Several high-profile mathematical problems have been solved in recent decades by computer-assisted proofs. Some philosophers have argued that such proofs are a posteriori on the grounds that some such proofs are unsurveyable; that our warrant for accepting these proofs involves empirical claims about the reliability of computers; that there might be errors in the computer or program executing the proof; and that appeal to computer introduces into a proof an experimental element. I argue that none of these arguments withstands scrutiny, (...) and so there is no reason to believe that computer-assisted proofs are not a priori. Thanks are due to Michael Levin, David Corfield, and an anonymous referee for Philosophia Mathematica for their helpful comments. Earlier versions of this paper were presented at the Hofstra University Department of Mathematics colloquium series, and at the 2005 New Jersey Regional Philosophical Association; I am grateful to both audiences for their comments. CiteULike Connotea Del.icio.us What's this? (shrink)

In recent decades, experimental mathematics has emerged as a new branch of mathematics. This new branch is defined less by its subject matter, and more by its use of computer assisted reasoning. Experimental mathematics uses a variety of computer assisted approaches to verify or prove mathematical hypotheses. For example, there is “number crunching” such as searching for very large Mersenne primes, and showing that the Goldbach conjecture holds for all even numbers less than 2 × 1018. There are “verifications” of (...) hypotheses which, while not definitive proofs, provide strong support for those hypotheses, and there are proofs involving an enormous amount of computer hours, which cannot be surveyed by any one mathematician in a lifetime. There have been several attempts to argue that one or another aspect of experimental mathematics shows that mathematics now accepts empirical or inductive methods, and hence shows mathematical apriorism to be false. Assessing this argument is complicated by the fact that there is no agreed definition of what precisely experimental mathematics is. However, I argue that on any plausible account of ’experiment’ these arguments do not succeed. (shrink)

Duncan Pritchard's version of the safety analysis of knowledge has it that for all contingent propositions, p, S knows that p iff S believes that p, p is true, and (the “safety principle”) in most nearby worlds in which S forms his belief in the same way as in the actual world, S believes that p only if p is true. Among the other virtues claimed by Pritchard for this view is its supposed ability to solve a version of the (...) lottery puzzle. In this paper, I argue that the safety analysis of knowledge in fact fails to solve the lottery puzzle. I also argue that a revised version of the safety principle recently put forward by Pritchard fares no better. (shrink)

Jody Azzouni has offered the following argument against the existence of mathematical entities: if, as it seems, mathematical entities play no role in mathematical practice, we therefore have no reason to believe in them. I consider this argument as it applies to mathematical platonism, and argue that it does not present a legitimate novel challenge to platonism. I also assess Azzouni's use of the ‘epistemic role puzzle’ (ERP) to undermine the platonist's alleged parallel between skepticism about mathematical entities and external-world (...) skepticism. I conclude that ERP fails to undermine this parallel. (shrink)

Mathematical apriorism holds that mathematical truths must be established using a priori processes. Against this, it has been argued that apparently a priori mathematical processes can, under certain circumstances, fail to warrant the beliefs they produce; this shows that these warrants depend on contingent features of the contexts in which they are used. They thus cannot be a priori. -/- In this paper I develop a position that combines a reliabilist version of mathematical apriorism with a platonistic view of mathematical (...) ontology. I argue that this view both withstands the above objection and explains the reliability of a priori mathematical warrant. (shrink)

The safety analysis of knowledge, due to Duncan Pritchard, has it that for all contingent propositions, p, S knows that p iff S believes that p, p is true, and (the “safety principle”) in most nearby worlds in which S forms his belief in the same way as in the actual world, S believes that p only if p is true. Among the other virtues claimed by Pritchard for this view is its supposed ability to solve a version of the (...) lottery puzzle. In this paper, I argue that the safety analysis of knowledge in fact fails to solve the lottery puzzle. I also argue that a revised version of the safety principle recently put forward by Pritchard fares no better. (shrink)

The Generality Problem for process reliabilism is to outline a procedure for determining when two beliefs are produced by the same process, in such a way as to avoid, on the one hand, individuating process types so narrowly that each type is instantiated only once, or, on the other hand, individuating them so broadly that beliefs that have different epistemic statuses are subsumed under the same process type. In this paper, I offer a solution to the problem which takes belief‐independent (...) processes to be functions that take as inputs information about distal states of affairs, and produce beliefs as outputs. Processes are individuated narrowly, so as to avoid the latter aspect of the Generality problem, but, by holding process tokens to be of the same type when they take perceptually equivalent scenes as inputs, and produce beliefs of the same kind as outputs, the former aspect of the problem is avoided too. Having argued that this method of typing process tokens solves the Generality Problem, I then argue that my solution does not fall prey to objections that have been, or might be, raised for similar proposals. (shrink)

