Cet article vise à montrer l’extrême proximité entre ces deux mathématiciens-philosophes que furent Gian-Carlo Rota et Gilles Châtelet disparus la même année. Au moins quatre points communs les relient : une philosophie romantique radicale ; une rigueur intellectuelle exemplaire ; une vision affine de la recherche mathématique ; une révolte intérieure exécrant tout sensus communis.This text shows the great proximity between two mathematicianphilosophers, Gilles Châtelet and Gian-Carlo Rota who both died in the same year 1999. There are at least four (...) common points amongst themselves: a radical romantic philosophy; an exemplary intellectual rigor; a great affinity in their vision of mathematical research; an inmost rebellion against all common sense. Quest’articolo tende a mostrare l’estrema prossimità tra due matematici-filosofi che furono Gian-Carlo Rota e Gilles Châtelet che ci hanno lasciato lo stesso anno. Ci sono almeno quattro punti comuni che le legano: una radicale filosofia romantica; un rigore intellettuale esemplare; una visione affine della ricerca matematica; una rivolta intima che esecra ogni senso comune.This article is in French. (shrink)

In his testamentary letter to Auguste Chevalier, Évariste Galois states that, in modern terminology, the smallest simple group has order 60. No proof of this statement survives in his papers, and it has been suggested that a proof would have been impossible using the methods available at the time. We argue that this assertion is unduly pessimistic. Moreover, one fragmentary document, dismissed as a triviality and misunderstood, looks suspiciously like cryptic notes related to this result. We give an (...) elementary proof of Galois’s statement, explain why it is likely that he would have been aware of the methods involved, and discuss the potential relevance of the fragment. (shrink)

ArgumentThis article offers a social history of the “Galois Affair,” which arose in 1831 when the French Academy of Sciences decided to reject a paper presented by an aspiring mathematician, Évariste Galois. In order to historicize the meaning of Galois's work at the time he tried to earn recognition for his research on the algebraic solution of equations, this paper explores two interrelated questions. First, it analyzes scholarly algebraic practices and the way mathematicians were trained in (...) the nineteenth century to reconstruct the scientific meaning of Galois's research. By emphasizing the historically contingent nature of scientific debates, the paper argues that to historicize the work of a mathematician, we must historicize the scientific world as well. The second part of this article goes further to seek a fresh perspective on Galois's short career by analyzing the cultural, social, and institutional dynamics that governed how the mathematical community came to recognize an aspiring mathematician as one of its own. (shrink)

In this paper, we wish to explore the role that textual representations play in the creation of new mathematical objects. We do so by analyzing texts by Joseph-Louis Lagrange (1736–1813) and Évariste Galois (1811–1832), which are seen as central to the historical development of the mathematical concept of groups. In our analysis, we consider how the material features of representations relate to the changes in conceptualization that we see in the texts.Against this backdrop, we discuss the idea that (...) new mathematical concepts, in general, are increasingly abstract in the sense of being detached from material configurations. Our analysis supports the opposite view. We suggest that changes in the material aspects of textual representations (i.e., the actual graphic inscriptions) play an active and crucial role in conceptual change.We employ an analytical framework adapted from Bruno Latour’s 1999 account of intertwined material and representational practices in the empirical sciences. This approach facilitates a foregrounding of the interconnection between the conceptual development of mathematics, and the construction, (re-)configuration, and manipulation of the materiality of representations. Our analysis suggests that, in mathematical practice, distinctions between the material and structural features of representations are not permanent and absolute. This problematizes the appropriateness of the distinction between concrete inscriptions and abstract relations in understanding the development of mathematical concepts. (shrink)

Vorwort über das Leben der Mathematiker.--Die Bernoullis.--Leonhard Euler und die Phantasie in der Mathematik.--Gauss und die Mathematik.--Geist und Character d'Alemberts.--Evariste Galois.--über Niels Henrik Abels.--Bernard Bolzanos Grösse.--Felix Klein, der Systematiker der Geometrie.--David Hilbert.

