Online research in philosophy

Hume's discussion of the idea of space in his Treatise on Human Nature is fundamental to an understanding of his treatment of such central issues as the existence of external objects, the unity of the self, the relation between certainty and belief, and abstract ideas. Marina Frasca-Spada's rich and original study examines this difficult part of Hume's philosophical writings and connects it to eighteenth-century works in natural philosophy, mathematics and literature. Focusing on Hume's discussions of the infinite divisibility of extension, the origin of the idea of space, geometry, and the notion of a vacuum, she shows that the central questions of Hume's 'science of human nature' - what does the 'science of human nature' reveal about the mind and its operations? what is experience? - underlie all of these discussions. Her analysis points the way to a reassessment of the central current interpretative problems in Hume studies.

The Protean Character of Mathematics SAUNDERS MAC LANE (Chicago) 1. Introduction The thesis of this paper is that mathematics is protean. ...

The author focuses on the tension "realism - idealism" in the philosophy of mathematics, but he does that from the perspective of a theoretical physicist. It is not only that one's standpoint in the philosophy of mathematics determines our understanding of the effectiveness of mathematics in physics, but also the fact that mathematics is so effective in physical sciences tells us something about the nature of mathematics.

'' FROM THE INTRODUCTION Among the aims of this book are: - The discussion of some important philosophical issues using the precision of mathematics.

We can modify Hume’s Principle in the same manner that George Boolos suggested for modifying Frege’s Basic Law V. This leads to the principle Small Hume. Then, we can show that Small Hume is interderivable with Hume’s Principle.

This paper contains answers to the following Five questions, posed by the editors are answered: (1) Why were you initially drawn to the foundations of mathematics and/or the philosophy of mathematics? (2) What example(s) from your work (or the work of others) illustrates the use of mathematics for philosophy? (3) What is the proper role of philosophy of mathematics in relation to logic, foundations of mathematics, the traditional core areas of mathematics, and science? (4) What do you consider the most neglected topics and/or contributions in late 20th century philosophy of mathematics? (5) What are the most important open problems in the philosophy of mathematics and what are the prospects for progress?

Several scholars have argued that Wittgenstein held the view that the notion of number is presupposed by the notion of one-one correlation, and that therefore Hume's principle is not a sound basis for a definition of number. I offer a new interpretation of the relevant fragments on philosophy of mathematics from Wittgenstein's Nachlass, showing that if different uses of ‘presupposition’ are understood in terms of de re and de dicto knowledge, Wittgenstein's argument against the Frege-Russell definition of number turns out to be valid on its own terms, even though it depends on two epistemological principles logicist philosophers of mathematics may find too ‘constructivist’.

Most contemporary philosophy of mathematics focuses on a small segment of mathematics, mainly the natural numbers and foundational disciplines like set theory. While there are good reasons for this approach, in this paper I will examine the philosophical problems associated with the area of mathematics known as applied mathematics. Here mathematicians pursue mathematical theories that are closely connected to the use of mathematics in the sciences and engineering. This area of mathematics seems to proceed using different methods and standards when compared to much of mathematics. I argue that applied mathematics can contribute to the philosophy of mathematics and our understanding of mathematics as a whole.

Hume describes the sciences as "noble entertainments" that are "proper food and nourishment" for reasonable beings (EHU 1.5-6; SBN 8).1 But mathematics, in particular, is more than noble entertainment; for millennia, agriculture, building, commerce, and other sciences have depended upon applying mathematics.2 In simpler cases, applied mathematics consists in inferring one matter of fact from another, say, the area of a floor from its length and width. In more sophisticated cases, applied mathematics consists in giving scientific theory a mathematical form and then explaining and predicting matters of fact by means of mathematics and the theory. Since Hume holds that, "All inferences from experience are . . . ..