Abstract
The present article is intended as a critical assessment of some basic assumptions underlying the analysis of modernism in mathematics in its relationship with the broader aspects of cultural modernism, especially in the period 1890–1930. It discusses the potential historiographical gains of approaching the history of mathematics in the periods under such a perspective and suggests that a fruitful analysis of the phenomenon of modernism in mathematics must focus not on the common features of mathematics and other contemporary cultural trends, but rather on the common historical processes that led to the dominant approaches in all fields.
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Notes
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Although more naturally seen as dealing with the history of physics, Galison’s book devotes considerable attention to Poincaré’s mathematics as well.
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The most prominent example that would come to mind is that of Luitzen J.E. Brouwer, whose doctoral advisor urged him to delete the more philosophical and controversial parts of the dissertation and to focus on the more mainstream aspects of mathematics that it contained. It was only somewhat later, as he became a respected practitioner of a mainstream mathematical domain, that he started publishing and promoting his philosophical ideas, and to devote his time and energies to developing new kinds of radical, intuitionistic mathematics. Brouwer promoted a kind of logic, later called “intuitionistic logic”, deviating from the mainstream but not implying a call to abandon classical logic, but rather to revert logic to a previous stage in its evolution, where no considerations of the actual infinite had (wrongfully and dangerously, from his perspective) made deep headway into mainstream mathematics. See (van Dalen 1999, 89–99). Another interesting case is that of Doron Zeilberger’s call, after a distinguished carrer in classical disciplines, for an abandonment of “Human-Supremacist”, “human-generated, and human-centrist ‘conceptual’ pure math mathematics” in favor of computer-generated, “experimental mathematics”.
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In an illuminating article about the use of the terms “classical” and “modern” by physicists in the early twentieth century, Staley (2005) addresses this difference from an interesting perspective. In his opinion, whereas in physics discussions about “classical” theories and their status were more significant for the consolidation and propagation of new theories and approaches than any invocation of “modernity”, in mathematics, different views about “modernity” were central to many debates within the mathematical community.
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A typical version of which appears in (Miller 2002).
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Similar in this respect, with an emphasis on mathematics, are the account presented in (Gamwell 2015).
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See also (Epple 1996).
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An alternative, but not very convincing, way to connect mathematics with the general phenomenon of modernism appears in (Everdell 1997), where Cantor and Dedekind are presented as the true (unaware) initiators of modernism because the way in which they treated the continuum in their mathematical work. See also (Pollack-Milgate 2021).
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One can find in Greenberg’s own texts support for such a view, but in other places he emphatically denied that his analysis was ever intended as anything beyond pure description. See, e.g., (Greenberg 1983): “I wrote a piece called ‘Modernist Painting’ that got taken as a program when it was only a description.”
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It is worth stressing, however, that the issue of self-criticism and the ability of an individual (or a collective for that matter) to effectively distance himself from the normative framework in which he functions in order to be self-critical and innovative is a truly complex one, when considered from a broader philosophical point of view. For a through discussion that examines the views of philosophers like Brandom, Friedman, Davidson, Habermas, Rorty, and others, see (Fisch and Benbaji 2011).
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A discussion of “purity” and its centrality in modernism, from a different perspective appears in (Cheetham 1991).
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Greenberg, of course, is not the only one to discuss modernism in terms of the processes that led to its rise, rather than by just providing a checklist of characteristic features. Also worthy of mention here is the work of Dan Albright (Albright 1997, 2000), who stresses the crisis of values in art that led to modernism. In his view, if in previous centuries, artists, writers, and musicians could be inherently confident about the validity of the delight and edification they provided to their audiences, during the twentieth century art found itself in a new and odd situation, plagued with insecurity. Faced with the crisis, radical claims about the locus of value in art were advanced in various realms at nearly the same time. The various radical modernist manifestoes thus produced reflect the need of the artist not only to create, as was always the case in the past, but also to promote new standards of value and to provide some new kind of justification to the very existence of art.
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(Galison 1990) presents an analysis that complements this view and locates the Bauhaus movement in relation with logical positivism, as part of Viennese modernism.
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(Janik 2001, 147–69) discusses the somewhat different relation between Hertz’s famous Introduction and the late Wittgenstein.
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Epple’s description of the intellectual background to fin-de-siècle Vienna also strongly relies on the classical study (Schorske 1980).
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In this context it is natural to stress that also Wittgenstein was born to a privileged and immensely wealthy Viennese family, who generously supported the likes of Gustav Klimt and Alfred Loos as well as the poets Georg Trakl and Rainer Maria Rilke. The circle of friends of the Wittgenstein family included many distinguished figures of the Viennese musical milieu, such as Johannes Brahms (Monk 1991).
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Acknowledgments
An earlier version of this paper was written about ten years ago and remained unpublished. Nevertheless, it was posted on my website, and it was read in its preliminary format and even cited in several places. I am glad and proud to be able to publish a fully revised and updated version in this volume dedicated to Jeremy Gray. I thank Jeremy for our interesting conversations on the topic of modernism and beyond, and, above all, for his decisive and lasting contribution to our discipline.
For enlightening discussions and comments on the topic of the present text, I would also like to thank Moritz Epple, Gal Hertz and Menachem Fisch. For important editorial comments that led to an improved final version of this text, I am thankful to Karine Chemla, Hongxing Zhang, and two anonymous reviewers.
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Corry, L. (2023). How Useful Is the Term ‘Modernism’ for Understanding the History of Early Twentieth-Century Mathematics?. In: Chemla, K., Ferreirós, J., Ji, L., Scholz, E., Wang, C. (eds) The Richness of the History of Mathematics. Archimedes, vol 66. Springer, Cham. https://doi.org/10.1007/978-3-031-40855-7_16
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