Abstract
We consider the concept of curve in the context of the transition from mathematical “modernity” to mathematical “modernism,” the transition defined, the article argues, by the movement from the primacy of geometrical to the primacy of algebraic thinking. The article also explores the ontological and epistemological aspects of this transition and the connections between modernist mathematics and modernist physics, especially quantum theory, in this set of contexts.
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Notes
- 1.
G. Tomlinson’s book on the prehistory of music, with the revealing title “A Million Years of Music,” confirms this view, even if sometimes against its own grain [64].
- 2.
I will be less concerned with general or point-set topology, which has a different and much longer history, extending, arguably, to the ancient Greek thinking, although my claim concerning the modernist algebraization of mathematics could still be made in this case. See A. Papadopoulos’ contribution to this volume for an illuminating discussion of the topological aspects of Aristotle’s philosophy, via Thom’s engagement with Aristotle [48].
- 3.
Finding a good term poses difficulties because such, perhaps more suitable, terms as “geometric algebra” and “algebraic geometry,” are already in use for designating, respectively, the Clifford algebra over a vector space with a quadratic form and the study of algebraic varieties , defined as the solutions of systems of polynomial equations. This object and this field, however, equally exemplify the modernist algebraization of mathematics.
- 4.
This may remain true in low-dimensional geometry or topology. I would argue, however, that spatial algebra is still irreducible there because one commonly converts topological operations into algebraic ones. This conversion in low dimensions was essential to the origin of algebraic topology. On the other hand, the recent development of low-dimensional topology, following, among others, W. Thurston’s work, from the 1970s on, is a more geometrically oriented trend that, to some degree, counters the twentieth-century modernist algebraic trends and returns to Riemann’s and Poincaré’s topological thinking, but only to a degree, because the algebraic structures associated with these objects remain crucial. Some of the most powerful (modernist) algebraic tools of algebraic topology and algebraic geometry have been used and sometimes developed during this more geometrical stage of the field. These areas have important connections to quantum field theory and then string theory, as in E. Witten’s work, which, especially in quantum field theory, are fundamentally algebraic, in part by virtue of their probabilistic nature.
- 5.
Descartes’ La Géométrie was originally published as an appendix to his Discourse on Method, and it was part of a vast philosophical agenda that encompassed mathematics [18].
- 6.
For a detail discussion of the subject, see the companion article by the present author [56].
- 7.
That, again, does not exclude either realist or causal interpretations of quantum mechanics or alternative theories of this behavior that are realist or causal. The so-called Bohmian mechanics is one example of such an alternative theory. Unlike quantum mechanics, however, Bohmian mechanics expressly violates the requirement of locality, which entered physics with relativity theory and which dictates that the instantaneous transmission of physical influences between spatially separated systems is forbidden.
- 8.
The so-called many-worlds interpretation of quantum mechanics, which aimed to resolve some of the paradoxes of the theory in a realist and causal way, does not affect this point, because this kind of material reality is still retained within each world involved, and there are no connections between these worlds.
- 9.
Thus, postmodernity was also epistemologically shaped by certain developments in mathematics and science, most of which are modernist in the present sense (e.g., [37]).
- 10.
