On the face of it, Locke rejects the scholastics' main tool for making sense of talk of God, namely, analogy. Instead, Locke claims that we generate an idea of God by 'enlarging' our ideas of some attributes (such as knowledge) with the idea of infinity. Through an analysis of Locke's idea of infinity, I argue that he is in fact not so distant from the scholastics and in particular must rely on analogy of inequality.

Published in 1903, this book was the first comprehensive treatise on the logical foundations of mathematics written in English. It sets forth, as far as possible without mathematical and logical symbolism, the grounds in favour of the view that mathematics and logic are identical. It proposes simply that what is commonly called mathematics are merely later deductions from logical premises. It provided the thesis for which _Principia Mathematica_ provided the detailed proof, and introduced the work of Frege to a wider (...) audience. In addition to the new introduction by John Slater, this edition contains Russell's introduction to the 1937 edition in which he defends his position against his formalist and intuitionist critics. (shrink)

Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one association between two sets. As is well known, all countable infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collectionAis properly included in a collectionBthen the ‘size’ ofAshould be less than the (...) ‘size’ ofB(part–whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this article I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers (Thabit ibn Qurra, Grosseteste, Maignan, Bolzano). Then, I review some recent mathematical developments that generalize the part–whole principle to infinite sets in a coherent fashion (Katz, Benci, Di Nasso, Forti). Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, based on the part–whole principle, envisaged by Bolzano (Kitcher). (shrink)

One argument that Leibniz employed to rule out the possibility of a world soul appears to turn on the assumption that the very notion of an infinite number or of an infinite whole is inconsistent. This argument was considered in a series of three papers published in The Leibniz Review: in the first, by Laurence Carlin, the argument was delineated and analyzed; in the second, by myself, the argument was criticized and rejected; in the third, by Richard Arthur, an attempt (...) was made to defend Leibniz’s argument against my criticisms. In the present paper, I take up the matter again in an attempt to clarify the issues involved and to defend my original criticisms of the argument against the objections raised by Arthur. (shrink)

Descartes notoriously characterizes substance in two ways: first, as an ultimate subject of properties ; second, as an independent entity. The characterizations have appeared to many to diverge on the definition as well as the scope of the notion of substance. For it is often thought that the ultimate subject of properties need not—and, in some cases, cannot—be independent. Drawing on a suite of historical, textual, and philosophical considerations, this essay argues for an interpretation that reconciles Descartes's two characterizations. It (...) proposes that both characterizations invoke a type of independence that obtains just in case there is no relation to another entity that holds by the nature of the entity in question. Even though the ultimate subject of properties is sometimes not independent in other respects, it satisfies this independence-by-nature condition. (shrink)

This paper aims to show that a proper understanding of what Leibniz meant by “hypercategorematic infinite” sheds light on some fundamental aspects of his conceptions of God and of the relationship between God and created simple substances or monads. After revisiting Leibniz’s distinction between (i) syncategorematic infinite, (ii) categorematic infinite, and (iii) actual infinite, I examine his claim that the hypercategorematic infinite is “God himself” in conjunction with other key statements about God. I then discuss the issue of whether the (...) hypercategorematic infinite is a “whole”, comparing the four kinds of infinite outlined by Leibniz in 1706 with the three degrees of infinity outlined in 1676. In the last section, I discuss the relationship between the hypercategorematic infinite and created simple substances. I conclude that, for Leibniz, only a being beyond all determinations but eminently embracing all determinations can enjoy the pure positivity of what is truly infinite while constituting the ontological grounding of all things. (shrink)

"Williams's masterful translation satisfies (at last!) a long-standing need. There are lots of good translations of Augustine's great work, but until now we have been forced to choose between those that strive to replicate in English something of the majesty and beauty of Augustine's Latin style and those that opt instead to convey the careful precision of his philosophical terminology and argumentation. Finally, Williams has succeeded in capturing both sides of Augustine's mind in a richly evocative, impeccably reliable, elegantly readable (...) presentation of one of the most impressive achievements in Western thought--Augustine's Confessions." --Scott MacDonald, Professor of Philosophy and Norma K. Regan Professor in Christian Studies, Cornell University. (shrink)

I explore Locke’s complex attitude toward the natural philosophy of his day by focusing on Locke’s own treatment of Newton’s theory of gravity and the presence of Lockean themes in defenses of Newtonian attraction/gravity by Maupertuis and other early Newtonians. In doing so, I highlight the inadequacy of an unqualified labeling of Locke as “mechanist” or “Newtonian.”.

