In his 1686 essay GI Leibniz undertook to reduce sentences to noun-phrases, truth to being. Such a reduction arose from his equating proof with conceptual analysis. Within limits Leibniz’s logical calculus provides a reasonable way of surmounting the dichotomy, thus allowing a reduction of hypothetical to categorical statements. However it yields the disastrous result that, whenever A is possible and so is B, there can be an entity being both A and B. Yet, Leibniz was in the GI the forerunner of 20th century combinatory logic, which (successfully!) practices - sometimes for reasons not entirely unlike Leibniz’s own grounds - reductions of the same kinds he tried to carry out.

In the parts I and II of this paper, the Author shows:1. how Leibniz’s arithmetico-intensional logical calculi of April 1679 can be completed and transformed in an intensional Boolean algebra (U, v, &:, -, e, -e) admitting, on the one hand, two different logical interpretations:li1: as a complete and consistent calculus of terms (properties) and syllogistic;li2: as a deontic first-order calculus and, on the other hand, two different arithmetical interpretations:ai1: as a numerical Boolean algebra (DM, lcm, ged, M/..., 1, M) of all divisors of a natural number M;ai2: as a numerlcal Boolean algebra (BA, lcbc, gcbc, A-..., 0, A) of all binary components of a natural number A.Arithmetical representations of negation of a term x and of combination (intensional conjunction) and alternative (intensional disjunction) of two or more terms are respectively M/x, lcm (lower common multiple) and gcd (greatest common divisor) in the first Boolean algebra (ai1) and A-x, lcbc (lower common binary composite) and gcbc (greatest common binary component) in the second one (ai2).2. that, in this context, each possible world U of 2ⁿ elements (terms, acts) can be defined on the basis on n elements of U choosen as “saturated” (intensionally maximal, but possible) in U or, inversely, on the basis of n elements of U choosen as “primitive” (intensionally minimal, but not-universal) in U. In fact, each possible element of U can be defined now as an alternative of saturated elements of U, now as a combination of primitive (opposite to saturated) elements of U; all combinations of saturated elements of U being equivalent to the impossible element (-e, non-entity, resp. impossible act) of U and an alternatives of elements of U being equivalent to the universal element (e, entity, resp. possible act) of U.In the arithmetical representation of each possible world U, the maximal number M (for ai1) or A (for ai2) represents the impossible (non existent) element of U. Now, each possihle world defined by n saturated (resp. primitive) elements can be automatically enlarged (restricted) by the introduction (suppression) of m new (old) saturated (resp. primitive) elements, producing a new possible world U’ where m impossible elements (centaur, pegasus, syren, unicorn, etc.) of U become possible elements of U’ or inversely.After this first type of arithmetical representation of logical calculi, where the terms are -as in Leibniz’s 1679 calculi- represented by natural numbers and the propositions by equations (for universal resp. prescriptive) or inequations (for particular, resp. permissive), in the part III the Author presents a second type of arithmetical representation where propositions are represented by natural numbers and the valid (classical or deontic) syllogisms by true arithmetical relations between the numbers of premisses and the number of conclusion. Here the entire syllogistic adopts the form of a multiplication table wherea syllogism is valid if and only if the lcm (in ai1 ) or the lcbc (in ai2) of the characterlstic numbers of the premisses is a multiple (in ai1) or a binary composite (in ai2) of the characteristic number of the conclusion.

L’histoire du texte des Régles pour la Direction de l’Esprit (Regulae) de Descartes est un peu singulière: non publié du vivant de Descartes, il n’a paru qu’en 1701, dans les Opera Posthuma d’Amsterdam. De façon plus significative, et contrairement aux autres traités cartésiens perdus, ce texte secret n’est jamais explicitement evoqué par Descartes, fût-ce au détour d’une correspondance. Par leur étroite dépendance vis à vis des mathématiques, les Regulae sont cependant un texte majeur, constitutives de la pensée de leur auteur dans ses années de jeunesse (1619-1628), et par là de toute la philosophie moderne. Descartes avait jugé le texte suffisamment important pour I’emmener à Stockholm, où il a été découvert apres sa mort, dans ses papiers.Entre les mathématiques et les Regulae, ce texte “éclatant et obscur” (J.P. Weber), il est ces trois types principaux de rapports croisés que nous tâcherons d’analyser: historiquement d’abord, quelles furent la formation et I’expérience mathématique du jeune Descartes, qui constituerent, à notre sens, I’armature conceptuelle du texte. Quelles sont ensuite les voies par lesquelles, dans les Regulae, Descartes a putransmuer cette expérience mathématique premiere à la fois en une pratique, une méthode, une théorie de la connaissance, enfin en une épistémologie assez radicalement neuve. Enfin, et prenant Descartes au sérieux nous examinerons à I’occasion cette question: quel est le sort réservé, de nos jours, à cette épistémologie cartésienne, en particulier confrontée aux mathématiques contemporaines?