Les Principes de la connaissance humaine sont l'occasion pour Berkeley de nier l'existence des idées générales abstraites. Il admet cependant l'existence d'idées générales, plus exactement d'idées déterminées à signification générale. C'est ainsi qu'il peut rendre compte de la généralité de certaines démonstrations. L'exemple choisi est celui de l'idée de triangle dans le cadre d'une démonstration géométrique. Mais peut-on également rendre compte de cette manière des démonstrations et des idées algébriques et notamment celle de quantité? In the Principles of human knowledge, (...) Berkeley clearly denies the existence of abstract general ideas. He assumes, however, the existence of general ideas or, more exactly, of determinate ideas with a general meaning. In this way, he explains the generality of certain demonstrations. He chooses the example of the idea of a triangle within a geometrical demonstration. But can he likewise account for algebraic demonstrations and ideas, and in particular for the idea of quantity? (shrink)

This essay demonstrates the inadequacy of contemporary substantivalist and relationist interpretations of quantum gravity hypotheses via an historical investigation of the debate on the underlying ontology of space in the seventeenth century. Viewed in the proper context, there are crucial similarities between seventeenth century theories of space and contemporary work on the ontological foundations of spacetime theories, and these similarities challenge the utility of the substantival/relational dichotomy by revealing a host of underlying conceptual issues that do not naturally align with (...) that distinction. (shrink)

Many historians of the calculus deny significant continuity between infinitesimal calculus of the seventeenth century and twentieth century developments such as Robinson’s theory. Robinson’s hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, (...) Robinson regards Berkeley’s criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley’s criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz’s infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz’s defense of infinitesimals is more firmly grounded than Berkeley’s criticism thereof. We show, moreover, that Leibniz’s system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz’s strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity. (shrink)

It is sometimes suggested that Berkeley adheres to an empirical criterion of meaning, on which a term is meaningful just in case it signifies an idea (i.e., an immediate object of perceptual experience). This criterion is thought to underlie his rejection of the term ‘matter’ as meaningless. As is well known, Berkeley thinks that it is impossible to perceive matter. If one cannot perceive matter, then, per Berkeley, one can have no idea of it; if one can have no idea (...) of it, then one cannot speak meaningfully of it. But if this is Berkeley’s position, then there is a puzzle, because Berkeley also explicitly claims that it is impossible to perceive/have ideas of minds. So if he is relying on a criterion on which terms get their meaning by referring to ideas, then, just as Berkeley rejects talk of material substance, so, too, must he reject talk of mental substance. Famously, however, Berkeley insists that there is no parity between the cases of material and mental substance. It is typically suggested that the disparity between matter and minds rests on the fact that although one cannot strictly speaking perceive minds, nonetheless Berkeley thinks that one can have experiential access to minds via reflection, and that this access allows for meaningful talk of minds. Of course, one can only have reflective experience of one’s own mind. But what of other minds, which one cannot reflectively experience? Here the usual tactic is to suppose that, although one cannot have direct reflective experience of other minds, nonetheless one can indirectly experience such minds via analogy to our own minds, and that this indirect experience grounds the meaningfulness of talk of other minds. In this paper, I argue that the reasoning behind Berkeley’s ‘likeness principle,’ that an idea can only be like another idea, can be generalized to argue against this experience-based account of our access to other minds. I claim instead that Berkeley allows for the meaningfulness of talk of other minds by expanding the criterion of meaning in a different way. I argue that Berkeley holds a criterion of meaning on which a term is meaningful just in case it signifies either an object of experience or an object that one has reason to posit on the basis of experience, i.e., an object that is necessary to explain our experiences. When an object is neither experienced nor explains our experiences, then and only then is Berkeley willing to reject it as meaningless. Thus he writes of “the word matter,” that “it is no matter whether there is such a thing or no, since it no way concerns us: and I do not see the advantage there is in disputing about we know not what, and we know not why” (Principles, §77.) The word is not meaningless merely because we do not know what matter might be; it is meaningless because we also do not know why it should be. Correspondingly, I argue that the term ‘mind’ is meaningful because although we have no experience of minds, nonetheless they play an important role in explaining our experiences. (shrink)

The paper provides an account of necessary truths in Berkeley based upon his divine language model. If the thesis of the paper is correct, not all Berkeleian necessary truths can be known a priori.

The instability inherent in the historical inventory of mathematical objects challenges philosophers. Naturalism suggests we can construct enduring answers to ontological questions through an investigation of the processes whereby mathematical objects come into existence. Patterns of historical development suggest that mathematical objects undergo an intelligible process of reification in tandem with notational innovation. Investigating changes in mathematical languages is a necessary first step towards a viable ontology. For this reason, scholars should not modernize historical texts without caution, as the use (...) of anachronistic notation tends to impede, rather than enhance, our ability to recognize the emergent nature of mathematical objects. (shrink)

There is a wide range of realist but non-Platonist philosophies of mathematics—naturalist or Aristotelian realisms. Held by Aristotle and Mill, they played little part in twentieth century philosophy of mathematics but have been revived recently. They assimilate mathematics to the rest of science. They hold that mathematics is the science of X, where X is some observable feature of the (physical or other non-abstract) world. Choices for X include quantity, structure, pattern, complexity, relations. The article lays out and compares these (...) options, including their accounts of what X is, the examples supporting each theory, and the reasons for identifying the science of X with (most or all of) mathematics. Some comparison of the options is undertaken, but the main aim is to display the spectrum of viable alternatives to Platonism and nominalism. It is explained how these views answer Frege’s widely accepted argument that arithmetic cannot be about real features of the physical world, and arguments that such mathematical objects as large infinities and perfect geometrical figures cannot be physically realized. (shrink)

The notion of indivisibles and atoms arose in ancient Greece. The continuum—that is, the collection of points in a straight line segment, appeared to have paradoxical properties, arising from the ‘indivisibles’ that remain after a process of division has been carried out throughout the continuum. In the seventeenth century, Italian mathematicians were using new methods involving the notion of indivisibles, and the paradoxes of the continuum appeared in a new context. This cast doubt on the validity of the methods and (...) the reliability of mathematical knowledge which had been regarded as established by the axiomatic method in geometry expounded by Aristotle’s younger contemporary Euclid. The teaching of indivisibles was banned within the Society of Jesus, the Jesuits. In England, indivisibles were used by the mathematician John Wallis, and there was an acrimonious and extended feud between Wallis and the philosopher Thomas Hobbes over legitimate methods of argument in mathematics. Notions of the infinitesimal were used by Isaac Newton and Gottfried Leibniz, and were attacked by Bishop Berkeley for the vagueness of the concept and the illegitimate reasoning applied to it. This article discusses aspects of these events with reference to the book Infinitesimal by Amir Alexander and to other sources. Also discussed are wider issues arising from Alexander’s book including: the changes in cultural sensibility associated with the growth of new mathematical and scientific knowledge in the seventeenth century, the changes in language concomitant with these changes, what constitutes valid methods of enquiry in various contexts, and the question of authoritarianism in knowledge. More general aims of this article are to widen the immediate mathematical and historical contexts in Alexander’s book, to bridge a gap in conversations between mathematics and the humanities, and to relate mathematical ideas to wider human and contemporary issues. (shrink)