Berkeley in his Introduction to the Principles of Human knowledge uses geometrical examples to illustrate a way of generating “universal ideas,” which allegedly account for the existence of general terms. In doing proofs we might, for example, selectively attend to the triangular shape of a diagram. Presumably what we prove using just that property applies to all triangles.I contend, rather, that given Berkeley’s view of extension, no Euclidean triangles exist to attend to. Rather proof, as Berkeley would normally assume, requires (...) idealizing diagrams; treating them as if they obeyed Euclidean constraints. This convention solves the problem of representative generalization. View HTML Send article to KindleTo send this article to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle. Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Find out more about the Kindle Personal Document Service.Berkeley and Proof in GeometryVolume 51, Issue 3RICHARD J. BROOK DOI: https://doi.org/10.1017/S0012217312000686Your Kindle email address Please provide your Kindle [email protected]@kindle.com Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Dropbox To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox. Berkeley and Proof in GeometryVolume 51, Issue 3RICHARD J. BROOK DOI: https://doi.org/10.1017/S0012217312000686Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Google Drive To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive. Berkeley and Proof in GeometryVolume 51, Issue 3RICHARD J. BROOK DOI: https://doi.org/10.1017/S0012217312000686Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Export citation Request permission. (shrink)

Physical boundaries and the earliest topologists. Topology has a relatively short history; but its 19th century roots are embedded in philosophical problems about the nature of extended substances and their boundaries which go back to Zeno and Aristotle. Although it seems that there have always been philosophers interested in these matters, questions about the boundaries of three-dimensional objects were closest to center stage during the later medieval and modern periods. Are the boundaries of an object actually existing, less-than-three-dimensional parts of (...) the object—that is, are solids bounded by two-dimensional surfaces, surfaces by one-dimensional “edges” or “physical lines”, edges by dimensionless “simples”? If not, how does a perfectly spherical object manage to touch a perfectly flat object—what part of the sphere is in immediate contact with the plane, if the sphere has no unextended parts? But if such parts be admitted, are we not then saddled with “actual infinities” of simples, lines, and surfaces spread throughout each continuous object—the boundaries of all the object’s internal parts? Does it help to say that these internal boundaries exist only “potentially”? (shrink)

Throughout history, almost all mathematicians, physicists and philosophers have been of the opinion that space and time are infinitely divisible. That is, it is usually believed that space and time do not consist of atoms, but that any piece of space and time of non-zero size, however small, can itself be divided into still smaller parts. This assumption is included in geometry, as in Euclid, and also in the Euclidean and non- Euclidean geometries used in modern physics. Of the few (...) who have denied that space and time are infinitely divisible, the most notable are the ancient atomists, and Berkeley and Hume. All of these assert not only that space and time might be atomic, but that they must be. Infinite divisibility is, they say, impossible on purely conceptual grounds. (shrink)

My topic in this paper will be Hume’s claim that we have no idea of a vacuum. I offer a novel interpretation of Hume’s account of our ideas of extension that makes it clear why those ideas cannot include any ideas of vacuums, and I distinguish my interpretation from prominent readings offered by other Hume scholars. An upshot of Hume’s account, I will argue, is his commitment to a remarkable and distinctly Humean view I call “perceptual relationism.” Perceptual relationism is (...) a fundamental characteristic of Hume’s “universe of the imagination,” and a manifestation of just how “loose and separate” the constituents of that inner universe are. Once we understand perceptual relationism and its entailments, we are in a better position to understand the rest of Hume’s sometimes puzzling remarks on space and the vacuum. (shrink)

This work analyzes two perspectives, Atomism and Infinite Divisibility, in the light of modern mathematical knowledge and recent developments in computer graphics. A developmental perspective is taken which relates ideas leading to atomism and infinite divisibility. A detailed analysis of and a new resolution for Zeno's paradoxes are presented. Aristotle's arguments are analyzed. The arguments of some other philosophers are also presented and discussed. All arguments purporting to prove one position over the other are shown to be faulty, mostly by (...) question begging. Included is a sketch of the consistency of infinite divisibility and a development of the atomic perspective modeled on computer graphics screen displays. The Pythagorean theorem is shown to depend upon the assumption of infinite divisibility. The work concludes that Atomism and infinite divisibility are independantly consistent, though mutually incompatible, not unlike the wave/particle distinction in physics. (shrink)

Eric Watkins has argued on philosophical, textual, and historical grounds that Kant’s account of causation in the first Critique should not be read as an attempt to refute Hume’s account of causation. In this paper, I challenge the arguments for Watkins’ claim. Specifically, I argue (1) that Kant’s philosophical commitments, even on Watkins’ reading, are not obvious obstacles to refuting Hume, (2) that textual evidence from the “Disciple of Pure Reason” suggests Kant conceived of his account of causation as such (...) a refutation, and (3) that none of Hume’s early German critics provided responses to this account that would have satisfied Kant. Watkins’ reading of Kant’s account of causation is thus more compatible with traditional views about Kant’s relationship to Hume than Watkins believes. (shrink)

If Tahiti suggested to theorists comfortably at home in Europe thoughts of noble savages without clothes, those who paid for and went on voyages there were in pursuit of a quite opposite human ideal. Cook's voyage to observe the transit of Venus in 1769 symbolises the eighteenth century's commitment to numbers and accuracy, and its willingness to spend a lot of public money on acquiring them. The state supported the organisation of quantitative researches, employing surveyors and collecting statistics to..

