Abstract
This chapter is an essay on the conceptual nature of Riemann’s thinking and its impact, as conceptual thinking, on mathematics, physics, and philosophy. In order to fully appreciate the revolutionary nature of this thinking and of Riemann’s practice of mathematics, one must, this chapter argues, rethink the nature of mathematical or scientific concepts in Riemann and beyond. The chapter will attempt to do so with the help of Deleuze and Guattari’s concept of philosophical concept. The chapter will argue that a fundamentally analogous concept of concept is also applicable in mathematics and science, specifically and most pertinently to Riemann, in physics, and that this concept is exceptionally helpful and even necessary for understanding Riemann’s thinking and practice, and creative mathematical and scientific thinking and practice in general.
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Notes
- 1.
I have considered physical concepts from this perspective in [27, pp. 2–11].
- 2.
- 3.
K. Ohshika’s chapter in this volume [20] is a notable exception as a reflection on the architecture of Riemann’s concepts.
- 4.
On Riemann’s subsequent developments of his geometrical ideas and their application to physics and beyond, see Athanase Papadopoulos’s contribution to this volume [22].
- 5.
Although the term “Mannigfaltigkeit” was not uncommon in German philosophical literature, including in Leibniz and Kant, it is worth noting that the German word for the Trinity is “Dreifaltigkeit,” thus, etymologically, suggesting a kind of “three-folded-ness,” which could not have been missed by Riemann, or, for that matter, Leibniz and Kant. See [20] on the use of the term “Mannigfaltigkeit” in Kant vs. Riemann.
- 6.
It is true that Riemann never considered or even mentioned this possibility, arguably, first expressly investigated by Poincaré, and it is not my aim to make a historical claim to the contrary. My point instead is that this possibility and, as will be explained below, the concept of topological space may be seen as conceptual implications of his argument. It is conceivable, especially given his concept of a Riemann surface, that Riemann entertained this type of idea, just as he (admittedly, expressly) entertained the idea that the reality underlying space may be discrete.
- 7.
Mathematics, science, and philosophy also involve the creation of compositions, found in artistic thought, and the latter may, conversely, involve planes of immanence and the creation of concepts, or planes of reference. For one thing, concepts thus defined are composed. More pertinently here, Riemann’s concept of manifold is compositional because it defines a manifold as composed of local spaces [26].
- 8.
Poincaré’s extensive (much more extensive than Riemann’s) philosophical works (e.g. [28, 29]), while influenced, as were his mathematical works in geometry, by Riemann’s Habilitation lecture, may be seen along these lines. I realize that this claim may be challenged, and make it with caution. I would, nevertheless, argue that Riemann’s philosophical thinking plays a greater constitutive role in his mathematical thinking than Poincaré’s philosophical thinking in his mathematical thinking. The situation is of course different when it comes to physics, which is a major part of Riemann’s and Poincaré’s mathematical thinking alike, and both made major contributions to physics.
- 9.
There were other, more extrinsic, reasons, beginning with the fact that Gauss was Riemann’s mentor and the chair of his Habilitation committee. Indeed, Gauss selected this topic among three proposed by Riemann (following the rules). The philosopher R. H. Lotze , a fervent opponent of non-Euclidean geometry, was a member of the philosophy faculty, to which Riemann’s Habilitation was presented. Later on, Lotze criticized Riemann’s approach anyway, as part of his general critique of non-Euclidean geometry (see [16, pp. 222–226] and [33, pp. 97–112]).
- 10.
As I qualified earlier, at least this is a workable and widely accepted view, widely but not universally. It has never been established definitively or, in any event, agreed upon whether such a determination is ever rigorously possible, as opposed to having a practically effective and possibly, within its proper limits, the best available theory or, as Poincaré would have it, “convention,” without making a real claim concerning “the reality underlying space” [31, p. 33]. Einstein had his doubts too, although he was ultimately inclined to accept the possibility of such a determination, at least in principle, as, it appears, was Riemann, but not Poincaré, with whose position Einstein, nevertheless, had to contend and which he tried to accommodate within his own (e.g. [11, pp. 324–328]).
- 11.
For the development of Hilbert’s ideas, as reflected in different editions of the Grundlagen, see [4]. In the first version of the book, Hilbert was closer to Riemann, and he later returned to a more Riemannian view of geometry in the wake of general relativity to which he made important contributions.
- 12.
One might challenge this argument on historical grounds because it would have been difficult, if not impossible, to present a concept such as that of manifold in axiomatic form at the time of Riemann’s lecture. That may be true. My point, however, is that Riemann’s alternative, conceptual-problematic rather than axiomatic-theorematic, thinking, could still be contrasted to that of Hilbert and lead to a different type of mathematical thinking. It is difficult to say how Riemann would have approached the foundations of geometry if he had means of axiomatizing his concepts. On the other hand, it is possible to argue, as I do here, on historical grounds, that Riemann, unlike his predecessors, Lobachevsky and Bolyai, was not pursuing an axiomatic approach to geometry.
- 13.
It would be similar to the three-dimensional sphere. As I explained, the currently dominant view or hypothesis (which appears to be confirmed by cosmological measurements) is that the universe is on average flat and is expanding.
- 14.
As most of his contemporaries, Riemann did not distinguish continuous and differentiable manifolds. It became eventually clear, however, that not all continuous (also called topological) manifolds are differentiable. There are topological manifolds with no differentiable structure, and some with multiple non-diffeomorphic differentiable structures. Thus, there is a continuum of non-diffeomorphic differentiable structures of \(\mathbf {R}^4\).
- 15.