We argue that Descartes’s theistic proofs in the ’Meditations’ are much simpler and straightforward than they are traditionally taken to be. In particular, we show how the causal argument of the "Third Meditation" depends on the intuitively innocent principle that nothing comes from nothing, and not on the more controversial principle that the objective reality of an idea must have a cause with at least as much formal reality. We also demonstrate that the so-called ontological "argument" of the "Fifth Meditation" (...) is best understood not as a formal proof but as an axiom, revealed as self-evident by analytic meditation. (shrink)

Reductio arguments are notoriously inconclusive, a fact which no doubt contributes to their great fecundity. For once a contradiction has been proved, it is open to interpretation which premise should be given up. Indeed, it is often a matter of great creativity to identify what can be consistently given up. A case in point is a traditional paradox of the infinite provided by Galileo Galilei in his Two New Sciences, which has since come to be known as Galileo’s Paradox. It (...) concerns the set of all numbers, N: since to every number there is a corresponding square, there are as many squares as numbers. But since there are non-squares between the squares, “all numbers, comprising the squares and the non-squares, are greater than the squares alone”, i.e. there must be fewer numbers in the set of all squares S than in N. Thus N is both equal to and greater than S. This is a contradiction, so one of the premises must be given up: the question is, which one? (shrink)

The article presents Leibniz's preoccupation (in 1675?6) with the difference between the notion of infinite number, which he regards as impossible, and that of the infinite being, which he regards as possible. I call this issue ?Leibniz's Problem? and examine Spinoza's solution to a similar problem that arises in the context of his philosophy. ?Spinoza's solution? is expounded in his letter on the infinite (Ep.12), which Leibniz read and annotated in April 1676. The gist of Spinoza's solution is to distinguish (...) between three kinds of infinity and, in particular, between one that applies to substance, and one that applies to numbers, seen as auxiliaries of the imagination. The rest of the paper examines the extent to which Spinoza's solution solves Leibniz's problem. The main thesis I advance is that, when Spinoza and Leibniz say that the divine substance is infinite, in most contexts it is to be understood in non-numerical and non-quantitative terms. Instead, for Spinoza and Leibniz, a substance is said to be infinite in a qualitative sense stressing that it is complete, perfect and indivisible. I argue that this approach solves one strand of Leibniz's problem and leaves another unsolved. (shrink)

Peter Anstey presents a thorough and innovative study of John Locke's views on the method and content of natural philosophy. Focusing on Locke's Essay concerning Human Understanding, but also drawing extensively from his other writings and manuscript remains, Anstey argues that Locke was an advocate of the Experimental Philosophy: the new approach to natural philosophy championed by Robert Boyle and the early Royal Society who were opposed to speculative philosophy. On the question of method, Anstey shows how Locke's pessimism about (...) the prospects for a demonstrative science of nature led him, in the Essay, to promote Francis Bacon's method of natural history, and to downplay the value of hypotheses and analogical reasoning in science. But, according to Anstey, Locke never abandoned the ideal of a demonstrative natural philosophy, for he believed that if we could discover the primary qualities of the tiny corpuscles that constitute material bodies, we could then establish a kind of corpuscular metric that would allow us a genuine science of nature. It was only after the publication of the Essay, however, that Locke came to realize that Newton's Principia provided a model for the role of demonstrative reasoning in science based on principles established upon observation, and this led him to make significant revisions to his views in the 1690s. On the content of Locke's natural philosophy, it is argued that even though Locke adhered to the Experimental Philosophy, he was not averse to speculation about the corpuscular nature of matter. Anstey takes us into new terrain and new interpretations of Locke's thought in his explorations of his mercurialist transmutational chymistry, his theory of generation by seminal principles, and his conventionalism about species. (shrink)

The Purpose of this paper is to ask how far Locke can be said to have anticipated modern theories of number, particularly the intuitionist theory of Brouwer and Heyting. It has in mind Mr Edward E. Dawson's statement that Locke's account of number was not merely ‘a good effort in his own day’ but that ‘what Locke had to say really was quite fundamental, and a good deal of modern mathematics assumes his position, either explicitly or implicitly’. Mr Dawson thinks (...) that some of the central notions of the intuitionist theory are already present in the Essay Concerning Human Understanding , II, xvi, ‘Of Number’. We should like to examine the view. (shrink)

Presents Newton's unifying idea of gravitation and explains how he converted physics from a science of explanation into a general mathematical system.

INTRODUCTION Ever since the beginning of the modern phenomenological movement disciplined attention has been paid to various patterns of human experience as ...

The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmatic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with (...) the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics. (shrink)

Volumes I and II provided a completely new translation of the philosophical works of Descartes, based on the best available Latin and French texts. Volume III contains 207 of Descartes' letters, over half of which have previously not been translated into English. It incorporates, in its entirety, Anthony Kenny's celebrated translation of selected philosophical letters, first published in 1970. In conjunction with Volumes I and II it is designed to meet the widespread demand for a comprehensive, authoritative and accurate edition (...) of Descartes' philosophical writings in clear and readable modern English. (shrink)

Mathematician and philosopher Hermann Weyl had our subject dead to rights.

