Keywords

1 Introduction

It is not uncommon in the historical and philosophical studies of mathematics to discuss mathematicians’ own philosophical reflections on mathematics. While such reflections will also be addressed in this chapter, its primary aim is different. It is to consider mathematicians’ working philosophy of mathematics as emerging in and defining actual mathematical thinking and practice, as exemplified in the three cases stated in my subtitle: Niels Henrik Abel and Évariste Galois, Nikolai Lobachevsky and Bernhard Riemann, and André Weil and Alexander Grothendieck. The chapter’s approach to their mathematical practice and to creative mathematical practice in general is parallel to that of Gilles Deleuze and Félix Guattari to creative philosophical practice, as defined by the invention of new concepts, with philosophical concepts given by them a particular definition, in part in juxtaposition to mathematical and scientific concepts (Deleuze and Guattari 1996). The working philosophy of mathematics this chapter considers, under the heading of “mathematical practice as philosophy,” is, analogously, that of the invention of new mathematical concepts. Against the grain of Deleuze and Guattari’s argument, mathematical concepts will be understood here in affinity with rather than in juxtaposition to philosophical concepts, as the latter are defined by Deleuze and Guattari. This invention can sometimes be collective, either as a product of collaboration, more common in twentieth and twenty-first century mathematics, or of independent work by different mathematicians resulting in the same concepts. Both situations have been less common in philosophy, where the invention of new concepts is rarely collective. No such invention is ever entirely individual, because every concept, no matter how innovative, has a history defined by other concepts from which a new concept emerges and without which it would not be possible. A new concept is always a product of an interplay of continuities with and breaks from preceding concepts, although the balance of continuities and breaks can be different. Galois’s and Riemann’s concepts, such as that of a group by Galois and that of a manifold [Manningfaltigkeit] by Riemann, both especially central to this chapter, were more radically innovative, but still not completely, because they had continuities with earlier concepts, such as (among others) those of Adrien-Marie Legendre in the case of Galois and those of Karl Friedrich Gauss in the case of Riemann.

This form of mathematical practice becomes especially important at a time when one or another part of mathematics and mathematical practice needs to be transformed, often in response to a crisis on one or another scale, from that defined by a single problem to that affecting a large area of mathematics. Such transformative events, leading to practice as philosophy, are also found in science, such as physics or biology, or of course in philosophy. One might argue that philosophy, especially as enacted in the work of creators of original and powerful concepts, such figures as Plato (ideas), René Descartes (cogito), Gottfried Leibniz (monads), or Immanuel Kant (things-in-themselves), is always practiced at the time of crisis, because each such philosopher always sees philosophy in crisis, to which such a concept responds. Or, to adopt Thomas Kuhn’s language (without fully subscribing to his argument, which is difficult to apply in philosophy and which requires qualifications in mathematics and science), philosophy as a creation of concepts only admits a “revolutionary” practice, with a “normal” practice having at most only auxiliary importance for the advancement of philosophy, say, by exploring or applying new concepts (Kuhn 2012). In mathematics or science, both revolutionary and normal practices appear to be equally important for its advancement, as Kuhn recognizes, although they tend to be in a more complex interplay than was argued for by Kuhn. Thus, overtly normal, in technical terms, mathematical practice can, by reinterpreting the concepts used, adopt a different form of this practice from that of the revolutionary mathematics it follows and thus introduce a new, “revolutionary,” form of mathematical practice.

Things are of course not so straightforward. There is more to a normal practice in philosophy than meets the eye. Institutional philosophical fields, such as analytic philosophy, are based on normal practice, which made important contributions to our understanding of philosophical concepts or theories, and, it might be added, of mathematical and scientific concepts in the analytic philosophy of mathematics or science. The distinction between the normal and the revolutionary, or the importance of each, is not unconditional and requires caution in any domain. There tends to be more continuity than the appearance of a break might suggest, or across a longer history with many revolutions. Thus are the relationships between Cantor’s set theory and category theory, especially in Grothendieck’s hands, those of a revolutionary break (a more common view) or that of continuity, a view advocated, for example, by Yu. I. Manin, who prefers to see the history of mathematics, as against, that of physics, in terms of continuity (Manin 2002, p. 8, 2019, pp. 129–130)? Perhaps they are equally both, in a complex interplay, in either field. The cases considered here, including that of Grothendieck, could be seen in terms of this interplay. They were revolutionary events insofar as they radically transformed the mathematical situation in relation to which they emerged. It may not always be possible, and perhaps is never ultimately possible, to fully track how new trajectories emerge from and against previous trajectories of thought. This limitation, however, leaves room for meaningful assessments of this emergence in terms of both continuities and breaks, in the case of breaks, I argue, in particular by virtue of the invention of new concepts or conglomerates of concepts, some new and some already in place, for example, conglomerates comprising and defining new theories, such as algebraic geometry, as transformed by Grothendieck.

There are other cases that would illustrate my argument in this chapter, for example, that of Richard Dedekind and Leopold Kronecker, a famous example of two opposing (mathematical) philosophical positions. The main reasons for considering the cases selected here is that the first two stand at the origin of what we now see as modern mathematics, respectively, modern algebra and geometry – two ways of mathematical thinking (broadly conceived, so as to include arithmetic in the first and topology in the second) and the relationships between them that governed the history of mathematics since Pythagorean mathematics to our own time. Both and their relationships were transformed by modern mathematics that emerged roughly around 1800, although some of its trends can be traced to the mathematics, such as analytic geometry and calculus, contemporaneous with the rise of modernity itself as a cultural formation, defined by a series of transformations – scientific, technological, economic, political, and philosophical – from, roughly, the sixteenth century on. Of course, when it comes to modern mathematics, too, the thinking and work of other figures, such as Gauss in the case of algebra (his geometrical ideas will be considered in conjunction with Riemann’s work), was important and some of this thinking and work will be noted as this chapter proceeds. As I argue here, however, two concepts, defining respectively Galois’s algebraic and Riemann’s geometrical thinking, that of a group and that of a manifold [Manningfaltigkeit] respectively, had an unmatched shaping influence on the subsequent developments of algebra and geometry, and in the case of a group, on both, and much of modern mathematics, as well as physics and other sciences, where, especially in physics, the concept of a manifold had played a role as well. Galois also introduced a related concept, that of a group action on another mathematical entity, such as a manifold in geometry or a Hilbert space in quantum physics, a concept that has been important in mathematics and science. Another concept invented by Riemann, that of a Riemann surface, has played a role in the development considered in this chapter, beginning with Riemann’s own work, including his invention of the concept of manifold.

While one of the most important and remarkable examples of bringing algebra and geometry together in algebraic geometry and, as such, the main reason for being considered here, the case of Weil and Grothendieck is different. It belongs to twentieth-century mathematics, with the work of Grothendieck having a unique impact on subsequent mathematics, rather than to the rise of modern algebraic geometry in the nineteenth century. This rise followed non-Euclidean geometry (and its effects on the development of projective geometry), on the one hand, and, and on the other, the theory of Abelian integrals, which led Riemann to his concept of a Riemann surface (mathematically, a complex manifold), thus making Riemann a key figure in this development as well. As will be seen, his ideas also had a more direct impact on both Weil’s and Grothendieck’s work, as did Galois’s theory.

I speak of “and” and hence “conjunction,” rather than the disjunctive “or,” in all three cases considered because, while my primary concern is on the (historically) second figure – Galois, Riemann, and Grothendieck – in each case, I am not merely juxtaposing, let alone opposing, these figures to each other, especially given the significance of their thinking for transforming mathematics. The accomplishments of Abel, Lobachevsky, and Weil were revolutionary events as well. Instead, while granting the differences between their thinking, my aim is to explore the shared grounding that gives rise to these differences, defining, and defined by, their mathematical practice.

Realizing that mathematics itself or its basic fields cannot be given a single definition, I shall state a general understanding of these fields which is sufficient for my purposes and is, I would contend, in accord with how they are generally viewed. I understand algebra as the mathematical formalization of the relationships between symbols, arithmetic as dealing specifically with numbers, geometry as the mathematical formalization of spatiality, especially in terms of measurement, and topology as the mathematization of the structure of spatial-like objects apart from measurements, through their continuity or discontinuity. The corresponding mathematical fields are algebra, number theory, geometry, and topology. Analysis deals with questions of limit and related concepts, such as continuity, differentiation, and so forth. There are multiple intersections between these fields, and fields that branch off them.

On the other hand, defining algebra as the mathematical formalization of the relationships between symbols makes it part of all modern mathematics. Geometrical and topological mathematical objects always have algebraic components, while algebraic objects may, but need not, have geometrical or topological components. The term algebra has other meanings, such as that (standing at the origins of algebra as a mathematical discipline) of the study of algebraic equations, or that of algebraic structures, such as groups. These forms of algebra are important to both algebra itself and other mathematical disciplines, such as geometry, topology, or analysis. In defining algebra as the mathematical formalization of the relationships between symbols, however, it is not only a matter of having an algebraic component as part of the mathematical structure of geometrical, topological, or analytical objects, but also and especially of defining these objects themselves algebraically.

My argument, then, gives preference to the role of conceptual thinking in mathematics rather than to logical propositions or calculations. Both logic and calculations are essential to mathematics, including conceptual thinking there, because the structure of mathematical concepts is logical, and because calculations require concepts and can lead to new concepts. While, however, fully respecting these features, this chapter views creative mathematical thinking as most essentially defined by the invention of new concepts, a view based on a specific definition or concept of concept and of a mathematical theory as an organized assemblage of concepts, as considered in Sect. 2. Mathematics, I argue, advances most by the inventions of new concepts, which defines what I call “mathematical practice as philosophy,” a mathematical philosophy. For this mathematical philosophy, especially in its emergence, could be analyzed in terms of a more general philosophy of mathematics, in part by means of nonmathematical concepts, as it is in this chapter.

The next section outlines a philosophical grounding of this chapter’s argument. The subsequent sections are devoted to the three case studies – Abel and Galois, Lobachevsky and Riemann, and Weil and Grothendieck – as concrete instantiations of this architecture as mathematical practice.

2 Philosophy: The Architecture of Concepts and Mathematical Practice as Philosophy

The concept of a mathematical concept to be proposed here is homomorphic to Deleuze and Guattari’s concept of a philosophical concept proposed in What is Philosophy? (Deleuze and Guattari 1996). This homomorphism is not an isomorphism. It preserves the differences (which would be difficult to avoid) between philosophical and mathematical or scientific concepts, and thus the disciplinary specificity of each field, in particular a greater role of logical propositions or formal and quantitative elements in mathematical and scientific concepts vis-à-vis philosophical ones. This role was emphasized by Deleuze and Guattari in their argumentation concerning the difference between and even juxtaposition of philosophical and mathematical or scientific concepts. As discussed below, in the present view (if not strictly in that of Deleuze and Guattari), this difference is also related to the sharpness of the definition of a concept. Deleuze and Guattari’s argumentation goes beyond arguing for the difference between philosophical and mathematical or scientific concepts. They see their concept of a philosophical concept as belonging only to philosophy in its defining features, while denying these features to mathematics and science, defined by logical propositions and formal mathematical or scientific elements, such as functions. “The concept,” they say, “belongs to philosophy and only to philosophy” (Deleuze and Guattari 1996, pp. 11–12, 33–34). They might have argued that the homomorphism proposed here is either too limited or locates (strictly) philosophical subconcepts within mathematical or scientific concepts, something to which they refer as an “interference” (Deleuze and Guattari 1996, pp. 217–218). I argue that this homomorphism is sufficiently rich to be useful and goes beyond such interferences, although the latter are sometimes found in mathematical or scientific concepts. Ultimately, this type of objection would not matter for my argument, insofar as the present definition of a mathematical concept is workable, as I hope it is, for our understanding of what mathematical concepts are and how they work in mathematics. It is this understanding, rather than Deleuze and Guattari’s philosophy, that is my concern, helped as my argument may be by theirs or those of other philosophical figures considered here, such as Kant.

In some respects, this chapter returns Deleuze and Guattari’s view of concepts to its mathematical origin, given that their concept of a concept was in part modeled on a mathematical concept, that of Riemann’s concept of manifold. In addition, as will be seen, part of the genealogy of their argument concerning philosophical concepts is in Abel’s and Galois’s, in my terms, mathematical practice as philosophy, considered in Deleuze’s earlier analysis of “ideas,” on which Deleuze and Guattari’s argument builds (Deleuze 1995, pp. 180–181). Riemann’s concept of manifold is, admittedly, not the only origin of their concept of a concept, which also has a philosophical genealogy. The mathematical genealogy of this concept is, however, important and paradigmatic historically, beginning with Plato, who, too, both modelled his philosophical concepts on mathematical ones and yet claimed an essential distinction between them (Negrepontis 2019, p. 24, Plotnitsky 2021a, pp. 649–651). Riemann’s ideas, as well as those of Galois and Abel, had a major impact on Deleuze and Guattari’s philosophy in general, especially, again, Riemann’s concept of manifold, beyond being a model for their concept of a concept. They saw this concept as heralding a revolutionary philosophical change. This view was helped by the French term “multiplicité” as a translation of Riemann’s Mannigfaltigkeit (which the English translation of Deleuze and Guattari’s works often renders, understandably but incorrectly, as “multiplicity”). They say: “It was a decisive event when the mathematician Riemann uprooted the manifold from its predicative state and made it a noun, manifold [multiplicité]. It marked the end of dialectics and the beginning of the typology and topology of manifolds” (Deleuze and Guattari 1987, p. 483; translation modified).

Of course, even if mathematical concepts were the only origins of new philosophical ideas or concepts (to the degree that such a single origin is ever possible), this would not challenge claims, such as those of Plato, Hegel, or Deleuze and Guattari, for the fundamental difference between mathematical and philosophical concepts, because these mathematical concepts could be transformed into philosophical concepts in this transition or translation. My argument, however, is that this difference may not be as fundamental as Deleuze and Guattari contend, insofar as arguably the most essential aspect in their definition of a concept, which makes it a composition of its components, a composition specific to each concept, equally defines a mathematical or scientific concept, in the present definition. As will be seen, another key feature, which makes a concept a problem, that they associate strictly with philosophical concepts, can be transferred, even if with qualifications, to mathematical and scientific concepts as well, at least to some of them, such as that of Galois’s concept of a group or Riemann’s concept of a manifold.

