Original explorers often see a puzzling conceptual landscape more vividly than jaded later travelers. This essay surveys several ways in which Descartes and Leibniz recognized descriptive problems within applied mathematics more clearly than later commentators have appreciated.

ABSTRACT This paper aims to show that intuitionistic Kripke models are a powerful tool for interpreting Kant’s ‘Critical Philosophy’. Part I reviews some old work of mine that applies these models to provide a reading of Kant’s second antinomy about the divisibility of matter and to answer several attacks on Kant’s antinomies. But it also points out three shortcomings of that original application. First, the reading fails to account for Kant’s second antinomy claim that matter is divisible ‘ad infinitum’ and (...) his first antinomy claim that empirical space extends only ‘ad indefinitum’. Secondly, in conformity with the ‘assertability’ heuristics for Kripke models, it attributes an assertability semantics to Kant. In fact, it attributes a pair of assertability semantics to Kant, but neither of these accords with modern assertabilism. And finally, one must wonder the propriety of using contemporary assertability and contemporary formal logic to save an eighteenth-century text. Part II goes historical to solve the problems about assertability. It outlines the Descartes–Leibniz dialectic about the infinite/indefinite distinction and demonstrates how each position reflects larger philosophical positions. It then demonstrates that Kant’s two versions of assertability and his version of the infinite/indefinite dichotomy emerge naturally from his attempts to resolve a tension in Leibniz’s philosophy. In light of this demonstration, Part III adapts the Kripke model interpretation so that it does reflect that dichotomy. It then refines the Kripke semantics in order to model some of Kant’s signature critical doctrines, and it shows how Kant’s critical version of the infinite/indefinite distinction differs from all those of his predecessors and from his own pre-critical version. It also addresses the question of using contemporary logical machinery to interpret Kant. Part IV turns the tables and uses what we have learned about Kant and modern semantics in order to address a pair of issues in the contemporary critical foundations of logic. (shrink)

In the paper I discuss the problem of the nature of the relationship between objects and their properties. There are three contexts of the problem: of comparison, of change and of interaction. Philosophical explanations of facts indicated in the three contexts need reference to properties and to a proper understanding of a relationship between them and their bearers. My aim is to get closer to this understanding with the use of some models but previously I present the substantialist theory of (...) object and shortly argue for its main theses. The two models enabling us the understanding of the subject–properties structure are: the plastic stuff model and the functional model. On the ground of the first a subject is compared to a piece of plastic stuff which is informed by different shapes. Properties are ways how a subject is, they “give” some “figure” to a subject. The core idea of the second model is that essences (performing the role of subjects) are immanent functional laws governing correlations of properties. As such they are similar to mathematical functions which are saturated by values. The relationship between a subject and properties can be grasped by analogy to such a saturation. (shrink)

A critical re-examination of the history of the concepts of space (including spacetime of general relativity and relativistic quantum field theory) reveals a basic ontological elusiveness of spatial extension, while, at the same time, highlighting the fact that its epistemic primacy seems to be unavoidably imposed on us (as stated by A.Einstein “giving up the extensional continuum … is like to breathe in airless space”). On the other hand, Planck’s discovery of the atomization of action leads to the fundamental recognition (...) of an ontology of non-spatial, abstract entities (Quine) for the quantum level of reality (QT), as distinguished from the necessarily spatio-temporal, experimental revelations (measurements). The elementary quantum act (measured by Planck’s constant) has neither duration nor extension, and any genuinely quantum process literally does not belong in the Raum and time of our experience. As Heisenberg stresses: “Während also die klassische Physik ein objectives Geschehen in Raum and Zeit zum Gegenstand hat, für dessen Existenz seine Beobachtung völlig irrelevant war, behandelt die Quantentheorie Vorgänge, die sozusagen nur in den Momenten der Beobachtung als raumzeitliche Phänomene aufleuchten, und über die in der zwischenzeit anschaulische physikalische Aussagen sinloss sind”. An admittedly speculative, hazardous conjecture is then advanced concerning the relation of such quantum ontology with the role of the pre-phenomenal continuum (Husserl) in the perception of macroscopically distinguishable objects in the Raum and time of our experience. Although rather venturesome, it brings together important philosophical issues. Coherently with recent general results in works on the foundations of QT, it is assumed that the linearity of quantum dynamical evolution does not apply to the central nervous system of living beings at a certain level of the evolutionary ramification and at the pre-conscious stage of subjectivity. Accordingly, corresponding to the onset of a non-linear dynamic evolution, a ‘primary spatial’ reduction is ‘continually’ taking place, thereby constituting the neural precondition for the experience of distinguishable macroscopic objects in the continuous spatial extension. While preventing the theoretically possible quantum superpositions of macroscopic objects from being perceivable by living beings, the ‘primary reduction’ has no effect on the standard processes concerning quantum level entities involved in laboratory man-made experiments. In this connection, an experimental check which might falsify the conjecture is briefly discussed. The approach suggested here, if sound, leads to a naturalization of that part of Kant’s Transcendental Aesthetics than can survive the Euclidean catastrophe. According to such naturalized transcendentalism, “space can well be transcendental without the axioms being so”, in agreement with a well-known statement by Boltzman. Finally, as far as QT is concerned, the conjecture entails that a scheme for quantum measurement of the von Neumann type cannot even ‘leave the ground’, vindicating Bohr’s viewpoint. A quantum theory of measurement, in a proper sense, turns out to be unnecessary and in fact impossible. (shrink)

