Philosophers of science increasingly recognize the importance of idealization: the intentional introduction of distortion into scientific theories. Yet this recognition has not yielded consensus about the nature of idealization. e literature of the past thirty years contains disparate characterizations and justifications, but little evidence of convergence towards a common position.

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)

Philosophers no longer argue whether Hume ever read Berkeley, yet some remain puzzled as to why so little of Berkeley appears in Hume's works. Professor Popkin has remarked that even “where Hume and Berkeley come closest to discussing the same subject or holding the same view, Hume neither uses Berkeley's terms nor refers to him.” An apparent exception to this generalization is Berkeley's doctrine ofminima sensibilia, for both philosophers use this term to denote indivisible sensible points, and both invoke such (...) points in order to show that sensible extension is not infinitely divisible. Yet it has been suggested that, although Berkeley and Hume employ the same terms, they nevertheless maintain different doctrines: Hume'sminimaare unextended, whereas Berkeley's are extended. (shrink)

Since both berkeley and hume are committed to the view that a line is composed of finitely many fundamental parts, They must find responses to the standard geometrical proofs of infinite divisibility. They both repeat traditional arguments intended to show that infinite divisibility leads to absurdities, E.G., That all lines would be infinite in length, That all lines would have the same length, Etc. In each case, Their arguments rest upon a misunderstanding of the concept of a limit, And thus (...) are not successful. Berkeley, However, Adds a further ingenious argument to the effect that the standard geometrical proofs of infinite divisibility misread the unlimited representational capacity of geometrical diagrams as a substantive feature of the objects that these diagrams represent. The article concludes that berkeley is right on this matter, And that the traditional proofs of infinite divisibility do not show what they are intended to show. (shrink)

I. Introduction Siris, Berkeley's last major work, is undeniably a rather odd book. It could hardly be otherwise, given Berkeley's aims in writing it, which are three-fold: 'to communicate to the public the salutary virtues of tar-water,'1 to provide scientific background supporting the efficacy of tar-water as a medicine, and to lead the mind of the reader, via gradual steps, toward contemplation of God.2 The latter two aims shape Berkeley's extensive use of contemporary natural science in Siris. In particular, Berkeley's (...) focus on what he calls fire (or aether or light) as a quasiuniversal 'cause' of natural change3 serves these purposes, for the 'activity' of the aether, in his view, can both explain the miraculous virtues of a certain medicine, i.e. tar-water, and reveal God's action and his divine order.4 Berkeley's corpuscular speculations, including his use of fire-theory, are not especially idiosyncratic as natural philosophy. In his theorizing, as Jessop and other have noted, he is heavily indebted to the work of Hermann Boerhaave, the Dutch chemist, botanist, and physician whose teachings were highly influential in mid-eighteenth century Britain.5 Boerhaave, along with other Dutch natural philosophers cited by Berkeley, assigned a central role in accounting for physio-chemical activity to fire, a subtle, insensible particulate substance, sometimes identified with light. (shrink)

A comprehensive historical overview of formalist ideas in the philosophy of mathematics.

It has been argued that Hume's denial of infinite divisibility entails the falsity of most of the familiar theorems of Euclidean geometry, including the Pythagorean theorem and the bisection theorem. I argue that Hume's thesis that there are indivisibles is not incompatible with the Pythagorean theorem and other central theorems of Euclidean geometry, but only with those theorems that deal with matters of minuteness. The key to understanding Hume's view of geometry is the distinction he draws between a precise and (...) an imprecise standard of equality in extension. Hume's project is different from the attempt made by Berkeley in some of his later writings to save Euclidean geometry. Unlike Berkeley, who interprets the theorems of Euclidean geometry as false albeit useful approximations of geometrical facts, Hume is able to save most of the central theorems as true. (shrink)