Une première période où les influences mêlées d’Alessandro Padoa et de Bertrand Russell s’exercent en France culmine avec les essais philosophiques de Jean Nicod. Une seconde période voit fleurir les travaux du mathématicien Jacques Herbrand ; avant de périr, il laisse son nom à un théorème fondamental. Suit une période de débats entre philosophes, mathématiciens et physiciens, stimulés en 1935 et 1937 par la tenue à Paris de deux congrès consacrés, totalement ou en partie, à la philosophie des sciences. Paulette (...) Février y esquisse une logique non classique où l’on postule l’existence de couples de propositions non composables pour ériger en principes les relations de Werner Heisenberg. Jean-Louis Destouches développe cette conception jusqu’à décrire comment édifier une théorie unifiante. La structuration des êtres mathématiques est l’objet d’études philosophiques d’Albert Lautman. Le rôle putatif de la notion de groupe en logique est interrogé. La notion de structure mathématique est l’objet de deux contributions : Marc Krasner généralise les conceptions d’Évariste Galois, attribuées à la logique et étendues à des langages infinitaires ; Nicolas Bourbaki, compte tenu de l’évolution des mathématiques, qualifie de structure ce que nous appelons aujourd’hui un modèle. (shrink)

Vorwort über das Leben der Mathematiker.--Die Bernoullis.--Leonhard Euler und die Phantasie in der Mathematik.--Gauss und die Mathematik.--Geist und Character d'Alemberts.--Evariste Galois.--über Niels Henrik Abels.--Bernard Bolzanos Grösse.--Felix Klein, der Systematiker der Geometrie.--David Hilbert.

A la croisée des chemins de la vulgarisation, de l'histoire et de la philosophie des sciences, l'auteur explore la nature de la connaissance et de la recherche via un mode d'exposition faisant largement appel à l'analogie. [extrait de la 4è de couv.].

We address the biological dynamics problem of the persistence of several species in conditions of non-existence of an equilibrium, including an example of stabilization by predation and the very controversial “competitive exclusion”. We give normal forms for various examples of such persistence and comments on the involved topology, which implies the presence of exceptional heteroclinic connections binding equilibria on the boundary.

The present work is a contribution to the understanding of the sempiternal problem of the “burden of factor two” implied by sexual reproduction versus asexual one, as males are energy consumers not contributing to the production of offspring. We construct a deterministic mathematical model in population dynamics where a species enjoys both sexual and parthenogenetic capabilities of reproduction and lives on a limited resource. We then show how polygamy implies instability of a parthenogenetic population with a small number of sexually (...) born males. This instability implies evolution of the system towards an attractor involving both populations. We also exhibit the analogy with a parasite/host system. (shrink)

We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an F_σ normal subgroup: in general, as a topological group, and under NIP, also in terms of Borel cardinality. This allows us to obtain similar results for arbitrary strong types defined on a single complete type over ∅. As an easy conclusion of our main theorem, we get the main result of [K. Krupiński, A. Pillay and T. Rzepecki, Topological dynamics (...) and the complexity of strong types, Israel J. Math.228 (2018) 863–932] which says that for any strong type defined on a single complete type over ∅, smoothness is equivalent to type-definability. We also explain how similar results are obtained in the case of bounded quotients of type-definable groups. This gives us a generalization of a former result from the paper mentioned above about bounded quotients of type-definable subgroups of definable groups. (shrink)

We introduce tame abstract elementary classes as a generalization of all cases of abstract elementary classes that are known to permit development of stability-like theory. In this paper, we explore stability results in this new context. We assume that [Formula: see text] is a tame abstract elementary class satisfying the amalgamation property with no maximal model. The main results include:. Theorem 0.1. Suppose that [Formula: see text] is not only tame, but [Formula: see text]-tame. If [Formula: see text] and [Formula: (...) see text] is Galois stable in μ, then [Formula: see text], where [Formula: see text] is a relative of κ from first order logic. [Formula: see text] is the Hanf number of the class [Formula: see text]. It is known that [Formula: see text]. The theorem generalizes a result from [17]. It is used to prove both the existence of Morley sequences for non-splitting and the following initial step towards a stability spectrum theorem for tame classes:. Theorem 0.2. If [Formula: see text] is Galois-stable in some [Formula: see text], then [Formula: see text] is stable in every κ with κμ=κ. For example, under GCH we have that [Formula: see text] Galois-stable in μ implies that [Formula: see text] is Galois-stable in μ+n for all n < ω. (shrink)

In this paper we investigate the properties of automorphism groups of countable short recursively saturated models of arithmetic. In particular, we show that Kaye's Theorem concerning the closed normal subgroups of automorphism groups of countable recursively saturated models of arithmetic applies to automorphism groups of countable short recursively saturated models as well. That is, the closed normal subgroups of the automorphism group of a countable short recursively saturated model of PA are exactly the stabilizers of the invariant cuts of the (...) model which are closed under exponentiation. This Galois correspondence is used to show that there are countable short recursively saturated models of arithmetic whose automorphism groups are not isomorphic as topological groups. Moreover, we show that the automorphism groups of countable short arithmetically saturated models of PA are not topologically isomorphic to the automorphism groups of countable short recursively saturated models of PA which are not short arithmetically saturated. (shrink)

