Edmund Gettier is Professor Emeritus at the University of Massachusetts, Amherst. This short piece, published in 1963, seemed to many decisively to refute an otherwise attractive analysis of knowledge. It stimulated a renewed effort, still ongoing, to clarify exactly what knowledge comprises.

Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel version of mathematical realism. (...) She answers the traditional questions and poses a challenging new one, refocusing philosophical attention on the pressing foundational issues of contemporary mathematics. (shrink)

Since Edmund L. Gettier reminded us recently of a certain important inadequacy of the traditional analysis of "S knows that p," several attempts have been made to correct that analysis. In this paper I shall offer still another analysis (or a sketch of an analysis) of "S knows that p," one which will avert Gettier's problem. My concern will be with knowledge of empirical propositions only, since I think that the traditional analysis is adequate for knowledge of nonempirical truths.

An argument that takes issue with the contemporary epistemological consensus that justification is distinct from knowledge, proposing instead that justified belief simply is knowledge, and arguing in detail that a belief is justified when ...

According to a captivating picture, epistemic justification is essentially a matter of epistemic or evidential likelihood. While certain problems for this view are well known, it is motivated by a very natural thought—if justification can fall short of epistemic certainty, then what else could it possibly be? In this paper I shall develop an alternative way of thinking about epistemic justification. On this conception, the difference between justification and likelihood turns out to be akin to the more widely recognised difference (...) between ceteris paribus laws and brute statistical generalisations. I go on to discuss, in light of this suggestion, issues such as classical and lottery-driven scepticism as well as the lottery and preface paradoxes. (shrink)

Mathematics depends on proofs, and proofs must begin somewhere, from some fundamental assumptions. For nearly a century, the axioms of set theory have played this role, so the question of how these axioms are properly judged takes on a central importance. Approaching the question from a broadly naturalistic or second-philosophical point of view, Defending the Axioms isolates the appropriate methods for such evaluations and investigates the ontological and epistemological backdrop that makes them appropriate. In the end, a new account of (...) the objectivity of mathematics emerges, one refreshingly free of metaphysical commitments. (shrink)

In this book one of the world's foremost philosophers of language presents his unifying vision of the field--its principal achievements, its most pressing current questions, and its most promising future directions. In addition to explaining the progress philosophers have made toward creating a theoretical framework for the study of language, Scott Soames investigates foundational concepts--such as truth, reference, and meaning--that are central to the philosophy of language and important to philosophy as a whole. The first part of the book describes (...) how philosophers from Frege, Russell, Tarski, and Carnap to Kripke, Kaplan, and Montague developed precise techniques for understanding the languages of logic and mathematics, and how these techniques have been refined and extended to the study of natural human languages. The book then builds on this account, exploring new thinking about propositions, possibility, and the relationship between meaning, assertion, and other aspects of language use. An invaluable overview of the philosophy of language by one of its most important practitioners, this book will be essential reading for all serious students of philosophy. (shrink)

Edmund Gettier is Professor Emeritus at the University of Massachusetts, Amherst. This short piece, published in 1963, seemed to many decisively to refute an otherwise attractive analysis of knowledge. It stimulated a renewed effort, still ongoing, to clarify exactly what knowledge comprises.

When is a belief or judgment justified? One might be forgiven for thinking the search for single answer to this question to be hopeless. The concept of justification is required to fulfil several tasks: to evaluate beliefs epistemically, to fill in the gap between truth and knowledge, to describe the virtuous organization of one’s beliefs, to describe the relationship between evidence and theory (and thus relate to confirmation and probabilification). While some of these may be held to overlap, the prospects (...) for fulfilling all may well seem poor. Furthermore the internalist requires that justification be an introspectible property of beliefs and a fundamental epistemic concept, while the externalist is often happy to ignore the concept altogether or at best regard it as an embarrassing add-on to their epistemology. In the light of this one might reasonably give up on justification altogether or adopt pluralist approach, denying that justification is any single property of beliefs of judgments. (shrink)

Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...) coherent (in an informal sense) and places the problem of reliability within the province of the philosophy of mathematics. (shrink)

In a series of papers, Don Fallis points out that although mathematicians are generally unwilling to accept merely probabilistic proofs, they do accept proofs that are incomplete, long and complicated, or partly carried out by computers. He argues that there are no epistemic grounds on which probabilistic proofs can be rejected while these other proofs are accepted. I defend the practice by presenting a property I call ‘transferability’, which probabilistic proofs lack and acceptable proofs have. I also consider what this (...) says about the similarities between mathematics and, on the one hand natural sciences, and on the other hand philosophy. (shrink)

Edmund Gettier is Professor Emeritus at the University of Massachusetts, Amherst. This short piece, published in 1963, seemed to many decisively to refute an otherwise attractive analysis of knowledge. It stimulated a renewed effort, still ongoing, to clarify exactly what knowledge comprises.

