the histories of algebra, set theory, and logic." Grattan-Guinness noted the "clever use made by Mehrtens of the reminiscences that he obtained from prominent figures of the time who are still alive." He concluded "His blend of technical competence, social sense, and historical scholarship lends his book quite special qualities among recent publications in the field."
In 1994 Joseph Dauben commented: "This is all borne out in an exemplary fashion in Herbert Mehrtens' recent analysis of the origins and development of lattice theory. As Mehrtens shows, lattice theory arose in many different ways, as a result of diverse motives and different approaches. The eventual acceptance of lattice theory and its emergence as a recognized branch of mathematics in the 1930's was a social process, he says, as well as a matter of technical mathematics."
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. He stated baldly that mathematics is not "a package of eternal, spiritual truths gradually unwrapping itself in the course of history" but rather that "mathematics is man-made; its vital basis is the social interaction of mathematicians in their scientific community." In his 1979 history of lattice theory the reviewers pointed out Mehrtens', then unusual, use of social factors in his historical research.
Together with Henk J. M. Bos and Ivo Schneider, Mehrtens edited the 1981 book Social History of Nineteenth Century Mathematics which was based on a workshop held in Berlin in 1979 and included 16 papers. The book included a statement of Mehrtens' own views on the social history of mathematics.
The book, and Mehrtens' contribution, were widely reviewed. The reviewer for Studia Leibnitiana commented "Mehrtens provides a good overview of the current state of research on sociological interpretations of the history of mathematics." and commended Mehrtens' "in-depth knowledge" in compiling the select bibliography. The reviewers for The Mathematics Teacher recommended the book "Since it discusses a new field in mathematics, this text provides invaluable reference material for young scholars. In terms of diversity, the background of individual contributors to this book is highly suitable. The articles by themselves are professional with an ample amount of originality." In a review for Science the historian of mathematics Judith Grabiner praised the papers by Dirk Struik, Philip Enros, Thomas Hawkins, David Bloor, and Mehrtens' bibliography, but considered that the other papers suffered from a variety of flaws. In a review for Isis, historian of science Kathryn Olesko commented "In his concluding seminal article, Mehrtens notes that there is a lack of guidelines for looking at the history of any branch of knowledge as an interwoven cognitive and social practice."
Mehrtens' contributions to the social history of mathematics were reviewed by Reinhard Siegmund-Schultze, Joseph Dauben, and by Lukas M. Verburgt
Mathematical modernity
In 1990 Mehrtens published his most original and influential work entitled Moderne—Sprache—Mathematik (Modernity—Language—Mathematics: A history of the controversy over the foundations of the discipline and the subject of formal systems). The book was 640 pages in length and contained seven chapters. 1. Prelude: The work of Gauss, Dedekind, Riemann, Peano, and Lobachevsky leading up to modernity; 2. Hilbert, Cantor, Zermelo, and Hausdorff and the modernity (formalist) approach to mathematics centred around set theory; 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". ... Whereas the "moderns", championed by David Hilbert, proclaimed logical consistency was the bedrock of mathematical truth, the "counter-moderns", including Felix Klein, Henri Poincaré and L. E. J. Brouwer, sought to anchor mathematical knowledge in intuition and the pre-linguistic activity of the mind."
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", "somewhat unstructured, even inchoate, [and] also highly ambitious and original ... This is history of mathematics that matters for historians and humanists, and indeed for anyone interested in the (post)modern intellectual scene.", "a very ambitious historical reconstruction", "a history of a long running debate within the mathematical community over the foundations of its science: should mathematics be "modern" or "counter-modern," should it be constructed from a finite number of assumptions and axioms, or be "intuitive?" and "a bold attempt to understand the debates over the foundations of mathematics in the early twentieth century in terms of a power struggle over the rules of mathematical discourse within the (mostly German) mathematical community."
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. Doubts whether Mehrtens' bipolar modernist/counter-modernist theory is sufficient to explain the complexities in the development of early 20th-century mathematics.