We investigate the properties of a modal algebra, more specifically, a non-distributive lattice with operators via Galois connections. Pairs of Galois are very common in mathematical environments, and, in this article, they appear as unary operators in lattices even without the distributivity. In a previous paper, Castiglioni and Ertola-Biraben studied the meet-complemented lattices with two modal operators for necessary □ and possible ◊. We observed that this pair of operators determines an adjunction. Then, we used Galois pairs (...) on the meet-complemented lattices, showing some properties of this structure that were already been proved in their paper, and some new laws non presented. Lastly, we define a new pair of operators that also constitute another Galois pair. (shrink)

The concept of Galois connection between power sets is generalized from the point of view of fuzzy logic. Studied is the case where the structure of truth values forms a complete residuated lattice. It is proved that fuzzy Galois connections are in one-to-one correspondence with binary fuzzy relations. A representation of fuzzy Galois connections by Galois connections is provided.

This paper is a continuation of investigations on Galois connections from [1], [3], [10]. It is a continuation of [2]. We have shown many results that link properties of a given closure space with that of the dual space. For example: for every -disjunctive closure space X the dual closure space is topological iff the base of X generated by this dual space consists of the -prime sets in X (Theorem 2). Moreover the characterizations of the satisfiability relation for (...) classical logic are shown. Roughly speaking our main result here is the following: a satisfiability relation in a logic L with, a countable language is a fragment of the classical one iff the compactness theorem for L holds (Theorems 3–8). (shrink)

We study relativized Lascar groups, which are formed by relativizing Lascar groups to the solution set of a partial type $$\Sigma $$. We introduce the notion of a Lascar tuple for $$\Sigma $$ and by considering the space of types over a Lascar tuple for $$\Sigma $$, the topology for a relativized Lascar group is (re-)defined and some fundamental facts about the Galois groups of first-order theories are generalized to the relativized context. In particular, we prove that any closed (...) subgroup of a relativized Lascar group corresponds to a stabilizer of a bounded hyperimaginary having at least one representative in the solution set of the given partial type $$\Sigma $$. Using this, we find the correspondence between subgroups of the relativized Lascar group and the relativized strong types. (shrink)

Nonclassical logics have played an increasing role in recent years in disciplines ranging from mathematics and computer science to linguistics and philosophy. _Generalized Galois Logics_ develops a uniform framework of relational semantics to mediate between logical calculi and their semantics through algebra. This volume addresses normal modal logics such as K and S5, and substructural logics, including relevance logics, linear logic, and Lambek calculi. The authors also treat less-familiar and new logical systems with equal deftness.

The concept of fuzzy Galois connections is defined on fuzzy posets with Bělohlávek's fuzzy Galois connections as a special case. The properties of fuzzy Galois connections are investigated. Then the relations between fuzzy Galois connections and fuzzy closure operators, fuzzy interior operators are studied.

The method based on the idea of Galois connection is well known. It facilitates investigations into similarities between mathematical structures, including isomorphisms between these structures, the highest degree of similarity. This idea is employed here and adapted so as to get to the core of aspects of the relationship between some metamathematical structures. The focus is put on the relation between traditional methodological orthodoxy based on the idea of proof , on the one hand, and on some alternative methodological (...) set-ups based on other ideas such as consistency or some forms of maximality, on the other hand. (shrink)

This paper presents a systematic investigation of fuzzy Galois connections in the sense of R. Bělohlávek [1], from the point of view of enriched category theory. The results obtained show that the theory of enriched categories makes it possible to present the theory of fuzzy Galois connections in a succinct way; and more importantly, it provides a useful method to express and to study the link and the difference between the commutative and the non-commutative worlds.

. The connection presented in this paper mirror-links two metamathematical structures, the finitary closure operators, and the compact consistency properties, in such a way that a specification of one structure induces a provably equivalent specification of the other.

