In ?155 of his New Theory of Vision Berkeley explains that a hypothetical ?unbodied spirit? ?cannot comprehend the manner wherein geometers describe a right line or circle?.1The reason for this, Berkeley continues, is that ?the rule and compass with their use being things of which it is impossible he should have any notion.? This reference to geometrical tools has led virtually all commentators to conclude that at least one reason why the unbodied spirit cannot have knowledge of plane geometry is (...) because it cannot manipulate a ruler or a compass. In this article I will show that such an interpretation is flawed. I will instead argue that Berkeley's understanding of Euclidian geometry was based on Isaac Barrow's account of the foundations of geometry. On this view geometrical objects are conceived in terms of the idealized motion that generates the objects of geometry. Consequently, that what the unbodied spirit cannot do in this context is to form an idea of motion rather than being unable to handle geometrical tools. 1All references to Berkeley are from, A. A. Luce and T. E. Jessop (eds.), The Works of George Berkeley, Bishop of Cloyne (London: Thomas Nelson and Sons, Ltd., 1948) The following abbreviations are used: An Essay Towards A New Theory of Vision, section x = New Theory x; Philosophical Commentaries, entry x = Commentaries x; Part I of A Treatise concerning the Principles of Knowledge, section x = Principles x. All other references to Berkeley's works are of the form The Works of George Berkeley, Bishop of Cloyne, volume x, page y = Works, x, y. (shrink)

The defense of common sense in Berkeley's Three Dialogues is, first and foremost, a defense of the gardener's claim to know this cherry tree, a claim threatened by both Cartesian and Lockean philosophy. Berkeley's defense of the gardener's knowledge depends on his claim that the being of a cherry tree consists in its being perceived. This is not something the gardener believes; rather, it is a philosophical analysis of the rules unreflectively followed by the gardener in his use of the (...) word 'exists'. It is by following these rules that the gardener gains knowledge of the cherry tree. Uncovering these deep connections between Berkeley's epistemology and his philosophy of language and placing them in the context of his critique of both Cartesian and Lockean philosophy will clarify Berkeley's strategy for bringing his reader back to common sense and practical engagement in the ordinary affairs of life. (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)

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)

Berkeley desarrolla su teoría de la visión en la obra de juventud Ensayo para una nueva teoría de la visión, que por lo general ha sido leída atendiendo sólo a sus aspectos científicos o perceptuales. En este artículo propongo una lectura distinta, que busca mostrar que el Ensayo no sólo atiende aspectos científicos sino, por el contrario, anticipa el inmaterialismo de obras posteriores. Esto lo hace porque Dios cumple un importante papel en él, lo cual se debe, entre otras cosas, (...) a que la teoría de la visión es desarrollada en función de Dios, pues de Él depende tanto la vista y los objetos visibles como el argumento del lenguaje visual. (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)

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 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.

It has been argued that there is a genuine conflict between the views of geometry defended by Hume in the Treatise and in the Enquiry: while the former work attributes to geometry a different status from that of arithmetic and algebra, the latter attempts to restore its status as an exact and certain science. A closer reading of Hume shows that, in fact, there is no conflict between the two works with respect to geometry. The key to understanding Hume's view (...) of geometry is the distinction he draws between two standards of equality in extension. (shrink)

The subject of this essay-based dissertation is Hume’s natural philosophy. The dissertation consists of four separate essays and an introduction. These essays do not only treat Hume’s views on the topic of natural philosophy, but his views are placed into a broader context of history of philosophy and science, physics in particular. The introductory section outlines the historical context, shows how the individual essays are connected, expounds what kind of research methodology has been used, and encapsulates the research contributions of (...) the essays. The first essay treats Newton’s experimentalist methodology in gravity research and its relation to Hume’s causal philosophy. It is argued that Hume does not see the relation of cause and effect as being founded on a priori reasoning, similar to the way in which Newton criticized non-empirical hypotheses about the causal properties of gravity. Contrary to Hume’s rules of causation, the universal law does not include a reference either to contiguity or succession, but Hume accepts it in interpreting the force and the law of gravity instrumentally. The second article considers Newtonian and non-Newtonian elements in Hume more broadly. He is sympathetic to many prominently Newtonian themes in natural philosophy, such as experimentalism, critique of hypotheses, inductive proof, and the critique of Leibnizian principles of sufficient reason and intelligibility. However, Hume is not a Newtonian philosopher in many respects: his conceptions regarding space and time, the vacuum, the specifics of causation, the status of mechanism, and the reality of forces differ markedly from Newton’s related conceptions. The third article focuses on Hume’s Fork and the proper epistemic status of propositions of mixed mathematics. It is shown that the epistemic status of propositions of mixed mathematics, such as those concerning laws of nature, is that of matters of fact. The reason for this is that the propositions of mixed mathematics are dependent on the Uniformity Principle. The fourth article analyzes Einstein’s acknowledgement of Hume regarding special relativity. The views of the scientist and the philosopher are juxtaposed, and it is argued that there are two common points to be found in their writings, namely an empiricist theory of ideas and concepts and a relationist ontology regarding space and time. (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)