Schrödinger logics are logical systems in which the principle of identity is not true in general. The intuitive motivation for these logics is both Erwin Schrödinger's thesis (which has been advanced by other authors) that identity lacks sense for elementary particles of modern physics, and the way which physicists deal with this concept; normally, they understandidentity as meaningindistinguishability (agreemment with respect to attributes). Observing that these concepts are equivalent in classical logic and mathematics, which underly the usual physical theories, we (...) present a higher-order logical system in which these concepts are systematically separated. A classical semantics for the system is presented and some philosophical related questions are mentioned. One of the main characteristics of our system is that Leibniz' Principle of the Identity of Indiscernibles cannot be derived. This fact is in accordance with some authors who maintain that quantum mechanics violates this principle. Furthermore, our system may be viewed as a way of making sense some of Schrödinger's logical intuitions about the nature of elementary particles. (shrink)

We formulate Suppes predicates for various kinds of space-time: classical Euclidean, Minkowski's, and that of General Relativity. Starting with topological properties, these continua are mathematically constructed with the help of a basic algebra of events; this algebra constitutes a kind of mereology, in the sense of Lesniewski. There are several alternative, possible constructions, depending, for instance, on the use of the common field of reals or of a non-Archimedian field. Our approach was inspired by the work of Whitehead, though our (...) philosophical stance is completely different from his. The structures obtained are idealized constructs underlying extant, physical space-time. (shrink)

Sir Isaac Newton (1642-1727) left a voluminous legacy of writings. Despite his influence on the early modern period, his correspondence, manuscripts, and publications in natural philosophy remain scattered throughout many disparate editions. In this volume, Newton's principal philosophical writings are for the first time collected in a single place. They include excerpts from the Principia and the Opticks, his famous correspondence with Boyle and with Bentley, and his equally significant correspondence with Leibniz, which is often ignored in favor (...) of Leibniz's later debate with Samuel Clarke. Newton's exchanges with Leibniz place their different understandings of natural philosophy in sharp relief. The volume also includes 'De Gravitatione', offered here in a corrected translation, which is crucial for understanding Newton's relation to his great predecessor Descartes. In a historical and philosophical introduction, Andrew Janiak examines Newton's philosophical positions and his relations to canonical figures in early modern philosophy. (shrink)

In previous essays (1973, 1975, 1977) I have praised Derrida's contributions to philosophical dialogue and also insisted on their limitations. The considerations raised in this present essay do not lead me to a position that is less ambivalent. Philosophy is a particular language-game. Like any other, it has its constitutive rules; or, perhaps better: its practice has certain distinctive features by means of which we recognize philosophizing and distinguish it from other linguistic activities. None of this can be set down (...) in the form of necessary and sufficient conditions, and there always will be large areas of controversy about the paradigms themselves. Nonetheless there are philosophers, and they generally acknowledge Plato, Aristotle, Leibniz, Hume, and even some of their colleagues as also being philosophers. This acknowledgement holds even where there is philosophical disagreement - as must inevitably be the case, in view of the dialogical character of philosophical discourse. Derrida occasionally enters into the dialogue, as do many others - poets, novelists, critics, diplomats, attorneys, cooks, barbers, and babysitters. These occasional contributions to philosophical dialogue differ in focus, in style, and in lack of self-referentiality from the works which constitute the main corpus of philosophy. Some might wish to say, as if the matter were paradoxical, that such contributions are both philosophical and not philosophical, or that they are neither philosophy nor not-philosophy. That might be as good a thing to say as anything. The practice of philosophy is complex and has many levels. Those who are acknowledged as its finest practitioners have a focus, a style, and a respect for where questions begin and end. Derrida does not share these qualities, and does not care to share them. That is no reason to ignore his work, but it is sufficient to explain why philosophers do not recognize it as a contribution to the central corpus of philosophy. (shrink)

It may be helpful to recall that here as elsewhere Leibniz' pluralism finds a complementary counterpart in Spinoza's monism. Spinoza never deserted the notion of a single world-system wherein existence exhausts all possibility, so that, sub specie aeternaitatis, the two are the same.