Krull [4] extended Galois theory to arbitrary normal extensions, in which the Galois groups are precisely the profinite groups. Metakides and Nerode [7] produced two recursively presented algebraic extensionsK⊂Fof the rationals such thatFis abelian,Fis of infinite degree overK, and the Galois group ofFoverK, although of cardinalityc, has only one recursive element. This indicated the limits of effectiveness for Krull's theory. Nerode suggested developing a natural effective version of Krull's theory.It is evident from the classical literature that the (...) free profinite group on denumerably many generators can be obtained effectively as the Galois group of a recursive extension of the rationals over a subfield. Nerode conjectured that it could be obtained effectively as the Galois group of the algebraic numbers over a suitable subfield. The case of finitely many generators was done non-effectively by Jarden [3]. The author believes that the denumerable case, as presented in §2, is also new classically. Using this result and the effective Krull theory, every “co-recursively enumerable” profinite group is effectively the Galois group of a recursively enumerable field of algebraic numbers over a recursive subfield. (shrink)

As a continuation of the work of the third author in [5], we make further observations on the features of Galois cohomology in the general model theoretic context. We make explicit the connection between forms of definable groups and first cohomology sets with coefficients in a suitable automorphism group. We then use a method of twisting cohomology (inspired by Serre’s algebraic twisting) to describe arbitrary fibres in cohomology sequences—yielding a useful “finiteness” result on cohomology sets.Applied to the special case (...) of differential fields and Kolchin’s constrained cohomology, we complete results from [3] by proving that the first constrained cohomology set of a differential algebraic group over a bounded, differentially large, field is countable. (shrink)

Intuitionistic propositional logic with Galois negations ( \(\mathsf {IGN}\) ) is introduced. Heyting algebras with Galois negations are obtained from Heyting algebras by adding the Galois pair \((\lnot,{\sim })\) and dual Galois pair \((\dot{\lnot },\dot{\sim })\) of negations. Discrete duality between GN-frames and algebras as well as the relational semantics for \(\mathsf {IGN}\) are developed. A Hilbert-style axiomatic system \(\mathsf {HN}\) is given for \(\mathsf {IGN}\), and Galois negation logics are defined as extensions of \(\mathsf (...) {IGN}\). We give the bi-tense logic \(\mathsf {S4N}_t\) which is obtained from the minimal tense extension of the modal logic \(\mathsf {S4}\) by adding tense operators. We give a new extended Gödel translation \(\tau \) and prove that \(\mathsf {IGN}\) is embedded into \(\mathsf {S4N}_t\) by \(\tau \). Moreover, every Kripke-complete Galois negation logic _L_ is embedded into its tense companion \(\tau (L)\). (shrink)

First, it is pointed out how the author's new differential Galois theory contributes to the understanding of the differential closure of an arbitrary differential field . Secondly, it is shown that a superstable differential field has no proper differential Galois extensions.

Hilbert algebra with a Hilbert-Galois connection, or HilGC-algebra, is a triple \(\left(A,f,g\right)\) where \(A\) is a Hilbert algebra, and \(f\) and \(g\) are unary maps on \(A\) such that \(f(a)\leq b\) iff \(a\leq g(b)\), and \(g(a\rightarrow b)\leq g(a)\rightarrow g(b)\) forall \(a,b\in A\). In this paper, we are going to prove that some varieties of HilGC-algebras are characterized by first-order conditions defined in the dual space and that these varieties are canonical. Additionally, we will also study and characterize the congruences (...) of an HilGC-algebra through specific closed subsets of the dual space. This characterization will be applied to determine the simple algebras and subdirectly irreducible HilGC-algebras. (shrink)

We present a way of topologizing sets of Galois types over structures in abstract elementary classes with amalgamation. In the elementary case, the topologies thus produced refine the syntactic topologies familiar from first order logic. We exhibit a number of natural correspondences between the model-theoretic properties of classes and their constituent models and the topological properties of the associated spaces. Tameness of Galois types, in particular, emerges as a topological separation principle. © 2011 WILEY-VCH Verlag GmbH & Co. (...) KGaA, Weinheim. (shrink)

In [E. Hrushovski, D. Palacín and A. Pillay, On the canonical base property, Selecta Math. (N.S.) 19(4) (2013) 865–877], Hrushovski and the authors proved, in a certain finite rank environment, that rigidity of definable Galois groups implies that [Formula: see text] has the canonical base property in a strong form; “internality to” being replaced by “algebraicity in”. In the current paper, we give a reasonably robust definition of the “strong canonical base property” in a rather more general finite rank (...) context than [E. Hrushovski, D. Palacín and A. Pillay, On the canonical base property, Selecta Math. (N.S.) 19(4) (2013) 865–877], and prove its equivalence with rigidity of the relevant definable Galois groups. The new direction is an elaboration on the old result that [Formula: see text]-based groups are rigid. (shrink)

Preface 1 Partially ordered sets 1.1 Posets introduced 1.2 Partial orders and strict orders 1.3 Maps between posets 1.4 Compounding maps 1.5 Order similarity 1.6 Inclusion posets as typical..

