Michael Potter presents a comprehensive new philosophical introduction to set theory. Anyone wishing to work on the logical foundations of mathematics must understand set theory, which lies at its heart. Potter offers a thorough account of cardinal and ordinal arithmetic, and the various axiom candidates. He discusses in detail the project of set-theoretic reduction, which aims to interpret the rest of mathematics in terms of set theory. The key question here is how to deal with the paradoxes that bedevil set (...) theory. Potter offers a strikingly simple version of the most widely accepted response to the paradoxes, which classifies sets by means of a hierarchy of levels. What makes the book unique is that it interweaves a careful presentation of the technical material with a penetrating philosophical critique. Potter does not merely expound the theory dogmatically but at every stage discusses in detail the reasons that can be offered for believing it to be true. Set Theory and its Philosophy is a key text for philosophy, mathematical logic, and computer science. (shrink)
This is a critical examination of the astonishing progress made in the philosophical study of the properties of the natural numbers from the 1880s to the 1930s. Reassessing the brilliant innovations of Frege, Russell, Wittgenstein, and others, which transformed philosophy as well as our understanding of mathematics, Michael Potter places arithmetic at the interface between experience, language, thought, and the world.
What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions.
Gottlob Freg was unquestionably one of the most important philosophers of all time. He trained as a mathematician, and his work in philosophy started as an attempt to provide an explanation of the truths of arithmetic, but in the course of this attempt he not only founded modern logic but also had to address fundamental questions in the philosophy of languageand philosophical logic. He is generally seen as one of the fathers of the analytic method, which dominated philosophy in English-speaking (...) countries for most of the twentieth century. His work is studied today not just for its historical importance, but also because many of his ideas are relevant to current debates in the philosophies of logic, language, mathematics and the mind. The Cambridge Companion to Frege provides a route into this lively area of research. (shrink)
Crispin Wright and Bob Hale have defended the strategy of defining the natural numbers contextually against the objection which led Frege himself to reject it, namely the so-called ‘Julius Caesar problem’. To do this they have formulated principles (called sortal inclusion principles) designed to ensure that numbers are distinct from any objects, such as persons, a proper grasp of which could not be afforded by the contextual definition. We discuss whether either Hale or Wright has provided independent motivation for a (...) defensible version of the sortal inclusion principle and whether they have succeeded in showing that numbers are just what the contextual definition says they are. (shrink)
del's appeal to mathematical intuition to ground our grasp of the axioms of set theory, is notorious. I extract from his writings an account of this form of intuition which distinguishes it from the metaphorical platonism of which Gödel is sometimes accused and brings out the similarities between Gödel's views and Dummett's.
Suggests that the recent emphasis on Benacerraf's access problem locates the peculiarity of mathematical knowledge in the wrong place. Instead we should focus on the sense in which mathematical concepts are or might be "armchair concepts" – concepts about which non-trivial knowledge is obtainable a priori.
[Michael Potter] If arithmetic is not analytic in Kant's sense, what is its subject matter? Answers to this question can be classified into four sorts according as they posit logic, experience, thought or the world as the source, but in each case we need to appeal to some further process if we are to generate a structure rich enough to represent arithmetic as standardly practised. I speculate that this further process is our reflection on the subject matter already obtained. This (...) suggestion seems problematic, however, since it seems to rest on a confusion between the empirical and the metaphysical self. /// [Bob Hale] Michael Potter considers several versions of the view that the truths of arithmetic are analytic and finds difficulties with all of them. There is, I think, no gainsaying his claim that arithmetic cannot be analytic in Kant's sense. However, his pessimistic assessment of the view that what is now widely called Hume's principle can serve as an analytic foundation for arithmetic seems to me unjustified. I consider and offer some answers to the objections he brings against it. (shrink)
One of Michael Dummett's most striking contributions to the philosophy of mathematics is an argument to show that the correct logic to apply in mathematical reasoning is not classical but intuitionistic. In this article I wish to cast doubt on Dummett's conclusion by outlining an alternative, motivated by consideration of a well-known result of Kurt Gödel, to the standard view of the relationship between classical and intuitionistic arithmetic. I shall suggest that it is hard to find a perspective from which (...) to arbitrate between the competing views. (shrink)
The use of digital presentation tools such as PowerPoint is ubiquitous; however we still do not know much about the persuasiveness of these programs. Examining the use of visual analogy and visual chronology, in particular, this paper explores the use of visual argumentation in a Keynote presentation by Al Gore. It illustrates how images function as an integrated part of Gores reasoning.
In this book, Michael Potter offers a fresh and compelling portrait of the birth and first several decades of analytic philosophy, one of the most important periods in philosophy’s long history. He focuses on the period between the publication of Gottlob Frege’s _Begriffsschrift _in 1879 and Frank Ramsey’s death in 1930. Potter--one of the most influential writers on late 19 th and early 20 th century philosophy--presents a deep but accessible account of the break with Absolute Idealism and Neo-Kantianism, specifically, (...) and more generally with many of the metaphysical preoccupations of philosophy’s preceding history. Potter’s focus is on philosophical logic and philosophy of mathematics, but he also relies heavily on important issues in metaphysics and meta-ethics to complete his story. The book provides an essential starting point for any student or philosopher attempting to understand Frege, Russell, Wittgenstein, and Ramsey as well as their interactions and their intellectual milieux. It will also be of interest to a great many philosophers today who want to illuminate the problems they work on by better knowing their origins. KEY FEATURES: 1. Discusses the interconnections of Frege, Russell and Wittgenstein—founding thinkers in the history of analytic philosophy—and also brings the neglected Frank Ramsey into this conversation, providing a unique focus and depth to an introductory text 2. Increases the general awareness of the importance of the history of analytic philosophy for today’s non-historical debates, giving the book appeal in all areas of analytic philosophy 3. Written by one of the most influential philosophers of logic and writers in the history of analytic philosophy 4. Written for upper-level undergraduates, guaranteeing widespread accessibility 5. Includes coverage of topics and issues neglected in competing publications, including Russell’s _Principles_, solipsism in the _Tractatus_, and the contributions of Frank Ramsey 6. Emphasizes the chronological development of authors’ views so as to provide a better understanding of their motivation. (shrink)