It would seem that "possibility," "concrete actuality," and "decision" are terms indispensable in describing my existence. It may also be that the meaning of no one of these three terms may be adequately conceived without reference to the other two. By preferring to follow Santayana, Mr. Eddins emphasizes concrete actuality. Now, as I read Santayana, existence like essence is a category, not strictly a "realm" of being, a category that we come to respect as we act and make decisions. It (...) should not, even in Santayana's materialism, be considered in abstraction from critical situations. Santayana's animal faith accepts the presence of a concrete actuality, the "realm" of matter conceived as a field of action. I might add that a human faith will accept our act of decision that makes its terms with the changing conditions constituting this realm. Thus I have asserted that "to exist is to have an unfinished history and a problematic future, the two being united in decision." When I think of my own existence or that of my friends, I bear this meaning more or less clearly in mind. My own existence is apparent to me every time I consider my past and my open future as the margins of a decision that is being formulated in my mind. Suppose now that someone convinces me that my decisions do not face an open future but stand to their future in a strictly chronological relation comparable, let us say, to the relation of earlier and later events as portrayed in an elementary text-book of history. Then I must face the difficulties latent in Leibniz' theology. (shrink)

It would seem that "possibility," "concrete actuality," and "decision" are terms indispensable in describing my existence. It may also be that the meaning of no one of these three terms may be adequately conceived without reference to the other two. By preferring to follow Santayana, Mr. Eddins emphasizes concrete actuality. Now, as I read Santayana, existence like essence is a category, not strictly a "realm" of being, a category that we come to respect as we act and make decisions. It (...) should not, even in Santayana's materialism, be considered in abstraction from critical situations. Santayana's animal faith accepts the presence of a concrete actuality, the "realm" of matter conceived as a field of action. I might add that a human faith will accept our act of decision that makes its terms with the changing conditions constituting this realm. Thus I have asserted that "to exist is to have an unfinished history and a problematic future, the two being united in decision." When I think of my own existence or that of my friends, I bear this meaning more or less clearly in mind. My own existence is apparent to me every time I consider my past and my open future as the margins of a decision that is being formulated in my mind. Suppose now that someone convinces me that my decisions do not face an open future but stand to their future in a strictly chronological relation comparable, let us say, to the relation of earlier and later events as portrayed in an elementary text-book of history. Then I must face the difficulties latent in Leibniz' theology. (shrink)

For this new edition, Roger Ariew has adapted Samuel Clarke's edition of 1717, modernizing it to reflect contemporary English usage. Ariew's introduction places the correspondence in historical context and discusses the vibrant philosophical climate of the times. Appendices provide those selections from the works of Newton that Clarke frequently refers to in the correspondence. A bibliography is also included.

It is usually stated that quantum mechanics presents problems with the identity of particles, the most radical position—supported by E. Schrödinger—asserting that elementary particles are not individuals. But the subject goes deeper, and it is even possible to obtain states with an undefined particle number. In this work we present a set theoretical framework for the description of undefined particle number states in quantum mechanics which provides a precise logical meaning for this notion. This construction goes in the line of (...) solving a problem posed by Y. Manin, namely, to incorporate quantum mechanical notions at the foundations of mathematics. We also show that our system is capable of representing quantum superpositions. (shrink)

We formulate Suppes predicates for various kinds of space-time: classical Euclidean, Minkowski's, and that of General Relativity. Starting with topological properties, these continua are mathematically constructed with the help of a basic algebra of events; this algebra constitutes a kind of mereology, in the sense of Lesniewski. There are several alternative, possible constructions, depending, for instance, on the use of the common field of reals or of a non-Archimedian field (with infinitesimals). Our approach was inspired by the work of Whitehead (...) (1919), though our philosophical stance is completely different from his. The structures obtained are idealized constructs underlying extant, physical space-time. (shrink)