These two concepts could, especially in modern understanding, be subsumed under the same concept. This is because all zero-dimensional manifolds, which are discrete manifolds in Riemann’s terms, are continuous (topological) manifolds. In fact they are also differentiable manifolds, because transition functions for them are constant functions, which are continuous and even differentiable. In modern terminology, the distinction between continuous and discrete manifolds in Riemann’s lecture would be interpreted as that of zero-dimensional manifolds and positive dimensional manifolds. I am grateful to Ken’ichi Ohshika for helping to clarify this point. It is not inconceivable that Riemann’s thought along similar lines, which would explain his choice of the term manifold for both discrete and continuous manifolds, although the term had a more general use at the time. (Georg Cantor, possibly influenced by Riemann’s lecture, initially referred to sets as Mannigfaltigkeiten but eventually switched to Mengen.) It is, however, difficult to be certain on the basis of his lecture or his other writings. I would argue that the difference between these two types of manifolds is still crucial, both in general and for Riemann, especially for his analysis of physical space and geometry. Riemann stressed the significance of the relationships between continuity and discontinuity for mathematics, physics, and philosophy (e.g. [32, pp. 515–524]; [9, pp. 77–80]).
- 16.
It is also worth recalling in this connection that Grothendieck’s initial primary areas of mathematical research concerned topological vector spaces, which suggests yet another genealogical line in the history of the (broadly) Riemannian problematics in question here.
- 17.
Thus, Ferreirós’s discussion of Riemann in [11, pp. 39–80] appears to me to displace Riemann’s thinking into the axiomatic and set-theoretical register, dominant in the wake of Cantor , a displacement arguably due to Ferreirós’s insufficient attention to the nature of Riemann’s mathematical concepts, to Riemann’s concept of mathematical concept. In fairness, Ferreirós does relate Riemanns view of axioms to his concept of “hypothesis” and distinguishes it from the understanding of axioms developed in the twentieth-century philosophy of mathematics and mathematical logic. It does not appear to me, however, that Riemann thinks either in terms of axioms or, especially, in terms of sets (of points), as Ferreirós contends, although it could be and subsequently has been translated into these terms (e.g. [22]). See also Note 12 above.
- 18.
It would be instructive on both counts, to consider, as part of this genealogy, Dedekind’s and Noether’s work in algebra, reflecting the impact of Riemann’s work on modern algebra, and even apart from his work on the distribution of primes and his famous hypothesis concerning the \(\zeta \)-function. See [19] on Noether’s work in this connection.
- 19.
For the discussion of some of these developments, see the chapter by V. Pambuccian, H. Struve, and R. Struve [21] and other chapters in the part of this volume that addresses later developments of Riemann’s work.
- 20.
The higher-dimensional spaces of superstring theory have been extensively discussed in literature and can be safely bypassed here, pertinent as their geometrical and topological features are. I would like, however, to mention a recent investigation, along quantum-informational lines, of the possibility that the reality underling space is discrete at the Planck scale, with a radical implication that the Lorentz invariance and hence special relativity is broken at the Planck scale as well [5] [27, pp. 259–262]. The article is also innovative mathematically in its use of geometric group theory, which emerged from Gromov’s realization, Riemannian in spirit, that mathematical objects, such as groups, defined in algebraic terms, can be considered as geometric objects and studied with geometric techniques. This argument is still hypothetical, however, as, again, are all arguments thus far to the effect that the reality underlying space is discrete. If one accept what may be called the strong Copenhagen view, following Bohr, this “reality” may be beyond conception altogether and, hence, be neither continuous nor discontinuous [27, pp. 11–22]. I return to this possibility below.
- 21.
I have discussed the connections between Riemann’s Habilitation lecture and quantum theory in [25].
- 22.
How our phenomenal experience of space emerges is separate question, psychological, physiological, or now neurological. Remarkably, Riemannian geometry is used in recent neurological research, as in the work of Jean Petitot (neurogeometry).
- 23.
Cf. [20], on the epistemological differences between Kant and Riemann.
- 24.
On some of these difficulties, see [12, pp. 31–42].
- 25.
Riemann offered important reflections on causality, which he linked to continuity [32, p. 522].
- 26.
We cannot conceive of entities that are simultaneously continuous and discontinuous, the difficulty handled in quantum mechanics by means of Bohr’s concept of complementarity . Complementarity reflects the fact that continuous and discontinuous quantum phenomena (defined by what is observed in measuring instruments) are always mutually exclusive, while quantum objects themselves, responsible for these phenomena through their impacts on measuring instruments, are, again, assumed to be beyond any representation or even conception, continuous or discontinuous. For a full treatment, see [27, pp. 107–172].
- 27.
Dirac’s famous equation also introduced spinors into physics. Although the name itself was coined (by Paul Ehrenfest ) in 1929, following Dirac’s theory, the concept, still enigmatic and uneasily suspended between geometry and algebra, existed in mathematics previously and was extensively studied by Cartan, for example. It belongs to the post-Riemannian evolution of geometrical thinking, also as extending beyond geometry, for example, to algebra, although spinors are important for geometry as well.
- 28.
On these connections, see [22].
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Acknowledgements
I am grateful to Franck Jedrzejewski, Ken’ichi Ohshika, and Athanase Papadopoulos for reading the original draft of the article and helpful suggestions for improving it.
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Plotnitsky, A. (2017). Comprehending the Connection of Things: Bernhard Riemann and the Architecture of Mathematical Concepts. In: Ji, L., Papadopoulos, A., Yamada, S. (eds) From Riemann to Differential Geometry and Relativity. Springer, Cham. https://doi.org/10.1007/978-3-319-60039-0_11
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