Gabriel Catren

Fernando Cukierman

GROTHENDIECK'S THEORY OF SCHEMES AND THE ALGEBRA-GEOMETRY DUALITY

Grothendieck's theory of schemes and the algebra-geometry duality

I. Introduction

In the XVIIth century, Fermat and Descartes independently invented analytic geometry, that is, the study of the correspondence between algebraic expressions and geometric figures by means of coordinate systems (for a history of analytic and algebraic geometry see Refs. [START_REF] Dieudonné | History of Algebraic Geometry[END_REF][START_REF] Dieudonné | The Historical Development of Algebraic Geometry[END_REF][START_REF] Houzel | La géométrie algébrique. Recherches historiques[END_REF]). This interplay between algebra and geometry is based on the fact that the solution set of an equation in the variables (x1, . . . , xn) defines a subset of the space spanned by these variables, i.e. the subset of points-the "figure"-whose coordinates solve the equation. Now, this correspondence between algebra and geometry was unsettled by the fact that it is not possible to associate a figure to certain equations, such as for instance the equation x 2 + y 2 + 1 = 0 (when we search for solutions (x, y) in R 2 ). This problem was bypassed by acknowledging that some equations might have solutions that do not belong to the "field of definition" of the equation, i.e. to the field of numbers in which the equation is formulated. The important fact in the present context is that the geometric "figure" that "schematizes" a given algebraic expression, far from being absolutely defined, depends on the field of numbers in which one searches for the solutions of the corresponding equation (see for instance Ref. [START_REF] Mclarty | There is No Ontology Here': Visual and Structural Geometry in Arithmetic[END_REF][385][386]). In this way, the relativity of the geometric figures with respect to the codomains in which one searches for the solutions obstructs the possibility of establishing a duality between algebra and geometry: a single algebraic expression can induce several figures, one for each possible codomain. This situation was described by Barr, McLarty, and Wells in the following terms: "[...] the unit circle is the set of all solutions in the x-y plane to the equation

x 2 + y 2 = 1
The set of solutions of this equation depends on the type of number that one understands x and y to refer to. [...] There are many more abstract types of numbers [than the integers, the rationals, the real, and the complex numbers] and each gives rise to its own circle. In the classical set theoretic framework, there is thus an integer circle, a rational circle, a real circle, a complex circle and myriads of others [...] [START_REF] Barr | Variable Set Theory[END_REF] The theory of schemes has completely refounded the field of algebraic geometry. This theory mainly emerged from the work of A. Weil, C. Chevalley, P. Cartier, M. Nagata, J.-P. Serre, and A. Grothendieck and it was mainly developed by the latter. 1 As we shall see, the theory of affine schemes permits to cope with the aforementioned relativity in a manner that allows to recover an algebra-geometry duality for general (commutative) rings. Grothendieck's theory accomplishes this task by means of a two-step procedure. Rather than fixing a single codomain, Grothendieck fully embraces the relativity of the figures associated to a family of polynomial equations with respect to the codomains in which one searches for their common solutions. By introducing the category theoretic notion of functor of solutions (or, in the dual geometric version, functor of points), Grothendieck takes into account the whole family of "figures" at once (Section V). Second, Grothendieck synthesizes the resulting "fan" of figures into a single geometric entity, namely the unique affine scheme associated to the original ring. This two-step description of the construction of an affine scheme articulates the two alternative standpoints from which scheme theory can be approached (for a formal demonstration of their equivalence see Ref. [START_REF] Demazure | Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs[END_REF]). According to the category theoretic description, a scheme is a functor that encodes the solutions to a system of polynomial equations in every possible codomain (or more generally the representations A → B of a ring A for every B in the category). From a geometric standpoint, schemes are ringed spaces. According to this description, a scheme is constructed by adding structure to an underlying set of points. More precisely, one starts with a set of points, then introduces a notion of locality by means of a topology, enriches this topological space by adding local information (local rings of regular functions) in order to finally glue these local pieces together into a global object (structure sheaf). 2 The notion of duality is a central concept in both mathematics and physics (see Ref. [START_REF] Atiyah | Duality in Mathematics and Physics[END_REF] for a general survey on the notion of duality, Refs. [START_REF] Johnstone | Stone spaces[END_REF]Chapt.6,Sect.4,[253][254][255][256][257][258][259][260][261][262][263][264][265][266]] and [START_REF] Porst | Concrete dualities[END_REF] for a mathematical approach, Refs. [START_REF] Awodey | Sheaf Representations and Duality in Logic[END_REF][START_REF] Catren | On the Notions of Indiscernibility and Indeterminacy in the Light of the Galois-Grothendieck Theory[END_REF][START_REF] Corfield | Duality as a category-theoretic concept[END_REF][START_REF] Halvorson | Logic in Philosophy of Science[END_REF][START_REF] Krömer | The Form and Function of Duality in Modern Mathematics[END_REF][START_REF] Lambek | The Influence of Heraclitus on Modern Mathematics[END_REF][START_REF] Lane | Duality for Groups[END_REF][START_REF] Nagel | The Formation of Modern Conceptions of Formal Logic in the Development of Geometry[END_REF] for philosophical-oriented analyses of this notion in modern mathematics). 3 Among the different instantiations of the mathematical notion of duality, the duality between algebra and geometry occupies a privileged place. Now, there are two alternative approaches for understanding the duality between algebra and geometry. In S. Germain's terms, we can either consider that "algebra is but written geometry" or that "geometry is but figured algebra" [START_REF] Germain | OEuvres philosophiques[END_REF]223]. According to the first description (that we shall call geometric approach), geometry provides the starting point. This means that we start with some kind of space X and use algebraic devicessuch as for instance functions or sections of a fiber bundle on X-to "observe" the space and try to separate its points by means of the properties defined by these "quantities". In this conceptual framework, a ring is an algebraic structure intended to be thought of as a ring of functions on a space. In Nestruev's terms, "smooth manifolds are 'worlds' whose observation can be carried out by means of smooth algebras" [99, ix]. 4 According to the second description (that we shall call spectral approach), algebra provides the starting point. This means that we start with an algebraic structure A (e.g. a Boolean algebra, a group, a ring, a C * -algebra, etc.) and we consider the conditions under which A can be faithfully realized (by means of what we shall generally call Gelfand transform) as an algebra of quantities (such as functions or sections) on a suitable space (the spectrum of A). 5 As we shall see, the remarkable fact is that the spectrum of the algebra A is a space that parameterizes a suitable collection of representations A → B of A. Hence, the spectral approach establishes a bridge between the algebra-geometry duality and what we shall call structure-semiotics duality, that is the duality between a structure A and the representations A → B of A in B. It is a striking fact that reconstruction theorems based on the same kind of algebra/structure-geometry/semiotics duality can be found in several areas of mathematics such as Boolean algebras (Stone duality), group representation theory (Pontryagin duality), algebraic geometry, Galois-Grothendieck theory, and C * -algebra theory. As we shall explain in Section N • III, all these dualities can be subsumed under the same general pattern: that of an algebraic structure A that is isomorphic to an algebra of quantities over a space that parameterizes a certain class of 2 In both descriptions, the notion of sheaf (introduced by J. Leray in the field of algebraic topology during the second World War [START_REF] Houzel | A Short History: Les debuts de la théorie des faisceaux[END_REF]) plays a central role, albeit in a different manner: whereas in the case of ringed spaces, a sheaf encodes the global organization of local pieces of information defined on the space (typically functions locally defined), in the functorial approach a sheaf defines a generalized space by specifying how this space is "probed" by test objects in a given category ([100, The basic idea of sheaves]; for an introduction to sheaf theory see Ref. [START_REF] Vaquié | Sheaves and Functors of Points[END_REF]). 3 It is worth noting that the common use of the term duality in physics and mathematics does not necessarily mean that the corresponding formal concepts are the same or even that the term is used in an homogeneous manner within each discipline. Whereas in mathematics a duality usually takes the form of an equivalence of categories [START_REF] Lambek | The Influence of Heraclitus on Modern Mathematics[END_REF], in physics the term duality is also used to denote a correspondence between different theories encoding some form of "physical equivalence" between them (e.g. between quantum field theories and string theories in the gauge/gravity dualities). In some cases, the use of the term in physics coincides with the mathematical notion. For instance, the wave-particle duality in quantum mechanics relies on the theory of Fourier transform, that is on the Pontryagin duality for abelian locally compact groups. For philosophical discussions of the notion of duality in physics see the different contributions to Ref. [START_REF]Dualities in Physics[END_REF]. 4 By following the physics-oriented description proposed in Ref. [START_REF] Nestruev | Smooth Manifolds and Observables[END_REF], we can understand the space X as a set of physical states, the "observables" f as measuring devices and the result f (x) of the evaluation as the outcome of the "observation" of the state x by means of f . 5 For an analysis of both the "geometric" and the "spectral" approaches to the algebra-geometry duality see Ref. [START_REF] Nestruev | Smooth Manifolds and Observables[END_REF].

representations of A. As we shall see, the main innovation introduced by Grothendieck's theory of affine schemes is that the spectrum of the algebra A (in this case a ring) parameterizes representations of A defined with respect to different codomains B. This extension of the semiotics of an algebraic structure justifies the distinction that we shall introduce between restricted reconstruction theorems (where the representations A → B defined with respect to a single codomain B are enough to recover the structure A) and general reconstruction theorems (where the structure A can be recovered by considering representations defined with respect to different codomains B).

Grothendieck's refoundation of algebraic geometry and the panoply of new notions that he introduced (such as schemes, topoi, motives, and stacks) constitute one of the most important paradigm shifts that occurred in mathematics during the XXth century. Moreover, Grothendieck's approach is strongly based on-and significantly contributed to develop-category theory and groupoid theory, thereby having a seminal impact on the "foundations of mathematics". In spite of its importance-and probably due to its technical complexity-the revolution launched by Grothendieck has not really entered as a proper subject into the realm of philosophy of mathematics (a philosophical-oriented introduction to scheme theory can be found in Ref. [START_REF] Mclarty | There is No Ontology Here': Visual and Structural Geometry in Arithmetic[END_REF] and references to scheme theory in philosophy of mathematics can be found in Refs. [START_REF] Parrochia | Philosophy and Mathematics[END_REF][START_REF] Szczeciniarz | For a Continued Revival of the Philosophy of Mathematics[END_REF][START_REF] Zalamea | Synthetic Philosophy of Contemporary Mathematics[END_REF]). This paper is-in the wake of the aforementioned references-an attempt to contribute to fill this gap. To do so, we shall provide a conceptual analysis of some aspects of Grothendieck's theory of affine schemes by mainly following the path described in the introduction to the treatise Éléments de Géométrie Algébrique by Grothendieck and Dieudonné [START_REF] Grothendieck | Éléments de Géométrie Algébrique I[END_REF] (see also Ref. [START_REF] Grothendieck | Introduction to functorial algebraic geometry, part 1: affine algebraic geometry, summer school in Buffalo[END_REF]). 6 The analysis of scheme theory that we shall propose is certainly partial and many important aspects of the subject will not be addressed here (like for instance the role played by scheme theory in bridging the gap between geometry and number theory-which gave rise to the field of arithmetic geometry-and the definition of general non-affine schemes). We shall mainly focus on the aspects of scheme theory that are relevant with respect to what we could generally call philosophy of (mathematical) dualities. To do so, we shall proceed as follows:

(1) We shall analyze the relation between the spectral approach (where we start with an algebraic structure) and the geometric approach (where we start with a geometric object) to the algebra-geometry duality (Section N • III). [START_REF] Allison | Kant's Transcendental Idealism: An Interpretation and Defense, Revised and Enlarged Edition[END_REF] We shall analyze the relation between the algebra-geometry dualities and (what we shall call) the structure-semiotics dualities (of which the syntax-semantics dualities provide particular cases) between an algebraic structure A and a collection of its representations A → B (Section N • II). (3) We shall characterize the twofold relevance of scheme theory with respect to the debate between syntactic and semantic conceptions of "scientific theories", namely (a) that it provides the guiding example for the generalization of the Stone duality (encoding the representation theory of propositional logic) to predicate logic (Section N • II), (b) that it entails an extended notion of the "semantics" of a "syntactic" structure A that includes "models" A → B defined with respect to different codomains B (Section N • V). (4) We shall analyze the general pattern underlying several representation theorems in different areas of mathematics (Boolean algebra theory, algebraic geometry, group representation theory, C * -algebras, Galois-Grothendieck theory, etc.) (Section N • III). ( 5) We shall establish a distinction between restricted reconstruction theorems (like the representation theorem associated to algebraic varieties, Section N • IV) and general reconstruction theorems (like the representation theorem associated to affine schemes, Section N • VI). [START_REF] Artin | Algebra[END_REF] We shall analyze the main formal tool that encodes Grothendieck's functorial extension of the semiotics of a mathematical structure, namely the Yoneda lemma.

We shall describe the two-fold conceptual importance of the Yoneda lemma (Section N • V), namely that it encodes (a) a transition from a substantivalist conception of mathematical objects to a relational one, (b) a domain extension from objects in a category (described by representable functors) to general functors (representable or not). [START_REF] Atiyah | Introduction to Commutative Algebra[END_REF] We shall stress the innovative aspect of the duality between rings and affine schemes with respect to previous instantiations of the algebra-geometry duality (Section N • VI), namely (a) the presence of points corresponding to extended and unextended irreducible subsets (see also Section N • VII), (b) the fact that the corresponding quantities on the spectrum of a ring have values in point-dependent codomains, (c) the original treatment of differential structures. (d) We shall extend Majid's Kantian-oriented analysis of the Pontryagin duality [START_REF] Majid | Principle of Representation-Theoretic Self-Duality[END_REF] to the duality between rings and affine schemes and use the resulting conceptual framework to address the difference between the notion of a (co)domain extension [START_REF] Bellomo | Domain Extension and Ideal Elements in Mathematics[END_REF][START_REF] Cantù | An argumentative approach to ideal elements in mathematics[END_REF][START_REF] Manders | Domain Extension and the Philosophy of Mathematics[END_REF] and the more general variations encoded in Grothendieck's notion of functor of points (Sections N • VII and VIII).

In order to make this article readable by the wider possible audience, we have tried to avoid as much technical details as possible by addressing the reader to the corresponding references. In several occasions, we have included some relevant technical precisions in footnotes. We assume a basic knowledge of algebraic notions (e.g. rings, ideals, etc.), the rudiments of algebraic geometry, and an elementary acquaintance with category theory. This material can be found in Ref. [START_REF] Artin | Algebra[END_REF], Ref.[57, Chap. I] and Refs. [START_REF] Awodey | Category Theory[END_REF][START_REF] Lane | Categories for the Working Mathematician[END_REF] respectively. All throughout the article, the term ring means commutative ring with unity.

II. The Structure-Semiotics Duality

The so-called syntax-semantics debate refers to the tension between the understanding of a scientific theory either in terms of syntactic structures or in terms of classes of models (see Refs. [START_REF] Halvorson | Scientific theories[END_REF][START_REF] Lutz | What Was the Syntax-Semantics Debate in the Philosophy of Science About?[END_REF][START_REF] Winther | The Structure of Scientific Theories, The Stanford Encyclopedia of Philosophy (Spring 2021 Edition[END_REF] and references therein). The syntactic approach (also known as the Received View ) was born in the framework of logical positivism and developed, mainly by Carnap and Hempel, as an attempt to formalize scientific theories in predicate (higher order) logic in a language-independent (or presentation-independent) manner [START_REF] Carnap | Foundations of Logic and Mathematics[END_REF][START_REF] Hempel | On the 'Standard Conception' of Scientific Theories[END_REF]. On the other side, the semantic approach started with the work of Tarski on formal semantics (model theory) and was notably developed from the 60's out of the criticism addressed to the syntactic view of theories by scholars like Putnam [START_REF] Putnam | What theories are not[END_REF], Suppe [START_REF] Suppe | Understanding Scientific Theories: An Assessment of Developments, 1969-1998[END_REF], Suppes [START_REF] Suppes | Representation and Invariance of Scientific Structures[END_REF], and Van Fraassen [START_REF] Van Fraassen | The Semantic Approach to Scientific Theories[END_REF][START_REF] Van Fraassen | On the Extension of Beth's Semantics of Physical Theories[END_REF].