La decouverte par le Dr. E. A. Fellmann de l'exemplaire des Principia Mathematica de Newton ayant appartenu a Leibniz et contenant des notes marginales manuscrites (datant vraisemblablement de 1689), apporte un temoignage direct de la premiere reaction de Leibniz devant l'oeuvre du savant anglais. La presente edition met le document original entre les mains du lecteur, en l'accompagnant des notes et commentaires objectifs indispensables (en allemand et en francais). Le rapport de Leibniz a Newton a souvent ete envisage (...) dans le cadre de la facheuse querelle de priorite dans l'invention du calcul differentiel et integral. Or Leibniz, prenant connaissance des Principia, n'y voit pas un traite purement mathematique: refusant d'enteriner la scission entre la mecanique et les lois de la pensee, il se soucie avant tout d'etablir une fondation metaphysique pour la dynamique. Les Marginalia ont l'originalite de nous presenter, non pas un Leibniz adversaire ou rival, mais un Leibniz lecteur de Newton, dispose a tirer le meilleur parti d'un systeme qui aurait pu, a bien des egards, lui sembler etranger, comme le confirmera la systemisation ulterieure du concept de vis visa. (shrink)

This addresses the transformations of metaphysics as a discipline, the emergence of analytical mechanics, the diverging avenues of 18th-century Newtonianism, the body-mind problem, and philosophical principles of classification in the life sciences. An appendix contains a critical edition and first translation into English of Newton's scholia from David Gregory's Estate on the Propositions IV through IX Book III of his Principia.

El capítulo introduce al debate sobre la naturaleza del espacio entre Leibniz y Clarke/Newton y a la posición que adopta Kant más tarde. En particular, se exponen los dos principales argumentos de Leibniz, basados en los Principios de Razón Suficiente e Identidad de Indiscernibles, en favor del relacionismo así como algunas respuestas de Clarke/Newton. También se presenta el argumento basado en la orientación del espacio que propuso Kant en 1768 para refutar el relacionismo de Leibniz. Se concluye con (...) una breve exposición del idealismo trascendental de Kant en su Primera Crítica. (shrink)

It is common to hold that our conscious experiences at a single moment are often unified. But when consciousness is unified, what are the fundamental facts in virtue of which it is unified? On some accounts of the unity of consciousness, the most fundamental fact that grounds unity is a form of singularity or oneness. These accounts are similar to Newtonian views of space according to which the most fundamental fact that grounds relations of co-spatiality between various points (or regions) (...) of a space is the fact that these points (or regions) are parts of the same single space. In this paper, I sketch and defend an alternative account of unity of consciousness. Very roughly, the view holds that experiences are unified when they are connected in the right way. In this respect, the view is analogous to Leibnizian views of space according to which the oneness of space emerges from certain conditions over spatial relations. The Leibnizian alternative has significant implications for our understanding of the metaphysics of conscious experience, the cognitive architecture of the mind and our assessment of the conditions under which unity of consciousness breaks. (shrink)

In the famous Correspondence with Clarke, which took place during the last year of Leibniz's life, Leibniz advanced several arguments purporting to refute the absolute theory of space and time that was held by Newton and his followers. The main aim of this book is to reassess Leibniz's attack on the Newtonian theory in so far as he relied on the principle of the identity of indiscernibles. The theological side of the controversy is not ignored but isolated and discussed (...) in the last three chapters, which deal with problems connected with the notions of omnipotence and omniscience. (shrink)