Riemann’s thinking and practice provide, arguably, the most famous example of the expressly conceptual approach to mathematics, especially in geometry, where his thinking, leading him beyond all previous approaches to geometry, including those leading to non-Euclidean geometry, abandoned Euclid’s axiomatic-theorematic method, previously dominant in foundational thinking concerning geometry. The axiomatic aspects of geometry were at most secondary for Riemann, as against Lobachevsky’s and Jànos Bolyia’s work, leading each to the discovery of non-Euclidean (hyperbolic) geometry. As I shall argue, however, conceptual thinking plays an essential role in the thinking of all figures considered in this chapter, although the relative balance between conceptual and other forms of mathematical thinking and practice, such as the axiomatic one, is different in each case. Lobachevsky in effect made axioms or postulates of geometry, ultimately all axioms and postulates (rather than only the fifth postulate of Euclid), hypotheses, based in conditional assumptions or conventions, and thus, in the first place, concepts. Intentionally or not, his discovery deprived these axioms and postulates of their status as self-evident or preexisting truths that needed to be assumed or discovered, rather than invented, as concepts and thus geometries always are in the present view. Ultimately, all axioms, not the least those of Euclid, are conceptual inventions, even when their origins as such are lost. Moreover, as Gödel’s theorems tell us, mathematical axioms, assumed to reduce the complex to the simple, are defined by the irreducible complexity that precedes the simple or what appears to be simple. Imre Lakatos, citing Hermann Weyl, referred to Cantor’s set-theoretical paradise (as David Hilbert famously dubbed it), as “a bold theoretical construction, and as such the very opposite of analytical self-evidence” (Lakatos 1980, p. 31, n. 3, Weyl 2021, p. 64). This is no less true of Euclid’s geometrical paradise.

It is, to reiterate, not my intention to diminish the role of logical or calculational aspects of mathematics or science, or to take anything away from the contribution of those philosophical investigations, such as those in the analytic philosophy of mathematics, that focus primarily on logical and propositional structures in mathematics or science (There are new powerful and rich approaches to logic in mathematics and beyond, such as homotopy type theory (Voevodsky et al. 2013; Corfield 2020). I would contend, however, that concepts still play an essential and unavoidable role in them). My aim instead is to place a proper emphasis on the role of concepts, especially in creative thinking in mathematics or science, but not only there, because working with already established concepts is indispensable in all mathematical or scientific practice. Besides, analytic philosophers, too, sometimes, give concepts their due, beginning with one of the founders of analytic philosophy, sometimes even seen as the founder of it, Gottlob Frege, who gave priority to the role of concepts in mathematics and philosophy. The unconditional opposition between the logical-propositional and conceptual structures, or between calculations and concepts, is not so easy to maintain, even in considering figures such as Galois and Riemann, or Dedekind. While their thinking was primarily governed by concepts vis-à-vis calculations, they were perfectly capable of and performed difficult calculations. And then, logic and calculations do involve concepts and lead to new concepts, as they did in the case of Abel, or more pointedly (as he insisted on their primacy) Kronecker. Calculations can also function as statements, sometimes even as philosophical statements, thus connected to concepts. (There are still other aspects of mathematical thinking, for example, a narrative one, the role of which, as constitutive rather than merely auxiliary, which has always been recognized, has received a considerable amount of attention during recent decades. For a representative collection of essays on the subject, see Doxiadis and Mazur (2012). For a full disclosure, the present author was among the contributors.) Calculations, too, involve narrative dimensions. My main emphasis in this article is, however, on concepts as, arguably, the primary creative vehicles of mathematics.

In the present understanding, then, extending that of Deleuze and Guattari beyond philosophy, concepts are essential and even primary vehicles of creative thinking in theoretical fields. What is a concept, then? First, a concept is not merely a generalization from particulars (which is commonly assumed to define concepts) or a general or abstract idea, although a concept may contain such generalizations or ideas, specifically abstract mathematical ideas in mathematics or physics, where these abstractions may also become concepts. A concept is a multicomponent structure, defined by the organization, composition, of its components, and some of these components may be concepts in turn. This is, however, only a general, abstract, definition. What meaningfully defines a concept is the specific character of its compositional structure, defined by both the nature of each component and the character of their relations within this composition, by how these components relate to each other in the structure of this concept. Such components may be more general or particular, and some of them are concepts in turn. This composition is unique in a new concept, and make this concept unique. The creative essence of concepts is in their compositional individuality, which is only possible by virtue of their generally compositional nature, but is defined by the specificity of this composition. All musical works are compositions, but each gives us different music. A group is composed of its elements, but in accordance to very specific rules, which give it its structure and role in mathematics in group theory and beyond, and then it multiplies further to still more specific concepts of a group, a finite group, a Lie group, an algebraic group, and so forth, and then into further concepts within each of these.

A concept is a multicomponent conglomerate of concepts, figures, metaphors, and so forth, which are conjoint in a composition, a relational organization of the components. It is rare for a concept to have only one component, and ultimately impossible to do so. As Deleuze and Guattari say, “there are no simple concepts. Every concept has components and is defined by them. … There is no concept with only one component” (Deleuze and Guattari 1996, p. 16). A single-component concept, such as an element of a group, is a product of a provisional suspension or cut-off of its multicomponent organization, such as that defining the concept of an element, usually taken as a primitive concept as it may be. In fact, there are always cut-offs in delineating a concept, cut-offs that result from assuming some of the components of this concept to be primitive entities whose structure is not specified. These components could, however, be specified by alternative delineations, leading to a new concept, containing a new set of more primitive components.

Consider as an example the mathematical concept of space, discussed in detail in connection with Riemann’s rethinking of the foundations of geometry in Sect. 4. Emerging historically at the intersection of mathematics, physics, philosophy, and general phenomenal intuition, and defined by such constitutive concept-components as point, line, plane, or distance, as philosophical, physical, and mathematical concepts, this concept and each of these components have a long history of definitions, modifications, transformations, and so forth. As a general phenomenal concept, it is arguably as old as is human consciousness (any concept requires consciousness to be defined as a concept). Kant did not see space and time as concepts, but rather as a priori forms of our phenomenal intuition. He had a point, even if one rejects (as many, including Riemann, have done) his a priori view of space and time, insofar as these phenomenal forms precede any concept of space and time our general phenomenal thinking can form, as Kant realized they can. As a geometrically representable physical concept, space is at least as old as ancient Greek geometry, well preceding Euclid, who does not define, although, arguably, implies space in the Elements. Its history as a mathematizable physical and eventually (that happened only in the nineteenth century) purely mathematical concept extends, to mark some of its junctures, from Euclid to Descartes (a coordinate space) to Riemann (a space defined as a manifold) to Felix Hausdorff (topological space) to Grothendieck (topos). For the first three figures, the concept of space qua space was, while mathematizable, still physical. The concept also has its algebraic aspects and history. After each transformation, such as that of defining it as (representable by) a manifold by Riemann, the new concept of space, became, to stay with Kuhn’s idiom, part of the normal practice in mathematics, or physics, during which it acquired further features and new definitions.

A concept, then, is an assemblage of its components connected by its composition, which define its specific individual nature. Some of its components may be unique as well but need not be. The components of a concept need not be new, but its composition must be new for a concept to be new. In this respect a concept is akin to a work of art, always defined by its composition, in accord with the ancient Greek meaning of poiein (to make), as making something new, in which the term poetry originates. A theory, then, or a disciplinary field, mathematical, scientific, or philosophical, such as geometry, classical physics, or Kant’s philosophy (philosophy, again, tends to be more individualized), becomes an organized assemblage of concepts. It is a product of a broader poiein, in mathematics or science, more commonly collective, although it may be initiated by a single new concept, such as that of Galois’s concept of a group or Riemann’s concept of a manifold. In the present view, however, a creative conceptual mimesis (which may borrow elements of previous concepts but combines them in a new way) is a technical mimesis of previous concepts, rather than a Platonist mimesis of some preexisting independent reality, assumed to govern this mimesis.

I am not equating theoretical concepts (or theories) with works of art, either in their compositional nature or in their functioning. For one thing, a work of art, as a composition, may but need not contain concepts, although so-called conceptual art does. In functioning, a concept can be used directly or with minor modifications in, again, in Kuhn’s terms, the normal practice in the corresponding theoretical field, especially mathematics and science. As noted, philosophy, at least a certain form of philosophy, may be associated, as it is in Deleuze and Guattari, with the invention of concepts as a revolutionary practice, which thus becomes its normal practice. In this respect, this practice is closer to that of art, which, while it may use a tradition, is compelled to be innovative in each work, but only in this respect, because the nature of the conceptual and artistic composition remains different. The relationships among mathematics, philosophy, and art have been an important and sometimes defining subject in the history of philosophy, from Plato on. It is, accordingly not surprising that art, as defined by artistic composition, versus philosophy, as defined by concepts, or versus mathematics and science, as defined by formal structure and logical propositions (and, in science the concept of reference), was an important part of Deleuze and Guattari’s argument for defining philosophy. Art is beyond my scope here. On the other hand, it is difficult to bypass the fact that Deleuze and Guattari deny that concepts in their sense are found in mathematics or science, a denial that can be misleading for understanding the nature of conceptual thinking in either domain.

They even declare that “it is pointless to say that there are concepts in science [including mathematics],” adding that “even when science is concerned with the same ‘objects’ [as philosophy] it is not from the viewpoint of the concept; it is not by creating concepts” (Deleuze and Guattari 1996, p. 33). This, in my view problematic or at least insufficiently qualified, assessment also includes Riemann’s thinking concerning manifolds and spatiality: “Riemannian functions of space [one presumes, those defining the mathematical structure of his manifolds], for example, tell us nothing about a Riemannian concept of space peculiar to philosophy: it is only to the extent that philosophy is able to create it we have a concept of a function” (Deleuze and Guattari 1996, p. 161). That may be true as concerns some philosophical “Riemannian” concepts of space, such as that of Henri Bergson (to whom Deleuze often refers in this connection). Perhaps this is what this statement primarily refers to, insofar as at stake in it is a Riemannian concept of space peculiar to philosophy and not Riemann’s concept of space as a mathematical concept. Still, it would be difficult to agree that “it is only to the extent that philosophy is able to create it we have a concept of a function.” The opposite appears to be the case: one can only have a function to the extent that mathematics was able to create its concept or the concept of function in the first place. This is certainly so in the case of those functions that appear in Riemann’s concept of manifold or in his analysis of functions of a complex variable, which led him to his concept of a Riemann surface. Deleuze and Guattari add irrational numbers as another example, “the concept, on the other hand, refers not to series of [rational] numbers [series defining irrational numbers mathematically] but to a string of ideas that are reconnected over a lacuna (rather than linked by continuation)” (Deleuze and Guattari 1996, p. 161). Is not, however, a new mathematical concept, such as that of an irrational number (any such concept) equally defined by a string of new ideas reconnected over a lacuna left by earlier ideas? Was it not how Dedekind invented his concept of real numbers? One could in principle argue, as Deleuze and Guattari might have, that a mathematical definition of a concept by putting together its components, such as that of a group as a particular organization of its elements, represents a philosophical, rather than mathematical, thinking within mathematics, and as such defines a philosophical concept that underlies this mathematical concept. As noted, Deleuze and Guattari see this situation, or more generally, the relationships between philosophical and mathematical or scientific thinking, or between either thinking and art, in terms of “interference” with each other (Deleuze and Guattari 1996, pp. 217–218). There may be philosophical or, again, artistic dimensions to mathematical or scientific thinking, especially in its intuitive part, involved, for example, in plausible reasoning (Polya 1990; Corfield 2006). It would, however, be difficult to limit this type of creation in this way, even given the exact nature of mathematical or scientific concepts and thinking, which distinguishes them from inexact but (philosophically) rigorous philosophical concepts and thinking. This difference between the exact and the inexact is invoked by Deleuze and Guattari, in part, it is worth noting, via Edmund Husserl’s analysis of the origin of geometry as an exact science vs. inexact, but, again, rigorous, philosophical thinking (Deleuze and Guattari 1987, pp. 407, 483; Husserl 1970). It may ultimately be impossible to strictly maintain this difference, as the exact and inexact can coexist and invade each other, but it is not possible to disregard it either, these gray areas notwithstanding. One might take advantage here of an astute and far-reaching remark of Heisenberg, aimed especially against Kant, in view of quantum physics, but leading Heisenberg to a broader point, which equally and even more directly applies in mathematics:

Any concepts or words which have been formed in the past through the interplay between the world and ourselves are not really sharply defined with respect to their meaning; that is to say, we do not know exactly how far they will help us in finding our way in the world. We often know that they can be applied to a wide range of inner or outer experiences, but we practically never know precisely the limits of their applicability. This is true even of the simplest and most general concepts like “existence” and “space and time.” Therefore, it will never be possible by pure reason to arrive at some absolute truth [in physics or even mathematics as Kant thought it might be].

The concepts may, however, be sharply defined with regard to their connections. This is actually the fact when the concepts become a part of a system of axioms and definitions which can be expressed consistently by a mathematical scheme. Such a group of connected concepts may be applicable to a wide field of experience and will help us to find our way in this field. But the limits of the applicability will in general not be known, at least not completely. (Heisenberg 1962, p. 92)

Kant is not as far from much of Heisenberg’s argument as Heisenberg thinks, and if Kant could have benefited from modern physics, especially quantum theory, or mathematics, beginning with non-Euclidean geometry, he might have agreed with Heisenberg as concerning the role of causality, which he strictly maintained but which was challenged by quantum theory. I shall discuss the question of Kant’s claim concerning our capacity to “arrive at absolute truth” in connection with Riemann’s views of space (a physical concept for Riemann) and geometry in Sect. 4. Kant’s analysis is also characterized by a search for a sharper definition and the limits of applicability of concepts, in part through establishing the connections between them, even if not as embedded in “a mathematical scheme.” It may well be, however, that, although, by virtue of establishing more rigorous connections between concepts, a definition of an inexact philosophical concept is sharper than in the case of our daily concepts, this definition cannot be as sharp as that of a mathematical concept. Such a mathematical concept can also be used in physics, although a physical concept that uses this mathematical concept may not be defined as sharply as a mathematical concept either.

If one adopts this view, as I do here, the difference between inexact philosophical and exact mathematical concepts will be maintained as essential and as defining mathematical concepts, on this point in accord with Deleuze and Guattari’s view. On the other hand, the compositional character, as equally defining both types of concepts, will be maintained, thus challenging Deleuze and Guattari’s unequivocal claim that “the concept belongs to philosophy and only to philosophy,” a claim that they appear to extend to the compositional character of concepts. It is conceivable that they would have accepted the view proposed here. Generally, however, they see the relationships between inexact philosophical and exact mathematical or scientific concepts or thinking as either operative alongside each other or in terms “interference” (in the positive sense of interfering wave fronts rather than in the sense of inhibition), as when “a philosopher attempts to create the concept of … a function (for example, a concept peculiar to Riemannian space or to irrational number)” (Deleuze and Guattari 1996, p. 217). By contrast, in the present view, creative mathematical and scientific thinking is defined by the creation of multicomponent mathematical or scientific concepts, compositionally, if not functionally, analogous or homomorphic to that of philosophical concepts in Deleuze and Guattari’s sense, with each concept defined by the specificity of its composition. Just as there are Plato’s ideas, Descartes’s cogito, and Leibniz’s monads, there are Gauss’s curvature, Riemann’s manifolds, Dedekind’s ideals, and Grothendieck’s topoi in mathematics. It may be that in mathematics and science such signatures below the concept are erased by history and the concepts themselves more commonly modified or translated by means of new concepts than in philosophy. But, apart from the fact that this is not always so, as in the case of Riemann surfaces or Hilbert spaces (redefined as each has been many times as a concept), this forgetting does not diminish the significance of the creation of concepts. These conceptual compositions are not isomorphic because of technical or, again, exact aspects of mathematical and scientific concepts, which enables a greater and perhaps maximal possible, even if, as Heisenberg says, never complete, sharpness of their definitions, which is demanded by and has defined the disciplinary nature of mathematics and science throughout their history.

