In a focused assessment of one of the founding members of the liberal tradition in philosophy and a self-proclaimed “Under-Labourer” working to support the scientific revolution of the seventeenth century, the author maps the full range of John Locke’s highly influential ideas, which even today remain at the heart of debates about the nature of reality and our knowledge of it, as well as our moral and political rights and duties. Comprehensive introduction to the full range of Locke’s ideas, providing (...) an up-to-date account that acknowledges issues raised by recent scholarship over the past decade A well-rounded perspective on one of the intellectual giants of the western philosophical tradition Provides detailed coverage of Locke’s two key works, _An Essay Concerning Human Understanding_ and _The Two Treatises of Government_. A sophisticated analysis by a highly respected academic A vital addition to the _Blackwell_ _Great Minds_ series. (shrink)

In a recent article, I have explored the historical, mathematical, and philosophical issues related to the new theory of numerosities. The theory of numerosities provides a context in which to assign numerosities to infinite sets of natural numbers in such a way as to preserve the part-whole principle, namely if a set A is properly included in B then the numerosity of A is strictly less than the numerosity of B. Numerosities assignments differ from the standard assignment of size provided (...) by Cantor’s cardinality assignments. In this talk, I generalize some specific worries emerging from the theory of numerosities to a line of thought resulting in what I call a ‘good company’ objection to Hume’s Principle (HP). The talk has four main parts. The first takes a historical look at 19th-century attributions of equality of numbers in terms of one-one correlations and argues that there was no agreement as to how to extend such determinations to infinite sets of objects. This leads to the second part where I show that there are countably infinite many abstraction principles that are ‘good’, in the sense that they share the same virtues of HP and from which we can derive the axioms of second order arithmetic. The third part connects this material to a debate on Finite Hume Principle between Heck and MacBride and states the ‘good company’ objection. Finally, the last part gives a tentative taxonomy of possible neo-logicist responses to the ‘good company’ objection and makes a foray into the relevance of this material for the issue of cross-sortal identifications for abstractions. (shrink)

Leibniz had a well-known argument against the existence of infinite wholes that is based on the part-whole axiom: the whole is greater than the part. The refutation of this argument by Russell and others is equally well known. In this note, I argue (against positions recently defended by Arthur, Breger, and Brown) for the following three claims: (1) Leibniz himself had all the means to devise and accept this refutation; (2) This refutation does not presuppose the consistency of Cantorian set (...) theory; (3) This refutation does not cast doubt on the part-whole axiom. Hence, should there be an obstacle to Gödel's wish to integrate Cantorian set theory within Leibniz' philosophy, it will not be this famous argument of Leibniz'. (shrink)

This book explicates Leibnizian analysis as a search for conditions of intelligibility, and reconsiders his use of principles and methods as well as his account of truth in this way. Via careful reading of well-known, lesser known, and previously unedited texts, it gives a more accurate picture of his philosophical intentions, as well as the relevance of his project to contemporary debate. Two case studies are included, one concerning logic and the other arithmetic; they illustrate a theory of intelligibility that (...) takes as its central notion "possibility for thought", a notion which allows Leibniz to escape certain traps of psychologism, the pseudo-ontology of empiricism, and the empty forms of logicism, and suggests new approaches for contemporary philosophy. "In this remarkable study, Grosholz and Yakira offer a fresh interpretive and conceptual angle on Leibniz's metaphysics. [...] this study deserves high marks for its subtlety, novelty, and creative insight into Leibniz's modes of inquiry as well as for its philosophical acumen." Annals of Science. (shrink)

Leibniz claims that nature is actually infinite but rejects infinite number. Are his mathematical commitments out of step with his metaphysical ones? It is widely accepted that Leibniz has a viable response to this problem: there can be infinitely many created substances, but no infinite number of them. But there is a second problem that has not been satisfactorily resolved. It has been suggested that Leibniz’s argument against the world soul relies on his rejection of infinite number, and, as such, (...) Leibniz cannot assert that any body has a soul without also accepting infinite number, since any body has infinitely many parts. Previous attempts to address this concern have misunderstood the character of Leibniz’s rejection of infinite number. I argue that Leibniz draws an important distinction between ‘wholes’– collections of parts that can be thought of as a single thing – and ‘fictional wholes’ – collections of parts that cannot be thought of as a single thing, which allows us to make sense of his rejection of infinite number in a way that does not conflict either with his view that the world is actually infinite or that the bodies of substances have infinitely many parts. (shrink)

Anyone who has pondered the limitlessness of space and time, or the endlessness of numbers, or the perfection of God will recognize the special fascination of this question. Adrian Moore's historical study of the infinite covers all its aspects, from the mathematical to the mystical.