We study the groups Gal L and Gal KP, and the associated equivalence relations EL and EKP, attached to a first order theory T. An example is given where EL≠ EKP. It is proved that EKP is the composition of EL and the closure of EL. Other examples are given showing this is best possible.

We revisit Kolchin's results on definability of differential Galois groups of strongly normal extensions, in the case where the field of constants is not necessarily algebraically closed. In certain classes of differential topological fields, which encompasses ordered or p‐valued differential fields, we find a partial Galois correspondence and we show one cannot expect more in general. In the class of ordered differential fields, using elimination of imaginaries in, we establish a relative Galois correspondence for relatively definable subgroups (...) of the group of differential order automorphisms. (shrink)

ZusammenfassungDer vorliegende Aufsatz verfolgt das Ziel, die Entwicklung der Galois-Theorie etwa von 1850 bis 1960 zu skizzieren. Hervorgehoben werden dabei diejenigen Entwicklungslinien, die mit der Funktionentheorie, der algebraischen Topologie und der Verallgemeinerung des Separabilitäts-Begriffs verknüpft sind. Es wird ein Ausblick auf die Galois-Theorie der kommutativen Ringe (nach M. Auslander & O. Goldman) und der Schemata (nach A. Grothendieck) gegeben.

Symmetric generalized Galois logics (i.e., symmetric gGl s) are distributive gGl s that include weak distributivity laws between some operations such as fusion and fission. Motivations for considering distribution between such operations include the provability of cut for binary consequence relations, abstract algebraic considerations and modeling linguistic phenomena in categorial grammars. We represent symmetric gGl s by models on topological relational structures. On the other hand, topological relational structures are realized by structures of symmetric gGl s. We generalize the (...) weak distributivity laws between fusion and fission to interactions of certain monotone operations within distributive super gGl s. We are able to prove appropriate generalizations of the previously obtained theorems—including a functorial duality result connecting classes of gGl s and classes of structures for them. (shrink)

We give a characterization of the fixed points and of the lattices of fixed points of fuzzy Galois connections. It is shown that fixed points are naturally interpreted as concepts in the sense of traditional logic.

We construct a Galois connection between closure and interior operators on a given set. All arguments are intuitionistically valid. Our construction is an intuitionistic version of the classical correspondence between closure and interior operators via complement.

In this paper, I will give a new characterisation of the spaces of complete theories of pseudofinite fields and of algebraically closed fields with a generic automorphism (ACFA) in terms of the Vietoris topology on absolute Galois groups of prime fields.

Apresentamos conceitos algébricos básicos e fundamentais como conjuntos ordenados, reticulados, álgebra de Boole e as TK-álgebras. Destacamos os espaços de Tarski, associados ao conceito de sistema dedutivo e sua apresentação quase topológica. Então, apresentamos a Lógica da Dedutibilidade, vinda da formalização lógica dos espaços de Tarski. A seguir, trazemos os pares de funções de Galois, que surgem em muitos tópicos da Matemática. Como resultado original, além de alguns desenvolvimentos teóricos, destacamos uma conexão de Galois com os espaços de (...) Tarski. (shrink)

In this work, an intuitionistic propositional logic with a Galois connection is introduced. In addition to the intuitionistic logic axioms and inference rule of modus ponens, the logic contains only two rules of inference mimicking the performance of Galois connections. Both Kripke-style and algebraic semantics are presented for IntGC, and IntGC is proved to be complete with respect to both of these semantics. We show that IntGC has the finite model property and is decidable, but Glivenko's Theorem does (...) not hold. Duality between algebraic and Kripke semantics is presented, and a representation theorem for Heyting algebras with Galois connections is proved. In addition, an application to rough L-valued sets is presented. (shrink)

Implicational tonoid logics and their relational semantics have been introduced by Yang and Dunn. This paper extends this investigation to implicational partial Galois logics. For this, we first define some implicational partial gaggle logics as special kinds of implicational tonoid logics called “implicational partial Galois logics.” Next, we provide Routley–Meyer-style relational semantics for finitary those logics.