Another difference, noted earlier, is, in Kuhn’s terms, in the nature of normal, rather than revolutionary, practice in each domain. In philosophy, normal practice consists primarily in understanding, interpreting, and commenting on concepts, while in mathematics and science, the normal practice consists primarily (while still reflecting on and interpreting concepts) in working with frames of reference and other mathematical or scientific formations, propositions, and so forth. It is true that for Deleuze and Guattari revolutionary philosophical practice is the primary form of philosophy. This is, however, not in conflict with the view of creative practice in mathematics or science advocated here, insofar as this practice is equally defined by the invention of concepts.

In addition, according to Deleuze and Guattari, a philosophical concept in their sense is also defined as a problem, as positing a problem along with and even as this concept. This view has a mathematical genealogy, although the concept of a problem well preceded mathematics and was adopted by it and given a mathematical specificity by ancient Greek mathematics. A problem in this sense is not something that, like a theorem, is derived from assumed axioms by means of logical rules, but is something that is posed, created, along with and as a concept. A theorem could be a problem in this sense, when it arises, from a mathematical concept, such as that of Riemann’s concept of manifold or Galois’s concept of a group. In Deleuze and Guattari’s understanding, however, insofar as it remains a philosophical concept, a problem persists rather than disappears in its solutions, which fact enables such a “concept-problem,” as it may be called, to remain “always new,” insofar as, as I understand it, its productive significance is retained in the history following its invention (Deleuze and Guattari 1996, p. 5). This is a complex conception, which is not easy to interpret consistently, let alone definitively. The view of concepts as “concept-problems” that I adopt as applicable to mathematical concepts may only be a version or a variation on the theme, rather than an interpretation, of Deleuze and Guattari’s understanding of concepts as problems. This view, however, appears to me to be workable in mathematics.

First of all, in this view, not all mathematical or scientific concepts as defined above are concept-problems, while all philosophical concepts in Deleuze and Guattari’s sense appear to be for them, which is, however, a secondary matter here. My point here is that some mathematical or scientific concepts are in a way analogous to philosophical concept-problems in Deleuze and Guattari. A new concept-problem, such as that of a group by Galois or a manifold by Riemann, would be introduced, first, just as a concept in order to solve an already existing problem, which defied a solution by means of already existing concepts or perhaps offered such a solution that was unsatisfactory for one or another reason, conceptual, technical, or aesthetic. Finding the algebraic solution to a polynomial equation (or proving the impossibility of doing so) in Galois’s case, or that of properly defining any possible geometry in Riemann’s case, was such a problem. Their way of thinking already reflects two approaches (predictably with gray areas between them or sometimes combining both) to solving a problem, especially a difficult problem: the first is essentially logical, which aims to solve it by means of logical and technical manipulations of previously existing concepts, and the second is essentially conceptual, which uses a new concept or set of concepts, although technical and logical manipulations remain unavoidable of course. Such a concept may be relatively simple, which is not to say trivial, such as that of a group, or (still more nontrivially) of an action of a group, or it may be more complex, such as that of a manifold, or very complex, such as those of Grothendieck. But, even when relatively simple, such a concept usually arises in response to a complex problem, as in the case of the concept of a group, invented by Galois to find an algebraic solution to a polynomial equation (and proving the impossibility of doing so), a very complex problem at the time. By be means of his concept of a group, Galois redefined this problem by establishing that whether such a solution exists or not is related to the structure of a group of permutations associated with the roots of the polynomial, known as the Galois group of this polynomial. Abel’s approach to this problem had elements of this thinking by posing the question whether a solution exists or not, given the structure of the equation, and finding that in general it does not, the finding now known as the Abel-Ruffini theorem or Abel’s impossibility theorem. Paolo Ruffini gave an incomplete proof in 1799, and Abel was the first to prove it in 1824.

As a concept-problem, however, a new concept does more than merely solving a given initial problem. It defines a new problem-posing field in which new types of actual problems emerge and are being solved, while the concept-problem that initiated and governs this field is not something that is “solved” so as to no longer be necessary for this field to continue to develop, which it does by defining new concepts and concept-problems (some of which are new forms of the original concept-problem). Doing so is a far-reaching form of mathematical practice as philosophy. Galois’s concept of a group not only gave the Abel-Ruffini theorem a more general grounding but also, now as a concept-problem, gave algebra a new problem-posing field, Galois theory. This type of invention of a new concept in response to a concrete problem not only in solving this problem but also making this concept a concept-problem, which leads to a new problem-posing field, can happen as an outcome of a deliberate strategy, as it was arguably the case in Grothendieck or (although it is more difficult be certain) Galois and Riemann, or it might happen contingently, in a complex interplay between chance and necessity. In this sense, such concept-problems give rise to theories, such as Galois theory, Riemann geometry, or Grothendieck’s topos theory.

Deleuze and Guattari would have probably seen such concept-problems as philosophical, even when they are operative in mathematics, while seeing mathematics as being about solutions of its problems, a relation that, as discussed in Sect. 3, Deleuze suggested earlier, via Albert Lautman and Jules Vuillemin, in speaking of ideas, rather than concepts, but conceived analogously, still using Abel’s and Galois’s thinking as a key model of this process (Deleuze 1995, pp. 163–164, 179–182). It is true that mathematics is commonly in pursuit of solutions of its problems, and there is often a philosophical dimension to defining such a new problem-posing field. However, in the present view, as just outlined (which I am, again, not attributing to Deleuze and Guattari), a concept-problem, such as a group, is mathematical. It still enables one to solve, or to advance the possibility of solving, concrete problems, beginning, in Galois’s case, with a problem that gave rise to the concept of a group – the problem of finding the algebraic solution to a polynomial equation – but it does so by embedding this original problem in a new mathematical concept-problem, which defines a new problem-posing mathematical field, or even a new field of mathematics, such as Galois theory in algebra and in effect much more. This concept-problem is, accordingly, mathematical, including, again, by virtue of the sharpness of its formal definition, possible for all mathematical concepts, but difficult to achieve outside mathematics.

That need not and here does not mean that such a concept-problem contains or fully determines a problem-posing field it creates, even if only as a potentiality, in a Platonist fashion. This field emerges in a contingent development, as a complex interplay of causal chains and chance occurrences. In general, in adopting a problem-posing versus axiomatic-theorematic view of mathematics, one is more likely to reject a Euclidean, essentially Platonist, view in which axioms are assumed as pregiven truths rather than as created as concepts, as axioms are from the present perspective, a distinction that, as discussed below, was brought into the foreground by the discovery of non-Euclidean geometry. To sharpen this difference, I shall adopt Lakatos’s distinction between a Euclidean theory (paradigmatically based on Euclidean geometry) and a quasi-empirical theory, which is how Lakatos sees mathematics. (“Quasi” reflects the fact that mathematics is “empirical” in dealing with mental rather than material phenomena.) The mathematical practice of figures considered here is quasi-empirical, in parallel with being problem-posing, rather than Euclidean and, hence, axiomatic-theorematic, although some of them, such as Abel, Weil, and Grothendieck might have adopted a more Euclidean ideology of mathematics. This split is not uncommon; and one might even argue, as Lakatos does, that no Euclidean ideology can avoid quasi-empirical practice, thus making a purely Euclidean practice impossible. According to Lakatos:

The methodology of a science is heavily dependent on whether it aims at a Euclidean or at a quasi-empirical ideal. The basic rule in a science which adopts the former aim is to search for self-evident axioms—Euclidean methodology is puritanical, antispeculative. The basic rule of the latter is to search for bold, imaginative hypotheses with high explanatory or ‘heuristic’ power, indeed, it advocates a proliferation of alternative hypothesis to be weeded out by severe criticism—quasi-empirical methodology is uninhibitively speculative.

The development of Euclidean theory consists of three stages: first the naïve prescientific stage of trial and error which constitutes the prehistory of the subject; this is followed by the foundational period which reorganizes the discipline, trims the obscure border, establishes the deductive structure of the safe kernel; all that is then left is the solution of problems inside the systems, mainly constructing proofs or disproofs of interesting conjectures. ([The discovery of] a decision method for theoremhood may abolish this stage altogether and put an end to the development.)

The development of a quasi-empirical theory is very different. It starts with problems followed by daring solutions, then by severe tests, refutations. The vehicle of progress is bold speculations, criticism, controversy between rival theories, problem shifts. Attention is always focused on the obscure border. The slogans are growth and permanent revolution, not foundations and accumulations of eternal truth.

The main pattern of Euclidean criticism is suspicion: Do the proofs really prove? Are the methods used too strong and therefore fallible? The main pattern of quasi-empirical criticism is proliferation of theories and refutations. (Lakatos 1980, pp. 29–30)

Every new concept, especially a concept-problem, is taking a chance (possibly, sacrificing one’s own interests to the rigor of thought) on the future of mathematics or science, or philosophy. The invention of new concepts, as “always new” concept-problems, has, according to Deleuze and Guattari, defined the creative practice in philosophy from the pre-Socratics on. Deleuze and Guattari may have been indebted most to Friedrich Nietzsche’s view of all true philosophy as “a philosophy of the future,” his subtitle to Beyond Good and Evil: A Prelude to a Philosophy of the Future (Nietzsche 1966). One can, I argue, take the same view of many new mathematical concepts, such as that of Galois’s concept of a group or Riemann’s concept of a manifold, although one cannot always know in advance which. Robert Langlands (of the Langlands program fame, a program in part grounded in Galois theory) compared Grothendieck to “Nietzsche’s Philosoph der Zukunft” (Langlands and Shelstad 2007, p. 486). This is no less true about Galois and Riemann, or Abel, Lobachevsky, and Weil, or Langlands, as the Langlands program proved.

3 Algebra: Abel and Galois

My argument in this section is indebted to that of Jules Vuillemin’s analysis, over half a century ago, of Abel and Galois, and their predecessors, such as Legendre and Gauss, and of algebra itself in La philosophie de l’algèbre (Vuillemin 1962) and that of Deleuze, building on Vuillemin’s analysis (Deleuze 1995). This is not to say that my argument follows that of Vuillemin, which has a strong structuralist orientation shaped by the Bourbaki program (the book is dedicated to, among others, Pierre Samuel, a member of Bourbaki), very different from the conceptual and quasi-empirical one of this chapter. Also, the project of Vuillemin’s book is to transpose the methods of algebra, such as those of Abel and Galois, into philosophy, which is not my aim either. My argument absorbs some aspects of Vuillemin’s argument within a conceptual orientation, as Deleuze does, inspired by Vuillemin and Lautman, again, in transposing the methods of Abel and Galois into his theory of ideas (a precursor of his and Guattari’s theory of concepts) and thus into philosophy. Nor, for the reasons explained in the preceding section in considering concept-problems, does my argument quite follow that of Deleuze either, but instead builds on and yet transforms it. According to Deleuze:

Abel elaborated a whole method according to which solvability must follow from the form of the problem. Instead of seeking to find out by trial and error whether a given equation is solvable [in radicals] in general, we must determine the conditions of the problem which progressively specify the fields of solvability in such a way that ‘the statement contains the seeds of the solution’. This is a radical reversal in the problem-solution relation, a more considerable revolution than the Copernican. It has been said [by Vuillemin] that Abel thereby inaugurated a new Critique of Pure Reason, in particular going beyond Kantian ‘extrinsicism’. This same judgement is confirmed in relation to the work of Galois: starting from a basic ‘field’ (R), successive adjunctions to this field (R’, R”, R”’...) allow a progressively more precise distinction of the roots of an equation, by the progressive limitation of possible substitutions. There is thus a succession of ‘partial resolvents’ or an embedding of ‘groups’ which make the solution follow from the very conditions of the problem: the fact that an equation cannot be solved algebraically, for example, is no longer discovered as a result of empirical research or by trial and error, but as a result of the characteristics of the groups and partial resolvents which constitute the synthesis of the problem and its conditions (an equation is solvable only by algebraic means—in other words, by radicals, when the partial resolvents are binomial equations and the indices of the groups are prime numbers). The theory of problems is completely transformed and at last grounded, since we are no longer in the classic master-pupil situation where the pupil understands and follows a problem only to the extent that the master already knows the solution and provides the necessary adjunctions. For, as Georges Verriest remarks, the group of an equation does not characterize at a given moment what we know about its roots, but the objectivity of what we do not know about them. Conversely, this non-knowledge is no longer a negative or an insufficiency but a rule or something to be learnt which corresponds to a fundamental dimension of the object. The whole pedagogical relation is transformed—a new Meno—but many other things along with it, including knowledge and sufficient reason. Galois’s progressive discernibility unites in the same continuous movement the processes of reciprocal determination and complete determination (pairs of roots and the distinction between roots within a pair). It constitutes the total figure of sufficient reason. (Deleuze 1995, pp. 179–180)

This is a radical transformation of mathematical practice as philosophy and, on this pattern, of philosophy itself, including the pedagogy of ideas or concepts, which philosophy has always been. The argument also implies that the pedagogy of mathematics, at least as it implicitly functions in mathematical practice (and possibly even in the actual teaching of mathematics, which is, however, a separate subject). Galois’s thinking has a more revolutionary significance for both Vuillemin and Deleuze. Nevertheless, according to Vuillemin, Abel transforms algebra, as the inventor of “a general method”:

In the first place, we analyze, in its most general form, a mathematical relation or a defined set of such relations, which make it possible to determine a property, of which we do not yet know whether or not we can attribute it to a class of beings: by example the character of being algebraically solvable, of being convergent, of being expressible by a defined number of functions of a certain class. In the second place, one considers the class of beings to which it is a question of attributing this property [...]; we analyze these beings from a general point of view, we define the relationships to which their nature allows them to be subjected. Finally, this double examination reveals the cases of incompatibility (demonstrations of impossibility) and, possibly, indicates the way to find the new required relationships in cases of possibility. (Vuillemin 1962, pp. 214; translation mine)

Thus, rather than looking for the solution to the equations of the fifth degree, one asks whether such a solution is possible, in the first place, for any polynomial equation. Abel’s new “idea of ​​a general method consists in giving a problem a form such that it is always possible to solve it” (Vuillemin 1962, pp. 209; translation mine). What Abel then shows is that the conditions for any equation to be solvable are not, in general, fulfilled once the degree of the equation is more than four. In this way, by reversing the method of his predecessors, which proceeded from the particular to the general, by generalizing the particular, to proceeding from the general to the particular (the general conditions of solubility of algebraic equations are satisfied for any specific equations of any degree less than five), Abel becomes one of the creators of the modern algebraic method, which Vuillemin sees as, in effect, structuralist in Bourbaki’s sense. Against Kant, who limited his construction of the proof of impossibility by positing the possibility of an experience as an exclusive criterion of knowledge, “general demonstrations, in the sense of Abel, change the modality of the proof. Particular demonstrations are actual [réelles]: they assume as their grounding principle the possibility of the experience given in the affect of sensation. The general proofs deal with the possible and start from the concept alone, disregarding the restrictive conditions of sensitivity” (Vuillemin 1962, p. 216; translation mine). This initiating concept is, thus, defined not by a generalization from particulars, but as a new structured entity. Mathematics needed Galois in algebra (a group) and then Riemann in geometry (a manifold) to be expressly rethought in this way. Abel thus introduced a new critique (in Kant’s sense of an analysis of concepts) of pure reason, at least in mathematics, but, as both Vuillemin and Deleuze argue, extendable beyond mathematics.

