Online research in philosophy

This paper is a critical analysis of Putnam’s “consistency objection,” an objection made against a particular reading of Wittgenstein’s philosophy of mathematics (“up-to-us-ism”). I show that Putnam’s objection presupposes a rather unlikely version of Wittgenstein’s “up-to-us-ism” and is unable to undermine a more likely anti-Platonist version. I also show that a companion argument, (the “something more” argument) is unable to overturn this more sophisticated anti-Platonist version of Wittgenstein’s up-to-us-ism. Along the way I try to clarify Wittgenstein’s anti-Plalonist account of mathematics, so that others do not repeat Putnam’s mistake.

THE FOUNDATIONS OF MATHEMATICS () PREFACE The object of this paper is to give a satisfactory account of the Foundations of Mathematics in accordance with ...

Hilary Putnam suggests that the essence of the realist conception of mathematics is that the statements of mathematics are objective so that the true ones are objectively true. An argument for mathematical realism, thus conceived, is implicit in Putnam's writing. The first premise is that within currently accepted science there are objective truths. Next is the premise that some of these statements logically imply statements of pure mathematics. The conclusion drawn is that some statements of pure mathematics are objectively true. A key principle assumed is that if one statement logically implies a second, then if the first is objectively true so is the second. A question about this principle is raised and answered. The problem with the argument is with the second premise.

In this paper, I will discuss, and to some extent criticize, Hilary Putnam’s views on ontology, recently summarized and defended in his Ethics without Ontology (2004). I will start out with a critical discussion of Putnam’s thesis of conceptual relativity. Then I will turn to what is the main issue in the book: the criticism of the focus on ontological matters in philosophical discussions of mathematics and ethics.

Professor Hilary Putnam has been one of the most influential and sharply original of recent American philosophers in a whole range of fields. His most important published work is collected here, together with several new and substantial studies, in two volumes. The first deals with the philosophy of mathematics and of science and the nature of philosophical and scientific enquiry; the second deals with the philosophy of language and mind. Volume one is now issued in a new edition, including an essay on the philosophy of logic first published in 1971.

Much recent discussion in the philosophy of mathematics has concerned the indispensability argument—an argument which aims to establish the existence of abstract mathematical objects through appealing to the role that mathematics plays in empirical science. The indispensability argument is standardly attributed to W. V. Quine and Hilary Putnam. In this paper, I show that this attribution is mistaken. Quine's argument for the existence of abstract mathematical objects differs from the argument which many philosophers of mathematics ascribe to him. Contrary to appearances, Putnam did not argue for the existence of abstract mathematical objects at all. I close by suggesting that attention to Quine and Putnam's writings reveals some neglected arguments for platonism which may be superior to the indispensability argument.