This paper argues that reliabilism can handle Gettier cases once it restricts knowledge producing reliable processes to those that involve a suitable causal link between the subject’s belief and the fact it references. Causal tracking reliabilism (as this version of reliabilism is called) also avoids the problems that refuted the causal theory of knowledge, along with problems besetting more contemporary theories (such as virtue reliabilism and the “safety” account of knowledge). Finally, causal tracking reliabilism allows for a response to Linda (...) Zagzebski’s challenge that no theory of knowledge can both eliminate the possibility of Gettier cases while also allowing fully warranted but false beliefs. (shrink)

Mathematical apriorists sometimes hold that our non-derived mathematical beliefs are warranted by mathematical intuition. Against this, Philip Kitcher has argued that if we had the experience of encountering mathematical experts who insisted that an intuition-produced belief was mistaken, this would undermine that belief. Since this would be a case of experience undermining the warrant provided by intuition, such warrant cannot be a priori.I argue that this leaves untouched a conception of intuition as merely an aspect of our ordinary ability to (...) reason. Thus the apriorist may still hold that some mathematical beliefs are warranted by intuition. (shrink)

The lottery problem is often regarded as a successful counterexample to reliabilism. The process of forming your true belief that your ticket has lost solely on the basis of considering the odds is, from a purely probabilistic viewpoint, much more reliable than the process of forming a true belief that you have lost by reading the results in a normally reliable newspaper. Reliabilism thus seems forced, counterintuitively, to count the former process as knowledge if it so counts the latter process. (...) -/- I offer a theory of empirical knowledge which, while being recognizably reliabilist, restricts empirical knowledge to cases in which the fact that p and the belief that p are causally connected. I show that this form of reliabilism solves the lottery problem, avoids the problems that beset the causal theory of knowledge, and show how it handles a number of problematic cases in the recent literature. (shrink)

In this article, I present a novel account of a priori warrant, which I then use to examine the relationship between a priori and a posteriori warrant in mathematics. According to this account of a priori warrant, the reason that a posteriori warrant is subordinate to a priori warrant in mathematics is because processes that produce a priori warrant are reliable independent of the contexts in which they are used, whereas this is not true for processes that produce a posteriori (...) warrant. Following this, I show how this difference explains why a priori warranting processes, such as proof (and mathematical intuition) can, and a posteriori warranting processes cannot, definitively establish mathematical results. I then show why, in the event of a conflict between a priori and a posteriori warranting processes, the former undermine, and cannot be undermined by, the latter. Finally, I explain the connection between apriority and mathematical necessity, but do so in a way that allows for the existence of contingent a priori knowledge. (shrink)

An unadorned form of process reliabilism (UPR) contends that knowledge is true belief, produced by a reliable process, undefeated by a more reliable process. There is no requirement that one know that one’s belief meets this requirement; that it actually does so is sufficient. An integral aspect of UPR, then, is the rejection of the KK thesis. One popular method of showing the implausibility of UPR is to specify a case where a subject satisfies all of UPR’s conditions on knowledge (...) but “clearly” fails to know. Since the subject satisfies all of UPR’s conditions on knowledge, but fails to know, the conditions for knowledge are not as UPR maintains. UPR’s analysis, it is alleged, leaves something out. That something is usually taken to be that the subject lacks appropriate evidence for his belief. This is the internalist counterexample to UPR. In this paper I argue that the internalist counterexample fails to refute UPR. (shrink)

Brings together an impressive collection of primary sources from ancient and modern philosophy. Arranged chronologically and featuring introductory overviews explaining technical terms, this accessible reader is easy-to-follow and unrivaled in its historical scope. With selections from key thinkers such as Plato, Aristotle, Descartes, Hume and Kant, it connects the major ideas of the ancients with contemporary thinkers. A selection of recent texts from philosophers including Quine, Putnam, Field and Maddy offering insights into the current state of the discipline clearly illustrates (...) the development of the subject. (shrink)

It is sometimes argued that mathematical knowledge must be a priori, since mathematical truths are necessary, and experience tells us only what is true, not what must be true. This argument can be undermined either by showing that experience can yield knowledge of the necessity of some truths, or by arguing that mathematical theorems are contingent. Recent work by Albert Casullo and Timothy Williamson argues (or can be used to argue) the first of these lines; W. V. Quine and Hartry (...) Field take the latter line. I defend a version of the argument against these, and other objections. (shrink)