While (independently) adopting the same “general method” as that of Abel, Galois brings a new, an even more revolutionary, dimension to this critique by his invention of the concept of a group. Some elements of group-theoretical thinking are found in Abel, or Legendre, earlier, but the concept carries Galois’s signature, and it was conceived in accord with a concept of concept as defined here. It was, moreover, accompanied by another, equally innovative and important concept, that of an action of a group on another structured entity, also defined as a concept, such as an algebraic field or vector space.

For Vuillemin, deeply steeped in Bourbaki’s ideology, the emphasis is on the idea of structure, and he sees the concept of a group as the first algebraic structure, introduced and analyzed as such, in the history of mathematics. Galois’s thinking, then, is both akin to that of Abel as concerns Abel’s philosophy of solving problems, as just considered, and reaches beyond Abel by bringing in the role of a structure and the concept of a group as an entity (we would now say a set) with a structure, a relationally organized multiplicity. Galois, for Vuillemin, becomes the first modern algebraist, who associates with solving a problem the analysis of the structure from which this problem arises, and which makes it possible or not to solve it, such as the problem of finding algebraic solutions of polynomial equations. Vuillemin’s structuralist perspective is not out of place. One cannot avoid structures, as all concepts are structures, compositions, and hence, merely dismiss structuralism (Corry 2003). My point here is the emergence of structures as concepts.

Indeed, as in the case of Abel’s thinking, one of Vuillemin’s main points is more about concepts than structures, keeping in mind that a concept is a structure, too, and conversely any given structure qua structure has to be defined and thus may be a concept, including in the present sense of problem-concepts, as it indeed was for Bourbaki. Vuillemin distinguishes two types of abstraction: the first is conventional, defined by an increasing generalization from the particular to the general, and the second, found in group theory, is conceptual in the present sense. It defines the differences between specific individual cases, because it constructs these cases, from the initial concept, such as that of a group. Vuillemin sees the second abstraction or, more precisely, “formalization” in terms of a double abstraction: the first is that of the formal presentation of the group, as a concept, in terms of algebraic symbols and their relations; the second is that defined by providing a method that makes it possible “to construct individuals, no longer in intuition according to imperfect [more arbitrary] schemes, but [determinately] in the concepts themselves” (Vuillemin 1962, pp. 288–289; translation mine).

Thus, one deals with an algebraically formalized conceptual practice, in accordance with the argument of this chapter, and redefines algebra accordingly, ushering in a new mathematical practice as philosophy. In Galois’s and (more extensively) in Abel’s case, one also deals with a parallel shift to the conceptual algebraization of analysis, as well as, in effect, geometry (where this shift was already initiated with the analytic geometry of Fermat and Descartes). The subsequent “returns” of geometry still use and are often based on this lateralization rather than dispensing with it (Plotnitsky 2021a). In noting that, in the wake of Galois, “modern mathematics is therefore regarded as based upon the theory of groups or set theory rather than upon differential calculus,” Deleuze adds: “Nevertheless, it is no accident that Abel’s method consisted in giving a problem a form such that it is always possible to solve it, concerned above all the integration of differential formulae” (Deleuze 1995, pp. 180–181).

Galois’s contribution was ultimately more fundamental conceptually, as based in the concept of a group as a multiplicity, in effect as a set, with a structure, an entirely new type of concept. As a concept-problem, this concept defines a new problem-posing field through the new mathematical formulations of existing problems (such as that of the solution of polynomial equations in radicals) and posing new problems, and the simultaneous establishing of their conditions of solvability. This new thinking changes the future of mathematics as a field of solving problems, because both these problems and the conditions of their solvability change, in particular, by asking under what conditions a given problem (such as that of finding the solution of polynomial equations in radicals) can be solved. Abel does something similar, but there is no concept of the type a group is. Galois, again, also creates the concept of an action of a group on another structural entity, such as a field or a manifold in geometry.

A momentous role of Galois’s concept of a group for algebra has been akin to that of Riemann’s concept of manifold for geometry. It may not, in retrospect, be surprising that, in the wake of both Galois and Riemann, with the work of, to mention some among the most prominent names, Felix Klein, Sophus Lie, Poincaré, Élie Cartan, and Weyl, the role of group theory – the action of a group on a manifold – became so crucial in geometry and elsewhere in mathematics or physics. Most elementary particle physics is now understood and practiced in terms of group representations and symmetries they define (irreducible representations correspond to elementary particles). The examples are numerous and will undoubtedly continue to accumulate. One of them is especially intriguing because, expectedly, it establishes remarkable connections between Galois theory and quantum field theory (QFT), specifically one of the most important and still controversial aspects of it, namely, the so-called renormalization. I would like, accordingly, to briefly comment on these connections in closing this section.

It was realized by the early 1930s that the computations provided by quantum electrodynamics (QED), the original form of QFT, were reliable only to a first order of perturbation theory. (Perturbation theory is a set of approximation methods using mathematical perturbation for describing a complex quantum system by using a simple one with a known solution and adding to it a “perturbing” Hamiltonian representing a small disturbance of that simple system.) These computations led to the appearance of infinities or divergences when one attempted to use the formalism for calculations that would provide closer approximations matching the experimental data. These difficulties were eventually handled through the renormalization procedure, which became and has remained a crucial part of QFT, because it still contains these divergencies, specifically certain divergent and, hence, mathematically illegitimate integrals. Roughly speaking, the renormalization procedure manipulates such infinite integrals by, at a certain stage of calculations, replacing them with finite integrals through artificial cut-offs. These cut-offs have no proper mathematical justification and are performed by putting in, by hand, experimentally obtained numbers that make these integrals finite, which removes the infinities (which happen to cancel each other) from the final results of calculations (e.g., Teller 1995, pp. 149–168.) At some point of the history of renormalization, a new concept (a concept-problem), that of the renormalization group, was introduced, with the work of Kenneth G. Wilson giving this concept its current form. The renormalization group is a mathematical technique, part of the so-called effective QFTs, that enables one to properly investigate the changes of a quantum system as represented (“viewed”) at different scales. It reflects the changes in the underlying force laws of QFT, changes due to the fact that the energy scale at which physical processes occur varies, with energy-momentum and resolution distance scales obeying the uncertainty relations. Actual computations are difficult and performed by means of various mathematical techniques, massively assisted by, and in fact impossible apart from, the use of very powerful computers. Conceptually, one evaluates the transformations in the formalism in accordance with Feynman’s diagrams, which are, however, only heuristic guides to such calculations in most views, including Richard Feynman’s own (e.g., Plotnitsky 2021b, pp. 300–302).

Quite unexpectedly, Galois theory can be brought up to recast the mathematics involved in a new and more rigorous form. The technical complexities are, again, formidable, even for most working in QFT, including among other stratospheric techniques, Grothendieck’s theory of motives. These complexities may, however, be bypassed here. My point is a remarkable fact itself of the role of a Galois group in renormalization theory. The idea was initially suggested by Pierre Cartier, who designated the corresponding Galois group a “cosmic Galois group,” in view, one suspects, of its potential significance in the ultimate QFT, possibly including gravity (Cartier 2001). It is sufficient to consider here the basic statement of how a specific (“motivic”) Galois group enters, given by Alain Connes and Matilde Marcolli’s article, which explains the essential point. The article and their related work also provide necessary formal definitions for a suitably informed reader, as the mathematics is formidable and requires advance levels of training. According to them:

The divergences of quantum field theory are a highly structured phenomenon [even beyond the renormalization group, which already structures them]. More precisely, they provide data that define an action of a specific “motivic Galois group” U* on the set of physical theories.

In particular, this exhibits the renormalization group as the action of a one parameter subgroup \( \mathbbm{G} \)a ⊂ U* of the above Galois group. …

The natural appearance of the “motivic Galois group” U* in the context of renormalization confirms a suggestion made by Cartier in [Cartier 2001], that in the Connes–Kreimer theory of perturbative renormalization one should find a hidden “cosmic Galois group” closely related in structure to the Grothendieck–Teichmüller group. The question of relations between the work of Connes–Kreimer, motivic Galois theory, and deformation quantization was further emphasized by Kontsevich in [Kontsevich 1999]. …

The “motivic Galois group” U* acts on the set of dimensionless coupling constants of physical theories, through the map of the corresponding group \( \mathbbm{G} \) to formal diffeomorphisms constructed in [Connes and Kreimer 2001].

This also realizes the hope formulated in [Connes 2003] of relating concretely the renormalization group to a Galois group. …

These facts altogether indicate that the divergences of Quantum Field Theory, far from just being an unwanted nuisance, are a clear sign of the presence of totally unexpected symmetries of geometric origin. (Connes and Marcolli 2004, pp. 4073–75) (See also Connes and Marcolli (2007, pp. 95–136) for a comprehensive exposition of these problematics).

The last conclusion is remarkable, because it counters the common sentiments (some of which have been more strongly negative) that have governed the attitude toward the divergencies of QFT and renormalization from 1930 on. In addition, it brings in yet another, highly complex, instance of the interplay between algebra and geometry into quantum theory. The physical meaning and implications of this mathematics have hardly been explored as yet, even though there were some developments along these lines since. My main point is, however, an unexpected and potentially far reaching role of Galois’s group, as a concept-problem in the present definition, in linking algebra, geometry (“symmetries of geometric origins” invoked by Connes and Marcolli), and fundamental physics. Moreover, this role changes our perspective on renormalization in QFT, which has appeared as inhibiting the relationships between mathematics and fundamental physics, but might reflect deeper aspects of these relationships, thanks to the concept-problem of a Galois group, which emerge along with the concept-problems of a group and an action of a group in the work of Galois.

I now move to Riemann and geometry. Galois, however, will make a return in the following section, via the role of Galois theory in algebraic geometry, specifically, Grothendieck’s étale cohomologies, another momentous impact of Galois’s way of thinking in mathematical practice as philosophy. Grothendieck’s theory of étale cohomologies was, actually, one of the motivations for his concept, concept-problem, or a motive, and motive theory, which, among its numerous applications, eventually led to the role, just considered, of motivic Galois groups in renormalization in QFT, a theory, again, indebted, on several counts, to Riemann’s concepts as well, as is in fact the concept of a cosmic Galois group. It might be added that the so-called anti-de-Sitter spaces, which are the Lorentzian analogies of Lobachevsky spaces, play an important role in developments of QFT theory and its extensions, such as string and brane theories – the holographic principle and Juan Maldacena’s AdS/CFT (anti-de-Sitter/conformal field theory). There are also deep connections between these ideas and motivic Galois’s groups, connections noted in (Kontsevich 1999), cited above, a key work on the role of motivic Galois theory in QFT.

4 Geometry: Lobachevsky and Riemann

My main focus in this section is Riemann’s mathematical practice, as the most explicit exemplification of my main thesis concerning mathematical practice as philosophy, as defined by the invention of concepts. While characterizing all of Riemann’s work, this philosophy is especially manifested in his rethinking of the foundations of geometry in his Habilitation lecture, “On the Hypotheses That Lie at the Foundations of Geometry” [Ueber die Hypothesen, welche der Geometrie zu Grunde liegen] (Riemann 1854). Lobachevsky’s thinking and work, however, not only, obviously, have their own revolutionary place in the history of geometry, but are important in the present context as well. The differences in their thinking and mathematical practice remain important. It would be difficult to argue that, unlike Gauss (expressly mentioned by Riemann in the lecture), Lobachevsky was a precursor of Riemann, and as often noted, it is doubtful that Riemann was familiar with Lobachevsky’s work. On the other hand, it would be difficult to argue either that Riemann was unaware of Lobachevsky’s discovery of non-Euclidean geometry (of the hyperbolic type), especially given that Gauss championed Lobachevsky’s work, as against that of Bolyai.

Specifically, I shall argue that, while, unlike Riemann, expressly addressing Euclid’s fifth postulate, Lobachevsky’s thinking implicitly challenges the axiomatic approach, as developed from Euclid on, to geometry at its very core. While not, like Riemann, overtly grounding geometry in hypotheses and concepts, Lobachevsky’s discovery in effect showed that the axioms of geometry in general, rather than only the fifth postulate, and, perhaps, all our axioms, including in physics, are hypotheses, based, however imperceptibly, in concepts, such as that of parallel lines in Euclidean geometry. The latter may be a notion arising from our general phenomenal intuition, which may still not be able to conceive how it is possible to have more than one parallel line in the same plane crossing a point outside this line. As a mathematical object, however, either is a concept, and the fact that there is one parallel line to a given one going through a point outside it is a hypothesis, either as concerns a physical space, with which the term geometry was associated at the time, or as concerns a possible mathematical geometry. Lobachevsky, thus, also (in accord with a key trend of modern mathematics), suggested a dissociation of geometry from physics, which is, arguably, one of the reasons why he, initially, referred to his hyperbolic geometry as imaginary geometry. Of course, he also thought that hyperbolic geometry may correspond to that of actual physical space on a broader scale, which, however, still allows for its mathematical independence. He gave his geometry or geometries other names, on which I shall comment below, merely noting now that his final name “pangeometry” may have been a reflection of the view, akin to that of Riemann, that what is mathematically shared is geometrical thinking. This thinking enables us to create new mathematical worlds, “out of the nothing,” as Bolyai famously said about his invention of hyperbolic geometry (Stäckel 1913, p. 86). These worlds are creations of thought from its insubstantial materials, “such stuff/As dreams are made on,” in William Shakespeare’s even more famous phrase (The Tempest, IV, 1, ll. 156–157). This thinking, then, can lead to either Euclidean or hyperbolic geometry, and possibly still other geometries, eventually an uncontainable multitude of geometries with no connections to physics, although for Lobachevsky, Bolyai, and Riemann the terms space and geometry still referred to physical space and its geometry.