In Ref. [56, §17.1.3], Halvorson and Tsementzis argued that rather than trying to privilege the syntactic side or the semantic side of the debate, one can focus on the conditions under which the dilemma can be sublated in the form of a syntax-semantics duality. This "dualization" of the syntax-semantics dilemma amounts to claim that neither the syntactic side not the semantic side have any privilege whatsoever over the other side, but that the two sides are related by adjoint functors that we shall call semantic functor and syntactic functor. Whereas the semantic functor sends a syntax to a suitable collection of models (i.e. to a collection containing "enough models"), the syntactic functor extracts the common syntactic structure from each collection of models. Like the case of the Stone duality for Boolean algebras shows (syntax-semantics duality for propositional theories 7 ), a pre-duality given by adjoint functors yields a proper duality when the adjunction defines an equivalence of categories between certain subcategories (see Ref. [65, §4, 253-269] and Ref. [START_REF] Porst | Concrete dualities[END_REF]). 8 This means that the the two possible compositions of the syntactic and the semantic functors yield the identity on objects modulo isomorphisms. This category theoretic description of the notion of duality can be summarized by means of the description according to which a duality is given by the fixed points of an adjunction [100, fixed point of an adjunction]. 9 As Halvorson and Tsementzis have stressed, the example provided by the Stone duality for Boolean algebras shows that the corresponding collections of models have to be endowed with a suitable geometric structure encoding the "topological relations between models" [START_REF] Halvorson | Categories of scientific theories[END_REF]411]. In this framework, as Lawvere writes, "models are points" in a topological space [START_REF] Lawvere | Continuously Variable Sets; Algebraic Geometry = Geometric Logic[END_REF]137]. The conceptual importance of this geometrization of the semantics is that it establishes a bridge between the syntax-semantics duality and the algebra-geometry duality, where the algebra is on the syntactic side and the geometry on the geometric side. Of course, we could legitimately question the generality of this relation between these two kinds of duality. As Halvorson and Tsementzis write, "we do not want to fall into the trap-all too common in twentieth-century philosophy-of being blinded by the glow of a shiny new piece of formal apparatus" [START_REF] Halvorson | Categories of scientific theories[END_REF]412]. In order to counterbalance this healthy skepticism, we can propose two pieces of evidence that suggest that this relation between the syntax-semantics duality and the algebra-geometry duality encodes some kind of fundamental meta-mathematical pattern or-in Lawvere's terms-"a kind of philosophical theorem in a soft mathematical guise." [START_REF] Lawvere | Functorial Semantics of Algebraic Theories[END_REF]16] (see also [START_REF] Lawvere | Functorial Semantics of Algebraic Theories[END_REF]Th.2,[START_REF] Lawvere | Adjointness in Foundations[END_REF]).

First, the attempts to extend the Stone duality to predicate logic by using topos theory also rely on a geometrization of the semantics [START_REF] Awodey | Sheaf Representations and Duality in Logic[END_REF][START_REF] Awodey | Sheaf representation for topoi[END_REF][START_REF] Awodey | Logic in Topoi: Functorial Semantics for Higher-Order Logic[END_REF][START_REF] Awodey | First-order logical duality[END_REF][START_REF] Breiner | Scheme representation for first-order logic[END_REF][START_REF] Lambek | Two sheaf representations of elementary toposes[END_REF]. Second, if we extend the logical meaning of the terms syntax and semantics to any morphism p : A → B in a category (where p might be looosely understood as a "model" in B of the "syntax" A), then we find similar reconstruction theorems entangling both dualities in several areas of mathematics like algebraic geometry, group-representation theory, C * -algebras theory, and Galois-Grothendieck theory. In order to introduce some terminological precision, we shall avoid using the logical terms syntax and semantics for the general case of an algebraic structure A and its representations A → B respectively. Instead, we shall use the term structure to denote any object A in a category C and the term semiotics to denote the representations of A with respect to other objects in C. Whereas the logical term semantics denotes the truth valuations of the corresponding propositional constants, the more general term semiotics denotes the representation of a structure A by means of morphisms into another structure B of the same kind. This means that the "values" that define each representation of A are not necessarily truth values, but rather values in more general mathematical structures (e.g. fields, rings, groups, etc.; see the column Representations in Table N • 1). In this sense, truth values provide a particular instantiation of the more general notion of quantity. It follows that the notion of syntax-semantics 7 A Boolean algebra A can always be understood as the Lindenbaum-Tarski algebra of a propositional theory T . In this case, the {0, 1}-models of T are the truth valuations of A. 8 Regarding this category theoretic understanding of the notion of duality, Lambek and Scott summarize (what they call in the wake of Lawvere) "'the unity of opposites' principle' by means of the following 'slogan': "Many equivalence and duality theorems in mathematics arise as an equivalence of fixed subcategories induced by a pair of adjoint functors" [START_REF] Lambek | Introduction to Higher Order Categorical Logic[END_REF][START_REF] Borceux | Galois theories[END_REF] (see also Ref. [START_REF] Lambek | The Influence of Heraclitus on Modern Mathematics[END_REF]). Regarding the syntacticsemantics adjunction (sometimes called structure-semantics adjunction) see Ref. [START_REF] Lawvere | Adjointness in Foundations[END_REF]. 9 As we shall see at the end of Section N • III, it might be the case that only the unit or the counit of the adjunction is a natural isomorphism. In such an intermediate case, the pre-duality is not strictly speaking a duality even if one of the two possible compositions of the adjoint functors does yield the identity modulo isomorphisms. duality in the strict logical sense of the term can be understood as a particular case of the structure-semiotics duality. In particular, the Stone representation theorem for Boolean algebras that encodes the duality between syntax and semantics for propositional theories (as well as its generalizations to predicate logic) can be understood as a particular case of what we could generally call representation theory (this point has been stressed by Halvorson in Ref. [START_REF] Halvorson | Logic in Philosophy of Science[END_REF]165]). In Fig. N • II.1, we have schematized the general duality between algebraic structures and their geometric semiotics, where the two adjoint functors are called structure functor and semiotic functor.

Algebraic Structures Geometric Semiotics

Structure Functor Semiotic Functor (II. [START_REF] Allais | Kant's One World: Interpreting 'Transcendental Idealism[END_REF] In what follows, we shall consider the relevance of Grothendieck's theory of affine schemes with respect to this entanglement between the structure-semiotics duality and the algebra-geometry duality. First, the theory of schemes provided the guiding example for the generalization of the Stone duality for propositional logic to predicate logic [START_REF] Awodey | Sheaf Representations and Duality in Logic[END_REF][START_REF] Awodey | Sheaf representation for topoi[END_REF][START_REF] Awodey | Logic in Topoi: Functorial Semantics for Higher-Order Logic[END_REF][START_REF] Awodey | First-order logical duality[END_REF][START_REF] Breiner | Scheme representation for first-order logic[END_REF][START_REF] Lambek | Two sheaf representations of elementary toposes[END_REF]. Whereas Grothendieck's representation theorem states (as we shall see) "that every ring is isomorphic to the ring of global sections of a sheaf of local rings," the sheaf representation theorem for toposes-in the version demonstrated by Lambek and Moerdijk [START_REF] Lambek | Two sheaf representations of elementary toposes[END_REF]-states that "every topos is isomorphic to the topos of global sections of a sheaf of local toposes" [9, 42 & 44]. In this sense, an understanding of Grothendieck's representation theorem for rings seems to be a necessary intermediate step to address the syntax-semantics duality for predicate logic. It is also worth noting-as Awodey and Forsell have stressed-that the relation between the syntax-semantics duality and the algebra-geometry duality provided the "leading idea" to extend the Stone duality from propositional to predicate logic. 10 Second, the duality between rings and affine schemes encodes an important enrichement in the notion of semiotics. Stone representation theorem shows that it is possible to recover a Boolean algebra A from a semantics containing models defined with respect to a unique codomain, namely the set {0, 1} endowed with the discrete topology [START_REF] Johnstone | Stone spaces[END_REF]. By contrast, the reconstruction of a ring A requires to consider representations A → B of A defined with respect to different codomains B. As we shall see, the definition of a scheme as a functor of solutions (of a family of polynomial equations) explicitly formalizes this extension in the definition of the semiotics of a mathematical structure. The remarkable fact is that the resulting semiotics can still be described by a unique geometric object, namely the corresponding affine scheme qua ringed space. In this sense, the equivalence between the definition of a scheme as a functor and its definition as a ringed space [START_REF] Demazure | Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs[END_REF] shows that the entanglement between the algebra-geometry duality and the structure-semiotics duality is still valid in the framework of Grothendieck's extended definition of the semiotic side.

It is also worth noting that in the geometric approach to the algebra-geometry duality, this functorial extension of the semiotics takes the form of a generalization of the notion of quantity on the space. Whereas in a restricted semiotics (defined by representations into a unique codomain) the quantities on the space takes theirs values in a structure that is constant all over the space, in the extended semiotic the quantities take theirs values in 10 In Awodey and Forsell's own terms: "In broad strokes, the leading idea [to obtain an extension of Stone duality for Boolean algebras (Stone duality) from classical propositional logic to classical first-order logic] is to take the traditional logical distinction between syntax and semantics and analyze it in terms of the classical mathematical distinction between algebra and geometry, with syntax corresponding to algebra and semantics to geometry" [START_REF] Awodey | First-order logical duality[END_REF].

structures that continuously vary over the space. This means that it is not only the value of a quantity that might change from point to point, but also the very structure to which these values belong. In other terms, the notion of continuous variation of a quantity on a space is enhanced to the notion of a continuously varying quantity valued in varying structures (we shall come back to this point in Section N • VI).

III. On Restricted and General Reconstruction Theorems

In philosophical terms, we can define the problem of representation as the problem of reconstructing a given structure from a multiplicity of representations, interpretations, or "models" of the structure. In figurative terms, we could say that the different representations provide different partial "perspectives" on the corresponding structure and that the latter can be faithfully recovered only when one considers a sufficient number of such perspectival representations. In order to cover the variety of mathematical examples we are interested in, we shall use a sufficiently general notion of representation, namely we shall understand any morphism p : A → B in a category C as a B-representation of the structure A. 11 By means of such maps the elements and operations of A are "represented" so to speak as elements and operations in B. By using Lawvere's terminology, we shall call the object B dualizing object of the set of representations A → B. 12 In turn, the set of B-representations of A defines what we shall call the B-spectrum of A (denoted Spec B (A)). The fact that all the elements of Spec B (A) are morphisms A → B allows us to introduce what we shall call Gelfand transform. The Gelfand transform is a map that associates to each element a ∈ A a B-valued function aB (a quantity in Lawvere's terms) on Spec B (A) defined by the Gelfand inversion formula

aB(p) = p(a), (III.1)
for every (p : A → B) ∈ Spec B (A). The Gelfand transform allows us to present the elements a of the mathematical structure A as quantities aB on the space Spec B (A) that parameterizes its B-representations p : A → B. In conceptual terms, the quantity aB encodes the different representations of a in all the B-representations of A. This means that the evaluation of a quantity aB on an element p : A → B of the spectrum Spec B (A) yields the representation of a in the B-representation p.

For the moment, the B-spectrum Spec B (A) of A is a set containing the different representations p : A → B of A in B. As it was stressed by Halvorson and Tsementzis for the propositional case given by the Stone duality (as well as its generalizations to predicate logic [START_REF] Awodey | Sheaf Representations and Duality in Logic[END_REF][START_REF] Makkai | Duality and Definability in First Order Logic[END_REF]), the formulation of a duality between a mathematical structure and its representations might require to equip the set of representations with some extra structure encoding the relations between the latter, typically the structure of a geometric object (see Refs.[56, §17.1.3], [55, §7]). 13 By doing so, the spectrum Spec B (A) containing the B-representations of A becomes a space. This geometrization of (what we could call) the B-semiotics of the structure A relies on the identification of the maps A → B (in the case of affine schemes, modulo an equivalence relation) to points of a space. By means of this identification between the semiotic notion of representation and the geometric notion of 11 In Cartier's terms, "[c]ategory theory is the mathematical expression of the idea of representation [...]" [START_REF] Cartier | A Mad Day's Work: from Grothendieck to Connes and Kontsevich. The Evolution of Concepts of Space and Symmetry[END_REF]398]. 12 Lawvere and Rosebrugh called formal dualization the involution operation (-) op : Cat → Cat that consists in reversing the arrows of a category in a merely formal manner (which means that it might not be always possible to understand the reversed arrows in terms of specific mappings). By fixing a dualizing object C, it is possible to define a "concrete duality" that assigns a reversed arrow C B → C A to each arrow A → B [START_REF] Lawvere | Sets for Mathematics[END_REF][START_REF] Van Fraassen | The Semantic Approach to Scientific Theories[END_REF][START_REF] Van Fraassen | On the Extension of Beth's Semantics of Physical Theories[END_REF]. 13 In Halvorson's terms: "[...] the semantic view was not wrong to treat theories as collections of models; rather, it was wrong to treat theories as nothing more than collections of models. Beginning with a syntactically formulated theory T , we can construct its class M od(T ) of models. But we have more information than just the collection of models: in particular, we have information about relations between these models. [...] [as the case of the Stone's duality theorem for Boolean algebras shows] we could rehabilitate the semantic view of theories by taking a theory to be a structured set of models, namely, a topological space of models" [START_REF] Halvorson | What Scientific Theories Could Not Be[END_REF][204][205].

point, the algebra-geometry dualities and the structure-semiotics dualities (of which the syntax-semantics dualities provide particular cases) get entangled. While the algebra is on the "syntactic" side of the duality, the geometry is on the "semantic" side.

Let's consider now the following reconstruction problem: is it possible to recover an algebraic structure A from its representations defined with respect to other (one or many) algebraic structures? Conceptually, a structure A can be reconstructed from its B-representations if the B-spectrum has enough points (or representations), i.e. if for any two elements a, a ∈ A there is at least a representation p in Spec B (A) such that aB(p) = a B (p). 14 The reconstruction theorems according to which a structure A can be reconstructed from the representations defined with respect to a unique dualizing object B will be called restricted reconstruction theorems. Examples of such restricted reconstruction theorems are provided by:

• the algebra-geometry duality for algebraic varieties [57, Th.3.2 (a), 17], 15 • the Stone duality for Boolean algebras [START_REF] Johnstone | Stone spaces[END_REF], 16 • the Gelfand duality for commutative C * -algebras [START_REF] Dixmier | C * -Algebras[END_REF][START_REF] Landsman | Mathematical Topics Between Classical and Quantum Mechanics[END_REF], 17 • the Pontryagin duality for abelian locally-compact groups and the associated theory of Fourier transform [START_REF] Pontryagin | The general topological theorem of duality for closed sets[END_REF] (see also [START_REF] Chevalley | Theory of Lie Groups[END_REF]), 18 • the Galois-Grothendieck duality [START_REF] Borceux | Galois theories[END_REF]Th.2.4.3,[START_REF] Chargois | L'influence de la théorie de Galois sur l'oeuvre de Grothendieck[END_REF], see also [START_REF] Catren | On the Notions of Indiscernibility and Indeterminacy in the Light of the Galois-Grothendieck Theory[END_REF] for a conceptualoriented analysis of this result). 19 In all these examples, the structures at stake are reconstructed by just considering their representations with respect to a unique dualizing object. 20 Now, it is not always possible to reconstruct an algebraic structure from its representations into a fixed codomain. Grothendieck's fundamental insight is that this problem can be bypassed by considering representations of A with respect to a whole category of dualizing objects B. Examples of such general reconstruction theorems are provided by:

• the algebra-geometry duality for affine schemes (that we shall analyze in Section N • IV), 21 • the sheaf representation theorems for toposes [START_REF] Awodey | Sheaf Representations and Duality in Logic[END_REF][START_REF] Awodey | Logic in Topoi: Functorial Semantics for Higher-Order Logic[END_REF][START_REF] Awodey | First-order logical duality[END_REF][START_REF] Breiner | Scheme representation for first-order logic[END_REF][START_REF] Lambek | On the sheaf of possible worlds[END_REF][START_REF] Lambek | Two sheaf representations of elementary toposes[END_REF].