This paper reexamines the historical debate between Leibniz and Newton on the nature of space. According to the traditional reading, Leibniz (in his correspondence with Clarke) produced metaphysical arguments (relying on the Principle of Sufficient Reason and the Principle of Identity of Indiscernibles) in favor of a relational account of space. Newton, according to the traditional account, refuted the metaphysical arguments with the help of an empirical argument based on the bucket experiment. The paper claims that Leibniz’s and (...) Newton’s arguments cannot be understood apart from the distinct dialectics of their respective positions vis-à-vis Descartes’ theory of space and physics. Against the traditional reading, the paper argues that Leibniz and Newton are operating within a different metaphysics and different conceptions of “place,” and that their respective arguments can largely remain intact without undermining the other philosopher’s conception of space. The paper also takes up the task of clarifying the distinction between true and absolute motion, and of explaining the relativity of motion implied by Leibniz’s account. The paper finally argues that the two philosophers have different conceptions of the relation between metaphysics and science, and that Leibniz’s attempt to base physical theory on an underlying metaphysical account of forces renders his account of physics unstable. (shrink)

The aim of this paper is to analyze Leibniz and Newton’s conception of space, and to point out where their agreements and disagreements lie with respect to its mode of existence. I shall offer a definite characterization of Leibniz and Newton’s conceptions of space. I will show that, according to their own concepts of substance, both Newtonian and Leibnizian spaces are not substantiva!. The reason of that consists in the fact that space is not capable of action. Moreover, (...) there is a sense in which space is relational, because their parts are individuated only by means of their mutual relations. (shrink)

Early modern foundations for mechanics came in two kinds, nomic and material. I examine here the dynamical laws and pictures of matter given respectively by Newton, Leibniz, and Kant. I argue that they fall short of their foundational task, viz. to represent enough kinematic behavior; or at least to explain it. In effect, for the true foundations of classical mechanics we must look beyond Newton, Leibniz, and Kant.

It is easy to get the impression that Newton and Leibniz do not see eye to eye on anything. Yet, as is so often the case, a closer look reveals that matters are much more complicated. Despite their disagreements, the two are frequently on the same side of central scientific and philosophical debates. This entry discusses some of the main agreements and disagreements between Newton and Leibniz, starting with their methodologies and then turning to their views on space, (...) motion, and gravitation. (shrink)

This book presents extracts from Leibniz's letters to Newtonian scientist Samuel Clarke.

Leibniz famously argues that there must be simple substances, since there are composites, and a composite is nothing but a collection of simples. I reconstruct Leibniz’s argument, showing that it relies on a commitment to mereological nihilism (i.e., the view that composites cannot be true beings). I show further that Leibniz endorses mereological nihilism as early as the 1680s and offers both direct and indirect support for this commitment: indirect support via the notion of unity and direct support via the (...) notion of persistence. I then assess the alignment of Leibniz’s mereological nihilism with his other commitments during the 1680s, including his potential commitment to corporeal substances. I argue that any viable interpretation of Leibniz’s commitment to corporeal substances is compatible with mereological nihilism, which provides a new perspective both on Leibniz’s developing theory of substance and on his mature theory of simple substance. (shrink)

In contrast with some recent theories of infinitesimals as non-Archimedean entities, Leibniz’s mature interpretation was fully in accord with the Archimedean Axiom: infinitesimals are fictions, whose treatment as entities incomparably smaller than finite quantities is justifiable wholly in terms of variable finite quantities that can be taken as small as desired, i.e. syncategorematically. In this paper I explain this syncategorematic interpretation, and how Leibniz used it to justify the calculus. I then compare it with the approach of Smooth Infinitesimal Analysis (...) (SIA), as propounded by John Bell. Despite many parallels between SIA and Leibniz’s approach —the non-punctiform nature of infinitesimals, their acting as parts of the continuum, the dependence on variables (as opposed to the static quantities of both Standard and Non-standard Analysis), the resolution of curves into infinitesided polygons, and the finessing of a commitment to the existence of infinitesimals— I find some salient differences, especially with regard to higher-order infinitesimals. These differences are illustrated by a consideration of how each approach might be applied to Newton’s Proposition 6 of the Principia, and the derivation from it of the v2/r law for the centripetal force on a body orbiting around a centre of force. It is found that while Leibniz’s syncategorematic approach is adequate to ground a Leibnizian version of the v2/r law for the “solicitation” ddr experienced by the orbiting body, there is no corresponding possibility for a derivation of the law by nilsquare infinitesimals; and while SIA can allow for second order differentials if nilcube infinitesimals are assumed, difficulties remain concerning the compatibility of nilcube infinitesimals with the principles of SIA, and in any case render the type of infinitesimal analysis adopted dependent on its applicability to the problem at hand. (shrink)