The Generality Problem for process reliabilism is to outline a procedure for determining when two beliefs are produced by the same process, in such a way as to avoid, on the one hand, individuating process types so narrowly that each type is instantiated only once, or, on the other hand, individuating them so broadly that beliefs that have different epistemic statuses are subsumed under the same process type. In this paper, I offer a solution to the problem which takes belief‐independent (...) processes to be functions that take as inputs information about distal states of affairs, and produce beliefs as outputs. Processes are individuated narrowly, so as to avoid the latter aspect of the Generality problem, but, by holding process tokens to be of the same type when they take perceptually equivalent scenes as inputs, and produce beliefs of the same kind as outputs, the former aspect of the problem is avoided too. Having argued that this method of typing process tokens solves the Generality Problem, I then argue that my solution does not fall prey to objections that have been, or might be, raised for similar proposals. (shrink)

Book reviewed:;Jerrold J. Katz, Language and OtherObjects;Book reviewed:;Jerrold J. Katz, Language and Other Abstract Objects;Book reviewed:;Jerrold J. Katz, Language and Other Abstract Objects;Book reviewed:;Jerrold J. Katz, Language and Other Abstract Objects;Book reviewed:;Jerrold J. Katz, Language and Other Abstract Objects;Book reviewed:;Jerrold J. Katz, Language and Other Abstract Objects;Book reviewed:;Jerrold J. Katz, Language and Other Abstract Objects;Book reviewed:;Jerrold J. Katz, Language and Other Abstract Objects;Book reviewed:;Jerrold J. Katz, Language and Other Abstract Objects;Book reviewed:;Jerrold J. Katz, Language and Other Abstract Objects.

Lottery puzzles involve an ordinary piece of knowledge which seems to imply knowledge of a so-called “lottery proposition,” which itself seems unknown: I might be said to know that I won’t be going on safari next year. But if I were to win the lottery, I would go, and I don’t know that I won’t win the lottery. Examples can be multiplied. Thus we seem left either with the paradoxical position of knowing certain ordinary propositions, but failing to know the (...) lottery propositions they imply, or else conceding to the skeptic. I present a version of reliabilism according to which empirical knowledge is true belief produced by a reliable process causally connecting belief and fact. According to this theory, if my ordinary belief and my belief in the lottery proposition are suitably connected to the facts that render them true, both count as knowledge. In cases where my ordinary belief and my belief in the lottery proposition are not suitably connected to the relevant facts, neither count as knowledge. Thus the paradoxical air of lottery puzzles is removed, and skepticism is avoided. (shrink)

According to what one might label the traditional view of proof in mathematics, proofs have the following characteristics. They are knowable a priori, the knowledge they provide is certain, rather than merely probable, they are surveyable, and, because of these other features, a mathematical proof is convincing to one who understands it. Opponents of this view typically drew their motivation not from the study of mathematics, but rather from a more general antipathy to apriority in epistemology and necessity in metaphysics (...) (Mill, Putnam and Quine all spring to mind here). Tymoczko (1979) provides a different sort of challenge, taking as its cue a development within mathematics itself, rather than being based on philosophical scruples over apriority or necessity. (shrink)

Michael Bishop and J.D.Trout have recently argued that analytic epistemology is incapable of incorporating insights from experimental psychology, and that while an acceptable epistemology should be normative, analytic epistemology lacks normativity. For these reasons, they urge that analytic epistemology should be replaced by what they call “ameliorative psychology”: a view that draws on empirical findings in psychology in order to help people become better reasoners. In this paper, I argue that analytic epistemology does not need to be replaced, as it (...) is indeed normative, and is quite capable of incorporating the insights of ameliorative psychology. (shrink)

The safety analysis of knowledge, due to Duncan Pritchard, has it that for all contingent propositions, p, S knows that p iff S believes that p, p is true, and (the “safety principle”) in most nearby worlds in which S forms his belief in the same way as in the actual world, S believes that p only if p is true. Among the other virtues claimed by Pritchard for this view is its supposed ability to solve a version of the (...) lottery puzzle. In this paper, I argue that the safety analysis of knowledge in fact fails to solve the lottery puzzle. I also argue that a revised version of the safety principle recently put forward by Pritchard fares no better. (shrink)

The safety analysis of knowledge, due to Duncan Pritchard, has it that for all contingent propositions, p, S knows that p iff S believes that p, p is true, and (the “safety principle”) in most nearby worlds in which S forms his belief in the same way as in the actual world, S believes that p only if p is true. Among the other virtues claimed by Pritchard for this view is its supposed ability to solve a version of the (...) lottery puzzle. In this paper, I argue that the safety analysis of knowledge in fact fails to solve the lottery puzzle. I also argue that a revised version of the safety principle recently put forward by Pritchard fares no better. (shrink)

Book reviewed:Jerrold J. Katz, Language and Other Abstract Objects.