In this chapter, John Locke's anti‐Cartesian stances on the difference between body and space, on whether the soul always thinks, on the possibility of thinking matter, all connect back to the basic opposition to Cartesian overreaching in regard to essences. The chapter presents a summary of Locke's anti‐Cartesianism, which seems to fit with his own representation of his Cartesian inheritance, which, notoriously, is that it is minimal, consisting only in anti‐scholasticism. The only acknowledgment that Locke wishes to give Descartes is (...) that reading his work helped him to shake off the influence of Aristotelianism. Descartes was not the only target of Locke's anti‐innatism, but he was surely a central one. Locke attacks the Cartesian doctrine of innate ideas because it poses a threat to his central empiricist commitments. Interestingly, he also sees the central tenets of Cartesian ontology as a threat. (shrink)

Descartes’s metaphysics posits a sharp distinction between two types of non-finitude, or unlimitedness: whereas God alone is infinite, numbers, space, and time are indefinite. The distinction has proven difficult to interpret in a way that abides by the textual evidence and conserves the theoretical roles that the distinction plays in Descartes’s philosophy—in particular, the important role it plays in the causal proof for God’s existence in the Meditations. After formulating the interpretive task, I criticize extant interpretations of the distinction. I (...) then propose an alternative at whose core is the idea that whereas the indefinite is a structural, iterative notion, designating the absence of an upper bound, the infinite is an ontic notion, signifying being in general, or what is, without qualification. (shrink)

It is well-known that Isaac Newton’s conception of space and time as absolute -- “without reference to anything external” (Principia, 408) -- was anticipated, and probably influenced, by a number of figures among the earlier generation of seventeenth century natural philosophers, including Pierre Gassendi, Henry More, and Newton’s own teacher Isaac Barrow. The absolutism of Newton’s contemporary and friend, John Locke, has received much less attention, which is unfortunate for several reasons. First, Locke’s views of space and time undergo a (...) dramatic evolution that mirrors the overall absolutist trend of the era. Second, there are good reasons to suppose that Locke was influenced late in this evolution by Newton’s Principia, which he read and reviewed shortly after its 1687 publication. It is even possible that Locke read or knew of the earlier and unpublished, though now famous, Newtonian tract De Gravitatione. Third, despite the influence of Newton, Locke’s retains a skeptical attitude concerning our empirical knowledge of absolute space and time. He is especially cautious about absolute time, bucking a widespread tendency to treat time as strongly analogous to space. Their disagreement about the measure of absolute time, we will suggest, reflects a deeper disagreement in the philosophy of science. While Locke counts as scientific knowledge only ideas and demonstrations derivable from immediate experience, Newton endorses a more theory-driven brand of empiricism, which holds that our best explanations of the phenomena provide genuine knowledge that goes beyond what we can directly observe. (shrink)

This chapter challenges Cantor’s notion of the ‘power’, or ‘cardinality’, of an infinite set. According to Cantor, two infinite sets have the same cardinality if and only if there is a one-to-one correspondence between them. Cantor showed that there are infinite sets that do not have the same cardinality in this sense. Further, he took this result to show that there are infinite sets of different sizes. This has become the standard understanding of the result. The chapter challenges this, arguing (...) that we have no reason to think there are infinite sets of different sizes. It begins with an initial argument against Cantor’s claim that there are infinite sets of different sizes and then proceeds, by way of an analogy between Cantor’s mathematical result and Russell’s paradox, to a more direct argument. (shrink)

Reductio arguments are notoriously inconclusive, a fact which no doubt contributes to their great fecundity. For once a contradiction has been proved, it is open to interpretation which premise should be given up. Indeed, it is often a matter of great creativity to identify what can be consistently given up. A case in point is a traditional paradox of the infinite provided by Galileo Galilei in his Two New Sciences, which has since come to be known as Galileo’s Paradox. It (...) concerns the set of all numbers, N: since to every number there is a corresponding square, there are as many squares as numbers. But since there are non-squares between the squares, “all numbers, comprising the squares and the non-squares, are greater than the squares alone”, i.e. there must be fewer numbers in the set of all squares S than in N. Thus N is both equal to and greater than S. This is a contradiction, so one of the premises must be given up: the question is, which one? (shrink)

One argument that Leibniz employed to rule out the possibility of a world soul appears to turn on the assumption that the very notion of an infinite number or of an infinite whole is inconsistent. This argument was considered in a series of three papers published in The Leibniz Review: in the first, by Laurence Carlin, the argument was delineated and analyzed; in the second, by myself, the argument was criticized and rejected; in the third, by Richard Arthur, an attempt (...) was made to defend Leibniz’s argument against my criticisms. In the present paper, I take up the matter again in an attempt to clarify the issues involved and to defend my original criticisms of the argument against the objections raised by Arthur. (shrink)