We investigate the structure common to causal theories that attempt to explain a (part of) the world. Causality implies conservation of identity, itself a far from simple notion. It imposes strong demands on the universalizing power of the theories concerned. These demands are often met by the introduction of a metalevel which encompasses the notions of 'system' and 'lawful behaviour'. In classical mechanics, the division between universal and particular leaves its traces in the separate treatment of cinematics and dynamics. This (...) analysis is applied to the mechanical theories of Newton and Leibniz, with some surprising results. (shrink)

Gottfried Wilhelm Leibniz, who was born in 1646 just before the end of the Thirty Years War and who died 1716, is surely one of the most bizarre and interesting of the early modern philosophers. He was an astonishing polymath, and responsible for some of the most advanced work in the sciences of his day—he was, for instance, the co-inventor along with Newton, of differential calculus, and is generally recognized as the greatest logician of the early modern period, responsible (...) for advances in logic not rivaled until the mid-nineteenth century. But this progressive aspect of Leibniz’s thought is paired by one that was more backward looking, deeply engaged with premodern forms of thinking that referred back through Medieval culture to the philosophy of ancient times. And alongside of his scientific advances, he is known for having created one of the most baroque and puzzling metaphysical systems in the history of philosophy—the so-called “Monadology”. For much of his life he was also absorbed in theological disputes that have now been long been forgotten, and generally thought of as alien to modern scientific modes of thought. (shrink)

In this paper I challenge the usual interpretations of Newton's and Leibniz's views on the nature of space and the relativity of motion. Newton's ‘relative space’ is not a reference frame; and Leibniz did not regard space as defined with respect to actual enduring bodies. Newton did not subscribe to the relativity of intertial motions; whereas Leibniz believed no body to be at rest, and Newton's absolute motion to be a useful fiction. A more accurate rendering (...) of the opposition between them, I argue, leads to a wholly different understanding of Leibniz's theory of space, one which is not susceptible to the objections Newton had raised against Descartes regarding the representation of motion. This in turn suggests a new approach for contemporary theory of space, one which neither hypostatizes space nor tries to reduce it to relations among actual things. * This work was generously supported by the National Endowment for the Humanities, with a Fellowship for College Teachers and Independent Scholars (FB-26897-89), and also by a sabbatical leave from my institution, Middlebury College. Iam very grateful to various members of faculty of York University for their appreciative reception of an earlier one-week-old version of this paper. ‘Relative Space in Newton and Leibniz’, read to the Department of Philosophy there in January 1990, and to Robert Rynasiewicz for criticisms of an extract read at the 1991 History of Science meeting. (shrink)