In all these examples the codomain of the representations parameterized by the corresponding spectra might be different for different points. Whereas in the case of restricted reconstruction theorems (e.g. Stone duality or Gelfand duality) the quantities defined by the elements of the algebra have their values in a constant algebraic structure (e.g. {0, 1} for the Stone duality and R for the Gelfand duality), in the theory of affine schemes the quantities have values in local rings that vary continuously on the spectrum. 14 From a logical standpoint, the property of having enough points is related to semantic completeness (i.e. that tautologies φ are theorems φ) [START_REF] Awodey | Sheaf Representations and Duality in Logic[END_REF]. See in particular the relation between Deligne's theorem and Godel's completeness theorem for first-order logic [START_REF] Mac Lane | Sheaves in Geometry and Logic. A First Introduction to Topos Theory[END_REF]523]. 15 That is, the isomorphism between a reduced and finitely generated commutative algebra A over an algebraically closed field K and the affine ring of K-valued regular functions on the algebraic variety defined by the K-algebra homomorphisms A → K [57, Th.3.2 (a), 17]. 16 That is, the isomorphism between a Boolean algebra B and the algebra of {0, 1}-valued functions on a (compact, totally disconnected, and Hausdorff) space S(B) called Stone space. The Stone space parameterizes the Boolean algebra morphisms B → {0, 1} (that we can understand as models or truthvaluations of the theory B) [START_REF] Johnstone | Stone spaces[END_REF]. 17 That is, the isomorphism between a (resp. unital) commutative C * -algebra A and the algebra of C-valued functions on a locally compact (resp. compact) and Hausdorff space Sp(A) called Gelfand spectrum. The Gelfand spectrum parameterizes the non-zero homomorphisms A → C (or characters) [START_REF] Dixmier | C * -Algebras[END_REF][START_REF] Landsman | Mathematical Topics Between Classical and Quantum Mechanics[END_REF]. 18 That is, the isomorphism between a locally-compact abelian group G and the algebra of U (1)-valued functions on the space Ĝ that parameterizes the characters G → U (1) [START_REF] Chevalley | Theory of Lie Groups[END_REF]. 19 That is, the isomorphism between a finite dimensional k-algebra A split by an algebraically closed field L (where L : k is a finite dimensional Galois extension of fields) and the algebra of L-valued functions on the Gal(L : k)-space that parameterizes the k-algebra homomorphisms A → L ([18, Th.2.4.3, 28], [START_REF] Catren | On the Notions of Indiscernibility and Indeterminacy in the Light of the Galois-Grothendieck Theory[END_REF]Sec.6]). 20 The dualizing objects that allows us to formulate the corresponding isomorphisms are K, {0, 1}, C, U (1), and L respectively. 21 That is, the isomorphism between a general ring A and the ring of global sections of the structure sheaf of local rings over the prime spectrum Spec(A) of A. In the Table N • 1 we include all the dualities mentioned thus far and specify both the mathematical structure that is being represented and the kind of representations used to recover it (see also Refs. [START_REF] Awodey | Sheaf Representations and Duality in Logic[END_REF] and [5, Table 4.3, 160]) 22 . It is a remarkable fact that all these important mathematical results are instances-up to contextual differences depending on each particular mathematical framework-of a common reconstruction protocol. In Figure N • III.2, we have summarized the general meta-mathematical pattern instantiated by these different algebra/structure-geometry/semiotics dualities. Given an algebraic structure A we can consider a certain class of representations of A, that is morphisms of the form A → B (for a fixed or a variable codomain B). In order to obtain a restricted (respectively, general) reconstruction theorem, the corresponding set (respectively, structured set) of representations has to be endowed with extra geometric structure. In general, there is a "semiotic" functor-that we have noted Spec(-) 23 -from the (opposite of the) category of algebraic structures to the corresponding category of geometric objects. Since any point in Spec(A) is a map of the form p : A → B, each element a in the algebraic structure A defines a quantity â on Spec(A) given by the Gelfand inversion formula â(p) = p(a). The quantity â encodes the representations of the element a ∈ A in all the representations parameterized by Spec(A). The reconstruction theorem takes the form of an isomorphism between the original algebraic structure A and an algebra of quantities on Spec(A) defined by means of a structure functor Γ(-). We could say that the elements of the structure A to be reconstructed are realized-by means of a Gelfand transform-as quantities on the "semiotic" space Spec(A) that parameterizes certain representations of A. The fact that this general pattern is at the basis of important reconstruction theorems in different areas of mathematics gives support to the thesis according to which this algebra/structuregeometry/semiotics duality is (as Lawvere states) "fundamental to mathematics" [75, 17] [79, 16]. 24 Algebraic Structure A

Algebra of Quantities Γ(Spec(A)) Semiotic Space Spec(A) Γ(-) Spec(-) Gelfand Transform ( ) Spectral Duality Geometric Duality (III.2)
In order to conclude this section, let's revisit the relation between the two approaches to the algebra-geometry duality, namely the spectral approach (in which one starts with an algebra) and the geometric approach (in which one starts with a geometric object). In Manin's terms, the "fundamental [algebra-geometry] duality that increasingly manifests itself in modern mathematics" assumes the following two symmetric forms ("with a different emphasis") [START_REF] Manin | Introduction to the Theory of Schemes[END_REF][START_REF] Awodey | Logic in Topoi: Functorial Semantics for Higher-Order Logic[END_REF]: "The space the ring of functions on the space" (geometric approach)

"The ring the spectrum of its ideals of certain type" (spectral approach) (III.3) We could say that these two descriptions of the algebra-geometry duality differ in the characterization of which structure is being represented or "observed". In the spectral approach, the algebra A plays the role of the represented object and each point of the space Spec(A) defines a particular representation of A. In the geometric approach, the points 23 Since we are also considering here the general case in which the codomain B can vary, we omit the subscript indicating the codomain of the representations. 24 In the article Adjointness in foundations, Lawvere states that the duality between commutative rings and the corresponding geometric entities is a particular case of a more general duality between what he calls the "Formal" (the side of the duality corresponding to the algebraic structure) and the "Conceptual" (the semiotic side of the duality) [START_REF] Lawvere | Adjointness in Foundations[END_REF]Sect.4] (see the discussions of this distinction in Refs. [START_REF] Rodin | Axiomatic Method and Category Theory[END_REF]Sect.5.7] and [START_REF] Marquis | The history of categorical logic: 1963-1977[END_REF]Sect.2.4]. In Lawvere's terms: "That pursuit of exact knowledge which we call mathematics seems to involve in an essential way two dual aspects, which we may call the Formal and the Conceptual. For example, we manipulate algebraically a polynomial equation and visualize geometrically the corresponding curve. [...] any attempt to formalize Foundations ['the study of what is universal in mathematics'] would be a description of this claimed 'duality' between the Formal and the Conceptual" [START_REF] Lawvere | Adjointness in Foundations[END_REF][START_REF] Ameriks | Recent Work on Kant's Theoretical Philosophy[END_REF][START_REF] Anel | The Geometry of Ambiguity. An Introduction to the Ideas of Derived Geometry[END_REF].

of the space Spec(A) are the "states" being "observed" by the "observables" (functions or sections) in the corresponding algebra of quantities. We could say that the Gelfand transform transforms each represented element a in A into an "observable" â on the corresponding space. 25 Now, the apparent symmetry between the two descriptions should not lead us to believe that they are strictly equivalent. Briefly, there exist geometric objects that cannot be constructed as the spectrum of an algebra. Let's consider this point in some detail.

A reconstruction theorem encodes the fact that the corresponding moduli space of representations contains "enough points/representations" to recover the represented algebraic structure. Technically, this means that the composition structure functor Γ(-) • semiotic functor Spec(-) in Figure N • II.1 yields the identity on the algebraic structures modulo isomorphisms (i.e. Γ(Spec(A)) A). But we could start with a geometric object X that is not in the image of the semiotic functor Spec(-) and consider the effect of the other possible composition of the two functors-i.e. the composition semiotic functor Spec(-)

• structure functor Γ(-)-acting on this gometric object. So, we start with the geometric object X, we extract an algebraic structure Γ(X) out of it and then consider the space Spec(Γ(X)) that parameterizes the corresponding representations of Γ(X). Now, even if we have a representation theorem of the form Γ(Spec(A)) A, it might not be necessarily the case that Spec(Γ(X))

X. In other terms, we can always use the functor Γ(-) to extract an algebraic structure Γ(X) out of the geometric object X, but X might not coincide with the moduli space Spec(Γ(X))

X that parameterizes the representations of Γ(X). In that case, the pre-duality given by the adjunction between the structure functor and the semiotic functor is not strictly speaking a duality (i.e. an equivalence of categories), even if we have a reconstruction theorem of the form Γ(Spec(A))

A. 26Hence, the spectral approach (that starts with an algebra) and the geometric approach (that starts with a geometric object) are equivalent only when the pre-duality between algebra and geometry is a duality. An instance of this scenario is provided by the notion of general (i.e. not necessarily affine) scheme (e.g. the scheme associated to a projective variety). Roughly speaking a scheme X is a locally ringed space such such each point has an open neighborhood where the scheme looks like an affine scheme (that is, like a scheme obtained as the spectrum of a ring) [START_REF] Hartshorne | Algebraic Geometry[END_REF][START_REF] Landsman | Mathematical Topics Between Classical and Quantum Mechanics[END_REF]. The composition Spec(Γ(X)) is related to the original scheme X by a unique morphism ψ : X → Spec(Γ(X)) that is an isomorphism only when X is affine [32, 4 & 7]. We could say that in this case there exists a pre-duality between the (opposite of the) category of rings and the category of schemes that restricts to a duality on affine schemes.

IV. Restricted Reconstruction Theorem in Algebraic Geometry

We shall now consider in more detail the restricted reconstruction theorem in classical algebraic geometry. This theorem only allows to recover certain kind of rings from a collections of their representations, namely reduced and finitely generated rings over algebraically closed fields (for the discussion proposed in this section we recommend the reader the introduction to Ref. [START_REF] Dieudonné | Algebraic Geometry[END_REF] and Ref. [START_REF] Manin | Introduction to the Theory of Schemes[END_REF]). A polynomial ring P .

= k[x1, . . . , xn] in n variables over a field k can be understood as a sort of "linguistic universe" in which it is possible to define predicates of the form ∃ n-tuples (a1, . . . , an) such that p(a1, . . . , an) = 0 (IV.1) with p ∈ P . 27 The field k plays here the role of what we could call in the wake of Kronecker domain of rationality [Rationalitätsbereich] (also called coefficient domain or domain of definition).

We shall now consider the truthmakers of such predicates, i.e. the values of the xi's that make the predicates true. In a first approach, it is natural to look for the truthmakers of a predicate in the domain of definition of the latter. However, there is no reason for such a restriction: the fact that a predicate is defined over a domain k does not mean that its truthmakers only exist in k. This distinction between the domain of rationality k in which a problem is formulated and the possible codomains of rationality wherein one can look for its solutions made possible the successive extensions of the number systems (from N to Z to Q to R to C) (see Refs. [START_REF] Bellomo | Domain Extension and Ideal Elements in Mathematics[END_REF][START_REF] Cantù | An argumentative approach to ideal elements in mathematics[END_REF][START_REF] Manders | Domain Extension and the Philosophy of Mathematics[END_REF] for a conceptual analysis of the notion of domain extension). 28 The minimal condition imposed on the possible codomains of definition is that they must be k-algebras, i.e. that their elements can be multiplied, added, and multiplied by scalars in k (otherwise, the polynomial expressions in the corresponding variables could not even be defined). In what follows, we shall restrict the possible codomains to k-algebras k that are fields. If T is a subset of P (i.e. a particular family of predicates defined over k), the n-tuples (a1, . . . , an) with ai ∈ k such that p(a1, . . . , an) = 0 for all p ∈ T will be called k -truthmakers of the family T . By using Grothendieck and Dieudonné's terminology, we shall also say that the solutions in k are k -valued points of the variety over k defined by the family T [START_REF] Grothendieck | Éléments de Géométrie Algébrique I[END_REF][START_REF] Atiyah | Introduction to Commutative Algebra[END_REF]. The set of k -truthmakers of the family T defines a subset (called an algebraic affine variety) of the affine n-space A n k over k (i.e. of the set of n-tuples of elements in k ). Now, the points in the variety also satisfies equations of the form i hipi = 0, where pi ∈ T and hi are polynomials in P . In order to get rid of this ambiguity regarding the polynomials that define the variety, we have to substitute the family T by the ideal t generated by T . The affine variety in A n k defined by an ideal t will be denoted Vt(k ). Now, an ideal t does not define an algebraic variety as such but rather an algebraic variety immersed in A n k (see for instance the example in Ref. [START_REF] Dieudonné | Algebraic Geometry[END_REF]236]). It is possible to "factor out" this immersion and pass to an intrinsic description by considering the quotient rings P/t. 29 Indeed, the quotient rings defined by different immersions of the same abstract variety are isomorphic. Passing from P to P/t amounts to identify different polynomials in P that coincide when evaluated on the variety, i.e. polynomials that differ in a function vanishing on the variety. In other terms, we are making abstraction from the fact that these polynomials might differ elsewhere. 30 Given an ideal t in P and a k-algebra k there is a natural bijection between the algebraic variety Vt(k ) and the k-algebras morphisms P/t → k defined by evaluating the polynomials in P/t on the corresponding point of the variety [START_REF] Grothendieck | Éléments de Géométrie Algébrique I[END_REF][START_REF] Atiyah | Duality in Mathematics and Physics[END_REF]. This means that each point of the variety Vt(k ) can be interpreted as a k -representation of P/t. We could say that the variety Vt(k ) encodes the k -representation theory of the algebraic structure P/t. We can now pose the following reconstruction problem: is it possible to recover the quotient ring P/t from its k -representations? As we have explained in Section N • III, the reconstruction of an algebraic structure from a collection of representations requires to 27 More precisely, a polynomial p ∈ P can be written in the form p(x1, . . . , xn) = i cix i , where ci ∈ k and x i are monomials of the form

x i = x i 1 1 x i 2 2 . . . x in
n with i = (i1, . . . , in) an n-tuple of non-negative integers. 28 By working with polynomial equations in one variable (which is a particular case of 0-dimensional algebraic geometry), Galois theory is essentially based on the distinction between the domain of definition (or coefficient domain) of the polynomial in question (i.e. a field containing the coefficients of the polynomial) and the possible codomains in which one can search for its solutions [START_REF] Catren | On the Notions of Indiscernibility and Indeterminacy in the Light of the Galois-Grothendieck Theory[END_REF]. According to Cartier, "Galois was certainly the first person to notice the polarity between equations and their solutions" [START_REF] Cartier | A Country of which Nothing is Known but the Name: Grothendieck and Motives[END_REF]290]. 29 For a discussion of the difference between extrinsic and intrinsic descriptions in terms of the functorial approach to algebraic geometry that we shall introduce below see Ref. [START_REF] Grothendieck | Introduction to functorial algebraic geometry, part 1: affine algebraic geometry, summer school in Buffalo[END_REF][START_REF] Chevalley | Theory of Lie Groups[END_REF][START_REF] Corfield | Duality as a category-theoretic concept[END_REF][START_REF] Deligne | Quelques idées maîtresses de l'oeuvre de A. Grothendieck[END_REF][START_REF] Demazure | Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs[END_REF][START_REF] De Risi | Mathematizing Space. The Objects of Geometry from Antiquity to the Early Modern Age[END_REF][START_REF] De Risi | Geometry and Monadology. Leibniz's Analysis Situs and the Philosophy of Space[END_REF][START_REF] Dieudonné | History of Algebraic Geometry[END_REF][START_REF] Dieudonné | The Historical Development of Algebraic Geometry[END_REF][START_REF] Dieudonné | Algebraic Geometry[END_REF]. 30 For instance, the polynomials x + y and x + x 2 coincide on the variety defined by the equation y = x 2 . In other terms, their difference is "invisible" on the parabola. introduce a Gelfand transform between the algebraic structure at stake (in this case the ring P/t) and an algebra of quantities on a moduli space of representations. In this case, the Gelfand transform assigns to every element [p] of the ring P/t a k -valued quantity p k on the variety Vt(k ) defined by the expression p k (α) = p(α) for α ∈ Vt(k ). The quantity p k encodes all the k -representations p(α) of [p] ∈ P/t. We could say that the Gelfand transform acts as a sort of prism, in the sense that it permits to associate to each element [p] ∈ P/t its k -spectrum, i.e. the set of k -representations of [p].

