The hexagonal structure for ‘the geometry of logical opposition’, as coming from Aristoteles–Apuleius square and Sesmat–Blanché hexagon, is presented here in connection with, on the one hand, geometrical ideas on duality on triangles (construction of ‘companion’), and on the other hand, constructions of tripartitions, emphasizing that these are exactly cases of borromean objects. Then a new case of a logical interest introduced here is the double magic tripartition determining the semi-ring ${\mathcal{B}_3}$ and this is a borromean object again, in the (...) heart of the semi-ring ${{\rm Mat}_{3}(\mathbb{B}_{\rm Alg})}$ . With this example we understand better in which sense the borromean object is a deepening of the hexagon, in a logical vein. Then, and this is our main objective here, the Post-Mal’cev full iterative algebra ${\mathbb{P}_4 = \mathbb{P}(\mathbb{F}_4)}$ of functions of all arities on ${\mathbb{F}_4}$ , is proved to be a borromean object, generated by three copies of ${\mathbb{P}_2}$ in it. This fact is induced by a hexagonal structure of the field ${\mathbb{F}_4}$ . This hexagonal structure is seen as precisely a geometrical addition to standard boolean logic, exhibiting ${\mathbb{F}_4}$ as a ‘boolean manifold’. This structure allows to analyze also ${\mathbb{P}_4}$ as generated by adding to a boolean set of logical functions a very special modality, namely the Frobenius squaring map in ${\mathbb{F}_4}$ . It is related to the splitting of paradoxes, to modified logic, to specular logic. It is a setting for a theory of paradoxical sentences, seen as computations of movements on the bi-hexagonal link among the 12 classical logics on a set of 4 values. (shrink)

As a significant extension of our previous calculus of logical differentials and moving logic, we propose here a mathematical diagram for specifying the emergence of novelty, through the construction of some “differentials” related to cohomological computations. Later we intend to examine how to use these “differentials” in the analysis of anticipation or evolution schemes. This proposal is given as a consequence of our comments on the Ehresmann–Vanbremeersch’s work on memory evolutive systems, from the two points of view which are characterization (...) of life and use of categorical modeling. It would not be possible to conceive of the “differentials” outside the frame of a MES. Furthermore, we hope that this diagram will be useful to exhibit the “emergence of sense” in discourses, and this idea is supported here by a brief examination of how a discourse could be seen as a living system. (shrink)

Descartes et Freud ont pour véritables soucis fonciers, l'un l'évidence, et l'autre l'étrangeté. Il est remarquable qu'ils ne déterminent ces notions qu'implicitement, de façons réellement paradoxales. Cela peut surprendre, surtout si l'on ajoute que leurs façons sont identiques. A partir de la décision cartésienne d'identifier le vrai à l'évident se trouve fondée la science ; et l'identification freudienne du vrai à l'étrange fonde la psychanalyse. La question de la relation entre science et psychanalyse porte donc, ici, sur le réel du (...) nouage paradoxal entre les deux déterminations du vrai pointées par Descartes et Freud, sur leur différence et leur inséparabilité. Plus spécifiquement, c'est d'abord du point de vue du rapport entre mathématique et psychanalyse que la question est approfondie. La visée du livre est de construire le principe de ce nouage, de ce nœud hétérogène. La ressource centrale est de considérer comme vitale, tant au jeu de l'évidence qu'à celui de l'étrangeté, la question de la lettre. (shrink)

The article affixes a resolutely structuralist view to Alain Badiou’s proposals on the infinite, around the theory of sets. Structuralism is not what is often criticized, to administer mathematical theories, imitating rather more or less philosophical problems. It is rather an attitude in mathematical thinking proper, consisting in solving mathematical problems by structuring data, despite the questions as to foundation. It is the mathematical theory of categories that supports this attitude, thus focusing on the functioning of mathematical work. From this (...) perspective, the thought of infinity will be grasped as that of mathematical work itself, which consists in the deployment of dualities, where it begins the question of the discrete and the continuous, Zeno’s paradoxes. It is, in our opinion, in the interval of each duality ― “between two ends”, as our title states ― that infinity is at work. This is confronted with the idea that mathematics produces theories of infinity, infinitesimal calculus or set theory, which is also true. But these theories only give us a grasp of the question of infinity if we put ourselves into them, if we practice them; then it is indeed mathematical activity itself that represents infinity, which presents it to thought. We show that tools such as algebraic universes, sketches, and diagrams, allow, on the one hand, to dispense with the “calculations” together with cardinals and ordinals, and on the other hand, to describe at leisure the structures and their manipulations thereof, the indefinite work of pasting or glueing data, work that constitutes an object the actual infinity of which the theory of structures is a calculation. Through these technical details it is therefore proposed that Badiou envisages ontology by returning to the phenomenology of his “logic of worlds”, by shifting the question of Being towards the worlds where truths are produced, and hence where the subsequent question of infinity arises. (shrink)