An early, very preliminary edition of this book was circulated in 1962 under the title Set-theoretical Structures in Science. There are many reasons for maintaining that such structures play a role in the philosophy of science. Perhaps the best is that they provide the right setting for investigating problems of representation and invariance in any systematic part of science, past or present. Examples are easy to cite. Sophisticated analysis of the nature of representation in perception is to be found already (...) in Plato and Aristotle. One of the great intellectual triumphs of the nineteenth century was the mechanical explanation of such familiar concepts as temperature and pressure by their representation in terms of the motion of particles. A more disturbing change of viewpoint was the realization at the beginning of the twentieth century that the separate invariant properties of space and time must be replaced by the space-time invariants of Einstein's special relativity. Another example, the focus of the longest chapter in this book, is controversy extending over several centuries on the proper representation of probability. The six major positions on this question are critically examined. Topics covered in other chapters include an unusually detailed treatment of theoretical and experimental work on visual space, the two senses of invariance represented by weak and strong reversibility of causal processes, and the representation of hidden variables in quantum mechanics. The final chapter concentrates on different kinds of representations of language, concluding with some empirical results on brain-wave representations of words and sentences. (shrink)

This thesis presents an interpretation of Immanuel Kant’s theoretical philosophy in which the notion of ‘force’ is of central importance. My analysis encompasses the full span of Kant’s theoretical and natural-scientific writings, from the first publication to the drafts of an unfinished final work. With a close focus on Kant’s texts, I explicate their explicit references to force, providing a narrative of the philosophical role and significance of force in the various periods of the Kantian oeuvre. This represents an intervention (...) into Kant scholarship that seeks to correct the marginal role accorded to ‘force’. The central problem that emerges through the thesis’ attention to force is: how to interpret the simultaneous separation and connection of physical and psychological forces in Kant’s mature, critical philosophy? Physical and psychological forces are strictly separated, and yet a common, ontological conception of force underpins these two domains. I show that this issue has its basis in a tradition of philosophical ‘dynamics’ stemming from Leibniz, which is examined in part one. The three parts of the thesis proceed chronologically through the Kantian oeuvre. Part one reconstructs the historical context of Leibnizian and Newtonian conceptions of force, and presents a narrative of the employment of force in Kant’s pre-critical writings, in their relation to the broad problematic of Leibniz’s dynamics. Part two explores the account of physical and psychological forces, and the common, ontological notion of force, in the major critical-period discussions. Part three presents the late works, namely the third Critique and the Opus postumum, as a ‘philosophy of force’, in which force, with all its necessary ambiguities, plays central roles in Kant’s late systematising endeavours. I conclude that the perspective of this thesis makes possible a new understanding of the nature and unity of Kant’s philosophical project. (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)

This thesis argues for the significance of G.W Leibniz’s concepts of ‘expression’, ‘force’ and ‘perspective’ to the writings of Walter Benjamin and Gilles Deleuze. By triangulating the philosophical projects of Benjamin, Deleuze and Leibniz, as has not yet been done, the thesis opens up new perspectives and provides new readings of all three. Designating a structure of relations in which every simple substance or monad serves as a ‘living mirror’ of the universe, Leibniz’s concept of ‘expression’ denotes virtual inclusion or (...) immanence. His concept of ‘force’ denotes the self-incurred drive that motivates the monad to action, while his ‘perspectivism’ defines the monads individuality through their infinite points of view on the world. Deleuze and Benjamin, I suggest, appropriate Leibniz’s concepts as part of their respective critiques of epistemology, which target Kant’s conception of experience as a hierarchic relation of representation, allowing them to redefine experience as non-hierarchal, de-centred and embodied. At the same time, for both, Leibniz’s philosophy serves to criticize historicist views of chronological time. Leibniz’s perspectivism is reformulated by Deleuze and Benjamin as part of their respective critical theories of the image, culminating in their later formulations of the ‘dialectical’ and ‘crystal’ image, respectively. The conclusion however, highlights the diverging paths formed by their returns to Leibniz. Benjamin develops a politically effective ‘historical perspectivism’ in which the discontinuity of history enables ‘true historical time’ to replace chronological time. Deleuze, on the other hand, opts for an ahistorical pure form of temporality, his ‘mannerist perspectivism’ describing a continuous, perpetually repeated ‘becoming’. (shrink)