Locke: Philosophy of Mathematics

The chapter emphasizes the complexity of the relations between philosophy and science in the eighteenth century, as they must be seen against the background that, in the early modern period, as in the preceding centuries, philosophy generally included physics or natural philosophy, mathematics, and metaphysics. Showing the variance in attitudes among Leibniz, Newton, and Locke on how to draw a line of division between metaphysics and physics with regard to a sample of topics, this chapter draws attention to the divergent (...) starting points of British and German metaphysical conceptions in the course of the eighteenth century. Tracing a trajectory of such conceptions in Britain – Berkeley, Hume, Cullen – and contrasting it with the position of Kant in Germany, this essay records the trembling or perhaps even crumbling of metaphysics caused by the growing tensions between metaphysics/philosophy and natural philosophy/science. In the end, this state of affairs led to a remarkably indifferent attitude toward metaphysics (Cullen) and, alternatively, to a radical critique of its epistemological foundations (Kant). (shrink)

John Locke’s treatment of quantity in the Essay Concerning Human Understanding is not nearly as extensive or as well-known as his treatment of quality and his distinction between primary and secondary qualities. Yet I contend that a close examination of Locke’s comments on quantity in the Essay reveals that he endorses a general theory of quantity that not only distinguishes quantities from qualities, but also plays several other important roles in his overall philosophy—particularly in his treatments of infinity and demonstrative (...) knowledge. Moreover, I argue that this theory does not appear out of thin air, but is in certain respects one and the same as the approach to quantity found in Aristotle and developed by medieval Aristotelian thinkers. Identifying the core tenets of this approach in the texts of Aristotle and medieval Aristotelians, and formulating them clearly enough to shed light on Locke’s theory of quantity in the Essay, is the principal task of this paper. (shrink)

Over Plato's Academy in ancient Athens, it is said, hung a sign: "Let no one ignorant of geometry enter here." Plato thought no one could do philosophy without also doing mathematics. In The Waltz of Reason, mathematician and philosopher Karl Sigmund shows us why. Charting an epic story spanning millennia and continents, Sigmund shows that philosophy and mathematics are inextricably intertwined, mutual partners in a reeling search for truth. Beginning with-appropriately enough-geometry, Sigmund explores the power and beauty of numbers and (...) logic, and then shows how those ideas laid the ground work for everything from the theory of a fair election, to modern conceptions of governance, cooperation, morality, and even of reason itself. Did you know, for example, that John Locke, author of some of the most important texts in the Western theory of government, was motivated in his work by his study of geometry? Or, that Locke was actually terrible at geometry, a seventeenth-century mathematical laughingstock? He was, a fact that might want us to think again about the logic of his life's work. And Locke wasn't the only one! Sigmund reveals how many of our modern ideas about what is true and what is reasonable are based on similarly shaky grounds. The economists and other thinkers who promulgated game theory, classical economics, and behavioral economics-the basis of so much in modern life-were not necessarily good at either math or philosophy, or both. The result is a remarkable book: accessible, funny, and wise, it tells an engrossing history of ideas that spins as dizzyingly and beautifully as a ballroom full of expert dancers. But it doesn't just celebrate the past. Instead, by making all these great ideas accessible to all, The Waltz of Reason empowers as it entertains, giving each of us the tools to ask, what do we know, how do we know it, and what do we want to do, with all these ideas? (shrink)

Locke is famed for his caution in speculative matters: “Men, extending their enquiries beyond their capacities and letting their thoughts wander into those depths where they can find no sure footing; ‘tis no wonder that they raise questions and multiply disputes”. And he is skeptical about the pretensions of natural philosophy, which he says is “not capable of being made a science”. And yet Locke is confident that “Our reason leads us to the knowledge of this certain and evident truth, (...) that there is an eternal, most powerful and most knowing being; which whether anyone will please to call God it matters not”. His certainty about the existence and attributes of God, I will argue, led him to surprisingly strong convictions about a deep and disputed problem at the intersection of seventeenth century metaphysics and natural philosophy: the absolute reality of space and time. Specifically, he based his absolutist conceptions of space and time on God’s literal omnipresence and eternity. Leibniz probably had Locke in mind when he inveighed in 1716 against “real absolute space, the idol of some modern Englishmen”. And Leibniz was right a decade earlier to voice through his mouthpiece Theophilus the suspicion that, despite Locke’s claim to know nothing about the substance of void space, “there are grounds for thinking you know more about it than you say or believe that you do. Some people have thought that God is the place of objects”. (shrink)