This study addresses a central theme in current philosophy: Platonism vs Naturalism and provides accounts of both approaches to mathematics, crucially discussing Quine, Maddy, Kitcher, Lakoff, Colyvan, and many others. Beginning with accounts of both approaches, Brown defends Platonism by arguing that only a Platonistic approach can account for concept acquisition in a number of special cases in the sciences. He also argues for a particular view of applied mathematics, a view that supports Platonism against Naturalist alternatives. Not only does (...) this engaging book present the Platonist-Naturalist debate over mathematics in a comprehensive fashion, but it also sheds considerable light on non-mathematical aspects of a dispute that is central to contemporary philosophy. (shrink)

Mathematicians often speak of conjectures as being confirmed by evidence that falls short of proof. For their own conjectures, evidence justifies further work in looking for a proof. Those conjectures of mathematics that have long resisted proof, such as Fermat's Last Theorem and the Riemann Hypothesis, have had to be considered in terms of the evidence for and against them. It is argued here that it is not adequate to describe the relation of evidence to hypothesis as `subjective', `heuristic' or (...) `pragmatic', but that there must be an element of what it is rational to believe on the evidence, that is, of non-deductive logic. (shrink)

This book is a massive study of epistemology and its connections with cognitive psychology, philosophy of mind, metaphysics, and philosophy of science. Its aim is to redirect the field of epistemology, which “should be a multidisciplinary affair, not the province of pure, a priori philosophy.”.

The rise of the field of “ experimental mathematics” poses an apparent challenge to traditional philosophical accounts of mathematics as an a priori, non-empirical endeavor. This paper surveys different attempts to characterize experimental mathematics. One suggestion is that experimental mathematics makes essential use of electronic computers. A second suggestion is that experimental mathematics involves support being gathered for an hypothesis which is inductive rather than deductive. Each of these options turns out to be inadequate, and instead a third suggestion is (...) considered according to which experimental mathematics involves calculating instances of some general hypothesis. The paper concludes with the examination of some philosophical implications of this characterization. (shrink)

Fitch's argument purports to show that if all truths are knowable then all truths are known. The argument exploits the fact that the knowledge predicate or operator is untyped and may thus apply to sentences containing itself. This article outlines a response to Fitch's argument based on the idea that knowledge is typed. The first part of the article outlines the philosophical motivation for the view, comparing it to the motivation behind typing truth. The second, formal part presents a logic (...) in which knowledge is typed and demonstrates that it allows nonlogical truths to be knowable yet unknown. (shrink)

In light of the failure of attempts to analyse knowledge as a species of justified belief, a number of epistemologists have suggested that we should instead understand justification in terms of knowledge. This paper focuses on accounts of justification as a kind of ‘would-be’ knowledge. According to such accounts a belief is justified just in case any failure to know is due to uncooperative external circumstances. I argue against two recent accounts of this sort due to Alexander Bird and Martin (...) Smith. A further aim is to defend a more traditional conception, according to which justification is a matter of sufficiently high evidential likelihood. In particular, I suggest that this conception of justification offers a plausible account of lottery cases: cases in which one believes a true proposition – for example that one's lottery ticket will lose – on the basis of probabilistic evidence.Send article to KindleTo send this article to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle. Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply. Find out more about the Kindle Personal Document Service.JUSTIFICATION AS ‘WOULD-BE’ KNOWLEDGEVolume 9, Issue 4Aidan McGlynnDOI: https://doi.org/10.1017/epi.2012.22Your Kindle email address Please provide your Kindle [email protected]@kindle.com Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Dropbox To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Dropbox. JUSTIFICATION AS ‘WOULD-BE’ KNOWLEDGEVolume 9, Issue 4Aidan McGlynnDOI: https://doi.org/10.1017/epi.2012.22Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Send article to Google Drive To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive. JUSTIFICATION AS ‘WOULD-BE’ KNOWLEDGEVolume 9, Issue 4Aidan McGlynnDOI: https://doi.org/10.1017/epi.2012.22Available formats PDF Please select a format to send. By using this service, you agree that you will only keep articles for personal use, and will not openly distribute them via Dropbox, Google Drive or other file sharing services. Please confirm that you accept the terms of use. Cancel Send ×Export citation Request permission. (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)

Goldman and Olsson ( 2009 ) have responded to the common charge that reliabilist theories of knowledge are incapable of accounting for the value knowledge has beyond mere true belief. We examine their “conditional probability solution” in detail, and show that it does not succeed. The conditional probability relation is too weak to support instrumental value, and the specific relation they describe is inessential to the value of knowledge. At best, they have described conditions in which knowledge indicates that additional (...) epistemic value is likely to be forthcoming in the future. We also argue that their motive analogy breaks down. The problem, we conclude, is that being produced by a reliable process is not sufficient for a belief to be justified. (shrink)

Ordinary mathematical proofs—to be distinguished from formal derivations—are the locus of mathematical knowledge. Their epistemic content goes way beyond what is summarised in the form of theorems. Objections are raised against the formalist thesis that every mainstream informal proof can be formalised in some first-order formal system. Foundationalism is at the heart of Hilbert's program and calls for methods of formal logic to prove consistency. On the other hand, ‘systemic cohesiveness’, as proposed here, seeks to explicate why mathematical knowledge is (...) coherent and places the problem of reliability within the province of the philosophy of mathematics. (shrink)

Easwaran has given a definition of transferability and argued that, under this definition, randomized arguments are not transferable. I show that certain aspects of his definition are not suitable for addressing the underlying question of whether or not there is an epistemic distinction between randomized and deductive arguments. Furthermore, I demonstrate that for any suitable definition, randomized arguments are in fact transferable.

Mathematical instrumentalism construes some parts of mathematics, typically the abstract ones, as an instrument for establishing statements in other parts of mathematics, typically the elementary ones. Gödel’s second incompleteness theorem seems to show that one cannot prove the consistency of all of mathematics from within elementary mathematics. It is therefore generally thought to defeat instrumentalisms that insist on a proof of the consistency of abstract mathematics from within the elementary portion. This article argues that though some versions of mathematical instrumentalism (...) are defeated by Gödel’s theorem, not all are. By considering inductive reasons in mathematics, we show that some mathematical instrumentalisms survive the theorem. (shrink)