In this paper I argue that inconsistencies in scientific theories may arise from the type of causality relation they—tacitly or explicitly—embody. All these seemingly different causality relations can be subsumed under a general strategy developed to defeat the paradoxes which inevitably occur in our experience of the real. With respect to this, scientific theories are just a subclass of the larger class of metaphysical theories, construed as theories that attempt to explain a (part of) the world consistently. All metaphysical theories (...) share a common structural backbone specificially designed to defeat paradoxes, their often wildly diverging ontological claims notwithstanding. This common structure shapes the procedures which govern the invention of ideas in the context of such theories, by codifying some onto-logical a priori assumptions regarding the consistency of reality into its bare conceptual framework. Causality plays a key rôle here, because it implies conservation of identity, itself a far from simple notion. It imposes strong demands on the universalising power of the theories concerned. These demands are often met by the introduction of a metalevel which encompasses the notions of ‘system’ and ‘lawful behaviour’. In classical mechanics, the division between universal and particular leaves its traces in the separate treatment of cinematics and dynamics. The fundamental backbone’s specific gestalt thus functions as a theory’s individual signature and paves the way to a comparative historical approach towards their study. An important part of my paper therefore explores the strong connections between paradoxes as they appear and are dealt with in ancient philosophy and their re-appearance in early modern natural philosophy and science. This analysis is applied to the mechanical theories of Newton and Leibniz, with some surprising results. (shrink)

L'idea di relazionare la dimensionalità dello spazio ad una legge fisica, contenuta nel primo scritto di Kant "Pensieri sulla veridica estima delle forze vive", svela un modo di guardare i rapporti tra Fisica e Matematica così nuovo ed originale che potè essere sviluppato e compreso nella sua plenitudine soltanto nel secolo XX. Ci riferiamo qui ala prospettiva aperta da Ehrenfest nel suo "In what way does it become manifest in the fundamental laws of physics that space has three dimensions?". In (...) questo saggio si cerca di comprendere alcune delle radici fisiche, filosofiche e teologiche dell’approccio di Kant a questo problema, e la sua inserzione nel dibattito delle idee che sostengono la concezione meccanicista del Mondo. In primo luogo, si sottolinea che da una lettura attenta del suo Gedanken von der wahren Schätzung der lebendigen Kräfte si può concludere che - al contrario di quanto viene normalmente creduto - Kant non riuscì a dimostrare la sua congettura che la tridimensionalità dello spazio possa essere fondata sulla legge di gravitazione di Newton, secondo la quale la forza tra due corpi va come l'inverso del quadrato della distanza che li separa; in realtà, egli si limitò a giustificare la tridimensionalità della estensione. Dopo aver considerato la distinzione fra estensione e spazio, si fa notare che la catena logica della congettura del giovane Kant, presuppone la forza come concetto basilare, intrinseco alle sostanze. È mediante questa forza che si stabiliscono le relazioni tra le sostanze, le quali, a sua volta, creano l'ordine su cui viene basato il concetto di spazio. Si deve badare che la coerenza del suo argomento richiede, in ultima analisi, concepire la percezione, cioè, l'interazione fra le sostanze e l'anima, come risultato della medesima forza di lunga portata. Inoltre, si dimostra che l'argomento di Kant non poteva essere costrutto su una intelaiatura puramente newtoniana nel 1747. Infatti, egli costruì sua dimostrazione a partire da una concezione del mondo basata sul legato di Cartesio (soprattutto l'idea di um ordine cosmico) e Leibniz (l'esistenza di una forza che precede l'estensione e l'idea di spazio relativo) alla quale sovrappone, in modo ibrido, la legge di gravitazione di Newton. Si argomenta che questa scelta fu fortemente influenzata da alcune idee teologiche del Giovane Kant che divergono da quelle di Cartesio, Newton e Leibniz. Fu questo carattere ibrido del suo approccio che permise a Kant di trattare la questione della dimensionalità dello spazio come un problema di fisica, ma, che, allo stesso tempo, a causa delle sue limitazioni intrinsiche, portò Kant a provare appena la tridimensionalità della estensione. (shrink)