In both the early modern period and in contemporary debates, philosophers have argued that there are analogies between mathematics and morality that imply that the ontology and epistemology of morality are crucially similar to the ontology and epistemology of mathematics. I describe arguments for the math‐moral analogy in four early modern philosophers (Locke, Cudworth, Clarke, and Balguy) and in three contemporary philosophers (Clarke‐Doane, Peacocke, and Roberts). I argue that these arguments fail to establish important ontological and epistemological similarities between morality (...) and mathematics. There are analogies between the two areas, but the disanalogies are more significant, undermining the attempt to confer on morality the same ontological and epistemological status that mathematics possesses. (shrink)

Locke famously claimed that morality was capable of demonstration. But he also refused to provide a system of demonstrative morality. This paper addresses the mismatch between Locke’s stated views and his actual philosophical practice. While Locke’s claims about demonstrative morality have received a lot of attention it is rare to see them discussed in the context of his general theory of demonstration and his specific discussions of particular demonstrations. This paper explores Locke’s general remarks about demonstration as well as his (...) claims about demonstration in natural philosophy, mathematics, and morality. Careful attention to these detailed discussions motivates a reevaluation of Locke’s views on demonstrative knowledge of morality. Specifically, while Locke did believe that some demonstrative moral knowledge might be in-principle available to us he also believed that facts about the difficulty of demonstration meant that this knowledge would in-practice be largely unattainable. (shrink)

A number of commentators have recently suggested that there is a puzzle surrounding Locke’s acceptance of Newton’s Principia. On their view, Locke understood natural history as the primary methodology for natural philosophy and this commitment was at odds with an embrace of mathematical physics. This article considers various attempts to address this puzzle and finds them wanting. It then proposes a more synoptic view of Locke’s attitude towards natural philosophy. Features of Locke’s biography show that he was deeply interested in (...) mathematical physics long before the publication of the Principia. This interest was in line with important developments in the Royal Society. It is argued that Locke endorsed a two-stage approach to natural philosophy which was consistent with an embrace of both natural history and mathematical physics. The Principia can be understood as consistent with this approach. (shrink)

There are at least three deep and yet creative tensions in John Locke's writings on the knowledge of the natural world. An exposition of these tensions provides the framework for this chapter. The chapter provides an account of the development of Locke's views from his early medical essays of the late 1660s to his last published writings on natural philosophy. The central locus for Locke's "philosophy of science" will be the Essay. Chymical medicine provided the main field in which Locke (...) was regularly involved in experimentation. Locke believed that geometry and arithmetic presented paradigm examples of the sort of demonstrative reasoning that would be involved in a science of nature, that is, a demonstrative natural philosophy. An important consequence of Locke's ideal of a corpuscular metric is an ongoing concern with metrology. By the early 1690s, Locke modified his views on natural philosophy in the light of the Newtonian achievement. (shrink)

This paper attempts to shed light on three sets of issues that bear directly on our understanding of Locke and Kant. The first is whether Kant believes Locke merely anticipates his distinction between analytic and synthetic judgments or also believes Locke anticipates his notion of synthetic a priori cognition. The second is what should we as readers of Kant and Locke should think about Kant’s view whatever it turns out to be, and the third is the nature of Kant’s justification (...) for the comparison he draws between his philosophy and Locke’s. I argue (1) that Kant believes Locke anticipates both the analytic-synthetic distinction and Kant’s notion of synthetic a priori cognition, (2) that the best justification for Kant’s claim draws on Locke’s distinction between trifling and instructive knowledge, (3) that the arguments against this claim developed by Carson, Allison, and Newman fail to undermine it, and (4) that Kant’s own justification for his claim is quite different from what many commentators have thought it was (or should have been). (shrink)

Qu{u2019}est-ce que l{u2019}expérience? Que sont les expériences dont peuvent s{u2019}occuper les mathématiques? Quelles sont les caractéristiques des mathématiques qui se soucient de l{u2019}expérience ou des expériences? Comment les discours mathématiques se nouent-ils avec les expériences qu{u2019}ils symbolisent, qu{u2019}ils prétendent parfois refléter ou seulement déterminer sans aucun souci de vérité {OCLCbr#BB}?. Les dix-sept chapitres de cet ouvrage abordent ces questions à partir des discours qu{u2019}ont tenu philosophes et mathématiciens depuis les Sophistes jusqu{u2019}à Thom en passant par Galilée, Hobbes, Locke, Diderot, d{u2019}Alembert, (...) Hume, Gergonne, Riemann, Houel, Nietzsche, Borel, Poincaré, Einstein, Lautman, Gonseth. Cet ouvrage s{u2019}adresse à un public de mathématiciens, de physiciens et de philosophes, d{u2019}étudiants, d{u2019}enseignants et de chercheurs, de personnes passionnées par les mathématiques, la philosophie et leurs histoires. (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 examines the views of seventeenth-century British philosophers on the notion of logic and demonstrative knowledge, particularly Francis Bacon, Thomas Hobbes, and John Locke, offering an overview of traditional Aristotelianism in relation to logic and describing Bacon's approach to demonstration and logic. It also analyzes the contribution of the Cambridge Platonists and evaluates the influence of Cartesianism. The chapter concludes that theorizing about logic and demonstrative knowledge followed an arc familiar from other branches of philosophy such as metaphysics or (...) the philosophy of science. (shrink)

