Laurence Bouquiaux

SUPERSPACE FORMULATION OF N=2 PSEUDOMECHANICS AND SUPERPOTENTIALS

The general formalism of N=2 pseudomechanics in the superspace is presented for one spatial dimension. Taking into account the (super) symmetries of the Lagrangian, an exhaustive classification of superpotentials into three categories is obtained. The first class contains the harmonic oscillator potential, the free particle constant potential, and the superposition of both with a linear potential. The second one contains the λ2/q2 potential and its superposition with the harmonic oscillator potential and a constant one. The third class contains all other potentials. Through Noether’s theorem, conserved quantities are associated with (super)symmetry properties, and, for each class, we, respectively, get the following superalgebras: osp(2,2) ⧠ sh(1), osp(2,2), and spl(1,1) ⧠ so(2).

Leibniz Against the Unreasonable Newtonian Physics

Leibniz does not make any explicit reference to Newton in this text, but he clearly aims at the Principia mathematica when he writes, “It is astonishing that there are those who now, in the great light of our age, hope to persuade the world of a doctrine so foreign to reason” (AG AG A&G 324). Attraction requires a means of communication that is “invisible, intangible, not mechanical”. He might as well have added,

Monads and chaos : the vitality of Leibniz's philosophy

Leibniz’s work resembles its author. A. Robinet has called it &dquo;an intellectual storm.&dquo; In its two hundred thousand pages of manuscript (most of it still unpublished) there are philosophical works that have nourished the thoughts of thinkers from generation to generation; mathematical texts of fundamental import (we all know of Leibniz as the founder or rather co-founder of infinitesimal calculus, but this triumph ought not to obscure his other contributions; for example, his being a precursor in the field of for-

On a "Mathematical Neo-Aristotelism" in Leibniz

line where all points are equivalent, and which can give only an inexact idea of what motion really is. For, as Y. Belaval again stresses, motion is not for Leibniz what it is for Descartes or Galileo, a state; it remains what it was for Aristotle, a process: “It is because Aristotle had seen something of these principles, I believe, that he concluded (he being in my opinion more profound than many people think) that there is needed some alteration besides change in place, and that matter is not similar to itself everywhere and does not remain invariable”14. Descartes’ time is made up of independent instants, his world consists of a succession of states of which, as Y. Belaval says, each one is immediately suspended from God, without owing anything to what it was itself in the preceding instant. Leibniz’ time, on the other hand, is a “living” time, a time which unfolds, a continuous time where each instant is different from all the others, where the present is burdened with the past and pregnant with the future15. Leibniz’ universe is a world where everything is in a perpetual state of flux, a world where everything is constantly transformed. There is no immobility, no rest. The world is full of souls, and the soul is perpetual restlessness. Substances are enveloped and developed, fold and unfold, are extended and drawn together, concentrated. “The body is in continuous change, like a river”16. “Souls continually advance and mature, like the world itself, of which they are the image, for nothing existing outside the universe can prevent it and the universe must necessarily go on advancing and developing”17. All this must, as it seems to me, incite us to consider circumspectly a certain interpretative tradition that makes of Leibniz the thinker of identity, of the reduction of becoming to being or the eradication of all temporality. Y. Belaval has summed up this aspect of leibnizianism very well. “In creation, each point, each instant is ‘characteristic’, individualised by the activity it houses and which makes a thing endure (...) 14. “(...) Aristoteles, profundior mea sententia, quam multi putant, iudicavit, praeter mutationem localem opus esse alteratione, nec materiam ubique sibi esse similem, ne maneat invariabilis”, De Ipsa Natura, GP IV, 514. Translation by Leroy E. Loemker, op. cit., 506. 15. Whereas the Cartesian world, “founded on an illusory motion, only calls for an illusory time, a dead time, where the present is absolutely not burdened with the past nor pregnant with the future”. Y. Belaval, Leibniz critique de Descartes, 426. 16. A Remond, GP III, 635. Translation by Leroy E. Loemker, op. cit., 658. 17. “Les âmes avancent et murissent continuellement, comme le monde lui-meme dont elles sont les images, car rien n’etant hors de l’univers qui le puisse empecher, il faut bien que l’univers avance continuellement et qu’il se developpe”. A Sophie, GP VII, 543. suppression of the idea of force, of an internal determination, which would be the true cause of motion, this motion is reduced to its trajectory, to an 168 ON A “MATHEMATICAL NEO-ARISTOTELISM” IN LEIBNIZ essentially restlessness. Bodies do not keep to a determined figure: like a river, or Theseus’ ship, an organism has as its only stable element, its guiding idea, its law of organisation. In species, individuals differ : there can be found no two leaves perfectly alike. Species are varied — and may have varied — to infinity. There is no repetition. A continual flux, an unwearied temporality. A Heraclitean vision of the world!”18.