A Study in the History and Philosophy of Mathematics
Author: Sebastien Gandon
Publisher: Palgrave Macmillan
One hundred years ago, Russell and Whitehead published their epoch-making Principia Mathematica (PM), which was initially conceived as the second volume of Russell's Principles of Mathematics (PoM) that had appeared ten years before. No other works can be credited to have had such an impact on the development of logic and on philosophy in the twentieth century. However, until now, scholars only focused on the first parts of the books – that is, on Russell's and Whitehead's theory of logic, set-theory and arithmetic. Sebastien Gandon aims at reversing the perspective. His goal is to give a picture of Russell's logicism based on a detailed reading of the developments dealing with advanced mathematics - namely projective geometry and the theory of quantity. This book is not only the first study ever made of the 'later' portions of PoM and PM, it also shows how this shift of perspective compels us to change our view of the logicist program taken as a whole.
This book collects 13 papers that explore Wittgenstein's philosophy throughout the different stages of his career. The author writes from the viewpoint of critical rationalism. The tone of his analysis is friendly and appreciative yet critical. Of these papers, seven are on the background to the philosophy of Wittgenstein. Five papers examine different aspects of it: one on the philosophy of young Wittgenstein, one on his transitional period, and the final three on the philosophy of mature Wittgenstein, chiefly his Philosophical Investigations. The last of these papers, which serves as the concluding chapter, concerns the analytical school of philosophy that grew chiefly under its influence. Wittgenstein’s posthumous Philosophical Investigations ignores formal languages while retaining the view of metaphysics as meaningless -- declaring that all languages are metaphysics-free. It was very popular in the middle of the twentieth century. Now it is passé. Wittgenstein had hoped to dissolve all philosophical disputes, yet he generated a new kind of dispute. His claim to have improved the philosophy of life is awkward just because he prevented philosophical discussion from the ability to achieve that: he cut the branch on which he was sitting. This, according to the author, is the most serious critique of Wittgenstein.
A long tradition, going back to Aristotle, conceives of logic in terms of necessity and possibility: a deductive argument is correct if it is not possible for the conclusion to be false when the premises are true. A relatively unknown feature of the analytic tradition in philosophy is that, at its very inception, this venerable conception of the relation between logic and necessity and possibility - the concepts of modality - was put into question. The founders of analytic philosophy, Gottlob Frege and Bertrand Russell, held that these concepts are empty: there are no genuine distinctions among the necessary, the possible, and the actual. In this book, the first of two volumes, Sanford Shieh investigates the grounds of this position and its consequences for Frege's and Russell's conceptions of logic. The grounds lie in doctrines on truth, thought, and knowledge, as well as on the relation between mind and reality, that are central to the philosophies of Frege and Russell, and are of enduring philosophical interest. The upshot of this opposition to modality is that logic is fundamental, and, to be coherent, modal concepts would have to be reconstructed in logical terms. This rejection of modality in early analytic philosophy remains of contemporary significance, though the coherence of modal concepts is rarely questioned nowadays because it is generally assumed that suspicion of modality derives from logical positivism, which has not survived philosophical scrutiny. The anti-modal arguments of Frege and Russell, however, have nothing to do with positivism and remain a challenge to the contemporary acceptance of modal notions.