The general question considered is whether and to what extent there are features of our mathematical knowledge that support a realist attitude towards mathematics. I consider, in particular, reasoning from claims such as that mathematicians believe their reasoning to be part of a process of discovery (and not of mere invention), to the view that mathematical entities exist in some mind-independent way although our minds have epistemic access to them.

Humans and other animals have an evolved ability to detect discrete magnitudes in their environment. Does this observation support evolutionary debunking arguments against mathematical realism, as has been recently argued by Clarke-Doane, or does it bolster mathematical realism, as authors such as Joyce and Sinnott-Armstrong have assumed? To find out, we need to pay closer attention to the features of evolved numerical cognition. I provide a detailed examination of the functional properties of evolved numerical cognition, and propose that they prima (...) facie favor a realist account of numbers. (shrink)

According to the evolutionary sceptic, the fact that our cognitive faculties evolved radically undermines their reliability. A number of evolutionary epistemologists have sought to refute this kind of scepticism. This paper accepts the success of these attempts, yet argues that refuting the evolutionary sceptic is not enough to put any particular domain of beliefs – notably scientific beliefs, which include belief in Darwinian evolution – on a firm footing. The paper thus sets out to contribute to this positive justificatory project, (...) underdeveloped in the literature. In contrast to a ‘wholesale’ approach, attempting to secure justification for all of our beliefs on the grounds that our belief-forming mechanisms evolved to track truth, we propose a ‘piecemeal’ approach of assessing the reliability of particular belief-forming mechanisms in particular domains. This stands in contrast to the more familiar attempt to transfer warrant obtained for one domain to another by showing how one is somehow an extension of the other. We offer a naturalist reply to the charge of circularity by appealing to reliabilist work on the problem of induction, notably Peter Lipton's distinction between self-certifying and non-self-certifying inductive arguments. We show how, for scientific beliefs, a non-self-certifying argument might be made for the reliability of our cognitive faculties in that domain. We call this strategy Humean Bootstrapping. (shrink)

The view that the subject matter of epistemology is the concept of knowledge is faced with the problem that all attempts so far to define that concept are subject to counterexamples. As an alternative, this article argues that the subject matter of epistemology is knowledge itself rather than the concept of knowledge. Moreover, knowledge is not merely a state of mind but rather a certain kind of response to the environment that is essential for survival. In this perspective, the article (...) outlines an answer to four basic questions about knowledge: What is the role of knowledge in human life? What is the relation between knowledge and reality? How is knowledge acquired? Is there any a priori knowledge? (shrink)

Excerpts from Robert Nozick's "Invariances" Necessary truths are invariant across all possible worlds, contingent ones across only some.

From an evolutionary standpoint, a default presumption is that true beliefs are adaptive and misbeliefs maladaptive. But if humans are biologically engineered to appraise the world accurately and to form true beliefs, how are we to explain the routine exceptions to this rule? How can we account for mistaken beliefs, bizarre delusions, and instances of self-deception? We explore this question in some detail. We begin by articulating a distinction between two general types of misbelief: those resulting from a breakdown in (...) the normal functioning of the belief formation system (e.g., delusions) and those arising in the normal course of that system's operations (e.g., beliefs based on incomplete or inaccurate information). The former are instances of biological dysfunction or pathology, reflecting “culpable” limitations of evolutionary design. Although the latter category includes undesirable (but tolerable) by-products of “forgivably” limited design, our quarry is a contentious subclass of this category: misbeliefs best conceived as design features. Such misbeliefs, unlike occasional lucky falsehoods, would have been systematically adaptive in the evolutionary past. Such misbeliefs, furthermore, would not be reducible to judicious – but doxastically1noncommittal – action policies. Finally, such misbeliefs would have been adaptive in themselves, constituting more than mere by-products of adaptively biased misbelief-producing systems. We explore a range of potential candidates for evolved misbelief, and conclude that, of those surveyed, onlypositive illusionsmeet our criteria. (shrink)

An extraordinary and challenging synthesis of ideas uniting Quantum Theory, and the theories of Computation, Knowledge and Evolution, Deutsch's extraordinary book explores the deep connections between these strands which reveal the fabric ...

A uniquely integrative work, this volume provides a much needed compilation of primary source material to researchers from basic neuroscience, psychology, developmental science, neuroimaging, neuropsychology and theoretical biology. * The ...