The ring P/t can be reconstructed from its k -representations if the Gelfand transform is injective. Now, it might be the case that the codomain k is such that the variety Vt(k ) does not have "enough points" (i.e. enough k -representations of P/t) to faithfully reconstruct P/t. For instance, if the field k does not contain all the roots of a polynomial p ∈ k[x], the algebra k[x]/(p) may not be faithfully represented as an algebra of quantities on the space Vt(k ). This means that different elements [f ] and [g] in P/t might define the same quantity on the space Vt(k ), thereby having the same k -spectrum. In such a case, the variety Vt(k ) does not "separate" [f ] and [g]. In the extreme case, there might be no k -representations of P/t. 31 In order to guarantee that the codomain provides enough representations of P/t so that this ring can be faithfully reconstructed, the codomain must be an algebraically closed field extension K of k. 32 If in addition we restrict the ideals under consideration to radical ideals (i.e. to ideals t such that t = √ t) 33 , then we obtain a bijective correspondence between algebraic objects (radical ideals of polynomials) and geometric objects (algebraic varieties) [START_REF] Hartshorne | Algebraic Geometry[END_REF]Cor.1.4,[START_REF] Anel | The Geometry of Ambiguity. An Introduction to the Ideas of Derived Geometry[END_REF].

A restricted reconstruction theorem can now be obtained by comparing two algebraic objects, namely the quotient ring P/t and a ring of quantities on the variety Vt(K) defined by means of the Gelfand transform. The corresponding reconstruction theorem states that the Gelfand transform establishes an isomorphism

P/t O(Vt(K)) (IV.2)
between the ring P/t that one wants to reconstruct and the ring O(Vt(K)) of regular functions 34 on the variety Vt(K) [57, Th.3.2 (a), 17]. Since P/t is reconstructed from the representations of P/t into a unique codomain (the algebraically closed field K), this isomorphism provides an example of a restricted reconstruction theorem. We could say that the algebraically closed field K provides enough representations of the ring P/t (i.e. enough common roots of the polynomials in t) so that P/t can be faithfully reconstructed from the collection of its K-representations.

It is worth noting that this algebra-geometry (or structure-semiotics) duality encodes a sort of "indeterminacy principle" that presents an analogy with the indeterminacy principle in quantum mechanics. Indeed, there is an inverse correlation between the unambiguous (or sharp) determination of an element in P/t and the unambiguous determination of a particular K-representation in the variety Vt(K). If one just considers a single K-representation of P/t-i.e. a single point in Vt(K)-, then the elements in the ring P/t remain maximally undetermined. Indeed, different elements [f ], [g] ∈ P/t might be 31 For example, there are no R-representations of the ring R[x]/(x 2 + 1), which means that there are no R-truthmakers of the predicate x 2 + 1 = 0. Of course, this does not mean that this predicate is false, but rather that the its truthmakers have to be searched in an algebraic extension of the codomain R. 32 The Hilbert Weak Nullstellensatz (weak theorem of zeros) states that if the codomain K is algebraically closed, then the only ideal t such that Vt(K) = ∅ is t = P . This theorem may be understood as the fundamental theorem of algebra for polynomials in several variables in the sense that it states that every family of polynomials generating an ideal smaller than the whole ring has at least one common zero in K n , i.e. it does not define the empty variety. 33 Let's recall that the radical √ t of an ideal t is the set of elements f ∈ P such that f n ∈ t for some natural number n. The Hilbert Nullstellensatz states that two ideals t1 and t2 define the same K-variety (being K an algebraically closed field) if and only if √ t 1 = √ t 2 . Therefore, ideals t1 and t2 with the same radical cannot be distinguished by just considering theirs K-representations. This explains why we have to restrict the algebraic side to radical ideals in order to obtain a reconstruction theorem. 34 A function f : X → K is regular at a point p ∈ X if f can be expressed as a rational function g/h on a neighborhood of p such that h is nowhere zero in U . A function f is regular on X if it is regular at every p ∈ X [START_REF] Hartshorne | Algebraic Geometry[END_REF][START_REF] Bellomo | Domain Extension and Ideal Elements in Mathematics[END_REF].

represented by the same "image" in a K-representation x ∈ Vt(K). Conversely, if the K-representation is completely "undetermined" (i.e. if one considers all the possible Krepresentations of the elements in P/t at once), then the elements in the ring may be faithfully determined. This remark permits to understand why each single [f ] ∈ P/t is in correspondence-via the Gelfand transform-with a quantity fK on the set of all possible K-representations of P/t. This leads to what we could call the "indeterminacy principle" associated to the algebra-geometry duality: whereas the sharp determination of a point in the space requires to consider a whole family of quantities, the sharp determination of a quantity requires to consider the values that it takes on all the points of the space. This "indeterminacy principle" could be understood as the mathematical incarnation of the phenomenological fact that an empirical object cannot be faithfully reconstructed from one of its profiles, i.e. from the single aspect that it offers to a subject placed in a particular position. Roughly speaking, a faithful determination of an object requires to turn around the object, i.e. to "undetermine" the viewpoint of observation. 35

V. From Sets of Points to Functors of Points

As we have explained before, the distinction between the domain of definition containing the coefficients of a family of polynomial predicates and the different possible codomains that might contain truthmakers of the former elicits a sort of relativity in algebraic geometry: we can associate to a quotient ring P/t different algebraic varieties, namely one algebraic variety Vt(k ) for each k-algebra k . As we have seen in the previous Section, it is possible to reconstruct certain rings from the algebra of regular functions on the corresponding varieties by choosing a suitable codomain, namely a k-algebra given by an algebraically closed field K. The decision to privilege a (well-chosen) particular codomain was notably fostered by A. Weil in his treatise Foundations of Algebraic Geometry in the form of what he called "universal domains" (algebraically closed fields of infinite transcendence degree) [START_REF] Weil | Foundations of Algebraic Geometry[END_REF] (regarding Weil's treatise see also Ref. [START_REF] Raynaud | André Weil and the Foundations of Algebraic Geometry[END_REF]). 36 Roughly speaking, the idea is to work within a codomain that is big enough so that it contains all that one might need for the purposes at stake.

The problem is that the reconstruction protocol used in classical (i.e. before Grothendieck) algebraic geometry only works for certain rings, namely reduced and finitely generated rings over algebraically closed fields. Now, in the framework of the refoundation of algebraic geometry associated to scheme theory, Grothendieck proposed a more general reconstruction protocol that works for general rings. The key point of Grothendieck's innovative strategy is that it does not "break" the aforementioned relativity by privileging a particular codomain, but rather fully embraces it by considering all the codomains at once [START_REF] Grothendieck | Introduction to functorial algebraic geometry, part 1: affine algebraic geometry, summer school in Buffalo[END_REF]. 37 According to Grothendieck's philosophy, a family of polynomial equations over k does not have a solution set defined in a privileged codomain, but rather (what we could call in the wake of Ref. [97, p.113]) a solution structured set composed of the sets of solutions in each k-algebra B for every B (also called variable set of solutions in Ref. [91, 35 This heuristic relation between the algebra-geometry duality and the indeterminacy principle in quantum mechanics has been explored in the series of papers [START_REF] Catren | On the Notions of Indiscernibility and Indeterminacy in the Light of the Galois-Grothendieck Theory[END_REF][START_REF] Page | Phase Symmetries of Coherent States in Galois Quantum Mechanics[END_REF][START_REF] Page | Towards a Galoisian lnterpretation of Heisenberg lndeterminacy Principle[END_REF]. In order to argue that this might be more than a mere analogy, it is worth noting that the position-momentum duality in quantum physics is provided by Fourier theory, that is-in group-theoretic terms-by the Pontryagin duality for locally-compact abelian groups [START_REF] Chevalley | Theory of Lie Groups[END_REF][START_REF] Pontryagin | The general topological theorem of duality for closed sets[END_REF]. Moreover, the symplectic structure of the corresponding phase space encodes the "internal" self-dual structure associated to the "external" duality between the position and the momentum representations (see Refs. [START_REF] Majid | Principle of Representation-Theoretic Self-Duality[END_REF][START_REF] Page | Towards a Galoisian lnterpretation of Heisenberg lndeterminacy Principle[END_REF][START_REF] Page | Phase Symmetries of Coherent States in Galois Quantum Mechanics[END_REF] for further discussions of this point). 36 Grothendieck and Dieudonné describe Weil's approach in the following terms: "C'est seulement à partir de 1940, avec la Géométrie algébrique 'abstraite' (c'est-à-dire sur un corps de base k quelconque, pouvant être de caractéristique = 0) développée surtout par Weil, Chevalley et Zariski, que l'idée du changement de base prend de l'importance dans un contexte plus général [...] Toutefois [...] chez Weil [la généralité de cette opération] est quelque peu masquée par le parti pris de se restreindre une fois pour toutes à n'envisager que des sous-corps d'un corps algébriquement clos 'assez grand' (le 'corps universel'), restant donc en apparence assez proche du point de vue classique où le corps C tenait ce role" [START_REF] Grothendieck | Éléments de Géométrie Algébrique I[END_REF][START_REF] Artin | Algebra[END_REF]. 37 See Ref. [START_REF] Reed | Figures of Thought: Mathematics and Mathematical Texts[END_REF]Sec.5.4] for a comparison between Weil's universal domains and Grothendieck's functorial approach.

14.3.2]

). 38 More formally, the "solution" to a family of polynomial equations over k will be defined as the "application" that sends any k-algebra B to the solution set in B in a functorial manner.

The central mathematical object in Grothendieck's approach is given by the so-called functor of solutions [START_REF] Grothendieck | Introduction to functorial algebraic geometry, part 1: affine algebraic geometry, summer school in Buffalo[END_REF] or (in the dual geometric version) functor of points. So, let us consider this notion with some detail (see also Refs. [START_REF] Catren | On the Notions of Indiscernibility and Indeterminacy in the Light of the Galois-Grothendieck Theory[END_REF][START_REF] Eisenbud | The Geometry of Schemes[END_REF][START_REF] Vaquié | Sheaves and Functors of Points[END_REF]). To do so, let's consider a general ring A. The idea is to assign to each possible codomain given by a ring B a "space" Spec B (A) whose underlying set will be given by HomRing(A, B). More precisely, we can formalize the fact that a ring A defines a different space Spec B (A) of B-representations A → B for each possible codomain B by associating to A a covariant functor

Spec (-) (A) : Ring → Set (V.1)
sending each ring B to the B-spectrum Spec B (A) = HomRing(A, B) of A. 39 By definition, the functor of solutions Spec (-) (A) of A is an object in the category [Ring, Set] of all functors from Ring to Set (where the morphisms are given by the natural transformations of functors). The Yoneda embedding associates to any ring A the functor (V.1) "represented" by A (regarding the Yoneda embedding and the Yoneda lemma, see for instance Ref. [10, §8.2 & 8.3, 160-166]). This means that the functor (V.1) can also be described as the covariant representable functor represented by the ring A (see for instance the discussion of representable functors proposed in Ref. [START_REF] Marquis | From a Geometrical Point of View[END_REF]Sect.3.2.2]). In particular, the ring P/t is the object that represents the "functor of solutions" Spec (-) (P/t), that is, the object that "classifies" the solutions of the equations defined by the ideal t in all the k-algebras k . The Yoneda lemma states that the set F (B) is in bijective correspondence with the set of natural transformations F (-) → Spec (-) (B), where F (-) is any (i.e. not necessarily representable) Set-valued functor on Ring. It follows from the Yoneda lemma that the Yoneda embedding is fully faithful, i.e. that this embedding defines a full subcategory of the category of Set-valued functors on Ring.

The conceptual importance of the Yoneda lemma is (at least) twofold. Firstly, the Yoneda lemma entails a shift of perspective from a substantivalist conception according to which the fundamental mathematical entities (of algebraic geometry in this case) are given by the objects of the corresponding category to a structuralist, relational, behavioral, or (in Manin's words [START_REF] Manin | Georg Cantor and His Heritage[END_REF]) sociological perspective according to which the fundamental entities are functors. As Mazur puts it, "[t]he lights are dimmed on mathematical objects and beamed rather on the corresponding functors; that is, on the networks of relationships entailed by the objects" [START_REF] Mazur | When is one thing equal to some other thing?[END_REF]. The main conceptual point here is that the functor Spec (-) (A) provides a good 'semiotic' substitute for the ring A. This means that Spec (-) (A) describes A in terms of the representations A → B of A defined with respect to all the other objects B of the category. If two objects have equal (i.e. isomorphic) functors encoding their representations, then the two objects are also equal (i.e. isomorphic).