The modernist aspects of Riemann ’s work, equally in Gray’s definition of modernism, pose difficulties for Gray, because Riemann preceded modernism by several decades [26, p. 5]. It is not a problem for the present argument, firstly, because the present view of modernism is different, and, secondly, because modernism is seen here as more continuous with modern mathematics from Fermat and Descartes on, a longer history in which Riemann ’s work is a decisive juncture. This continuity is recognized by Gray, but it seems to worry him because it disturbs the stricter chronology he considers. The present view emphasizes, in part following G. W. F. Hegel , the conceptual over the chronological, even in historical considerations. Gray, in addition, appears to see the axiomatic, not central for Riemann (in contrast to Hilbert), rather than the conceptual, as more characteristic of modernism. In the present view, modernism is more about concepts and their history than about the chronology of events or developments, such as those associated with the spreading of modernist thinking or practices. This chronology cannot of course be disregarded, but a concept or a form of practice in a given field can precede a chronologically defined state of this field, with which this concept or practice would be in accord. This accord is not an “anticipation” but a determinate quality of a concept or a form of practice. Riemann’s concepts and practice are modernist, in the present (or, with some differences, Gray’s) definition, and a similar claim could be made, helped by his revolutionary algebraic thinking, concerning Galois. The degree or even the existence of such an accord, or to what degree this accord reflects the understanding of this concept by its inventor, is a matter of interpretation, which could be contested. Riemann’s thinking has complexities when it comes to the role of algebra there because of the topological and geometrical aspects of his thinking, which often take the center stage, while algebra, when still present, appears in a supporting role. This is, however, only so in a more narrow or technical sense, as opposed to the broader sense assumed here as defining modernism. Riemann ’s work, as noted, is defined by the joint workings of geometry, topology, algebra, and analysis in his mathematics, added by philosophical and physical, aspects of his thinking. Hilbert made major contributions in all these areas as well (apart from topology), but one does find the same type of fusion of different fields dealing with a given subject that one finds in Riemann , as in the case of Riemann surfaces .
- 11.
The nature of these connections and, in part correlatively, the effectiveness of using the term modernism, specifically by Mehrtens and Gray, have been questioned, for example, by S. Feferman [24] and L. Corry [15]. While both articles (that of Feferman is a review of Gray’s book) make valid points, I don’t find them especially convincing on either count, in part because their engagement with modernist art is extremely limited and because neither considers the epistemological dimensions of modernism, which are, in my view, important in addressing these connections. For an instructive counter argument to Mehrtens, challenging his historical claims, specifically those concerning F. Klein, see [6].
- 12.
Weyl’s classic book had undergone several editions, some of them with significant revisions. I cite here the last edition.
- 13.
Intriguingly, Cohen ultimately thought that the hypothesis was likely to be false [12, p. 151].
- 14.
I borrow the juxtaposition between Hilbert’s remark and Euclid’s definition of a point from G. E. Martin [40, p. 140], who, however, only states this juxtaposition without interpreting it.
- 15.
Classical statistical physics introduces certain complications here, which are, however not essential because the behavior of individual constituents of the systems considered there is governed by the deterministic laws of classical mechanics. In quantum mechanics, even elementary individual objects (the so-called elementary particles) can only be handled probabilistically, and in the present view, their behavior is beyond representation or even conception.
- 16.
See [3, pp. 523–524] on Grothendieck’s use of the term “multiplicity,” which is, on the one hand, specific (close to what is now called “orbifold”), and on the other hand, is clearly chosen to convey the multiple, plural nature of the objects considered. This is also true concerning Riemann’s concept of manifold. I would argue that Riemann and Grothendieck share thinking in terms of multiplicities as their primary mathematical philosophy, a modernist trend that is especially pronounced in their thinking. As will be seen, this philosophy, manifested already in Grothendieck’s early work in functional analysis, drives his use of sheaves and category theory (both concepts of the multiple), and then his concept of topos. Nothing is ever single. Everything is always positioned in relation to a multiplicity, is “sociological,” and is defined and studied as such, which is itself a trend characteristic of modernism.
- 17.
While, the concept of “qualitative” is of much interest in the context of this article, it would require a separate treatment. I might note, however, that, while the qualitative could be juxtaposed to the quantitative, it has more complex relationships with the algebraic, which is not the same as the quantitative, just as the geometrical is not the same the qualitative. Still the genus of a surface, which is a number and thus is quantitative, is important in a qualitative approach to its topology or geometry. See note 4 above.
- 18.