Leibniz's dispute with Newton over the physico-mathematical theories expounded in the Principia Mathematica have long been identified as a crucial episode in the history of science. Dr. Bertoloni Meli examines several hitherto unpublished manuscripts in Leibniz's own hand illustrating his first reading of and reaction to Newton's Principia. Six of the most important manuscripts are here edited for the first time. Contrary to Leibniz's own claims, this new evidence shows that he had studied Newton's masterpiece before publishing (...) An Essay on the Causes of Celestial Motions. This article, representing his response to Newton, is included here in English translation. Dr. Bertoloni Meli analyses the important implications of this episode on a variety of themes ranging from priority claims to the mathematization of nature in the seventeenth century. Besides providing a careful study of Leibniz's style and strategy, the author examines how our perception of Newton's achievement is affected and the reception of the rival theories by the mathematical community around 1700. "Bertoloni's book provides a very detailed and deep analysis of Leibniz's calculus and dynamics by focusing on a consistent and important set of previously unknown manuscripts..." Niccolo Guicciardini, Universita de Bologna "...Equivalence and Priority is a major contribution to our understanding of the development of mathematical physics in the late seventeenth and early eighteenth century." Daniel Garber, University of Chigago. (shrink)

Newton (1643-1727) es considerado una de las más importantes figuras de la historia de la ciencia moderna. Así como Descartes es considerado el fundador de la moderna filosofía de la subjetividad, Newton es visto como el fundador de la moderna ciencia matemática de la naturaleza, en particular de la mecánica, que ha operado alrededor de dos siglos de manera paradigmática y a la que solemos caracterizar como "física clásica". Aunque Newton no es en modo alguno el único (...) innovador de la ciencia moderna -al lado de él participaron también otros como Copémico, Kepler, Galileo, Descartes-, logró, empero, merced a la introducción de la fuerza de la gravitación, unificar diferentes formas de movimiento, como los terrestres -la caída libre, el movimiento del péndulo, el lanzamiento- y los celestes, -los movimientos de los planetas los cuales con Galileo aún permanecían desconectados unos al lado de otros y demostrarlos como una modificación de la misma fuerza fundamental y dar, gracias a ello, una estructura unitaria, sistemática a la teoría del movimiento. Ningún otro sistema representa de una forma tan fiel el concepto de paradigma de Kuhn, como la mecánica newtoniana. (shrink)

Newton and Leibniz had profound disagreements concerning metaphysics and the relationship of mathematics to natural philosophy, as well as deeply opposed attitudes towards analysis. Nevertheless, or so I shall argue, despite these deeply held and distracting differences in their background assumptions and metaphysical views, there was a considerable consilience in their positions on the status of infinitesimals. In this paper I compare the foundation Newton provides in his Method Of First and Ultimate Ratios (sketched at some time between (...) 1671 and 1684, and published in the Principia of 1687) with that provided independently by Leibniz in his unpublished manuscript De quadratura arithmetica (1675-6) as well as in later writings. I argue that both appeal to a version of the Archimedean Axiom to underwrite their use of infinitesimal techniques, which must be interpreted as a shorthand for rigorously finitist methods. (shrink)

Our present knowledge in the field of dynamical systems, information theory, probability theory and other similar domains indicates that the human brain is a complex dynamical system working in a strong chaotic regime in which random processes play important roles. In this environment our mental life develops. To choose a logically ordered sequence from a random or almost random stream of thoughts is a difficult and energy consuming task. The only domain in which we are able to do this with (...) a full success is mathematics. Leibniz’s life ambition was to extend this success, with the help of what he called _characteristica universalis_, to other areas of human activity. The belief that this is possible lies at the basis of Leibniz’s rationalist system. Reasoning within his system, Leibniz claimed that also fundamental laws of physics can be deduced from the “first principles”. Just as linguistic or conceptual units are at the basis of the _charactersistica universalis_, his monads are responsible for physical activity of material bodies. When this rationalistic strategy is applied to the philosophy of space and time, it leads to their radically relational conception. Leibniz’s rationalistic approach to philosophy and science arouses out sympathy but it was Newton’s mathematical-empirical method that turned out to be effective in human endeavour to understand the functioning of the physical world. Successes of the Newtonian method compel us to revise our concept of rationality. (shrink)