The second important conceptual consequence of the Yoneda lemma is that it introduces an ontological extension of the domain of entities at stake, namely an extension from the objects of the initial category to any functor in [Ring, Set], be it representable or not. Indeed, the Yoneda embedding is not essentially surjective, i.e. not every functor 38 In Grothendieck and Dieudonné's own terms: "[o]n y fait abstraction des propriétés spéciales aux solutions du système [p(x1, ..., xn) = 0 for all p ∈ T ] dans l'espace particulier k n d'oú l'on est parti, pour considérer, pour chaque k-algébre k , l'ensemble des solutions de [ce système] dans k n , et la façon dont cet ensemble varie avec k [...] L'idée de 'variation' de l'anneau de base que nous venons d'introduire s'exprime mathématiquement sans peine grâce au langage fonctoriel [...]" [START_REF] Grothendieck | Éléments de Géométrie Algébrique I[END_REF][START_REF] Artin | Algebra[END_REF]. 39 On morphisms, the functor Spec (-) (A) sends each morphism B f -→ C to the morphism Spec f (A) :

Spec B (A) → Spec C (A) given by (A g -→ B) → (A f •g ---→ C).
In particular, when the ring A is the k-algebra P/t defined by an ideal t of polynomials, the functor (V.1) sends each k-algebra B to the solution set in B of the corresponding polynomials (i.e. B → Vt(B)) and each morphism h : B → B is sent into Vt(h) : Vt(B) → Vt(B ) where Vt(h)(a1, . . . , an) = (h(a1), . . . , h(an)).

in [Ring, Set] is representable. 40 Now, the Yoneda lemma guarantees that even nonrepresentable functors F (-) can be interpreted as functors of points, i.e. in terms of morphisms to other objects. More precisely, the Yoneda lemma states that the elements in the set F (B) are in bijection with the natural transformations F (-) → Spec (-) (B), i.e. with (what we could call) the Spec (-) (B)-representations of F (-). Even if the elements in F (B) for F a non-representable functor cannot be interpreted in terms of morphisms in Ring, the Yoneda lemma guarantees that these elements can still be interpreted as representations of F (-) in the functor category [Ring, Set]. The trick to do so, is to substitute B by its functor of points Spec (-) (B). It is also worth noting that the description of the functor F in terms of its representations only uses morphisms to representable functors. Roughly speaking, the representable functors "generate" the whole functor category [Ring, Set]. 41 In the terms used by Schreiber, the representable functors Spec (-) (B) can be understood as "test objects" out of which "generalized objects" can be constructed. A "generalized object" F (-) is a non-representable element of the functor category [Ring, Set] that is completely defined by its "representations" into the test objects, i.e. by the sets Hom [Ring, Set] (F (-), Spec (-) (B)) of Spec (-) (B)-representations of F (-) for every Spec (-) (B) (see for instance Refs. [112,[START_REF] Hodges | Model Theory[END_REF] and [START_REF] Lawvere | Taking Categories Seriously[END_REF][START_REF] Bird | Kant's Theory of Knowledge. An Outline of One Central Argument in the Critique of Pure Reason[END_REF]).

VI. General Reconstruction Theorem in Algebraic Geometry

As discussed in the previous section, the functor of points Spec (-) (A) on the category of rings is a fully faithful substitute for the ring A. Since each set Spec B (A) may be thought of as a space, the functor of points establishes a correspondence between rings and a whole family of spaces. In Grothendieck and Dieudonné's terms, "[...] on n'a plus affaire à un 'objet' bien déterminé comme dans la Géométrie algébrique classique, mais à une 'famille d'objets' variable avec [B]" [START_REF] Grothendieck | Éléments de Géométrie Algébrique I[END_REF][START_REF] Awodey | Logic in Topoi: Functorial Semantics for Higher-Order Logic[END_REF]. Now, is it possible to recover an algebra-geometry duality between the ring A and a single space? Is it possible to establish an isomorphism between A and a ring of quantities on a single "moduli space" parameterizing representations of A? To do so, one should be able to unify the family of B-spectra Spec B (A) for all B into a single topological space (that we shall denote) Spec(A). Then, and in order to obtain an algebra-geometry duality, we have to introduce quantities on this space. As we shall see, these quantities are given by sections of a sheaf on Spec(A) endowed with a suitable topology (called structure sheaf and denoted O Spec(A) ). The geometric object (Spec(A), O Spec(A) ) resulting from this construction is called affine scheme. 42 Briefly, an affine scheme associated to a ring A is an underlying set Spec(A) endowed with a topology and a sheaf of quantities O Spec(A) on it. As we shall see, the stalks of the sheaf are local rings. Hence, an affine scheme is a locally ringed space. 43 We shall now briefly describe the salient features of this construction.

First, the underlying set of points of an affine scheme is defined by the field-valued representations of the ring, i.e. by representations of the form A → k for k a field. Now, 40 The utilization of the Yoneda embedding sometimes requires to study the conditions under which a given functor is representable [START_REF] Grothendieck | Fondements de la Géométrie Algébrique[END_REF]. One important feature of representable functors is that they have the sheaf property with respect to certain Grothendieck topologies (the subcanonical topologies) on the corresponding category. These ideas were used by Grothendieck in Ref. [START_REF] Grothendieck | Fondements de la Géométrie Algébrique[END_REF] for the construction of various important spaces, like for instance the Hilbert and Picard schemes. The strategy used by Grothendieck was to first study the relevant functor, and to prove then that the functor is representable. 41 This characterization of the functor category [Ring, Set] as "generated" by the representable functors is formalized by the result according to which any non-representable functor F is a colimit (i.e., roughly speaking, the result of a pasting) of representable functors (see for instance Ref. [START_REF] Mac Lane | Sheaves in Geometry and Logic. A First Introduction to Topos Theory[END_REF][START_REF] Friedman | Kant on Concepts and Intuitions in the Mathematical Sciences[END_REF]). The category [Ring, Set] can then be understood as the free co-completion of the category of rings. We could also say that the representable functors are "dense" so to speak in [Ring, Set]. 42 In Refs. [START_REF] Eisenbud | The Geometry of Schemes[END_REF][START_REF] Hartshorne | Algebraic Geometry[END_REF], the term spectrum is used to denote affine schemes (Spec(A), O Spec(A) ), that is topological spaces of the form Spec(A) endowed with sheaves of quantities O Spec(A) . Here we prefer to use the term spectrum only for the underlying set Spec(A) and called the corresponding ringed space affine scheme (like Dieudonné does in Ref. [START_REF] Dieudonné | History of Algebraic Geometry[END_REF]). 43 More generally, a scheme (not necessarily affine) is a ringed space (X, O X ) that is locally isomorphic to an affine scheme.

the field-valued representations of a ring are in a certain sense redundant since for instance t : A → k and t : A t -→ k → K (where K is a field extension of k) would count as different points. This redundancy can be removed by introducing an equivalence relation for kvalued representations A → k for variable k, namely that two such representations are equivalent if they have the same kernel [START_REF] Grothendieck | Éléments de Géométrie Algébrique I[END_REF][START_REF] Awodey | Logic in Topoi: Functorial Semantics for Higher-Order Logic[END_REF][START_REF] Awodey | First-order logical duality[END_REF]. Since a field is an integral domain, the kernel p = ker(t) of a k-representation t : A → k is a prime ideal of A and the integral domain A/p is isomorphic to a subring of k. All in all, the underlying set of the affine scheme associated to a ring A will be given by the set of prime ideals of A, where the latter encode the field-valued representations of A modulo the aforementioned redundancy.

This definition of the underlying set of points of an affine scheme constitutes a first surprising feature of Grothendieck's construction. In the classical theory of algebraic varieties, the underlying set of points of an algebraic variety is in bijective correspondence with the maximal ideals of the coordinate ring [57, Th.3.2(b), 17]. In turn, prime ideals are in bijective correspondence with the irreducible closed subsets of the algebraic variety. Therefore, the underlying set Spec(A) of an affine scheme includes not only the standard points corresponding to maximal ideals but also the so-called generic points (introduced by the Italian algebraic geometers at the beginning of the XXth century) that correspond to irreducible closed subsets (see for instance [START_REF] Hartshorne | Algebraic Geometry[END_REF][START_REF] Lawvere | Taking Categories Seriously[END_REF]). 44 In a certain sense, Grothendieck's theory of schemes bridges the gap between what De Risi calls the classical (Greek) geometry of figures on the one hand and the modern geometry of space on the other [START_REF] De Risi | Mathematizing Space. The Objects of Geometry from Antiquity to the Early Modern Age[END_REF]. According to De Risi, Greek geometry deals with individual figures that "are not embedded in any spatial background " since "the concept of space itself is altogether missing from the cultural background of Antiquity" [START_REF] De Risi | Mathematizing Space. The Objects of Geometry from Antiquity to the Early Modern Age[END_REF][START_REF] Ameriks | Recent Work on Kant's Theoretical Philosophy[END_REF][START_REF] Anel | The Geometry of Ambiguity. An Introduction to the Ideas of Derived Geometry[END_REF]). 45 In this context, no figure plays a privileged role to the detriment of the others. In Cartier's terms, "For Euclid there are geometric figures, and a point is merely one element of a figure, the most elementary, perhaps, since it is assumed to have no dimension: neither length, nor breadth, nor thickness. Figures generate one another. A given line D is defined by the property of passing through two given points P and Q, but a point P is defined as the intersection of two lines D and ∆. [...] it is not in the spirit of Euclid to regard a line as a set of points. A figure in the sense of Euclid is more than a simple set of points." [START_REF] Cartier | A Mad Day's Work: from Grothendieck to Connes and Kontsevich. The Evolution of Concepts of Space and Symmetry[END_REF] By contrast-and according to the modern conception that began with Leibniz's analysis situs [START_REF] De Risi | Geometry and Monadology. Leibniz's Analysis Situs and the Philosophy of Space[END_REF] and became the common definition by the dawn of the XIXth century-, modern geometry is the science of space. 46 In this last framework, a space is understood as a set of points equipped with additional geometric structures (e.g. a topology, a connection, a metric, etc.) and the geometric figures are now understood as subsets of such space. In this framework, a particular kind of irreducible figure plays a privileged role to the detriment of the other irreducible figures, namely the unextended points that belong to the underlying set of the space. In a certain sense, the notion of affine scheme is diagonal to De Risi's distinction. On the one hand, an affine scheme is a set equipped with 44 A standard argument to justify the identification between the points of an affine scheme and the prime ideals of the corresponding ring (rather than just the maximal ideals) is that a ring homomorphism ϕ : A → B does not induce a well-define map Specm(B) → Specm(A) since ϕ -1 (b) is not necessarily a maximal ideal in A (where Specm(B) denotes the set of maximal ideals of B). However, the inverse image of a prime ideal under a ring homomorphism ϕ : A → B is a prime ideal of A. 45 According to Grant, "[t]here is nothing in Euclid's geometry to suggest that he assumed an independent, infinite, three-dimensional, homogeneous space in which the figures of his geometry were located. In a purely geometric sense, such a space would have been superfluous because every geometric figure has its own internal space" [START_REF] Grant | Much ado about nothing: Theories of space and vacuum from the middle ages to the scientific revolution[END_REF]16]. In Space-Time-Matter, also Weyl establishes a contrast between "a true geometry, [understood as] a doctrine of space itself" and "almost everything else that has been done under the name of geometry, a doctrine of the configurations that are possible in space" [START_REF] Weyl | Space-Time-Matter[END_REF][START_REF] Page | Towards a Galoisian lnterpretation of Heisenberg lndeterminacy Principle[END_REF]. 46 Regarding the difference-for instance-between Euclid's Elements and Hilbert's Grundlagen der Geometrie see Ref. [START_REF] Mueller | Philosophy of mathematics and deductive structure in Euclid's 'Elements[END_REF][START_REF] Barr | Variable Set Theory[END_REF][START_REF] Bellomo | Domain Extension and Ideal Elements in Mathematics[END_REF]. additional structure (namely, a topology and a sheaf of quantities), thereby providing an example of the modern understanding of the notion of space. However, the underlying set of "points" does not only include the standard unextended points associated to maximal ideals, but also one generic point for each irreducible closed subset. Hence, the underlying set of the space is a set of both extended and unextended irreducible figures. An affine scheme can therefore be understood as an underlying set of irreducible figures endowed with an additional geometric structure. The irreducible figures-rather than being understood as subsets of a preexisting space qua set of unextended points-form the "skeleton" of the space. 47 In order to obtain a reconstruction theorem for a ring A we need to define a Gelfand transform sending each element in A to a quantity on Spec(A). As we have explained in Section N • III, the Gelfand transform is defined by an inversion formula (III.1) that defines the value of a quantity â associated to a ring element a ∈ A at a point p by using the fact that this point is itself a representation of A. So, we shall define a Gelfand transform that sends each element a ∈ A to the quantity â on Spec(A) given by â(p) = tp(a) where tp : A → A/p is the quotient map associated to the point p. As we said before, the map tp can be redundantly understood as a field-valued representation of A with respect to different fields. Here we shall consider tp as a field-valued representation of A with values in the smallest field in which A/p can be embedded, namely the fraction field κp (also called residue field ) of the integral domain A/p. Since the A/p → κp is an inclusion, we shall consider that the quantity â is a κp-valued quantity on Spec(A). The fact that the fields κp depend on the points p in Spec(A) on which we evaluate the quantity â encodes the fact that the space Spec(A) parameterizes representations of A into different codomains. 48 According to the general pattern schematized in Figure III.2, we want to use this Gelfand transform to define an isomorphism between the ring A and an algebra of quantities on Spec(A). The main obstruction is that the Gelfand transform a → â that we have just defined is not injective if A has nilpotent elements (i.e. elements a such that a n = 0 for some n). Since a nilpotent element a is contained in every prime ideal [START_REF] Atiyah | Introduction to Commutative Algebra[END_REF]Prop.1.8,[START_REF] Anel | Topo-logie[END_REF], it defines a constant zero function on Spec(A), i.e. â(p) = tp(a) = 0 for every p ∈ Spec(A). This means that all the nilpotent elements are in the kernel of the Gelfand transform. It follows that the quantities â only allows us to recover the reduced ring A/N associated to A, namely A quotiened by the ideal N of nilpotent elements (called nilradical ).

In standard differential calculus, a function f (x) on a space M is completely determined by the values it takes at every x ∈ M . This means that the information encoded in its derivatives is (as the name indicates) "derived" from the functional expression that defines its values at every point. Now, the obstruction to the injectivity to the Gelfand transform given by the nilpotent elements in A can be bypassed by providing an enhanced definition of the quantities on Spec(A). In scheme theory, this is done by defining a quantity not only by means of its punctual values, but also by specifying from scratch (rather than "deriving" it) the differential behavior of the "quantity" around each point. This can be formally done by introducing quantities on Spec(A) with values in germs of sections of a sheaf.

In order to define such a sheaf, we have to introduce a topology on the set Spec(A). The coarsest topology that does the job is the so-called Zariski topology. 49 The open sets 47 It is worth noting that the set of ideals of a ring has the structure of a lattice (where the order relation is given by the inclusion), i.e. of an ordered set with finite join (or operation or sum) and meet (and operation or intersection). This lattice yields the "skeleton" of the variety in the sense that it encodes the relations of incidence and specialization between the different subvarieties. 48 Let's consider for instance Spec(Z) and let's denote Zp a prime ideal of Z. Any integer n ∈ Z defines a quantity n on Spec(Z) such that n(Zp) = class of n modulo p, which is an element of the field Fp = Z/Zp. Hence, the function n takes values in a different field Fp for each prime p (see for instance Ref. [START_REF] Manin | Introduction to the Theory of Schemes[END_REF]Fig.1.4,[START_REF] Awodey | Logic in Topoi: Functorial Semantics for Higher-Order Logic[END_REF]). 49 For certain purposes it is necessary to refine the topology by passing for instance to the étale topology or the fppf (fidélement plat de présentation finie) topology, which are Grothendieck topologies. This of this topology are given by the complement of algebraic sets. 50 The so-called principal open sets U f = Spec(A) -V (f ) obtained by removing the biggest possible algebraic sets (i.e. the algebraic sets V (f ) defined by the zeros of a single function f / ∈ p) are the smallest neighborhoods of the point p and form a basis of the Zariski topology. Now that the set Spec(A) was endowed with a topology, the next step is to define a sheaf of quantities on Spec(A). The corresponding sheaf is called structure sheaf and denoted O Spec(A) . 51 It can be shown that the stalk O Spec(A) (p) of the structure sheaf at p (i.e. the set of germs of sections of the sheaf at p) is given by the local ring Ap [57, Prop.2.2, 71] (regarding the notion of germ see for instance Ref. [START_REF] Mac Lane | Sheaves in Geometry and Logic. A First Introduction to Topos Theory[END_REF]Sect.II.5,[START_REF] Lane | Duality for Groups[END_REF][START_REF] Mac Lane | Sheaves in Geometry and Logic. A First Introduction to Topos Theory[END_REF][START_REF] Majid | Principle of Representation-Theoretic Self-Duality[END_REF]). 52 An affine scheme is a locally ringed space (Spec(A), O Spec(A) ), i.e. a topological space Spec(A) endowed with a structure sheaf of rings O Spec(A) such that the stalk at each point p is the local ring Ap.