On narrative in mathematics, see [19]. Of particular interest in the present context, as part of the history leading to the modernist algebraization of mathematics, is B. Mazur’s contribution there, which offers a discussion of L. Kronecker’s “dream, vision, and mathematics” in “Visions, Dreams, and Mathematics” [41]. It might be added that Kronecker’s “dream, vision, and mathematics,” also decisively shaped those full-fledged modernist ideas of Weil. It may also be connected to Grothendieck’s work. See the article by A’Campo et al for a suggestion concerning this possibility, as part of a much broader network, opened by Grothendieck’s work on Galois theory (“the absolute Galois group”), which confirms Galois’ work as a key juncture of the trajectory leading from modernity to modernism in mathematics [2, p. 405, also n. 12]. These themes could be conceptually linked to quantum field theory, via M. Kontsevitch’s work on the “Cosmic Galois Group” (Cartier 2001), noted below (note 19). The article by A’Campo et al is also notable for a remarkable narrative trajectory of Grothendieck’s work it traces. This confirms the role of narrative as part of mathematics itself and the philosophy of mathematics rather than only of the history of mathematics, a key theme of Mazur’s and other articles in [19]. The present author’s contribution to this volume deals with the epistemology of narrative, along the lines of this article [53].
- 19.
For an extensive historical account of the history of complex function theory, only mentioned in passing here, see [10], which considers at length most key developments conjoining geometry and complex analysis, from Cauchy to Riemann and then of Riemann ’s work [10, pp. 189–213, 259–342]. Intriguingly, the algebra of quantum field theory found the way to use Riemann’s algebraic work, his work and his hypothesis concerning the ζ-function (one of the greatest, if not the greatest, of yet unsolved problems of mathematics). The ζ-function plays an important role in certain versions of higher-level quantum field theory. See P. Cartier’s discussion, which introduces an intriguing idea of the “Cosmic Galois group” [11] and A. Connes and M. Marcoli’s book [14], which explores the role of Riemann ’s differential geometry in this context. The latter is a long and technical work in noncommutative geometry, which uses Grothendieck’s motive cohomology theory, but see p. 10 for an important definition of “the Riemann-Hilbert correspondence.” This is yet another testimony to the fact that much of modernism in mathematics and even in physics takes place along the trajectory or again, a network of trajectories between Riemann and Grothendieck. See note 16.
- 20.
One of his important, but rarely considered, contributions is his work on Teichmüller space, the genealogy of which originates in Riemann’s moduli problem, powerfully recast by Grothendieck in his framework. Especially pertinent in the present context is the idea of a “Teichmüller curve” and then Grothendieck’s recasting of it, another manifestly modernist incarnation of the idea of curve, via Riemann. Conversely, the theory provided an important case for Grothendieck to use his new technology. Étale cohomology came next. This is yet another modernist trajectory extending from Riemann and Grothendieck. For an excellent account, see A’Campo et al. [1].
- 21.
The discussion to follow is partly adopted from [54, pp. 265–274]. My argument here is essentially different, however.
- 22.
I qualify by “unavoidably” because we can sometimes define by an experiment what will happen in classical physics, say, by rolling a ball on a smooth surface, as Galileo did in considering inertia. In this case, however, we can then observe the ensuing process without affecting it. This is not so in quantum physics, because any new observation essentially interferes with the quantum object under investigation and defines a new experiment and a new course of events. Only some observations do in classical physics.
- 23.
I do not refer by this statement to the trend known as “constructivism” in the foundational philosophy of mathematics, from intuitionism on, relevant as it may be, in part given Kant’s influence. I use the term “constructivist” more generally.
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Acknowledgements
I am grateful to Athanase Papadopoulos for reading the original draft of the article and helpful suggestions for improving it, and for productive exchanges on the subjects considered here.
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Plotnitsky, A. (2019). On the Concept of Curve: Geometry and Algebra, from Mathematical Modernity to Mathematical Modernism. In: Dani, S.G., Papadopoulos, A. (eds) Geometry in History. Springer, Cham. https://doi.org/10.1007/978-3-030-13609-3_5
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