The landscape of geometrical thinking changed even more radically following Riemann’s Habilitation lecture, given in 1854 but first published by Dedekind in 1868, 2 years after Riemann’s death (On Riemann’s subsequent developments of his geometrical ideas and their application to physics, see essays in Ji et al. (2017), where the earlier (longer) version of this discussion of Riemann was published (Plotnitsky 2017). The present version revises the argument offered there on several key points). The lecture offered a radical rethinking of space and geometry, as against the preceding history of both, from before Euclid to the discovery of non-Euclidean geometry. This rethinking was based on the concept of manifold or manifoldness [Mannigfaltigkeit], a major mathematical innovation. Establishing the possibility of non-Euclidean geometry by Lobachevsky and Bolyai was a great discovery, with profound implications for mathematics, physics, and philosophy, and indeed culture. Riemann’s thinking and argumentation departed even more radically from Euclidean geometrical thinking concerning space and geometry than those of his predecessors, including, Lobachevsky and Bolyai, or even Gauss, Riemann’s teacher and, arguably, the main precursor as concerns his geometrical thinking. The question of the fifth postulate of Euclid played no role in Riemann’s argument. Riemann pursued a different way of thinking, following Gauss’s ideas concerning the curvature of two-dimensional surfaces (Riemann also mentions Johann F. Herbart as an influence or guidance, but, even leaving aside the fact that Riemann was ambivalent and sometimes critical of Herbart’s philosophy, this influence does not appear to me ultimately that important, especially in the present context). Riemann’s thinking was also, and correlatively, problem-posing, or quasi-empirical, rather than axiomatic-theorematic, as axioms of geometry were neither Riemann’s starting point nor figured significantly, if at all, in his investigation, in the way they did in that of Lobachevsky or Bolyai. As I argue here, however, this juxtaposition is not unconditional. The proximities between their thinking is significant, and the fact that Riemann almost certainly knew about their findings is not insignificant either. Lobachevsky’s thinking, I argue, had important conceptual and problem-posing aspects. Riemann’s approach, however, through the invention of a new concept, concept-problem, that of manifold, establishing a new problem-posing field (extending beyond geometry), led him beyond a single alternative to Euclidean geometry to an uncontainable multiplicity of geometries and, in principle, to an even greater multiplicity of possible spaces, because some among them would not admit geometry. The latter circumstance, not considered by Riemann, became important for the development of topology, influenced by Riemann otherwise, beginning with his concept of Riemann surfaces, which played a major role in the emergence of topology, as well as algebraic geometry.

The implications for physics, extending those of the discovery of a single non-Euclidean geometry, were dramatic as well. Riemann’s contribution to physics in the lecture was nearly as revolutionary as his contribution to mathematics, extending already major changes of thinking concerning space brought about by the discovery of non-Euclidean geometry. Before this discovery, one and only one geometry would be available for a geometrical description of physical space (which is, again, how the term space was understood at the time) – if one assumes that space or, in Riemann’s astute phrase, “the reality underlying space” could be mathematically described geometrically (Riemann 1854, p. 33). This had been a grounding assumption of classical physics, from Galileo and Newton on, or of modern philosophy, from Descartes on. Kant’s epistemology of phenomena (referring to appearances or representations constructed by our minds) versus noumena or things-in-themselves (referring to how things, material or mental, exist independently of how we perceive or represent them) qualified this assumption (Kant 1997). While important, this qualification did not change the essential import of this assumption for physics. Kant’s epistemology does not affect measurements that allow us to ascertain the observable properties of space, specifically whether the observable space is Euclidean (flat) or non-Euclidean (curved), or is three-dimensional (the only space we can phenomenally imagine) or not. Possible deviations from the flatness of physical space could be established indirectly by using measuring instruments, while our phenomenal experience of space could still be Euclidean. Riemann reflected on this situation in his lecture, leaving the question of the ultimate nature of physical space or of the reality underlying space to future physics. Kant, by contrast, assumed that physical space is Euclidean and three-dimensional, or at least that it was unlikely to be anything else. He allowed that such alternatives, specifically that space may have a dimension higher than three, are logically possible in his early (1747) work, but rejected the idea as improbable, on theological grounds (Kant 2012; de Bianchi and Wells 2015, Pesic 2007, pp. 3–4, 7).

Theological grounds would not have deterred Riemann, who was originally trained in theology, which might have even helped his thinking concerning space and geometry, including his concept of manifold. 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. Nor would they have been likely to deter Lobachevsky, which is, however, a separate subject, except for the fact that he only considered mathematically two and three dimensional space. It was, however, mathematically crucial that his geometry was three-dimensional, as was that of Bolyai, but it appears not that of Gauss, who thus came short of this discovery, even if he ever reached the establishment of the two-dimensional hyperbolic plane. In Riemann’s view, that space was a three-dimensional Euclidean manifold was a hypothesis, reasonably well confirmed by the measurements performed at the time, but a hypothesis nevertheless, the truth of which cannot be ascertained by reason alone, as Kant thought possible.

In any event, the situation dramatically changed, mathematically, physically, and philosophically, and even culturally, with the discovery of non-Euclidean geometry. That actual physical space may not be Euclidean (the three-dimensional nature of space was, again, rarely contested) made this discovery a major event, even though no measurements made at the time showed any deviation from Euclidean geometry. The possibility was momentous enough. In reality, this challenge was even more radical, including vis-à-vis Kant, because it went beyond questioning of the nature of physical space and geometry to that of our axiomatic presuppositions in general. Even if Lobachevsky himself only realized this radical nature and implications of this challenge to some degree (which was, I would surmise, likely), his discovery revealed that the fifth postulate and, by implication, all axioms or, in the first place, definitions are conceptual constructions based on our general phenomenal intuition but reaching beyond it as mathematical concepts. Axioms are not preexisting truths but are assumptions based in conceptual constructions. This difference became more radical and pronounced with Cantor’s concept of the continuum. A given set of these constructions allows one to define a consistent geometrical system, that of Euclidean geometry. But an alternative set of axioms, such as that without any postulate concerning parallel lines (now known as “neutral geometry” or “absolute geometry”) or that in which the fifth postulate is replaced by the postulate that there are at least lines parallel to a given line crossing a point outside this line (hyperbolic geometry), implies another consistent geometry. Mathematically, it is a matter of decision (a preferable category to that of choice) in which of them or possibly in several one pursues one’s mathematics, a decision the nature of which is ultimately not mathematical. Physically, any such geometry becomes a hypothesis concerning the nature of physical space, and as such is subject to testing, verification, qualification, refinement, and so forth. These considerations, although not all of them were pursued by Lobachevsky, give us an insight into his geometrical designations – imaginary geometry, as a construct of mathematical imagination, under the assumption that physical space and geometry are Euclidean, and then pangeometry, which, as Athanase Papadopoulus notes, refers to “general geometry,” which included all geometries known at the time, including spherical geometry in two-dimensions (Papadopoulos 2010, pp. 229–231). (There were no known elliptical geometries in three dimensions before Riemann.) Pangeometry may, arguably, also be seen as referring to geometrical thinking itself.

Physically, there is yet another possibility inherent in Lobachevsky’s discovery, which does not appear to have been entertained by Lobachevsky himself, but was expressly considered by Poincaré, who rediscovered one of Eugenio Beltrami’s models for hyperbolic geometry, which helped to prove that it was equiconsistent with Euclidean geometry. (Lobachevsky might have had some intuitions concerning this equiconsistency as well [Papadopoulos 2010, p. 250].) The decision concerning the geometry of physical space may be arbitrary (not the same as random!), in other words, a convention. Whether it actually is, as Poincaré thought, is a matter of debate, which was given a new impetus with general relativity, discovered in 1915, after Poincaré’s death (in 1913). Nevertheless, the possibility, which is inherent in Lobachevsky’s discovery, is significant. His discovery might indeed be seen as more radical than that of Copernicus, insofar as it would be parallel to the assumption that the decision between the geocentric and heliocentric view of the solar system is a mathematical convention (something, ironically, the Church wanted Galileo to say, admittedly on different grounds). In view of general relativity, the heliocentric view is a convention, much more convenient mathematically as it is.

Of course, Lobachevsky’s (or Bolyai’s) decision to consider the fifth postulate as independent of, but consistent with, those of “general geometry” and to consider the hyperbolic alternative as an equiconsistent system, could not emerge without technical mathematical work. And yet, as in the case of Abel’s and Galois’s rethinking the problem of the solution of the polynomial equations in radicals, it was an entirely new way of posing the problem of geometry (as that of consistency of the set of axioms defining geometry, axioms that are mathematical concepts) and thus also a new concept of geometry. That of Riemann was still more radical, especially as concerns the multiplicity of possible geometries, which required an invention of a new concept-problem and new mathematics, those of manifolds. As a result, Riemann also gave a firmer conceptual grounding and more radical dimensions to the problem of geometry. Before I turn to Riemann, I would like to comment on the following, in my view, misleading, assessment of Lobachevsky’s and Bolyai’s discovery by Jeremy Gray and José Ferreirós:

The new geometry posed a radical challenge to Euclidean geometry, because it denied traditional geometry its best claim to certainty, to wit, that it was the only logical system for discussing geometry at all. It also exploited the tension known to experts between the concepts of straightest and shortest. But in other ways it was conventional. It offered no new definitions of familiar concepts such as straightness or distance, it agreed with Euclidean geometry over angles, it merely offered a different behaviour of parallel lines based on a difference in the distant behaviour of straight lines. Its proponents did not offer a skeptical conclusion. Bolyai and Lobachevsk[y] did not say: “See, there are two logical but incompatible geometries, so we can never know what is true.” Instead, they held out the hope that experiments and observations would decide. The epistemological price people would have to pay if astronomical observations had come down in favour of the new geometry would, in a sense, have been slight: it would have been necessary to say that straight lines have an unexpected property after all, but one only detectable at very long distances. To be sure, many of the theorems of geometry would then have to be reworked, and their familiar Euclidean counterparts would appear only as very good approximations. But that is broadly comparable to the situation Newtonian mechanics found itself in after the advent of special relativity. (Gray and Ferreirós 2021)

That may be technically correct (although some qualifications would still be necessary), as is the claim that their view was that experiment and observation would decide which describes physical space, a view that Riemann held as well, with deep implications for physics. This assessment, however, misses the radical epistemological and ontological implications of the way of thinking thus brought about concerning our “basic” assumptions, axioms, postulates, and so forth, as no longer basic, but rather as constructions of thought the validity of which can be challenged in physics and in mathematics itself, as for example, when it comes to considering Newtonian physics, as axiomatic. It may be that hyperbolic geometry as such does not challenge physics in the way general relativity (which might have been the main example in this regard for Gray and Ferreirós) and quantum mechanics did as physical theories, but the way of thinking in which hyperbolic geometry was discovered suggests the possibility of questioning the “axioms” of physics. Einstein would have agreed: he did so in questioning our basic assumptions concerning the kinematics of motion in special relativity, or the special status of inertial systems in general relativity. On the other hand, he was unwilling to give up causality and especially realism as basic philosophical “axioms,” which led to his refusal to accept quantum theory as a fundamental theory (a theory describing the ultimate constitution of nature).

The boldness and radical nature of Lobachevsky’s thinking, or that of Bolyai, and their implications for our thought in physics is missing in Gray and Ferreirós’s assessment. If one can create a mathematical “universe out of nothing,” it is no longer easy to assume for modern physics, as a mathematical-experimental science, which one of these mathematical worlds corresponds best to the physical world, and perhaps ultimately none does, as quantum mechanics may imply (Plotnitsky 2021b). It is this spirit that matters more than technicalities invoked by Gray and Ferreirós. They give more credit to Riemann, while, however, still underappreciating the radical nature of his thinking, including as concerns the point of the possible (or impossible) relationships between a given mathematical world and the physical world. Riemann offers a much better way of looking at the connection between radical thinking in mathematics and physics: “Investigations like the one just made here [concerning foundations of geometry], which begin from general concepts, can only serve to insure that this work is not hindered by unduly restricted concepts and that progress in comprehending the connection of things is not obstructed by traditional prejudices” (Riemann 1854, p. 33). Riemann refers to his own mathematics as allowing physics to break with “unduly restricted practice” and “traditional prejudices,” such as those of Euclidean geometry or Newtonian mechanics. But the point is more general and concerns all concepts that become “unduly restricted” unquestionable axioms by forgetting their human origins and history.

Riemann’s argument, again, implied that an infinite number of possible geometries could be associated with physical space or “the reality underlying space.” Any such association is a hypothesis (this is one, the most direct, of the meanings of the “hypotheses” of Riemann’s title) and as such is subject to testing, verification, qualification, refinement, and so forth. These processes can rule out some among possible geometries or require different geometries at different scales, as indeed happens in modern physics. Thus, with the help of Einstein’s general relativity, we know reasonably well certain local physical geometries, say, the one, curved, in the vicinity of the solar system, and even more global geometries, say, that (on average flat) in the Milky Way. Or, more rigorously, the corresponding argumentation works well in physics and astronomy as things stand now. It is, however, much more difficult to be sure concerning the ultimate geometry of the Universe, although the current data seems to suggest that it is, on the average, flat as far as we can observe it. Locally, space could be curved by gravity or contain a singularity, such as a black hole, in accordance with general relativity. It is also possible that there are other Universes with different geometries. Riemann did not envision this possibility, which arises from quantum considerations concerning the Big-Bang origin of the Universe that we observe. However, his discovery of the infinite number of possible geometries is in accord with this idea, the genealogy of which goes back to Leibniz’s concept of compossible worlds, although for Leibniz there existed, created by God, only one world, the best one, in which we live.

The significance and impact of Riemann’s lecture for mathematics, physics, and philosophy were immense. This impact was delayed until its publication, in 1868, 2 years after Riemann’s death, and 14 years after it was presented in 1854, although some of Riemann’s key ideas contained there became known and had their impact earlier. Riemann opens as follows:

As is well known, geometry presupposes the concept of space, as well as assuming the basic principles for construction in space. It gives only nominal definitions of these things, while their essential specification appears in the form of axioms. The relationship between these presuppositions is left in the dark; we do not see whether, or to what extent, any connection between them is necessary, or a priori whether any connection between them is even possible.