It is worth unpacking here the geometric meaning of the algebraic notion of local ring. The spectrum of the local ring Ap is given by Spec(Ap) = {q ∈ Spec(A)/q ⊆ p} [START_REF] Gortz | Algebraic Geometry I. Schemes[END_REF][START_REF] Grant | Much ado about nothing: Theories of space and vacuum from the middle ages to the scientific revolution[END_REF]. This means that the localization from Spec(A) to Spec(Ap)

(1) makes abstraction of the inner structure of the irreducible variety p, i.e. it considers p as if it were just an unextended point lacking any internal structure, 53 (2) and only focus on the subvarieties of Spec(A) containing p (i.e. it only keeps the ideals contained in p).

In Mumford's terms, the localization A → Ap allows us to focus on the directions "normal" to Vp [START_REF] Mumford | The Red Book of Varieties and Schemes, Second Expanded Edition[END_REF][START_REF] Putnam | What theories are not[END_REF]. Since we have only kept the prime ideals contained in p, the ideal p is the unique maximal ideal of Ap (this explains why Ap is called a local ring). Roughly speaking, Spec(Ap) contains one unextended point (the maximal ideal p) and all the extended subschemes containing it. 54 We shall now define an enhanced Gelfand transform G that is injective, i.e. that distinguishes the different nilpotent elements. To do so, let's note that the field-valued representations A → A/p → κp associated to each point p ∈ Spec(A) coincides with the map A → Ap → Ap/pAp = κp (i.e., that the operations of localization and passing to the quotient commute [START_REF] Gortz | Algebraic Geometry I. Schemes[END_REF]Rem.2.13,[START_REF] Grant | Much ado about nothing: Theories of space and vacuum from the middle ages to the scientific revolution[END_REF]). We shall now define an enhanced Gelfand transform that only retains the localization map λp : A → Ap in the last composition. In other terms, the enhanced Gelfand transform sends each a ∈ A to an enhanced quantity ã on Spec(A) given by ã(p) = λp(a). Since the local ring Ap is the stalk of the structure sheaf at p, the enhanced Gelfand transform G sends each element a ∈ A to a quantity on means that the construction of a space associated to a given ring is not canonical, since requires to choose a particular topology on the set of points. 50 The closed set V (S) of the Zariski topology defined by a subset S ⊂ A is given by the algebraic "figure" defined by S (i.e. by the points p ∈ Spec(A) such that f (p) = 0 for all f ∈ S). 51 For each open set U ⊂ Spec(A), we take O Spec(A) (U ) as the set of regular sections s : U → p∈U Ap, where s(p) ∈ Ap for all p ∈ U and s is locally induced by fractions a/b with a, b ∈ A and b = 0 on the corresponding open sets [START_REF] Hartshorne | Algebraic Geometry[END_REF][START_REF] Lambek | On the sheaf of possible worlds[END_REF]. Formally, a regular section is given by pairs (U, x/y) modulo the equivalence relation (U, x/y) ∼ (V, z/h) if x/y = z/h on U ∩ V . In order to understand the restriction to regular sections, it might be useful to remember that not every possible family of germs defines a section of a sheaf. The extra needed condition that a family of germs {ux ∈ Ox} x∈U has to satisfy is that for all x ∈ U there exists a neighborhood W ⊂ U containing x and a section t ∈ O(W ) such that the germs defined by the section t on every y ∈ W are the elements uy in the family [START_REF] Shafarevich | Basic Algebraic Geometry 2, Second Revised and Expanded Edition[END_REF][START_REF] Cassirer | Philosophie der Symbolischen Formen. Dritter Teil: Phänomenologie der Erkenntnis[END_REF]. It is worth noting that this is also the extra condition used to define the sheaf associated to a given pre-sheaf [START_REF] Hartshorne | Algebraic Geometry[END_REF][START_REF] Husserl | Ideen zu einer reinen Phänomenologie und phänomenologischen Philosophie. Erstes Buch: Allgemeine Einführung in die reine Phänomenologie[END_REF]. 52 The local ring Ap is the ring obtained by adding inverses to all the ring elements f ∈ A that are not in p, i.e. the ring Ap = a f , f / ∈ p of rational functions well-defined on p. Conceptually, the algebraic localization A → Ap entails a geometric localization, namely a localization to the locus in which it is not possible to divide by elements in p. 53 This results from the fact that the localization removes the prime ideals containing p, i.e. defining subvarieties of p. 54 It is worth noting that the notion of local ring is "orthogonal" to the notion of coordinate ring: "[...] for a prime ideal p ⊂ A the passage from A to Ap cuts out all prime ideals except those contained in p. The passage from A to A/p cuts out all prime ideals except those containing p" [START_REF] Gortz | Algebraic Geometry I. Schemes[END_REF][START_REF] Grant | Much ado about nothing: Theories of space and vacuum from the middle ages to the scientific revolution[END_REF]. Conceptually, whereas the notion of local ring makes abstraction of the inner structure of Vp and only focuses on its immersion in the ambient variety (the "normal" directions), the notion of coordinate ring makes abstraction of the immersion and only focuses on the regular functions on the resulting "abstract" variety Vp (the "internal" directions). Spec(A) whose possible values at p are given by the germs of the sections of the structure sheaf. The specification of a germ of a section at a point p contains much more information than the mere value of the section at p since two sections having the same germ at a point p necessarily coincide on an open neighborhood of p (which is not necessarily true if the two sections only have the same value at p). The important point is that the enhanced Gelfand transform G is injective. 55 This means that the enhanced quantities ã distinguish the nilpotent elements in the ring A. This construction shows that the nilpotent elements encode differential (would could also say "germinal ") structures around the points in Spec(A). Conceptually, the fact that the enhanced Gelfand transform keeps the localization operation A → Ap and discards the quotient by the prime ideal p means that one is making abstraction from the internal structure of the corresponding irreducible set but not of its immersion in the ambient space (see footnote N • 54). The idea of infinitesimal structures corresponding to non-reduced rings (i.e. rings with nilpotent elements) is central in scheme theory and was used by Grothendieck to develop deformation theory and infinitesimal calculus on schemes (see Refs. [START_REF] Grothendieck | Fondements de la Géométrie Algébrique[END_REF] and [START_REF] Grothendieck | [END_REF]Exp.14]). 56 In particular, the consideration of non-reduced rings provided the formal tools required to deal with important geometric notions such as tangency, multiplicity of an intersection, and infinitesimal deformations [START_REF] Manin | Introduction to the Theory of Schemes[END_REF][START_REF] Awodey | First-order logical duality[END_REF][START_REF] Barr | Variable Set Theory[END_REF][START_REF] Bellomo | Domain Extension and Ideal Elements in Mathematics[END_REF].

The construction of an affine scheme (Spec(A), O Spec(A) ) associated to a ring A allows to prove a reconstruction theorem for general rings. It can be shown that the restriction of the structure sheaf to a distinguished open set 57 If in particular we consider the element f = 1, then U f = Spec(A)-V (1) = Spec(A) and A1 = A. It follows that O Spec(A) (Spec(A)) A, i.e. that A is isomorphic to the ring of global sections of the structure sheaf. The central difference with respect to the restricted reconstruction theorem of classical algebraic geometry (for reduced and finitely generated rings over algebraically closed fields) is that the reconstruction of a general ring A requires to consider representations of A into different codomains Ap.

U f is O Spec(A) (U f ) = A f [57, Prop.2.2, 71].
The algebra-geometry duality for general rings takes the form of an equivalence of categories between the opposite of the category of rings and the category of affine schemes 58 :

Ring op Af f Γ(-) Sch(-) (VI.1)
This equivalence of categories is defined by the functor Sch(-) that sends a ring A to the affine scheme (Spec(A), O Spec(A) ) and by the global section functor Γ(-) that sends an affine scheme (Spec(A), O Spec(A) ) to the ring O Spec(A) (Spec(A)) of global sections of its structure sheaf. In the terms introduce in Section N • II (see Fig. N • II.1), the functor Sch(-) is the semiotic functor that sends a ring to the locally ringed space that encodes its general representation theory. In turn, the global section functor Γ(-) is the structure 55 If G(a) = 0 then a/1 = 0 ∈ Ap for every prime p. This means that for each p there exists bp ∈ A -p such that a bp = 0. It follows that a = 0 since the ideal of all b ∈ A such that a b = 0 is not contained in any prime ideal and hence has to contain 1 ∈ A. 56 From a historical standpoint, it is worth noting that an important antecedent of Grothendieck's work on infinitesimal structures was provided by Weil's article Théorie des points proches sur les variétés differentiables [START_REF] Weil | Théorie des points proches sur les variétés differentiables[END_REF]. 57 This means that the notion of locality encoded by the notion of local ring A f obtained by localizing A at f ∈ A is the algebraic counterpart of the topological notion of locality encoded by the distinguished open sets U f of the Zariski topology. It is worth noting that the notations A f and Ap are somehow confusing since A f localizes to the open set U f where we can divide by f and Ap localizes to the point p where we cannot divide by the elements f in p. 58 A morphism of affine schemes (X, O X ) → (Y, O Y ) is given by (i) a map of topological spaces

π : X → Y and (ii) a morphism of sheaves π : O Y → π * O X , where the pushforward π * O X is given by π * O X (U ) = O X (π -1 (U )) for U an open set of Y .
functor that extracts the algebraic structure (the ring) whose representation theory is encoded by the corresponding locally ringed space. As we have briefly explained in this section, this algebraic structure-geometric semiotics duality relies on a precise dictionary between algebraic and geometric notions (see Table N • 2).

Table 2.

Algebra Geometry

Ring

VII. The Magic Fan

In the last section, we have obtained an algebra-geometry duality given by an equivalence of categories between a category of algebraic objects-rings A-and a category of geometric objects-affine schemes (Spec(A), O Spec(A) )-. Equipped as we are with these conceptual and formal tools, let's come back now to the starting point of the whole discussion, namely to the fact that a system of polynomial equations over a field of definition k defines different figures depending on the codomain in which one searches for their solutions (see also the discussion in Ref. [START_REF] Cartier | A Country of which Nothing is Known but the Name: Grothendieck and Motives[END_REF][290][291][292] and Ref. [START_REF] Deligne | Quelques idées maîtresses de l'oeuvre de A. Grothendieck[END_REF][START_REF] Awodey | First-order logical duality[END_REF][START_REF] Barr | Variable Set Theory[END_REF]).

As we have shown in Section N • V, an ideal t of polynomial equations in P .

= k[x1, . . . , xn] defines a functor of solutions Spec (-) (P/t) that sends each k-algebra B to the set of solutions of the corresponding equations in B. As we have seen in Section N • IV, each solution defines a morphism P/t → B, which means that the functor of solutions Spec (-) (P/t) is a representable functor. In other terms, the quotient ring P/t is the object that classifies the solutions of the equations in the ideal t (see Section N • V). Now, the algebra-geometry duality between rings and affine schemes-i.e. the equivalence of categories (VI.1)-implies that HomRing(P/t, B) Hom Af f (Sch(B), Sch(P/t)). In other terms, there is a bijective correspondence between each B-solution of the corresponding equations and each morphism of schemes ϕ : Sch(B) → Sch(P/t). As Cartier writes: "A solution of the 'system of equations' [t], with the 'domain of constants' [k], with values in the 'universal domain' [B] corresponds to a morphism ϕ from [Sch(B)] to [Sch(P/t)]. [...] Admirable simplicityand a very fruitful point of view-but a complete change of paradigm!" [START_REF] Cartier | A Country of which Nothing is Known but the Name: Grothendieck and Motives[END_REF]291] Hence, the existence of an algebra-geometry duality between rings and affine schemes means that the functor of solutions that encodes the solutions of an ideal of polynomial equations in the different possible codomains B can be dually understood as a functor of points that sends each affine scheme Sch(B) to the set Hom Af f (Sch(B), Sch(P/t)) of (what we shall call) Sch(B)-shaped generalized points of Sch(P/t). A morphism of affine schemes ϕ : Sch(B) → Sch(P/t) can be understood as a generalization of the characteristic functions χa : { * } → X defined by each element a in a set X given by * → a, that is of the functions that "point" to the different elements or "points" of X (or, in figurative terms, the functions that present each "point" of the set X as a concrete incarnation of the "universal set-theoretic point" given by the singleton { * }). When the codomain B is a field extension k of k, the space Spec(k ) contains a single point and the maps Sch(k ) → Sch(P/t) are indeed similar to the characteristic functions χa : { * } → X. The difference with respect to the set-theoretic situation is that now we have different "punctual" schemes of the form Sch(k) corresponding to different fields k. 59More generally, a B-representation of P/t (where the k-algebra B is not necessarily a field) can be geometrically "schematized" as a morphism Sch(B) → Sch(P/t) that "points" to a Sch(B)-shaped generalized point of Sch(P/t).

We could say that in the theory of schemes the notion of point is subjected to two generalizations. Firstly, the notion of point is generalized from unextended irreducible subsets (maximal ideals) to general irreducible subsets (prime ideals). As we have explained in Section N • VI, the underlying set of an affine scheme Sch(A) contains all the prime ideals of A, be them maximal (unextended points) or not (generic points). Now, both kinds of points (unextended and generic) describe representations of the original ring A into fields. The second generalization in the definition of a point amounts to pass from points associated to such field-valued representations to generalized points associated to more general (k-algebra)-valued representations. This means that the generalized points of a scheme Sch(P/t) are not only given by morphisms Sch(k ) → Sch(P/t) from "punctual" schemes of the form Sch(k ), but also by more general Sch(B)-shaped generalized points Sch(B) → Sch(P/t) where B is not necessarily a field. It is worth noting that whereas a general scheme Sch(A) is not defined by the underlying set of points of the corresponding topological space (e.g. the different schemes Sch(k) and Sch(k ) have the same underlying set of points), a scheme is defined by its functor of points, i.e. by the structured set of its Sch(B)-shaped generalized points for every ring B.

The scheme Sch(P/t) over the field of definition k encodes (what Grothendieck calls) a "fan" of schemes over field extensions k of k. Let's note first that Sch(P/t) is endowed with a morphism to the scheme Sch(k) dual to the inclusion k → P/t. Schemes X equipped with a morphism to another scheme S are called S-schemes or relative schemes over S (and the morphism X → S is called structure morphism) [START_REF] Grothendieck | Éléments de Géométrie Algébrique I[END_REF]226]. Given another morphism S → S over S, we can pullback a relative scheme X over S to a relative scheme X ×S S over S . This fundamental operation of the theory of schemes is called base change. In particular, one can pullback Sch(P/t) along the morphism Sch(k ) → Sch(k) dual to the inclusion k → k , thereby obtaining a scheme over Sch(k ). In this description, the Sch(k )-points of Sch(P/t) can be identified with the sections of the projection Sch(P/t) × Sch(k) Spec(k ) → Sch(k ) [START_REF] Deligne | Quelques idées maîtresses de l'oeuvre de A. Grothendieck[END_REF][START_REF] Awodey | First-order logical duality[END_REF]. All in all, a scheme Sch(P/t) over Sch(k) (or, equivalently, a scheme associated to a ring that is a k-algebra) can be understood as a "fan" of schemes, one for each field extension k of k. In this way, the scheme Sch(P/t) defined by a system of polynomial equations over k unifies the different "figures" associated to the solutions of these equations in the different field extensions k of k into a single geometric object.