In his last book about Locke’s philosophy, E. J. Lowe claims that Frege’s arguments against the Lockean conception of number are not compelling, while at the same time he painstakingly defines the Lockean conception Lowe himself espouses. The aim of this paper is to show that the textual evidence considered by Lowe may be interpreted in another direction. This alternative reading of Frege’s arguments throws light on Frege’s and Lowe’s different agendas. Moreover, in this paper, the problem of singular sentences (...) of number is presented, and Frege’s and Lowe’s views are confronted with it. (shrink)

In a crucial passage of the second-edition Transcendental Deduction, Kant claims that the concept of motion is central to our understanding of change and temporal order. I show that this seemingly idle claim is really integral to the Deduction, understood as a replacement for Locke’s “physiological” epistemology (cf. A86-7/B119). Béatrice Longuenesse has shown that Kant’s notion of distinctively inner receptivity derives from Locke. To explain the a priori application of concepts such as succession to this mode of sensibility, Kant construes (...) the mind as receptive to its own activity. As Longuenesse understands Kant’s response to Locke, “motion” becomes little more than a metaphor for the action of understanding on inner sense. For Michael Friedman, in contrast, this passage evidences Kant’s deep concern with the foundations of Newtonian science. He reads it as a reference to inertial motion, the standard by which temporal intervals are measured. I show that Longuenesse’s and Friedman’s interpretations are in fact complementary. Both Locke and Kant are deeply concerned with the quantification of time. So Longuenesse is right that §24 is meant to supplant Locke’s account, and Friedman correctly takes it as the foundation of Kant’s account of the application of quantitative concepts to time. Kant aims, specifically, to explain cognition of time as continuous magnitude. However, Longuenesse leaves Kant without an answer to Locke’s challenge that continuous magnitude cannot be understood on the basis of discrete magnitude. And on Friedman’s view Kant’s account of the understanding’s activity presupposes, and thus cannot explain, the achievements of the mathematical sciences (such as the representation of temporal continuity). On my reading, Kant’s key contention is that we must regard the segregation of temporally ordered representation into units as the work of the understanding, rather than (as Locke views it) of sensibility. By giving the understanding this role, Kant succeeds where he takes Locke to fail. I show how the power to unify temporal representation can be ascribed to the understanding on the basis, not of assumed mathematical or scientific knowledge, but of its characterization as the power of judgment. (shrink)

In Warranted Christian Beliet Alvin Plantinga claims that “The Enlightenment looked askance at testimony and tradition; Locke saw them as a preeminent source of error.” Locke, Plantinga suggests, is the “fountainhead” of this stance. This is importantly wrong about Locke and Locke”s views, and an examination of the views of Locke’s much admired friend and slightly older contemporary, Robert Boyle, reveals that the claim is mistaken about him as well, reinforcing the view that Plantinga is in general mistaken about the (...) intellectual milieu in which Locke wrote. In this paper I consider the views of Locke and Boyle on demonstration, observation, experiment, and testimony with a view to showing what, in the case of science and religion, their views actually were. For Locke I draw mainly on the Essay, while for Boyle I draw heavily on the MSS in the Royal Society Library, as well as on the printed works. (shrink)

Historians of modern philosophy have been paying increasing attention to contemporaneous scientific developments. Isaac Newton's Principia is of course crucial to any discussion of the influence of scientific advances on the philosophical currents of the modern period, and two philosophers who have been linked especially closely to Newton are John Locke and Immanuel Kant. My dissertation aims to shed new light on the ties each shared with Newtonian science by treating Newton, Locke, and Kant simultaneously. I adopt Newton's philosophy of (...) geometry as the starting point of investigation, for here I believe we have a constructive means by which to assess Locke and Kant's relationship to Newton, In particular, I defend the thesis that the justification Newton, Locke and Kant offer for applying geometrical principles to nature is central to understanding their respective ties to a Newtonian science characterized by the intermingling of mathematics and experiment, Although little is said by Locke in regard to a mathematical approach to nature, I hope to show that his interpretation of the origins of our geometrical ideas has a close affinity to Newton's own characterization of geometry, leading us to reexamine the extent of Locke's 'Newtonianism.' Kant famously attempts to bridge the gap between geometry and the empirical world by establishing space as a "pure form of intuition." My discussion of Kant's Newtonian approach to nature centers on the imagination, and I argue that the mediating work completed by this faculty in geometrical construction and experience in general is equally important to understanding Kant's application of geometry to the empirical realm, In the end, I hope my treatment of the strategies employed by Newton, Locke, and Kant to account for a mathematical-experimental method of natural philosophy sheds further light on the importance of Newton to the progress of modern philosophy. (shrink)