From Euclid to Legendre … this darkness has been dispelled neither by the mathematicians nor the philosophers who have concerned themselves with it. The reason [Grund] for this is undoubtedly because the general concept of multiply extended magnitudes [Grösse], which includes spatial magnitudes, remains completely unexplored. I have therefore first set myself the task of constructing the concept of a multiply extended magnitude from general notions of magnitude. It will be shown that a multiply extended magnitude is susceptible of various metric relations, so that space constitutes only a special case of a triply extended magnitude. From this, however, it is a necessary consequence that the theorems of geometry cannot be deduced from general notions of magnitude, but that those properties that distinguish space from other conceivable triply extended magnitudes can only be deduced from experience. Thus arises the problem of seeking out the simplest data from which the metric relations of space can be determined, a problem that by its very nature is not completely determined, for there may be several systems of simple data that suffice to determine the metric relations of space; for the present purposes, the most important system is that laid down as a foundation of geometry by Euclid. These data are—like all data—not logically necessary, but only of empirical certainty, they are hypotheses; one can therefore investigate their likelihood, which is certainly very great within the bounds of observation, and afterwards decide on the legitimacy of extending them beyond the bounds of observation, both in the direction of the immeasurably large [Unmessbargrosse] and in the direction of the immeasurably small [Unmessbarkleinen]. (Riemann 1854, p. 23; translation modified)

These introductory reflections are already profound and far-reaching, and Riemann develops them much further in the lecture. Riemann is, again, not interested in the axiomatic approach to geometry and appears to doubt its effectiveness. In contrast to most previous works on non-Euclidean geometry, the parallel postulate is not the starting point of his investigation. The reason for the ineffectiveness of the axiomatic approach is that the concepts of space and of measuring distances in space, necessary for geometry, are not adequately defined. This is what leaves “the relationships between [axioms] … in the dark. We do not see whether, or to what extent, any connection between them is necessary, or a priori whether any connection between them is even possible.” This questioning of the axioms of geometry was unusual at Riemann’s time, and it was much deeper than customary doubts concerning the parallel postulate, in any event no longer assumed as such by the time of Riemann’s lecture in view the discovery of hyperbolic geometry by Lobachevsky and Bolyai, of which, if not of their actual work, Riemann, as I suggested, was likely to be aware. As discussed earlier, however, their argumentation was not restricted to merely axiomatic thinking and, in effect, questioned the latter. In any event, only Gauss, rather than either Lobachevsky or Bolyai, was expressly invoked in the lecture (There were other 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, Lotze criticized Riemann’s approach anyway, as part of his general critique of non-Euclidean geometry. See Laugwitz (1999, pp. 222–226) and Russell (1897, pp. 97–112)).

Gauss does not appear to have been able to establish a possible existence (in the sense of logical consistency) of non-Euclidean geometry, because, unlike Lobachevsky and Bolyai, he did not do this for the three-dimensional case. In spirit, however, Gauss’s work on the geometry of two-dimensional surfaces and his concept of curvature as intrinsic to a given surface was closer to and must have influenced Riemann’s thinking. For Riemann, Gauss was pursuing a trajectory of thought better suited for the foundations of geometry and more fruitful for its conceptual development. In his famous theorem (theorema egregium), Gauss proved that the curvature of a surface, which he defined as well, was intrinsic to the surface. More precisely, the theorem states that Gaussian curvature of a surface does not change if one only bends the surface but does not stretch it. This means that curvature can be fully determined by measuring angles and distances on the surface itself, without considering the way it is embedded in the ambient (three-dimensional) Euclidean space, which makes the Gaussian curvature an intrinsic invariant of a given surface. Gauss’s theorem thus suggests a two-dimensional conceptual architecture of geometry akin to that of Riemann, who was able to give this architecture a great generality. It suggests that one could see the surface as an independent curved manifold, a new concept and a new philosophy emerging in mathematical practice, and then to generalize this concept to higher dimensions via “the concept of a multiply extended magnitude,” mentioned in this passage and the concept of manifold, which is what Riemann did. These concepts also helped him to generalize to higher dimensions Gauss’s concept of curvature. To do so required yet another new concept, that of the tensor of curvature, and a new form of differential calculus, tensor calculus on manifolds, a generalization of differential calculus, another great invention of Riemann. This calculus was fully developed later (sometimes under the name of absolute calculus), and it played a major role in Einstein’s general relativity.

The independence of the curvature or of geometry of a given manifold of any dimension from its embedding also allows one to determine, at least in principle, intrinsically whether the space thus represented is flat or curved, which is crucial for determining the nature of the physical space we inhabit. For example, as noted, space is curved in the immediate vicinity of the solar system, because of the gravity of the Sun, planets, and other material entities there (or locally around other stars), but, given the current experimental data, appears to be on the overage flat on the scale of the observable Universe. It has never been established definitively or, in any event, agreed upon whether such a determination is rigorously possible, at any scale, as opposed to having a practically effective and possibly, within its proper scope, the best available theory or, as Poincaré would have it, convention, without making a truth-claim concerning “the reality underlying space” (Riemann 1854, p. 33).

Riemann’s thinking was clearly conceptual and/as problem-posing rather than axiomatic-theorematic, quasi-empirical rather than Euclidean: he grounds mathematics, as well as physics, in hypotheses and concepts, concept-problems, such as the concept of manifold and its subconcepts, or associated concepts (such as distance, curvature, and tensor). The problem of space and geometry was seen by Riemann as that of finding concepts that are necessary to define space and to give it a geometry. A general concept of manifold is assumed to be applicable to any possible space, while specific manifolds define different spaces or subspaces. Thus, as foundational thinking, that of Riemann is different from that conforming to Hilbert’s concept of “foundations” [Grundlagen] in the sense of axiomatic foundations, an idea that Hilbert also tried to apply to physics. One of the problems of his famous 1900 list (Problem 6) was that of the development of (mathematized) axiomatic foundations, system(s) of axioms, for physics, on the model of his own Foundations of Geometry (Hilbert 1979). That such a system is possible is a hypothesis. It became clear subsequently that the existence of such a system is a hypothesis even in mathematics, and in view of Gödel’s incompleteness theorems of 1931, any such system cannot be proven to be free of contradiction, once it is large enough to include arithmetic, as most geometries are. While Hilbert’s foundational thinking aimed to bring physics closer to mathematics, by giving physics an axiomatic form, that of Riemann brings mathematics closer to physics by grounding it in hypotheses and concepts, rather than in axioms, by making it quasi-empirical.

The parallel postulate is, again, never mentioned in the lecture, although Euclidean geometry is invoked as the geometry of “flat space,” merely a particular and very special case of geometry, where metrical relations take an especially simple form, defined by the Pythagorean theorem. Euclidean geometry is no longer the starting point of Riemann’s thinking concerning foundations of geometry, which has been noted, although rarely appreciated as a consequence of Riemann’s mathematical thinking as defined by concepts. The starting point of geometry is measurement: the concept of the metrical relation and its instantiations characterizing given spaces and their geometry. The character of this relation is a hypothesis on which a geometry could be based, a hypothesis to be tested physically to establish whether it is in accord with actual physical space. Non-Euclidean geometry (Riemann, again, does not use the term) is introduced as a possible case of geometry, that of a curved, rather than flat, space of either negative or positive curvature, defined by Riemann by the corresponding type of quadratic form determining the metric. Riemann, thus, introduced a more general concept, concept-problem, of geometry, which also allows for a rigorous conceptual grounding to non-Euclidean geometry of either negative or positive curvature. He also introduced non-Euclidean geometry of positive curvature, elliptical geometry. Eclipsed, as it may have been, by Riemann’s overall achievement in the lecture, this was a major mathematical discovery with important cosmological implications, as in anticipating the idea, later considered by Einstein, that the universe may be unbounded and yet finite (As noted, 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). Spherical geometry, known previously, was a two-dimensional example of geometry of positive curvature, and no three or higher dimensional ones were established before Riemann.

Riemann’s thinking may, thus, be seen as returning to pre-Euclidean thinking and to the original meaning of the ancient Greek word geometry as geo-metry – the way of measuring distances and volumes of figures on the surface of the Earth and extending this measurement to volumes of three- or even higher dimensional figures. In order to do so and to have geometry, two concepts are necessary, that of space and that of measurement of space. Axioms, if one needs them, can come later. Besides, as Riemann clearly implies from the outset of his lecture, the axioms of geometry require concepts first, an understanding that, as I said, was implicit in Lobachevsky’s work. This does not mean that one cannot productively pursue an axiomatic approach to geometry, but only that one cannot do so without concepts. Hilbert, again, did so, and he used and invented concepts, but as part of an axiomatic program. This is, however, not how Riemann thought about geometry. The question for Riemann, as for Galois, is always how to define mathematical entities in terms of their conceptual architecture reflecting the problems they respond to, which led them to defining new problem-posing fields by the corresponding concept-problem, that of a manifold and a group, respectively. Besides mathematical axioms are, first, mathematical concepts, and sometimes concept-problems.

Riemann defined the concept of space, again, understood as physical space (as against the concept of manifold, which is mathematical), as a three-dimensional physical instantiation of the concept of continuous manifold, or as something that can be, in an idealized way, represented mathematically by this concept, in accordance with the hypotheses that he assumed as likely given the experimental data then available. While a manifold, as defined by Riemann, may be either discrete or continuous, the concept of continuous manifoldness has a richer and more complex mathematics, and most of Riemann’s lecture is devoted to it. Technically, continuous manifolds considered by Riemann were differentiable manifolds, which means that one can define differential calculus on them. Indeed, they are metrical manifolds, now called Riemannian, which allow for the concept of distance between any two points and thus for geometry (there were no discrete geometries yet. 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 nondiffeomorphic differentiable structures. Thus, there is a continuum of nondiffeomorphic differentiable structures of ℝ4. I shall, however, speak of continuous manifolds, following Riemann and his juxtaposition between continuous and discrete manifoldness. In modern use, the term manifold more customarily refers to continuous (but not necessarily differentiable) manifolds, although one also refers to discrete manifolds, which have topological dimension zero. Discrete and continuous manifolds do not appear to have that much in common and may be seen as different concepts. These two concepts could, in modern understanding, be subsumed under the same concept. It is conceivable that Riemann thought along similar lines, which would explain his choice of the term manifold for both discrete and continuous manifolds. It is difficult to be certain on the basis of his lecture or his other writings, because the term had a more general use at the time. Thus, Cantor, initially referred to sets as Mannigfaltigkeiten but eventually switched to Mengen. In any event, the difference between these two types of manifolds is crucial, in general and for Riemann, especially for his analysis of physical space. Riemann stressed the significance of the relationships between continuity and discontinuity for mathematics, physics, and philosophy (e.g., [Riemann 1991, pp. 515–524; Ferreirós 2006, pp. 77–80]).

Riemann’s concept of continuous manifoldness was a new concept of geometrical multiplicity. It is a multiplicity of local subspaces, most specifically those, “neighborhoods,” associated with each point, out of which a given space is composed. Riemann defines first the concept of “n-dimensional magnitude,” which allows one to determine a position in a manifold by n numerical determinations, in the same way a position is determined by coordinates in the Euclidean space of n dimensions. A crucial and even the single most defining feature of the concept of (continuous) manifold is that it is conceived as infinitesimally Euclidean. This makes a continuous manifold into a conglomerate of local, continuously connected, open neighborhoods around each point. The concept of neighborhood, assumed to be infinitesimally flat (Euclidean), is a component-concept of the concept of manifold. The concept of manifold as composed out of local neighborhoods is extendable to a more general concept of topological space, in which case local neighborhoods need no longer be Euclidean and can be defined with a great degree of generality. It is true that Riemann did not define a concept of topological space as such, although it is conceivable, especially given his concept of a Riemann surface, that he entertained this type of idea, just as he, expressly, entertained the idea that “the reality underlying space” may be discrete at a very small scale. In any event, the conceptual architecture defining topological spaces is a nearly inherent generalization of Riemann’s conceptual architecture of a manifold as a “space” composed of neighborhoods. Riemann surfaces, too, were recast in terms of manifolds. Weyl, who clearly considered a Riemann surface to be a concept, was the first to define them as manifolds (Weyl 2013). Riemann, however, must have realized that they were manifolds, and the concept of a Riemann surface was part of the genealogy of the concept of manifold.

In the case of Riemannian manifolds, while each neighborhood is infinitesimally Euclidean, the manifold as a whole is, in general, not, except in the special case of Euclidean manifolds. A manifold may be negatively or positively curved, and, which is another major innovation of Riemann, this curvature can also be variable. Riemann defined the metric form as a quadratic differential function of coordinates

$$ 1/\left(1+\frac{\alpha }{4}\sqrt{\sum d{x}^2}\right) $$

which is the only real formula in his lecture (discounting the coordinate expression for the line element). Riemann also assumed that the transition from one local coordinate system to another was differentiable. Thus, he de facto considered differentiable manifolds with positive metrics, Riemannian manifolds. In modern terms, each such a manifold has a differentiable section of positive-definite quadratic forms on the tangent-bundle. While, however, modern technical language can bring out deeper mathematical aspects of Riemann’s concepts, it can also obscure and displace how Riemann thought, a displacement sometimes found in twentieth-century English translations of Riemann, including his Habilitation lecture.

Although it was Riemann’s theory of continuous or differentiable manifolds that had the greatest impact, the concept of discrete manifolds was important for Riemann’s argument, and it is important for the modern understanding of both spatiality and geometry in mathematics, physics, and philosophy. While a discrete manifold has topological dimension zero, it may still be seen as multiply extended, if defined as forming a very fine lattice with very small intervals between points, which can be “filled,” as it were, to form a continuous space of the corresponding topological dimensions. It is also possible to introduce metrical relations for discrete manifolds. This concept is important in the context of the relationships between physical forces in nature and the nature of space or, to return to Riemann’s terms, the physical “reality underlying space,” although Riemann was cryptic on such metrical relations. Crucially, however, he allows for the possibility that the physical “reality underlying space” might be “a discrete manifold” (Riemann 1854, p. 33). This possibility had been entertained even before Riemann and has been often considered since. Also, mathematically, finite geometries were beginning to be developed, usually in more axiomatic ways, around Riemann’s time as well, later on also under the impact of his geometrical thinking (For some of these developments, see Pambuccian et al. (2017) and other articles in Ji et al. (2017)). Riemann saw the relationships between continuity and discontinuity as foundationally central to mathematics, physics, and philosophy (e.g., [Riemann 1991, pp. 515–524; Ferreirós 2010, pp. 77–80]), a view confirmed by the subsequent developments in the foundations of mathematics, from Dedekind and Cantor on, and quantum physics. The latter uses continuous (technically, again, differentiable) mathematics, that of Hilbert spaces over ℂ, to predict, in probabilistic terms, irreducibly discrete phenomena, that is, phenomena that are not, and that possibly cannot be, assumed to be connected to each other by a continuous physical process. The genealogy of this mathematics is still Riemannian, given that Riemann has already envisioned, along with discrete manifolds, infinite-dimensional ones. Tensor calculus, too, is a key aspect of quantum mechanics, reflecting crucial features of quantum phenomena. In sum, beyond Riemann’s contribution to physics in his lecture itself, which, as noted, was nearly as momentous as his contribution to mathematics, his mathematical concepts have immense impact on the subsequent development of physics.