As it is noted in Ref. [49, §17, 17-18], the scheme Sch(P/t) only depends on the ring structure of the k-algebra P/t. The fact that P/t is also a k algebra translates geometrically into the fact that the scheme Sch(P/t) is endowed with an additional structure, namely the structure morphism to Sch(k). Hence, we can associate an affine scheme Sch(A) to any ring A, be it a k-algebra or not (as we have done in Section N • VI). Now, for any ring A there is a canonical morphism Z → A given by n → 1 + 1 + ... + 1 where the unity 1 ∈ A is added n times (which means that Z is an initial object in the category of rings). Therefore, every scheme Sch(A) is canonically endowed with a structure morphism Sch(A) → Sch(Z) dual to Z → A. An absolute scheme is a Sch(Z)-scheme, that is a scheme over (what Grothendieck calls) the absolute base Sch(Z) [START_REF] Grothendieck | Recoltes et Semailles[END_REF]. In turn, the morphism of rings Z → Z/pZ (with p a prime number) induces a morphism of affine schemes Sch(Z/pZ) → Sch(Z). If we take the pullback of Sch(A) → Sch(Z) along this last map we obtain a scheme over Sch(Z/pZ), that is a scheme of characteristic p [START_REF] Gortz | Algebraic Geometry I. Schemes[END_REF][START_REF] Manin | Georg Cantor and His Heritage[END_REF]. In this was, an absolute scheme Sch(A) can be understood-in Grothendieck's terms-as a "magic fan" that "gives birth" to a scheme of characteristic p for each prime number p. It is worth quoting here Grothendieck's own description of this "magic fan": "La notion de schéma est la plus naturelle, la plus 'évidente' imaginable, pour englober en une notion unique la série infinie de notions de 'variété' (algébrique) qu'on maniait précédemment (une telle notion pour chaque nombre premier). [...] De plus, un seul et même 'schéma' (ou 'variété' nouveau style) donne naissance, pour chaque nombre premier p, à une 'variété (algébrique) de caractéristique p' bien déterminée. La collection de ces différentes variétés des différentes caractéristiques peut alors être visualisée comme une sorte d''éventail (infini) de variétés' (une pour chaque caractéristique). Le 'schéma' est cet éventail magique, qui relie entre eux, comme autant de 'branches' différentes, ses 'avatars' ou 'incarnations' de toutes les caractéristiques possibles. Par là-même, il fournit un efficace 'principe de passage' pour relier entre elles des 'variétés', ressortissant de géométries qui jusque là étaient apparues comme plus ou moins isolées, coupées les unes des autres. A présent, elles se trouvent englobées dans une 'géométrie' commune et reliées par elle" [START_REF] Grothendieck | Recoltes et Semailles[END_REF].

Regarding the distinguished role played by the ring Z in the category of rings, let us conclude this section with an important remark (see for instance Ref. [START_REF] Vaquié | Sheaves and Functors of Points[END_REF]445] and Ref. [START_REF] Demazure | Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs[END_REF][START_REF] Artin | Algebra[END_REF]). As we have argued all throughout this article, the reconstruction of a ring A from its representations A → B requires to consider representations with respect to a varying codomain B. This "generalized semiotics" of the ring A can still be encoded in a single geometric object, namely the affine scheme (Spec(A), O Spec(A) ). As we have explained, the fact that the codomain of the representations can vary translates geometrically into the fact that the elements in A define global sections of the structure sheaf with values in point-dependent codomains. Now, once we have the geometric notion of affine scheme at our disposal, it is possible to define a unique codomain given by an affine schemenamely, the so-called affine line A 1 := Sch(Z[x])-such that the elements in A are in bijective correspondence with the A 1 -valued functions on Spec(A). 60 In this way, we have described a sort of dialectical process that moves back and forward from the presupposition 60 To see that this is indeed the case, let's note that the ring Z[x] of polynomials in one variable with entire coefficients is such that A Hom Ring (Z[x], A), where each a ∈ A defines a morphism φa : Z[x] → A given by φa(p) = p(a). The duality between rings an affine schemes-i.e. the fact of a unique codomain (like the algebraically closed fields in the algebraic geometry of affine varieties; see Section N • IV) to the consideration of a whole category of codomains, and then back to the realization that in the end any ring A is in bijective correspondence with the A1-valued functions on its prime spectrum Spec(A). As we have stressed above, this last step can be accomplished only when we already have at our disposal the notion of affine scheme.

VIII. A Kantian Interpretation of the Reconstruction Theorems

In the wake of the Kantian analysis of the Pontryagin duality for locally-compact abelian groups proposed by the mathematician Majid [START_REF] Majid | Principle of Representation-Theoretic Self-Duality[END_REF], we shall now consider the mathematical problem of representation from a perspective inspired by Kant's phenomenonnoumenon distinction. 61 The interpretation of this distinction is a central topic in Kant's scholarship (see for instance Refs. [START_REF] Allison | Kant's Transcendental Idealism: An Interpretation and Defense, Revised and Enlarged Edition[END_REF][START_REF] Bird | Kant's Theory of Knowledge. An Outline of One Central Argument in the Critique of Pure Reason[END_REF][START_REF] Guyer | Kant and the Claims of Knowledge[END_REF][START_REF] Prauss | Kant und das Problem der Dinge an Sich[END_REF][START_REF] Strawson | The Bounds of Sense[END_REF]). In what follows we do not intend to take a stand in these debates. We shall rather use the interpretation of Kant's distinction that is best adapted to the conceptual analysis of the problem at stake, that is the problem of reconstructing a mathematical structure A from its representations A → B.

In The Critique of Pure Reason [66, B295/A236-B315/A260], Kant introduced the central difference between the things in themselves or noumena and the phenomena of human experience. According to the reading of Kant's distinction that we shall endorse here (which follows to a certain extent the so called two-aspect interpretation defended by Allison [START_REF] Allison | Kant's Transcendental Idealism: An Interpretation and Defense, Revised and Enlarged Edition[END_REF], Bird [START_REF] Bird | Kant's Theory of Knowledge. An Outline of One Central Argument in the Critique of Pure Reason[END_REF], and Prauss [START_REF] Prauss | Kant und das Problem der Dinge an Sich[END_REF] among others 62 ), the noumena and the phenomena do not refer to two kinds of entities but rather to the same entity considered from two different standpoints, namely the entity as it appears to subjects equipped with a particular cognitive (perceptual and conceptual) apparatus and the entity "considered as it is in itself" [Ding an sich selbst betrachten], that is, independently of any subjective cognitive framing. Whereas a thing in itself is independent of any subject, its phenomenalisation (i.e. the way according to which it appears for a subject) depends on the particular cognitive apparatus of the subject, that is, on what we could call its transcendental type. We could then say that subjects of different transcendental types B and B (e.g. a human being and a tick) constitute different phenoumena AB and A B out of the same thing in itself A.

In order to adapt this conceptual framework to the subject of this article, we shall endorse the readings of Kant's terminology (mainly associated to the two aspect interpretations) that enrich the phenomenon-noumenon distinction with a further distinction, namely the distinction between the phenomena of experience and the subjective representions [Vorstellungen] of these phenomena (regarding this distinction see Ref. [START_REF] Stang | Kant's Transcendental Idealism, The Stanford Encyclopedia of Philosophy[END_REF]Sect.3.1] and references therein). We could say, in Husserl's terms (see for instance Ref.[64, §41, 74; §44, 82]), that the phenomena of experience are not given to a subject all at once, but rather "onesidedly", that is through a multiplicity of perspectival profiles, aspects or "adumbrations" [Abschaltungen]. According to these definitions, the subjective experience of a phenomenon is given by one-sided representations of the phenomenon's adumbrations. By changing the relative position between the phenomenon and the subject, the latter can gain access to the different adumbrations of the phenomenon. All in all (and according to the particular interpretative framework endorsed here) the Kantian analysis of experience gives rise to a threefold structure composed of:

that Hom Ring (Z[x], A)
Hom Af f (Spec(A), A 1 )-allows us to translate this bijection into a bijection between A and the A 1 -valued functions on Spec(A). 61 In Ref. [START_REF] Lawvere | Tools for the Advancement of Objective Logic: Closed Categories and Toposes[END_REF]Sect.1], Lawvere provided a related Hegelian-oriented attempt to provide a philosophical interpretation of the duality between "space" and "quantity". 62 The two-aspect interpretation arised as an alternative with respect to the so-called two-world interpretation mainly defended by Strawson [START_REF] Strawson | The Bounds of Sense[END_REF] (see also related stances in Ref. [52, V.15, 333] and Ref. [START_REF] Van Cleve | Problems from Kant[END_REF]). Regarding the tension between these two interpretational frameworks see notably Refs. [START_REF] Allais | Kant's One World: Interpreting 'Transcendental Idealism[END_REF][START_REF] Ameriks | Recent Work on Kant's Theoretical Philosophy[END_REF][START_REF] Robinson | Two Perspectives on Kant's Appearances and Things-in-Themselves[END_REF][START_REF] Walker | Kant on the Number of Worlds[END_REF].

(1) the things in themselves (i.e. considered independently of the subjective conditions under which they appear), (2) the phenomena or the things as they appear to subjects of a given transcendental type, (3) the sets of perspectival representations that a subject can have of the corresponding phenomenon.

In order to apply this Kantian-oriented conceptual framework to the problem of reconstructing a mathematical structure from its representations, let's establish a precise dictionary between philosophical concepts and mathematical notions (see Table N • VIII). Each map A → B will be understood-in agreement with the terminology that we have used thus far-as a "representation" of the "thing in itself" A defined with respect to the "transcendental" codomain B. In this sense, the codomains B provide what we could metaphorically call the "subjective" side of the representations. In order to avoid any subjectivist overtones, we shall refer to B by means of the term (inspired by both Kant and Kronecker) transcendental codomain of rationality. We could say that each map A → B represents A by means of a particular "image" in B, namely the image (in the mathematical sense of the term) of the map. For each codomain B, there is a set SpecB(A) = HomC(A, B) of B-representations of A. By taking into account that the term spectrum comes for the Latin for revelation, apparition, or manifestation, we could say that Spec B (A) encodes all the ways according to which A can "appear" (i.e. it can be re-presented or modeled) in the transcendental codomain of rationality B. According to this description, each thing in itself A defines a multiplicity of spectra Spec B (A), Spec B (A), Spec B (A)... (one for each transcendental codomain B, B , B ...). We could then say that each thing in itself A defines a structured set of representations or a variable set (where we are here borrowing the mathematical terminology used in Refs. [START_REF] Mumford | The Red Book of Varieties and Schemes, Second Expanded Edition[END_REF][START_REF] Serre | Faisceaux Algébriques Coherents[END_REF], [START_REF] Mclarty | There is No Ontology Here': Visual and Structural Geometry in Arithmetic[END_REF]386], and [START_REF] Barr | Variable Set Theory[END_REF]). This means that the set of representations induced by a given thing in itself A is a collection of sets of the form SpecB(A) parameterized by the objects B in the category C, where each set SpecB(A) in the collection includes all the B-representations A → B of A. As we have seen in Section N • V, this assignation of a set SpecB(A) to each B in the category C is functorial. This means that morphisms B → C between transcendental codomains of rationality in C translate into morphisms SpecB(A) → SpecC (A) between the corresponding sets of representations.

Here we have two possible scenarios corresponding to what we have called restricted and general reconstruction theorems. First, it might be possible to faithfully reconstruct Ai.e. to reconstruct A as "it is in itself"-by only considering representations of A defined with respect to a single transcendental codomain of rationality B (restricted reconstruction theorem). In philosophical terms, this means that nothing is lost when passing from the thing in itself A to its B-phenomenalisation, i.e. to the phenomenalization of A with respect to the transcendental codomain of rationality B. Now, representations defined with respect to a single transcendental codomain might not be enough to reconstruct A as "it is in itself". If the space SpecB(A) does not have "enough points", it might be the case that different elements a, a ∈ A define-via the Gelfand transform-the same quantity on SpecB(A). In that case, the representations parameterized by the space SpecB(A) cannot "separate" the elements a and a . Since-as we have seen-the obstruction to such a reconstruction is given by the kernel of the Gelfand transform, the quotient AB = A/ker(GA,B) could be called B-phenomenalization of A. Conceptually, the B-phenomenonalization of A encodes what can be reconstructed from A when one only considers representations of A with respect to the transcendental domain of rationality B. In classical algebraic geometry, the lack of "enough points" can be bypassed by adding more points, that is by considering representations into an algebraically closed extension K of the domain of definition k of the corresponding system of equations (Section

N • IV).
In the terms of model theory, this algebraically closed extension of the domain of definition k can be understood as an existential closure of k [START_REF] Hodges | Model Theory[END_REF]362] (see also Refs. [START_REF] Bellomo | Domain Extension and Ideal Elements in Mathematics[END_REF][START_REF] Manders | Domain Extension and the Philosophy of Mathematics[END_REF]).

The extension of a given (co)domain of rationality by the adjunction of new "ideal elements" provides an important mechanism in the development of mathematics. We could say that these adjunctions elicit an expansion of the different transcendental regimes of "pure intuition" 63 , like for instance the regime provided by the understanding of numbers as formal entities intended to count an aggregate of things-or, as Kant writes in the Prolegomena, as concepts formed "through successive additions of units in time" [67, 4:283, §10, 35]. 64 According to Hilbert, "[t]his procedure of introducing ideal elements is one of the most important mathematical methods, the application of which is always repeated up to the highest parts of mathematics" [START_REF] Hilbert | Die Rolle von idealen Gebilden [The role of ideal elements[END_REF][START_REF] Mclarty | How Grothendieck Simplified Algebraic Geometry[END_REF]. The possibility of extending a given domain of rationality and enriching the corresponding forms of "pure intuition" entails a particular form of (what Friedman calls in his reading of Carnap, Reichenbach, and Schlick) "relativization of the a priori" [42, I.3]. In the particular case treated here, this means that the arithmetic and geometric forms of "pure intuition" (e.g. natural numbers, Euclidean space), far from being fixed once and for all (i.e. far from being endowed with a perennial form of aprioricity as it seems to be the case in Kant 65 ), are subjected to changes elicited by the very evolution of mathematics. The fact that domain extensions force to modify what in a given historical context takes the form of an a priori form of "pure intuition" explains the difficulties encountered in performing such extensions. 66 These difficulties left theirs traces in the denominations chosen to design the new elements 63 Regarding the Kantian notion of pure intuition see Ref. [START_REF] Friedman | Kant on Concepts and Intuitions in the Mathematical Sciences[END_REF]. 64 Regarding the concept of domain extension, see the corresponding texts by Hilbert [START_REF] Hilbert | Über das Unendliche[END_REF][START_REF] Hilbert | Die Rolle von idealen Gebilden [The role of ideal elements[END_REF], the discussion of "ideal elements" proposed by Cassirer [START_REF] Cassirer | Philosophie der Symbolischen Formen. Dritter Teil: Phänomenologie der Erkenntnis[END_REF]III.4.4], and the discussion of this topic in contemporary philosophy of mathematics [START_REF] Bellomo | Domain Extension and Ideal Elements in Mathematics[END_REF][START_REF] Cantù | An argumentative approach to ideal elements in mathematics[END_REF][START_REF] Manders | Domain Extension and the Philosophy of Mathematics[END_REF]. 65 For instance, Kant expresses in a letter to A.W. Rehberg the "puzzlement" he feels about an irrational number like √ 2: "The understanding is not even in a position to assume the possibility of an object √ 2, since it cannot adequately present the concept of such a quantity in an intuition of number, and would even less anticipate that such a quantity could be given a priori." [START_REF] Kant | Philosophical Correspondence[END_REF][168][169]. 66 As Cassirer writes in Substance and Function: "The difficulties encountered in the introduction of every new type of number-of the negatives and the irrationals as well as the imaginaries,are easily explained if we consider that, in all these transformations, the real basis of numerical assertions seems more and more to disappear. Enumeration, in its most fundamental sense, could be immediately shown to be 'real' by means of sensible objects and therefore valid" [START_REF] Cassirer | Substanzbegriff und Funktionsbegriff. Untersuchungen über die Grundfragen der Erkenntniskritik[END_REF][START_REF] Halvorson | What Scientific Theories Could Not Be[END_REF].