Locke notoriously included number amongst the primary qualities of bodies and was roundly criticized for doing so by Berkeley. Frege echoed some of Berkeley's criticisms in attacking the idea that ‘Number is a property of external things’, while defending his own view that number is a property of concepts. In the present paper, Locke's view is defended against the objections of Berkeley and Frege, and Frege's alternative view of number is criticized. More precisely, it is argued that numbers are assignable (...) to pluralities of individuals. However, it is also argued that Locke went too far in asserting that ‘Number applies itself to ... everything that either doth exist, or can be imagined’. (shrink)

One of the most frequently discussed passages from Locke's An Essay Concerning the Human Understanding is that which occurs in IV.vii.9, where he writes:… the Ideas first in the Mind, ‘tis evident, are those of particular Things, from whence, by slow degrees, the Understanding proceeds to some few general ones; which being taken from the ordinary and familiar Objects of Sense, are settled in the Mind, with general Names to them. Thus particular Ideas are first received and distinguished, and so (...) Knowledge got about them: and next to them, the less general, or specifick, which are next to particular. For abstract Ideas are not so obvious or easie to Children, or the yet unexercised Mind, as particular ones. If they seem so to grown Men, ’tis only because by constant and familiar use they are made so. For when we nicely reflect upon them, we shall find, that general Ideas are Fictions and Contrivances of the Mind, that carry difficulty with them, and do not so easily offer themselves, as we are apt to imagine. For example, Does it not require some pains and skill to form the general Idea of a Triangle, for it must be neither Oblique, nor Rectangle, neither Equilateral, Equicrural, nor Scalenon; but all and none of these at once. In effect, it is something imperfect, that cannot exist; an Idea wherein some parts of several different and inconsistent Ideas are put together.I shall not pretend that what Locke is claiming in this passage is wholly clear, nor shall I try to defend the views expressed in it. My intention is to show that what he says here has been widely misinterpreted. It has become common to treat the contents of the passage as aberrant; as presenting a ridiculous variant on his main thesis on abstraction, or that very thesis elaborated with a foolish rhetorical flourish. But what is said in this passage is not something said elsewhere, nor is it an alternative to something said elsewhere. Here, and nowhere else, Locke shows awareness of implications for his doctrine of abstraction, of views he holds about simple ideas. (shrink)

Dept. of Philosophy. Paper copy at Leddy Library: Theses & Major Papers - Basement, West Bldg. / Call Number: Thesis1987 .F355. Source: Masterss International, Volume: 40-07, page: . Thesis --University of Windsor , 1987.

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)

In his Essay concerning Human Understanding, John Locke explicitly refers to Newton’s Philosophiae naturalis principia mathematica in laudatory but restrained terms: “Mr. Newton, in his never enough to be admired Book, has demonstrated several Propositions, which are so many new Truths, before unknown to the World, and are farther Advances in Mathematical Knowledge” (Essay, 4.7.3). The mathematica of the Principia are thus acknowledged. But what of philosophia naturalis? Locke maintains that natural philosophy, conceived as natural science (as opposed to natural (...) history), would give us demonstrations of the necessary connection between the (ultimately, simple) ideas constitutive of our complex ideas of various natural kinds of substances (e.g., gold). Indeed Locke goes so far as to suggest that a completely adequate natural science would also realize (perhaps, per impossibile) the goal of transforming the corpuscularian hypothesis into knowledge by demonstrating the necessary connection between the ‘microstructure’ (primary qualities of insensible corpuscles) of a particular natural kind of substance (e.g., gold) and the ideas of secondary qualities constitutive of the complex idea of that kind of substance. Locke’s conclusion concerning the possibility of the development of a natural science thus conceived is pessimistic: In vain therefore shall we endeavor to discover by our Ideas, (the only true way of certain and universal knowledge,) what other Ideas are to be found constantly joined with that of our complex Idea of any Substance: since we neither know the real Constitution of the minute Parts, on which their Qualities do depend; not, did we know them could we discover any necessary connexion between them, and any of their Secondary Qualities: which is necessary to be done, before we can certainly know their necessary co-existence (Essay, 4.3.14). It is understandable that, with such a conception of the science of nature, Locke found little of it in Newton’s Principia. In this paper, I further explore what might, perhaps with some hyperbole, be termed Locke’s ‘disappointment’ with the Prinicipia as a contribution to natural science. In particular, I argue that Locke’s adherence to the idealist epistemology of the Way of Ideas entails that mathematics cannot lend its certainty as a scientia to natural philosophy. Consequently, he finds more mathematics than natural philosophy in the Principia. (shrink)