In last year’s Review Gregory Brown took issue with Laurence Carlin’s interpretation of Leibniz’s argument as to why there could be no world soul. Carlin’s contention, in Brown’s words, is that Leibniz denies a soul to the world but not to bodies on the grounds that “while both the world and [an] aggregate of limited spatial extent are infinite in multitude, the former, but not the latter, is infinite in respect of magnitude and hence cannot be considered a whole”. Brown (...) casts doubt on this interpretation—or rather, he begins by questioning its adequacy as an interpretation of a central passage, only to concede its essential correctness as an interpretation of Leibniz’s position, and to turn his attack on the latter itself. In this note I shall argue that Brown underestimates the subtlety of Leibniz’s views on the infinite, and that Carlin is basically correct in his suggestion that the infinite magnitude of the world is what precludes it from having even the phenomenal unity that a body does. (shrink)

Throughout his early writings, Leibniz was concerned with developing an acceptable account of God's relationship to the created world. In some of these early writings, he endorsed the idea that this relationship was similar to the human soul's relationship to the body. Though he eventually came to reject this idea, theanima mundi thesis remained the topic of several essays and correspondences during his career, culminating in the correspondence with Clarke. At first glance,Leibniz's discussions of this thesis may seem less important (...) in comparison to others, since it might seem like a topic which is far removed from what are regarded as his most important philosophical doctrines. I hope to show in what follows that such a view is mistaken. The large amount of attention Leibniz paid to this thesis is a sure indication of its importance to him. Further, as we shall see, his discussions of this thesis tum on some of his most interesting metaphysical topics, including the development of his thinking about the actual infinite, the structure of organic wholes, and the relationship between God and the created universe. In what follows, I examine these discussions chronologically, from the De Summa Rerum (1675-6), to the correspondence with Clarke (1715-6). (shrink)

Leibniz had a well-known argument against the existence of infinite wholes that is based on the part-whole axiom: the whole is greater than the part. The refutation of this argument by Russell and others is equally well known. In this note, I argue (against positions recently defended by Arthur, Breger, and Brown) for the following three claims: (1) Leibniz himself had all the means to devise and accept this refutation; (2) This refutation does not presuppose the consistency of Cantorian set (...) theory; (3) This refutation does not cast doubt on the part-whole axiom. Hence, should there be an obstacle to Gödel's wish to integrate Cantorian set theory within Leibniz' philosophy, it will not be this famous argument of Leibniz'. (shrink)

These selections from Le système du monde, the classic ten-volume history of the physical sciences written by the great French physicist Pierre Duhem (1861-1916), focus on cosmology, Duhem's greatest interest. By reconsidering the work of such Arab and Christian scholars as Averroes, Avicenna, Gregory of Rimini, Albert of Saxony, Nicole Oresme, Duns Scotus, and William of Occam, Duhem demonstrated the sophistication of medieval science and cosmology.

Pierre Gassendi, who did not like nonsense, said of the idea of infinity: ‘if someone calls something "infinite" he attributes to a thing which he does not grasp a label which he does not understand’. Gassendi’s is a harsh judgement for, surely, we all do quite cheerfully and successfully use the concept of infinity, and in a variety of contexts. Yet if Gassendi’s judgement is too hard it is easy enough to have sympathy with his claim. For it is a (...) perennial fact that we never, in Descartes’s phrase, seem to have an ‘adequate idea’ of infinity. Nor is this just because it is an abstract noun like friendship or strength, for it retains this familiar lack of adequacy when it appears in its adjectival or adverbial forms: infinite space, infinite power, infinitely large, infinitely good. It is not my intention in this paper to offer a philosophical account of this familiar state of affairs, though perhaps what I shall have to say will throw some little light on the matter. It is rather to explore how discussions of such questions take us into issues at the heart of the foundations of modern philosophy, and specifically, into the great debate which I will refer to by the usual title as that between the Rationalists and the Empiricists, of whom the protagonists are traditionally identified as Descartes on the one side and Locke on the other. It would not be out of place for somebody to say in response to that famous contrast that either it is hackneyed or else it is mistaken. It is hackneyed because we all know that Descartes and Locke represent contrasting traditions in modern philosophy and there is nothing new to be said about it. It is mistaken because, as a matter of fact, it is simplistic to set them up as dogmatic exponents of their respective schools. There are rationalist elements in Locke’s Essay, especially in Book IV, and there is a strong empiricist element in Descartes, especially in his science. Those emphasizing the former, Webb for example in the last century and Aaron in this, have underlined the place of intuition and demonstration in Locke’s account of knowledge. Descartes’s empirical leanings have been noted in his account of the role of experiment in the natural sciences. There is of course no denying these aspects of their philosophies. But my path will be more revisionist than supportive of such readings of their work. I shall argue that the dominant (though not the only) strain in Descartes is a rationalist one and that Locke was keenly aware of this and strongly hostile to it. On the other side, whilst Locke was impressed by much of Descartes’s presentation of knowledge, and borrowed heavily from it, he never looks tike subscribing at all to the central rationalist doctrines, and indeed saw his work as a major refutation of them. In all of this his account of our idea of infinity plays an exemplary role. But before we reach Locke we should go back to Descartes. (shrink)

In 1675, Leibniz elaborated his longest mathematical treatise he everwrote, the treatise ``On the arithmetical quadrature of the circle, theellipse, and the hyperbola. A corollary is a trigonometry withouttables''. It was unpublished until 1993, and represents a comprehensive discussion of infinitesimalgeometry. In this treatise, Leibniz laid the rigorous foundation of thetheory of infinitely small and infinite quantities or, in other words,of the theory of quantified indivisibles. In modern terms Leibnizintroduced `Riemannian sums' in order to demonstrate the integrabilityof continuous functions. The (...) article deals with this demonstration,with Leibniz's handling of infinitely small and infinite quantities,and with a general theorem regarding hyperboloids. (shrink)