(e.g. irrational numbers, imaginary numbers, ideal elements). Now-in Hilbert's terms-, "[t]he expression 'ideal elements' is only justified from the point of view of the original system. In the new system, we no longer distinguish between real [wirklichen] and ideal [idealen] elements" [START_REF] Hilbert | Die Rolle von idealen Gebilden [The role of ideal elements[END_REF][START_REF] Mclarty | There is No Ontology Here': Visual and Structural Geometry in Arithmetic[END_REF]. It follows-as Bernays states with respect to "the problem of infinity"-that "the sharp distinction between the intuitive [Anschaulichen] and the nonintuitive [Nicht-Anschaulichen] [...] can apparently not be drawn so strictly [...]" [16,[START_REF] Hodges | Model Theory[END_REF]. In particular, the extensions of the successive number systems and the projectivization of Euclidean space by adding new points at infinity constitute two paradigmatic examples of domain extensions that entail new forms of (arithmetic and geometric) intuition. Now, Grothendieck's groundbreaking insight is that the reconstruction of a general ring A might require to consider a notion of variation of the codomain of the representations of A more general than the notion of field extension. The "relativization of the a priori" encoded in the notion of domain extension remains limited and the reconstruction of a general ring from its representations makes necessary to perfom a more general relativization of the a priori. We could say in Kantian terms that the reconstruction of a thing A as "it is in itself" might require not only to consider representations defined with respect to a single transcendental codomain of rationality (or with respect to a suitable extension of it), but also to consider representations defined with respect to a whole category of codomains. As Grothendieck and Dieudonné write in the introduction to the Éléments de Géométrie Algébrique I, the strategy that consists in considering solutions in an extension of the field of definition of the system of equations worked "too well" since it caused mathematicians to lose sight of the fundamental idea of a "change of the base field" in all its generality. 67 Grothendieck's notion of a functor of solutions associated to a ring P/t-or the dual notion of a functor of points represented by the affine scheme Sch(P/t)-introduced for the first time an extended kind of variations that includes a "fan" of transcendental codomains of rationality not necessarily related by domain extensions. Whereas the relativity associated to domain extensions ultimately leads to the consideration of a single domain that is "big enough" (like Weil's universal domains), the functorial relativity envelops domains that cannot be nested into one another.

Galois theory can be understood as a sort of intra-mathematical reflection on the "relativization of the a priori" encoded in the notion of domain extension for the particular case given by sequences of (normal and separable) field extensions k → k of a field k defined by the adjunctions of new "ideal elements" (e.g., the extension C = R[i]) [START_REF] Borceux | Galois theories[END_REF][START_REF] Catren | On the Notions of Indiscernibility and Indeterminacy in the Light of the Galois-Grothendieck Theory[END_REF]. As we have just argued, Grothendieck's "relative point of view" pushes even further the Galoisian "relativization of the a priori". In this sense, we could say that Grothendieck can indeed be considered (as he claims) as a legitimate heir of "Galois' heritage" (regarding the influence of Galois on Grothendieck see Ref. [START_REF] Chargois | L'influence de la théorie de Galois sur l'oeuvre de Grothendieck[END_REF]). 68  67 In Grothendieck and Dieudonné's own terms, "[...] dès le début du XVIII e siècle, et systématiquement à partir de Monge et de Poncelet, on associe à un système [d'équations polynomiales] à coefficients réels le même système dont on ne cherche plus seulement les solutions dans R I , mais bien dans l'espace complexe correspondant C I , utilisant le fait que R est un sous-corps de C. Cette idée se montra très féconde, du fait que les propriétés des êtres algébriques étudiés se simplifiaient considérablement par cette 'extension' du corps de base; en fait on peut même dire que cette 'extension' réussit en un certain sense trop bien, car l'avantage additionnel de disposer, sur le corps C, de la puissante théorie des fonctions analytiques fut cause que pendant tout le XIX e siècle on cessa pratiquement de considérer d'autres systèmes [d'équations polynomiales] que ceux à coefficients complexes (ou dans des sous-corps de C tels que les corps de nombres algébriques); ce qui conduisit à perdre de vue l'idée fondamentale du 'changement du corps de base' sous sa forme générale [...]" [START_REF] Grothendieck | Éléments de Géométrie Algébrique I[END_REF][START_REF] Anel | Topo-logie[END_REF][START_REF] Artin | Algebra[END_REF]. 68 In Recoltes et Semailles, Grothendieck writes: "[...] le premier dans la lignée de mes 'frères de tempérament' [...] c'est Evariste Galois. Dans sa courte et fulgurante vie, je crois discerner l'amorce d'une grande vision-celle justement des 'épousailles du nombre et de la grandeur', dans une vision géométrique nouvelle. J'évoque ailleurs dans Récoltes et Semailles comment, il y a deux ans, est apparu en moi cette intuition soudaine: que dans le travail mathématique qui à ce moment exerçait sur moi la fascination la plus puissante, j'étais en train de 'reprendre l'héritage de Galois'. Cette intuition, rarement évoquée depuis, a pourtant eu le temps de mûrir en silence. La réflexion rétrospective sur mon oeuvre que je poursuis depuis trois semaines y aura sûrement encore contribué. La filiation la plus directe que je crois reconnaître à présent avec un mathématicien du passé, est bien celle qui me relie à Evariste Galois" [START_REF] Grothendieck | Recoltes et Semailles[END_REF].

IX. Conclusion

In this article we have tried to contribute to what we could call the "philosophy of (mathematical) duality" [START_REF] Catren | On the Notions of Indiscernibility and Indeterminacy in the Light of the Galois-Grothendieck Theory[END_REF][START_REF] Corfield | Duality as a category-theoretic concept[END_REF][START_REF] Halvorson | Logic in Philosophy of Science[END_REF][START_REF] Krömer | The Form and Function of Duality in Modern Mathematics[END_REF][START_REF] Lambek | The Influence of Heraclitus on Modern Mathematics[END_REF] by analyzing a major event in the history of this notion, namely the construction of a geometric object (an affine scheme) associated to any (commutative) ring. Let's stress the main argumentative thread of the proposed analysis:

(1) In Section N • III, we have described a general meta-mathematical pattern that reappears in different algebra-geometry dualities in many areas of mathematics (e.g. mathematical logic, algebraic geometry, group representation theory, C *algebra theory, Galois-Grothendieck theory). In all these results, a mathematical structure A (e.g. a Boolean algebra, a ring, a group, a C * -algebra, etc.) is reconstructed from its representations A → B with respect to other similar structures B. The underlying reconstruction protocol (schematized in Figure III.2) is always the same: the mathematical structure A is recovered as (i.e. is isomorphic to) an algebra of quantities Γ(Spec(A)) defined by means of a Gelfand transform A → Γ(Spec(A)) on a moduli space Spec(A) that parameterizes some of its representations.

(2) We have argued that this general meta-mathematical pattern exhibits an entanglement between an algebra-geometry duality and (what we have called) a structuresemiotics duality (of which the syntax-semantics dualities for propositional and predicate logic are particular cases) (Section N • II). This entanglement between these two kinds of duality results from the fact (already stressed by Halvorson and Tsementzis for the case of the Stone duality [START_REF] Halvorson | What Scientific Theories Could Not Be[END_REF][START_REF] Halvorson | Categories of scientific theories[END_REF]) that the formulation of a duality between a "theory" and its "models" requires to endow the set of "models" with an extra geometric structure encoding theirs relations (which means that the geometry is on the semiotic side of the duality). (3) In Section N • IV, we have reviewed a particular case of the algebra/structuregeometry/semiotics duality, namely the (restricted) reconstruction theorem of classical algebraic geometry. This result states that certain kind of rings (reduced and finitely generated rings over algebraically closed fields) can be reconstructed from its representations into a unique codomain. (4) In Section N • V, we have introduced Grothendieck's theory of affine schemes as an attempt to obtain a reconstruction theorem in algebraic geometry that works for general commutative rings. This can be done by extending the semiotic side of the structure-semiotics duality to representations defined with respect to a whole family of codomains (general reconstruction theorems). This extension of the semiotic side of the duality is encoded in the category theoretic notion of functor of points. (5) As we have explained in Section N • VI, Grothendieck's functorial extension of the structure-semiotics duality threatens the possibility of casting this duality into the form of an algebra-geometry duality since we have now a different space SpecB(A) for each codomain of rationality B. Now, Grothendieck showed that it is still possible to construct a single geometric object encoding the generalized semiotics of the original ring A, namely the affine scheme (Spec(A), O Spec(A) ). ( 6) In Section N • VI, we have proposed a conceptual-oriented introduction to the construction of the affine scheme (Spec(A), O Spec(A) ) associated to a ring A. The proposed presentation stresses the salient features that differentiates Grothendieck's construction from previous forms of the algebra-geometry duality, namely:

• that the "points" of the underlying space Spec(A) encode all the irreducible sub-schemes (and not just the irreducible sub-schemes given by the unextended points),

• that the quantities ã on the space Spec(A) defined by the ring elements a ∈ A have values in point-dependent codomains, • that these values are given by germs of sections that encode differential structures around the points. [START_REF] Atiyah | Introduction to Commutative Algebra[END_REF] We have finally proposed a Kantian-oriented interpretation of both the restricted and the general reconstruction theorems. According to this interpretation, Grothendieck's construction can be understood as a further generalization of the "relativization of the a priori" encoded in the notion of domain extension to more general variations of the "transcendental codomains of rationality".

In order to conclude, let us note that this transition from a restricted set of representations to a general sheaf of representations defined on a whole category of transcendental codomains is just the first step in a series of far-reaching generalizations. These generalizations proceed by modifying the domain category and/or the codomain category of the functor of points. First, we can pass from functors of points with values in the category of sets, i.e. from functors of the form Spec (-) (A) : Ring → Set, to functors with values in the category of groupoids (stacks) or in the category of simplicial sets (higher stacks). The next generalization-which is the grounding idea of the field of derived geometry-amounts to pass from functors over the category of rings to functors over the category of differential graded algebras, thereby giving rise to the theory of derived stacks (see for instance Refs. [START_REF] Anel | The Geometry of Ambiguity. An Introduction to the Ideas of Derived Geometry[END_REF][START_REF] Toen | Derived Algebraic Geometry[END_REF][START_REF] Vezzosi | What is ... a Derived Stack?[END_REF]). This final note suggests a possible direction for the continuation of the research project started in this article, namely that of upgrading the proposed analysis of the philosophical (and in particular mathematical) problem of reconstructing a structure from its representations to the formal framework defined by the two aforementioned generalizations.

A

  Affine scheme (Spec(A), O Spec(A) ) Point in Spec(A) corresponding Prime ideal p to an irreducible subscheme (generic points) Point in Spec(A) corresponding Maximal ideal m to an irreducible unextended subscheme (standard points) Local ring Ap Stalk O Spec(A) (p) of germs of sections on p Global section ã of the sheaf O Spec(A) Ring element a ∈ A with values in point-dependent codomains Ap

Table 1 .

 1

			Object	Representations
		Duality
			(algebraic structure)	(geometric semiotics)
		Stone	Bool. Alg. A	Bool. alg. morph.
		(syntax-semantics duality		A → {0, 1}
		for propositional logic)
		Classical	Ring k[x1, . . . , xn]/t	k-alg. homo.
		Algebraic	(reduced and finitely generated
	Restricted	Geometry	over an alg. closed field K)	k[x1, . . . , xn]/t → K
	Reconstruction
		Gelfand duality	Commutative	Characters
	Theorems	(observables-states duality)	C * -algebra A	A → C
		Pontryagin duality	Locally Compact	Characters
		(Fourier theory,	Abelian	G → U (1)
		position-momentum duality	Group G
		in quantum mechanics)
		Galois-Grothendieck	k-alg. A split by K	k-alg. homo.
				A → K
		Affine Schemes	Ring A	A → Ap
	General		(with no restrictions)	(Ap local ring)
	Reconstruction
		Sheaf representations
	Theorems	for toposes	Topos E	Functor E → Ep
		(syntax-semantics dualities
		for predicate logic)		(Ep local topos)

Table 3 .

 3

	Kantian Concepts	Mathematical Notions
	Thing in Itself	A
	Transcendental Codomain of Rationality	B
	B-Representations of A	A → B
		SpecB(A) = HomC(A, B)
	Space of B-Representations
		endowed with a topology
	B-Phenomenalization of A	AB = A/ker(GA,B)

Other introductions to the theory of schemes may be found in Refs.[START_REF] Dieudonné | Algebraic Geometry[END_REF][START_REF] Dieudonné | Fondements de la Géométrie Algébrique Moderne[END_REF][START_REF] Eisenbud | The Geometry of Schemes[END_REF][START_REF] Hartshorne | Algebraic Geometry[END_REF][START_REF] Manin | Introduction to the Theory of Schemes[END_REF][START_REF] Mumford | The Red Book of Varieties and Schemes, Second Expanded Edition[END_REF][START_REF] Shafarevich | Basic Algebraic Geometry 2, Second Revised and Expanded Edition[END_REF]. See also Ref.[START_REF] Mclarty | How Grothendieck Simplified Algebraic Geometry[END_REF] for a very short introduction to scheme theory and Ref.[START_REF] Zalamea | Grothendieck. Una guía a la obra matemática y filosófica[END_REF] for an encyclopedic description of Grothendieck's work (and in particular Chapter N •

for scheme theory). Ref.[START_REF] Mclarty | There is No Ontology Here': Visual and Structural Geometry in Arithmetic[END_REF] provides a nice introduction to the twofold nature of schemes, namely as functors (that encode solutions to families of polynomial equations in different codomains) and as ringed spaces (topological spaces equipped with sheaves of rings).

Majid describes this transformation from an element being "observed" into an "observable" in the following terms: "This map [the Gelfand transform] literally turns the tables and says that the object a being seen by p can instead be viewed as an observer â (corresponding directly to a) seeing p with the same value p(a) = â(p)"[START_REF] Majid | Principle of Representation-Theoretic Self-Duality[END_REF] 397].

In mathematical terms, both the unit and the counit of an adjunction are natural isomorphisms when the adjunction defines an equivalence of categories[START_REF] Lane | Categories for the Working Mathematician[END_REF][START_REF] Marquis | From a Geometrical Point of View[END_REF]. But we can have an intermediate situation in which only the unit or the counit of the adjunction is a natural isomorphism.

Even if the underlying sets of such schemes contain all a single point, the structure sheaf on the point is different for different k. In Ref.[START_REF] Grothendieck | Éléments de Géométrie Algébrique I[END_REF][START_REF] Awodey | Logic in Topoi: Functorial Semantics for Higher-Order Logic[END_REF], the points associated to field-valued representations are called geometric points.

Acknowledgments

The research leading to these results has received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013 Grant Agreement N • 263523) and from the Laboratoire International Associé (LIA) Identities, Forces, Quanta (CNRS, France).