I provide an interpretation of Hume’s argument in Treatise 1.2 Of the Ideas of Space and Time that finite extensions are only finitely divisible (hereafter Hume’s Finite Divisibility Argument). My most general claim is that Hume intends his Finite Divisibility Argument to be a demonstration in the Early Modern sense as involving the comparison and linking of ideas based upon their intrinsic contents. It is a demonstration of relations among ideas, meant to reveal the meaningfulness or absurdity of a given (...) supposition, and to distinguish possible states of affairs from impossible ones. It is not an argument ending in an inference to an actual matter of fact. Taking the demonstrative nature of his Finite Divisibility Argument fully into account radically alters the way we understand it. Supported by Hume’s own account of demonstration, and reinforced by relevant Early Modern texts, I follow to its logical consequences, the simple premise that the Finite Divisibility Argument is intended to be a demonstration. Clear, abstract ideas in Early Modern demonstrations represent possible objects. By contrast, suppositions that are demonstrated to be contradictory have no clear ideas annexed to them and therefore cannot represent possible objects—their ‘objects,’ instead, are “impossible and contradictory.” Employing his Conceivability Principle, Hume argues that there is a clear idea of a finite extension containing a finite number of parts and therefore, finitely divisible extensions are possible. In contrast, the supposition of an infinitely divisible finite extension is “absurd” and “contradictory” and stands for no clear idea. Consequently, Hume deems this supposition “impossible and contradictory,” that is, without meaning and therefore, descriptive of no possible object. This interpretation allays concerns found in the recent literature and helps us better understand what drives Hume’s otherwise perplexing argument in the often neglected or belittled T 1.2. (shrink)

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)

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)

Although Hume's analysis of geometry continues to serve as a reference point for many contemporary discussions in the philosophy of science, the fact that the first Enquiry presents a radical revision of Hume's conception of geometry in the Treatise has never been explained. The present essay closely examines Hume's early and late discussions of geometry and proposes a reconstruction of the reasons behind the change in his views on the subject. Hume's early conception of geometry as an inexact non-demonstrative science (...) is argued to be a consequence of his attempt to discredit geometrical proofs of infinite divisibility of extension by anchoring the meaning of geometrical concepts in inherently inexact qualitative measurement procedures. This measurement-based attack on the exactness and certainty of geometry is analyzed and shown to be both self-refuting and inconsistent with the general epistemological framework of the Treatise. The revised conception of geometry as a demonstrative science in the first Enquiry is then interpreted as Hume's response to the failure of his earlier attempt to discredit geometrical proofs of infinite divisibility of extension. (shrink)

This paper presents a taxonomy of divisibilism, a philosophical perspective advocating for the infinite divisibility of continua. The taxonomy is founded on various conceptualizations of indivisibles, enabling the identification of two types of divisibilism: ‘moderate’ and ‘strong’. The former denies indivisibles as constituent parts of magnitudes, whereas the latter rejects indivisibles as even intrinsic elements (such as limits or junctions) of magnitudes. The paper proceeds to demonstrate how Gregory of Rimini falls into the second category, utilizing geometry and non-entitism as (...) primary tools to dismiss the very possibility of the existence of indivisibles. Within the framework of this ‘strong divisibilism’, the paper also examines Gregory’s interpretation of mathematical items (e.g. points, lines, surfaces) against the backdrop of one of his main sources, that is, Augustine. Lastly, the paper delves into Gregory’s interpretation of the fundamental Aristotelian divisibilist claim, namely, that a continuum is infinitely divisible. It highlights how Gregory’s stance ultimately conflicts with Aristotelian premises, particularly concerning his assertion that the division of all continua is made into infinitely many actual parts. (shrink)

Hume’s account of the origin and nature of our ideas of space and time is generally thought to be the least satisfactory part of his empiricist system of philosophy. The main reason is internal in that the account is judged to be inconsistent with Hume’s fundamental principle for the relationship between senses and cognition, the copy principle. This paper defends Hume against the inconsistency objection by offering a new systematic interpretation of Hume on space and time and illuminating more generally (...) the role of the copy principle in his philosophy. Humes Theorie des Wesens und des Ursprungs unserer Vorstellungen von Raum und Zeit wird generell zu den am wenigsten befriedigenden Teilen seiner empiristischen Philosophie gezählt. Der Hauptgrund dafür ist werkimmanent: Die Raum- Zeit-Theorie einerseits und Humes „copy principle“ andererseits – d.h. dasjenige Fundamental-Prinzip, das die Relation zwischen unseren Sinnen und unserem Denken regelt – werden als miteinander inkonsistent erachtet. Dieser Beitrag bietet eine neue, systematische Interpretation der Raum-Zeit-Lehre Humes und eine umfassendere Darstellung der Rolle des „copy principles“ in seiner Philosophie an. Auf diese Weise wird Hume gegen den Vorwurf der Inkonsistenz verteidigt. (shrink)