The final implication of Riemann’s approach I would like to mention, in part, again, as a bridge to the next section, is that it allows one to define a geometrical or topological space (in modern terminology) not as a set of points continuously connected to each other, but in terms of open neighborhoods around each point. As explained above, following this way of thinking, topology describes a given space not only in terms of its points, continuously connected to each other, but also and most essentially in terms of its open neighborhoods around each point. These neighborhoods are subspaces of this space, the idea that underlies Riemann’s concept of manifold, in this case giving each neighborhood a Euclidean geometry. The approach enabled Riemann, again, following Gauss’s theory of curvature of two-dimensional surfaces, to define manifolds of any dimension in terms of its inner properties rather than in relation to the ambient Euclidean space, where a manifold could be placed against the flat Euclidean background. It is true that, if one appeals, as is usual even in considering Riemann, to open sets, this concept retains the concept of a set (of points) as a primitive concept. Thus, Ferreirós’s analysis displaces Riemann’s thought into the axiomatic and set-theoretical register, dominant in the wake of Cantor (Ferreirós 2006, pp. 39–80). This displacement reflects Ferreirós’s insufficient attention to the structure of Riemann’s mathematical concepts, or in the first place, to the concept of mathematical concept found in Riemann’s practice. Riemann did not think either in terms of axioms or in terms of sets of points, as Ferreirós contends, although his concepts could be translated into these terms (e.g., [Pambuccian et al. 2017]). Riemann thinks in terms of spaces composed of spaces. Riemann’s way of thinking concerning manifolds, however, also suggests a possibility of thinking of, and even defining, a space (as a mathematical concept) in terms of its relations to other spaces, in effect categorically, which allows one to use this structure as more primordial by replacing covering a space by open sets (of points) with covering it by open spaces. A topological space becomes a collection of open spaces as subspaces with algebraic rules for the relationships between them. This way of defining a space in its relation to other spaces (as opposed to their constitution as sets of points) extends to Grothendieck’s topos theory, inspired, however, primarily by the idea of a “covering space,” originating in Riemann’s theory of Riemann surfaces.

5 Algebraic Geometry: Weil and Grothendieck

I begin with a brief summary of the concept of curve, discussed in detail in [Plotnitsky 2019], which this section partly follows within the context and argument defined by the present chapter. By expressly making geometry algebra, or just about (this claim needs qualifications, considered below), the analytic geometry of Fermat and Descartes in effect gave mathematics its independence from physics, thus, initiating modern mathematics as a project dealing with ideal objects independent from material objects in nature. I qualify by “in effect” because it took about two centuries for this independence to get its momentum around the time of Gauss, Abel, and Galois, as discussed in Sect. 3. Analytic geometry did so because the equation corresponding to a curve, say, X2 – 1 = 0 for the corresponding parabola, could be studied as an algebraic object, independently of its geometrical representation or its connection to physics, which eventually enabled mathematics to define algebraic varieties over finite fields and thus as discrete objects. A curve becomes, in its composition, defined by its equation, divested from its representational geometrical counterpart. Nor does it any longer geometrically idealize the reality exterior to it. It only represents itself. The equation, as algebra, also defines a separation of a mathematical curve or surface from any curve or surface found in the world, thus also revealing a deeper essence of all mathematics. When one says in mathematics, “consider a curve X,” one separates it from every curved object in the world. This separation of algebra from material reality and geometry (as ordinarily understood) also enables it to define a discrete curve, or can make a curve a surface, or a surface a curve, give it an even more complex spatial algebraic architecture, or a yet more complex conceptual composition, such as that of a moduli space, the Teichmüller space (also the Teichmüller curve), Grothendieck’s scheme and representable-functors, and so forth. We are yet far from exhausting the problem-posing field defined by the concept-problem of a Riemann surface, which is also a curve over ℂ (e.g., A’Campo et al. 2016). This view also transforms our view of geometrical objects in the same way: modern geometry, too, separates or liberates its objects and itself from daily phenomenology, even if not as completely as algebra does.

The concept of a Riemann surface is one of Riemann’s great contributions to modern mathematics, which shaped several of its areas, most especially here algebraic geometry. In Papadopoulos’s account:

In his doctoral dissertation, Riemann introduced Riemann surfaces as ramified coverings of the complex plane or of the Riemann sphere. He further developed his ideas on this topic in his paper on Abelian functions. This work was motivated in particular by problems posed by multi-valued functions w(z) of a complex variable z defined by algebraic equations of the form

$$ f\left(w,,,z\right)=0, $$

where f is a two-variable polynomial in w and z.

Cauchy, long before Riemann, dealt with such functions by performing what he called “cuts” in the complex plane, in order to obtain surfaces (the complement of the cuts) on which the various determinations of the multi-valued functions are defined. Instead, Riemann assigned to a multi-valued function a surface which is a ramified covering of the plane and which becomes a domain of definition of the function such that this function, defined on this new domain, becomes single-valued (or “uniform”). Riemann’s theory also applies to transcendental functions. He also considered ramified coverings of surfaces that are not the plane. (Papadopoulos 2017, p. 240)

The concept of a Riemann surface gains much additional depth and richness when considered in terms of his concept of a manifold. Riemann, as noted, did not do so himself, although he undoubtedly realized that Riemann surfaces were manifolds, and they have likely been part of the genealogy of the concept of manifold. Riemann’s surfaces were first expressly defined as manifolds by Weyl in The Concept of a Riemann Surface (Weyl 2013). It is an intriguing question whether Riemann himself thought of Riemann surfaces as curves, but it would not be surprising if he had. Weyl undoubtedly did, although the point does not figure significantly in his book, focused, topologically, on the surface nature of Riemann surfaces. This may be surprising. But then, Weyl was not an algebraic geometer. The work of Émile Picard, a key figure in the history of algebraic geometry would be more exemplary in considering this point (Picard 1891–1896; Papadopoulos 2017).

Weyl’s argumentation leading him to his definition is an application of a principle akin to Joseph Silverman and John Tate’s principle, “Think Geometrically, Prove Algebraically”:

It is also possible to look at polynomial equations and their solutions in rings and fields other than ℤ or ℚ or ℝ or ℂ. For example, one might look at polynomials with coefficients in the finite field Fp with p elements and ask for solutions whose coordinates are also in the field Fp. You may worry about your geometric intuitions in situations like this. How can one visualize points and curves and directions in A2 when the points of A2 are pairs (x, y) with x, y∈Fp? There are two answers to this question. The first and most reassuring is that you can continue to think of the usual Euclidean plane, i.e., ℝ2, and most of your geometric intuitions concerning points and curves will still be true when you switch to coordinates in Fp. The second and more practical answer is that the affine and projective planes and affine and projective curves are defined algebraically in terms of ordered pairs (r, s) or homogeneous triples [a, b, c] without any reference to geometry. So, in proving things one can work algebraically using coordinates, without worrying at all about geometrical intuitions. We might summarize this general philosophy as: Think Geometrically, Prove Algebraically.” (Silverman and Tate 2015, p. 277)

Affine and projective planes and curves can be defined without any reference to our phenomenal intuition, which grounds the geometrical thinking Silverman and Tate refer to. Even in these more intuitively accessible cases, we think algebraically, too, by using what may be called “spatial algebra.” “Spatial algebra” refers to algebraic structures that mathematically define geometrical or topological objects and reflect their proximity to ℝ3 and mathematical spatial objects there that are close to our phenomenal intuition. This proximity may be, and commonly is, left behind in rigorous mathematical definitions and treatments of such objects, beginning with ℝ3 itself. The same type of algebra may also be used to define mathematical objects that are not available to our phenomenal intuition at all. Among examples are a projective line (a set of lines through the origin of a vector space, such as ℝ2, with projective curves defined algebraically, as algebraic varieties) and an infinite-dimensional Hilbert space (the points of which are typically square-integrable functions or infinite series, although a Euclidean space of any dimension is technically a Hilbert space, too). Spatial algebra is an algebraization of spatiality that makes it rigorously mathematical, as opposed to something that is phenomenally intuited spatially, even in the case of more conventional spatial objects in ℝ3. In addition, as Silverman and Tate state, spatial algebra retains its connection to geometrical thinking because analogies with ℝ3 continue to remain useful and even indispensable. Weyl’s argument concerning “thinking geometrically” about analytic forms is also a manifestation of the spirit of Riemann’s mathematical practice, in which geometry and algebra, indeed geometry, topology, algebra, and analysis, come together in a complex mixture of the rigorous and the intuitive, algebraic and spatial-algebraic, mathematical and physical, and mathematical and philosophical. Riemann’s work on his ζ-function and number theory reflect this mix as well, as his greatest contribution to algebra, but still not without geometry to do it. According to Weyl:

It was pointed out … that one’s intuitive grasp of an analytic form [an analytic function to which a countable number of irregular elements have been added] is greatly enhanced if one represents each element of the form by a point on a surface \( \mathcal{F} \) in space in such a way that the representative points cover \( \mathcal{F} \) simply and so that every analytic chain of elements of the form becomes a continuous curve on \( \mathcal{F} \). To be sure, from a purely objective point of view, the problem of finding a surface to represent the analytic form in this visual way may be rejected as nonpertinent; for in essence, three-dimensional space has nothing to do with analytic forms, and one appeals to it not on logical-mathematical grounds, but because it is closely associated with our sense perception. To satisfy our desire for pictures and analogies in this fashion by forcing inessential representation of objects instead of taking them as they are could be called an anthropomorphism contrary to scientific principles. (Weyl 2013, p. 16)

Weyl resolves this “problem” not only by suggesting, as did Silverman and Tate, that one might think geometrically, while proving algebraically, but also that one can think “spatial-algebraically” (along with intuitively geometrically), by joining Riemann’s concepts of Riemann surface and manifolds. Weyl says:

However, these reproaches of the pure logician are no longer pertinent if we pursue the other approach, already hinted at, in which the analytic form is a two-dimensional manifold to which all the ideas of continuity that we meet in ordinary geometry may be applied. To the contrary, not to use this approach is to overlook one of the most essential aspects of the topic. (Weyl 2013, p. 16)

Weyl will proceed, in the spirit of Gauss and Riemann, to his definition of a two-dimensional manifold and then Riemann’s surface, intrinsically, rather than in relation to its ambient three-dimensional space. This concept enables one to define a Riemann surface as a curve over ℂ. This intrinsic and abstract view of a Riemann surface was often forgotten by Riemann’s followers, especially at earlier stages of the history of the concept. According to Papadopoulos, who follows Klein’s assessment:

Riemann not only considered Riemann surfaces as associated with individual multi-valued functions or with meromorphic functions in general, but he also considered them as objects in themselves, on which function theory can be developed in the same way as the classical theory of functions is developed on the complex plane. Riemann’s existence theorem for meromorphic functions with specified singularities on a Riemann surface is also an important factor in this setting of abstract Riemann surfaces. Riemann conceived the idea of an abstract Riemann surface, but his immediate followers did not. During several decades after Riemann, mathematicians (analysts and geometers) perceived Riemann surfaces as objects embedded in three-space, with self-intersections, instead of thinking of them abstractly. They tried to build branched covers by gluing together pieces of the complex plane cut along some families of curves, to obtain surfaces with self-intersections embedded in three-space. (Papadopoulos 2017, p. 242)

Hence, Weyl (whom Papadopoulos cites) contends: “Thus, the concept ‘two-dimensional manifold’ or ‘surface’ will not be associated with points in three-dimensional space; rather it will be a much more general abstract idea,” in effect a spatial-algebraic one. Weyl’s position concerning the nature of mathematical reality is a different matter. As is clear from his philosophical writings (e.g., Weyl 2021), Weyl was ultimately a realist in mathematics and physics alike, notwithstanding his major contribution to quantum mechanics (which radically challenged realism), not the least by making group and group representation theories central there. Weyl continues as follows:

If any set of objects (which will play the role of points) is given and a continuous coherence between them, similar to that in the plane, is defined we shall speak of a two-dimensional manifold. Since all ideas of continuity may be reduced to the concept of neighborhood, two things are necessary to specify a two-dimensional manifold:

  1. 1.

    to state what entities are the “points” of the manifold;

  2. 2.

    to define the concept of “neighborhood.” (Weyl 2013, pp. 16–17)

This means one should define entities, such as points, lines, neighborhoods, by using algebraic symbols and relationships among them, without referring to objects in the world represented by ordinary language, even if still using this language, as the concept of a Riemann surface as a curve and then its avatars such as Gromov’s concept of pseudoholomorphic curve (a smooth map from a Riemann surface into an almost complex manifold) exemplify. Its connections to our phenomenal sense of surface are primarily intuitive, as when it comes to the idea of continuity, for example, as defined by Weyl here, in terms of the concept of neighborhood. In any event, a Riemann surface is not a curve in any phenomenal sense.

This multifaceted nature of Riemann surfaces equally and often jointly defined the history of complex analysis, the main initial motivation for Riemann’s introduction of the concept of a Riemann surface, and the history of algebraic curves, both building on this concept, and other developments, for example, in abstract algebra and number theory, including Riemann’s work on the ζ-function and the distribution of primes. All these developments were unfolding during the period between Riemann and Weyl, whose book initiated the treatment of the concept of Riemann surface that defines the present-day understanding of the concept. This history joins several “Riemanns” – the Riemann of the concept of manifold, the Riemann of the concept of Riemann surfaces, the Riemann of complex analysis. A few more Riemanns could be added. This multiple history shaped algebraic geometry, eventually leading to the work of Weil and Grothendieck, to which I now turn, although Weyl’s analysis of the idea of a Riemann surface will make yet another important appearance, via, remarkably, Galois theory.

