What is perception doing in mathematical reasoning? To address this question, I discuss the role of perception in geometric reasoning. Perception of the shape properties of concrete diagrams provides, I argue, a surrogate consciousness of the shape properties of the abstract geometric objects depicted in the diagrams. Some of what perception is not doing in mathematical reasoning is also discussed. I take issue with both Parsons and Maddy. Parsons claims that we perceive a certain type of abstract object. Maddy claims (...) (at least at one time claimed) that perception provides the basis for intuition of mathematical sets. 1 Mathematical reasoning with diagrams 2 Do we perceive abstract objects? 3 Do we perceive mathematical sets? (shrink)

Can visual thinking be a means of discovery in elementary analysis, as well as a means of illustration and a stimulus to discovery? The answer to the corresponding question for geometry and arithmetic seems to be ‘yes’ (Giaquinto [1992], [1993]), and so a positive answer might be expected for elementary analysis too. But I argue here that only in a severely restricted range of cases can visual thinking be a means of discovery in analysis. Examination of persuasive visual routes to (...) two simple theorems (Rolle, Bolzano) shows that they are not ways of discovering the theorems; the type of visual thinking involved can never be used to discover analytic theorems of a certain generality. The hypothesis that visual thinking is never a means of discovering the existence or nature of the limit of some infinite process is considered, and a likely counter-example is set out. It is still possible that restricted theorems can be discovered visually: an example from Littlewood is examined in detail and not found wanting. Even when visual thinking is not a means of discovery it can provide the idea for a proof in a direct way; an example is presented (Intermediate Value Theorem). In conclusion: it may be possible to discover theorems in elementary real analysis by visual means, but only theorems of a restricted kind; however, visual thinking in analysis can be very useful in other ways. (shrink)

About G.M.A Grube's translations of Plato: "Unmistakably superior: more lucid, more accurate, more readable. Above all, they’re lucidly adorned, unpretentious, and in translating Plato that counts a good deal. The prose is, as English prose, persuasive, cogent, and as eloquent as it can be without departing from the text. --William Arrowsmith.

Bazzocchi disposes the text of the Tractatus in a user-friendly manner, exactly as Wittgenstein's decimals advise. This discloses the logical form of the book by distinct reading units, linked into a fashioned hierarchical tree. The text becomes much clearer and every reader can enjoy, finally, its formal and literary qualities.

_Philosophy of Mathematics_ is an excellent introductory text. This student friendly book discusses the great philosophers and the importance of mathematics to their thought. It includes the following topics: * the mathematical image * platonism * picture-proofs * applied mathematics * Hilbert and Godel * knots and nations * definitions * picture-proofs and Wittgenstein * computation, proof and conjecture. The book is ideal for courses on philosophy of mathematics and logic.

Everyone appreciates a clever mathematical picture, but the prevailing attitude is one of scepticism: diagrams, illustrations, and pictures prove nothing; they are psychologically important and heuristically useful, but only a traditional verbal/symbolic proof provides genuine evidence for a purported theorem. Like some other recent writers (Barwise and Etchemendy [1991]; Shin [1994]; and Giaquinto [1994]) I take a different view and argue, from historical considerations and some striking examples, for a positive evidential role for pictures in mathematics.

Plato's Meno and Phaedo are two of the most important works of ancient western philosophy and continue to be studied around the world. The Meno is a seminal work of epistemology. The Phaedo is a key source for Platonic metaphysics and for Plato's conception of the human soul. Together they illustrate the birth of Platonic philosophy from Plato's reflections on Socrates' life and doctrines. This edition offers new and accessible translations of both works, together with a thorough introduction that explains (...) the arguments of the two dialogues and their place in Plato's thought. (shrink)

Bazzocchi disposes the text of the Tractatus in a user-friendly manner, exactly as Wittgenstein's decimals advise. This discloses the logical form of the book by distinct reading units, linked into a fashioned hierarchical tree. The text becomes much clearer and every reader can enjoy, finally, its formal and literary qualities.

This dissertation explores the reasons why structured graphics have been largely ignored in the representation and reasoning components of contemporary theories of axiomatic systems. In particular, it demonstrates that for the case of modern logic and geometry, there are systematic forces in the intellectual history of these disciplines which have driven the adoption of sentential representational styles over diagrammatic ones. These forces include: the changing views of the role of intuition in the procedures and formalisms of formal proof; the historical (...) contrast between the universalist subject matter of logic and the specific subject matter of geometry; and the ways in which both logic and geometry were affected by the changes that swept through mathematics during the nineteenth century. This dissertation traces the effects of each of these influences in the evolution of logic and geometry from antiquity to the early twentieth-century work of David Hilbert. For each discipline, it examines the historical factors impacting the creation and development of important diagrammatic systems, the reasons for the adoption or abandonment of these systems, and the philosophical context and debates surrounding the use of these systems. It concludes by arguing that the history of diagrammatic representations in logic and geometry cannot be adequately understood without essential reference to their philosophical background. (shrink)

Wittgenstein's work remains, undeniably, now, that off one of those few philosophers who will be read by all future generations.