The present paper discusses a proposal which says,roughly and with several qualifications, that thecollection of mathematical truths is identical withthe set of theorems of ZFC. It is argued that thisproposal is not as easily dismissed as outright falseor philosophically incoherent as one might think. Some morals of this are drawn for the concept ofmathematical knowledge.

Russian translation of Gettier E. L. Is Justified True Belief Knowledge? // Analysis, vol. 23, 1963. Translated by Lev Lamberov with kind permission of the author.

There are three sections in this paper. The first is a philosophical discussion of the general problem of reasoning under limited deductive capacity. The second sketches a rigorous way of assigning probabilities to statements in pure arithmetic; motivated by the preceding discussion, it can nonetheless be read separately. The third is a philosophical discussion that highlights the shifting contextual character of subjective probabilities and beliefs.

Recently, certain philosophers of mathematics (Fallis [1997]; Womack and Farach [(1997]) have argued that there are no epistemic considerations that should stop mathematicians from using probabilistic methods to establish that mathematical propositions are true. However, mathematicians clearly should not use methods that are unreliable. Unfortunately, due to the fact that randomized algorithms are not really random in practice, there is reason to doubt their reliability. In this paper, I analyze the prospects for establishing that randomized algorithms are reliable. I end (...) by arguing that it would be inconsistent for mathematicians to suspend judgement on the truth of mathematical propositions on the basis of worries about the reliability of randomized algorithms. (shrink)

Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favor of another approach--naturalism. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.

Sherrilyn Roush defends a new theory of knowledge and evidence, based on the idea of "tracking" the truth, as the best approach to a wide range of questions about knowledge-related phenomena. The theory explains, for example, why scepticism is frustrating, why knowledge is power, and why better evidence makes you more likely to have knowledge. Tracking Truth provides a unification of the concepts of knowledge and evidence, and argues against traditional epistemological realist and anti-realist positions about scientific theories and for (...) a piecemeal approach based on a criterion of evidence, a position Roush calls "real anti-realism." Epistemologists and philosophers of science will recognize this as a significant original contribution. (shrink)

In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of approaches to new thinking about the philosophy of mathematics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing (...) real mathematics, to the use of analogy, the prospects for a Bayesian confirmation theory, the notion of a mathematical research programme and the ways in which new concepts are justified. His inspiring book challenges both philosophers and mathematicians to develop the broadest and richest philosophical resources for work in their disciplines and points clearly to the ways in which this can be done. (shrink)

In this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically, and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of new ways to think philosophically about mathematics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing real mathematics, (...) to the use of analogy, the prospects for a Bayesian confirmation theory, the notion of a mathematical research programme, and the ways in which new concepts are justified. His highly original book challenges both philosophers and mathematicians to develop the broadest and richest philosophical resources for work in their disciplines, and points clearly to the ways in which this can be done. (shrink)

Throughout the 20th century, an enormous amount of intellectual fuel was spent debating the merits of a class of skeptical arguments which purport to show that knowledge of the external world is not possible. These arguments, whose origins can be traced back to Descartes, played an important role in the work of some of the leading philosophers of the 20th century, including Russell, Moore and Wittgenstein, and they continue to engage the interest of contemporary philosophers. (e.g., Cohen 1999, DeRose 1995, (...) Hill 1996, Klein 1981, Lewis 1996, McGinn 1993, Nozick 1981, Schiffer 1996, Unger 1975, Williams 1996) Typically, these arguments make use of one or more premises which the philosophers proposing them take to be intuitively obvious. Beyond an appeal to intuition, little or no defense is offered, and in many cases it is hard to see what else could be said in support of these premises. A number of authors have suggested that the intuitions undergirding these skeptical arguments are universal – shared by everyone (or almost everyone) who thinks reflectively about knowledge. In this paper we will offer some evidence indicating that they are far from universal. Rather, the evidence suggests that many of the intuitions epistemologists invoke vary with the cultural background, socioeconomic status and educational background of the person offering the intuition. And this, we will argue, is bad news for the skeptical arguments that rely on those intuitions. The evidence may also be bad news for skepticism itself – not because it shows that skepticism is false, but rather because, if we accept one prominent account of the link between epistemic intuitions and epistemic concepts, it indicates that skepticism may be much less interesting and much less worrisome than philosophers have taken it to be. (shrink)

Several philosophers have used the framework of means/ends reasoning to explain the methodological choices made by scientists and mathematicians (see, e.g., Goldman 1999, Levi 1962, Maddy 1997). In particular, they have tried to identify the epistemic objectives of scientists and mathematicians that will explain these choices. In this paper, the framework of means/ends reasoning is used to study an important methodological choice made by mathematicians. Namely, mathematicians will only use deductive proofs to establish the truth of mathematical claims. In this (...) paper, I argue that none of the epistemic objectives of mathematicians that are currently on the table provide a satisfactory explanation of this rejection of probabilistic proofs. (shrink)