Weil’s and Grothendieck’s work on algebraic geometry are among their greatest contributions and are remarkable cases of mathematical practice as philosophy, as the extension of the concept of algebraic variety over a finite field and the study of such objects by the standard tools of algebraic topology, in particular homotopy and cohomology theories. These theories had been previously proven to be very effective for the study of complex algebraic varieties. The project of extending them to finite fields was initiated by Weil by bringing together algebra, geometry, and number theory, in which he was a true heir of Fermat (and he probably saw himself as one), as well as of Kronecker (in this case, Weil expressly saw himself as one). Riemann remains a key presence in the history leading to Weil’s work in algebraic geometry, first of all, again, given his concepts of a Riemann surface and a covering space, but also by virtue of introducing the concept of a discrete manifold in his Habilitation lecture. Gino Fano, one of the founders of finite geometry, belonged to the Italian school of geometry (1880s–1930s), contemporary with and an important part of the history of mathematical modernism (Fano was a student of Federigo Enriques), strongly influenced by Riemann. Representatives of the Italian school (quite a few of them, even counting only major figures) made major contributions to many areas of geometry, especially algebraic geometry, which were an important part of a rich and complex history leading to Weil’s work on algebraic geometry, culminating in his 1946 Foundations of Algebraic Geometry, with the second edition published in 1962 (Weil 1946, 1962). The book is distinguished by many important technical contributions, which would be difficult to properly consider here. My main focus, defined by the relationships between Weil’s and Grothendieck’s philosophies of algebraic geometry, is Weil’s concept of a “universal domain,” assumed by him to be necessary. This concept distinguishes his approach and, thus, his mathematical practice as philosophy from the previous approaches by requiring that in addition to the base field of definition of the objects (algebraic varieties) considered, an assumption of an (algebraically closed) “universal domain” encompassing all fields that may appear in any constructions made over the base field. The Platonist implications of this claim are tempered by his view that such a domain need not be unique. Weil argued that the theorems of algebraic geometry will remain the same regardless of the choice of this domain if it has an infinite transcendence degree and is closed. Grothendieck will abolish this idea, without a return to any previous thinking, on his way to solving a problem posed by Weil, or rather making it a concept-problem. While suggesting a new problem-posing field, Weil did not define the corresponding concept as such, only suggested it as a possibility. Grothendieck’s redefinition of this problem as a concept-problem was a result of his mathematical practice as philosophy as different from that of Weil, the difference, arguably, most essentially defined philosophically by abandoning Weil’s assumption of a universal domain.

As one of his greatest and most visionary contributions, Weil eventually (in 1949) suggested that a cohomology theory for algebraic varieties over finite fields, now known as Weil cohomologies, could be developed by analogy with the corresponding theories for complex algebraic varieties or topological manifolds in general. Weil’s motivation was a set of conjectures (these go back to Gauss), known as the Weil conjectures, concerning the so-called local ζ-functions, which are the generating functions derived from counting numbers of points on algebraic varieties over finite fields. These conjectures, Weil thought, could be attacked by means of a proper cohomology theory, although he did not propose such a theory, as a corresponding set of concepts, only posed it as a problem, without such a concept. It was not a concept-problem or set of concept-problems. But the idea led Grothendieck to such a concept-problem and indeed a set of such concept-problems, and a new mathematical philosophy entailed by mathematical practice, a philosophy different, on several counts, from that of Weil that led him to this problem.

To develop such a concept, one needed, first, a proper topology, finding which was nontrivial because the objects in question are topologically discrete. A “native” topology that could be algebraically defined by then, known as Zariski’s topology, did not work, because it had too few open sets. The decisive ideas came from Grothendieck, helped by the sheaf-cohomology theory and category theory, known as “cohomological algebra,” by then the standard tool of algebraic topology. Using these tools, a hallmark of Grothendieck’s thinking throughout his career, and his previous concepts, such as that of “scheme,” led him to topos theory and étale cohomology, as a viable candidate for Weil’s cohomology, which it had proven to be. Soon thereafter, it also led him to a more general concept-problem of motivic cohomology, mentioned earlier in conjunction with motivic Galois groups, but the subject requires a separate discussion. By using étale cohomology, Grothendieck (with Michael Artin and Jean-Louis Verdier) and Pierre Deligne (Grothendieck’s student) were able to prove Weil’s conjectures, and then Deligne found and proved their generalization. Grothendieck’s key, extraordinary, insight, also extending what I call spatial algebra in a radically new direction, was to generalize, in terms of category theory, the concept of “open set,” beyond a subset of the algebraic variety, which was possible because the concept of sheaf and of the cohomology of sheaves could be defined by any category, rather than only that of open sets of a given space. Étale cohomology is defined by this type of replacement, specifically by using the category of étale mappings of an algebraic variety, which become “open subsets” of the finite unbranched covering spaces of the variety, a vast generalization of Riemann’s concept of a covering space (Grothendieck’s initial primary area of mathematical research was topological vector spaces, which suggests yet another genealogical link to Riemann, in this case, again, to his concept of manifold). Grothendieck was also building on related ideas of Jean-Pierre Serre. Part of the origin of this generalization was the fact that the fundamental group of a topological space, say, again, a Riemann surface, could be defined in two ways: it can either be defined more geometrically, as a group of the sets of equivalence classes of the sets of all loops at a given point, with the equivalence relation given by homotopy, or it can be defined even more algebraically, as a group of transpositions of covering spaces. In this second, algebraic, definition, it is analogous to the Galois group of the algebraic closure of a field, as Serre was first to consider for finite fields, importantly for Grothendieck’s work on étale cohomology, a concept that, thus, has its genealogy in both Galois’s and Riemann’s thought. The connection had been established in the case of Riemann surfaces long before then and found in Weyl’s book on a Riemann surface:

The group of cover transformations, regarded as an abstract group, expresses purely and completely everything in the relation between the normal covering surface \( \overline{\mathcal{F}} \) and the base surface \( \mathcal{F} \) which has the character of analysis situs [topology]. This group is also called the Galois group of \( \overline{\mathcal{F}} \). It is in fact the analog of the Galois group of a normal algebraic field (of finite degree) over a base field. (Weyl 2013, p. 58)

In a way, this passage defines the trajectory of this paper by connecting the mathematical practice as philosophy of Galois, Riemann, and Grothendieck.

For the moment, Grothendieck’s concept of étale mappings gives a sufficient number of additional open sets to define adequate cohomolgy groups for some coefficients, for algebraic varieties over finite fields. For complex varieties, one recovers the standard cohomology groups with coefficients in any constructible sheaf. The category of étale mapping is a topos, a concept that is, for now, the most abstract form of spatiality, at least as spatial algebra. As became apparent later, étale cohomologies could be defined for most practical uses in simpler settings. The concept of topos remains crucial, however, especially in the present context, because it can be seen as the concept of a covering space over a Riemann surface converted into the (spatial-algebraic) concept of topology of the surface itself, and then generalized to any algebraic variety. The concept of topos also came to play a major role in mathematical logic. The subject cannot, however, be addressed here, except by noting that mathematical logic has radical epistemological implications concerning the nature of mathematical reality, to which the concept of topos, as a logical concept, may be connected. On the other hand, Grothendieck’s use of topoi in algebraic geometry is ontological rather than logical. His philosophical position concerning the nature of mathematical reality remains somewhat unclear, for example, as concerns whether it conforms or not to mathematical Platonism. My focus for the moment are the mathematical technologies that the concept of topos, whatever its ontological status, enables, such as étale cohomology. Such technologies may suggest a possible break with the possibility of the ultimate ontological description of mathematical reality, again, assuming that any ultimate reality independent of human thought, say, again, of the type considered in physics, is even possible in mathematics, as opposed to what can be specified by one’s thought, possibly in one’s unconscious (Plotnitsky 2019, 2020). I shall return to this subject in closing.

It would not be possible to present topos theory here in its proper abstractness and rigor, sometimes prohibitive even for those who are trained in the field of contemporary algebraic geometry. The essential philosophical nature of the concept, may, however, be sketched. First, very informally, consider the following way of endowing a space with a structure, generalizing the definition of topological space in terms of open subsets, as mentioned above. One begins with an arbitrarily chosen space, X, potentially any given space, which may initially be left unspecified in terms of its properties and structure. What would be specified are the relationships between spaces applicable to X, such as mapping or covering one or a portion of one, by another. This structure is the arrow structure Y → X of category theory, where X is the space under consideration and the arrow designates the relationship(s) in question. One can also generalize the notion of neighborhood or of an open subspace of (the topology of) a topological space in this way, by defining it as a relation between a given point and space (a generalized neighborhood or open subspace) associated with it. This procedure enables one to specify a given space not in terms of its intrinsic structure (e.g., a set of points with relations among them) but “sociologically,” throughout its relationships with other spaces of the same category, say, that of algebraic varieties over a finite field of characteristic p (Manin 2002, p. 7). Some among such spaces may play a special role in defining the initial space, X, and algebraic structures, such as homotopy and cohomology, as Riemann in effect realized in the case of covering spaces over Riemann surfaces, again, one of the inspirations for Grothendieck’s concepts of a topos and more specifically étale topos.

To make this a bit more rigorous (albeit still informal), I shall briefly sketch the key ideas of category theory. It was introduced as part of cohomology theory in algebraic topology in 1940 and, as I said, later extensively used by Grothendieck in his approach to cohomological algebra and algebraic geometry, eventually leading him to the concept of topos. Category theory considers multiplicities (which need not be sets) of mathematical objects conforming to a given concept, such as the category of differential manifolds or that of algebraic varieties, and the arrows or morphisms, the mappings between these objects that preserve this structure. Studying morphisms allows one to learn about the individual objects involved, often to learn more than we would by considering them only or primarily individually. Categories themselves may be viewed as similar objects, in relating which one speaks of “functors” rather than “morphisms.” Topology relates topological or geometrical objects, such as manifolds, to algebraic ones, as in the case of homotopy and cohomology groups. Thus, in contrast to geometry, topology, almost by its nature, deals with functors from categories of topological objects, such as manifolds, to those of algebraic objects, such as groups. Mathematical practice (as philosophy) becomes that of functoriality: it no longer primarily concerns objects but categories of objects.

A topos in Grothendieck’s sense is a category of spaces and arrows over a given space, used for the purpose of allowing one to define richer algebraic structures associated with this space, as explained above. There are additional conditions such categories must satisfy, but this is not essential at the moment. To give one of the simplest examples, for any topological space S, the category of sheaves on S is a topos. The concept of topos is, however, very general and extends far beyond spatial mathematical objects (thus, the category of finite sets is a topos); it replaces the latter with a more algebraic structure of categorical and topos-theoretical relationships between objects. On the other hand, it derives from the properties of and categorical relationships between properly topological objects. The conditions, mentioned above, that categories that form topoi must satisfy have to do with these connections.

Beyond enabling the establishing of a new cohomology theory for algebraic varieties, as considered above, topos theory allows for such esoteric constructions as nontrivial or nonpunctual single-point “spaces” or, conversely, spaces (topoi) without points (first constructed by Deligne), sometimes slyly referred to by mathematicians as “pointless topology.” Philosophically, this notion is far from pointless, especially if considered within the overall topos-theoretical framework. In particular, it amplifies a Riemannian idea that “space,” defined by its relation to other spaces, is a more primary object than a “point” or, again, a “set of points.” Space becomes a Leibnizean, “monadological” concept, insofar as points in such a space (when it has points) may be seen as a kind of monad, thus also giving a nontrivial structure to single-point spaces. These monads are elemental but structured entities, “spaces,” rather than structureless entities (classical points), or at least are defined by structures associated to them (Cartier 2001). Naturally, this appeal to monads is metaphorical. Leibniz’s monads are elemental souls, the “atoms” of soul-ness. One might, however, say, getting more mileage out of the metaphor, that the space thus associated with a given point is the soul of this point. In other words, not all points are alike insofar as the mathematical nature of a given point may depend on the nature or structure of the space or topos to which it belongs or with which it is associated. This approach gives a much richer architecture to spaces with multiple points, and one might see (with caution) such spaces as analogous to Leibniz’s universe composed by monads. It also allows for different (mathematical) universes associated with a given space, possibly a single-point one, in which case a monad and a universe would coincide. Grothendieck’s topoi are possible universes, possible worlds, or compossible worlds in Leibniz’s sense, without assuming, like Leibniz (in dealing with the actual world), the existence of only one of them, the best possible.

One might also think of this ontology as an assembly, a “democracy,” of ontologies (Grothendieck’s concept of topos is, again, ontological, rather than logical), in the absence of any encompassing ontology. Topoi are multiple universes, defined ontologically, in the absence of a single encompassing domain, such as that found in Weil’s “imperial” approach. According to David Reed’s summary:

Grothendieck’s constructions [beginning with schemes replacing the varieties of Weil] are far from the ‘palaces’ which Weil suggest belong to algebraic geometers by birthright. Rather than refurbishing and renewing the old constructions he has instead created an entirely new type of architecture and in the process is forced to make extensive use of the ‘makeshift algebraic constructions’ Weil carefully thought to avoid placing reliance upon. Both Weil and Grothendieck seek generality in order to be able to analyze rigorously geometric situations to which the older algebraic geometers paid little attention. This generality permits consideration of ‘degenerate’ cases, objects with singularities, ‘non-reduced’ objects and the like. Whereas Weil sought this generality through a large ‘domain’ in which the result of his constructions could be contained, Grothendieck seeks generality by expanding the category of objects under considerations until a certain ‘self-sufficiency’ is acquired, i.e. until a natural reflexivity can be found so that the objects in the category can be related to each other and operations on them can be undertaken without going outside the original category. Furthermore, the objects do not require a predetermined ‘fixed point’ outside of the category for their specification. … This relativity also extends to the set theoretical foundations of the theory. Grothendieck introduces a set theoretic ‘universe’ which provides the sets used in his constructions and then studies those properties which remain invariant under a change of universe! (Reed 1995, pp. 131, 177n.36)

One, thus, encounters two very different (mathematical) ontological philosophies and, hence, forms of mathematical practice as philosophy – that of unification or encompassment and that of multiplicity, relational, but without a single encompassing domain. Grothendieck’s generality or, better (Reed is right to shift to this term), relativity of construction is a kind rigorous mathematical instantiation or model of the capacity of mathematics as a human endeavor, for which Riemann’s work on foundations of geometry is, arguably, the main modern precursor. This model is that of the creation of mathematical realities (possibly as mental ontologies, of which Grothendieck’s topoi are examples), on the basis of, but creatively transforming, previously established realities, in the absence of an ultimate underlying or encompassing Platonist reality. Phenomenally, it is indeed a kind of creation of universes (now plural!) out of nothing, as Bolyai said, insofar as it is a creation of new mathematical worlds out of elements of the stuff of thinking. It is the same “stuff as dreams are made on,” the stuff of all our thinking, conscious and unconscious, given a rigorous mathematical structure, rather than the worlds created by matter from its “dark materials,” as John Milton aptly described them, admittedly, referring to God’s power to create “worlds” out of his dark materials (Paradise Lost, Book 2, l. 916).

On the other hand, our thought and thus these mathematical worlds are ultimately created from and by these dark materials, and thus, by everything: they are the creations of our bodies and brains, which are the products, creations, of evolutionary history of life, in turn made possible by the whole history of the universe, composed of its dark materials, from the Big Bang on. It is true that our thought is likely to reach little of how all this was made possible. We certainly do not and possibly cannot know or even conceive of how, by means of what material processes, our bodies and brains make thinking and, hence, mathematics, or science, philosophy, and art, possible. That, however, still leaves us plenty to think about and create in these fields, including in dealing with the nature and practices of thinking. There is still so much in them to be invented and so much that we need from them in our life beyond them.

6 Cross-References