We are reliable about logic in the sense that we by-and-large believe logical truths and disbelieve logical falsehoods. Given that logic is an objective subject matter, it is difficult to provide a satisfying explanation of our reliability. This generates a significant epistemological challenge, analogous to the well-known Benacerraf-Field problem for mathematical Platonism. One initially plausible way to answer the challenge is to appeal to evolution by natural selection. The central idea is that being able to correctly deductively reason conferred a (...) heritable survival advantage upon our ancestors. However, there are several arguments that purport to show that evolutionary accounts cannot even in principle explain how it is that we are reliable about logic. In this paper, I address these arguments. I show that there is no general reason to think that evolutionary accounts are incapable of explaining our reliability about logic. (shrink)

Late ancient Platonists and Aristotelians describe the method of reasoning to first principles as "analysis." This is a metaphor from geometrical practice. How far back were philosophers taking geometric analysis as a model for philosophy, and what work did they mean this model to do? After giving a logical description of analysis in geometry, and arguing that the standard (not entirely accurate) late ancient logical description of analysis was already familiar in the time of Plato and Aristotle, I argue that (...) Plato, in the second geometrical passage of the "Meno" (86e4-87b2), is taking analysis as a model for one kind of philosophical reasoning, and I explore the advantages and limits of this model for philosophical discovery, and in particular for how first principles can be discovered, without circularity, by argument. (shrink)

This article examines the current justifications of deductive inferences, and finds them wanting. It argues that this depends on the fact that all such justification take no account of the role deductive inferences play in knowledge. Alternatively, the article argues that a justification of deductive inferences may be given in terms of the fact that they are non-ampliative, in the sense that the content of the conclusion is merely a reformulation of the content of the premises. Some possible objections to (...) this view are discussed and found inadequate. (shrink)

Scientific realism is the position that the aim of science is to advance on truth and increase knowledge about observable and unobservable aspects of the mind-independent world which we inhabit. This book articulates and defends that position. In presenting a clear formulation and addressing the major arguments for scientific realism Sankey appeals to philosophers beyond the community of, typically Anglo-American, analytic philosophers of science to appreciate and understand the doctrine. The book emphasizes the epistemological aspects of scientific realism and contains (...) an original solution to the problem of induction that rests on an appeal to the principle of uniformity of nature. (shrink)

The incredible achievements of modern scientific theories lead most of us to embrace scientific realism: the view that our best theories offer us at least roughly accurate descriptions of otherwise inaccessible parts of the world like genes, atoms, and the big bang. In Exceeding Our Grasp, Stanford argues that careful attention to the history of scientific investigation invites a challenge to this view that is not well represented in contemporary debates about the nature of the scientific enterprise. The historical record (...) of scientific inquiry, Stanford suggests, is characterized by what he calls the problem of unconceived alternatives. Past scientists have routinely failed even to conceive of alternatives to their own theories and lines of theoretical investigation, alternatives that were both well-confirmed by the evidence available at the time and sufficiently serious as to be ultimately accepted by later scientific communities. Stanford supports this claim with a detailed investigation of the mid-to-late 19th century theories of inheritance and generation proposed in turn by Charles Darwin, Francis Galton, and August Weismann. He goes on to argue that this historical pattern strongly suggests that there are equally well-confirmed and scientifically serious alternatives to our own best theories that remain currently unconceived. Moreover, this challenge is more serious than those rooted in either the so-called pessimistic induction or the underdetermination of theories by evidence, in part because existing realist responses to these latter challenges offer no relief from the problem of unconceived alternatives itself. Stanford concludes by investigating what positive account of the spectacularly successful edifice of modern theoretical science remains open to us if we accept that our best scientific theories are powerful conceptual tools for accomplishing our practical goals, but abandon the view that the descriptions of the world around us that they offer are therefore even probably or approximately true. (shrink)

Hilary Kornblith argues for a naturalistic approach to investigating knowledge. Knowledge, he explains, is a feature of the natural world, and so should be investigated using scientific methods. He offers an account of knowledge derived from the science of animal behavior, and defends this against its philosophical rivals. This controversial and refreshingly original book offers philosophers a new way to do epistemology.

Initially proposed as rivals of classical logic, alternative logics have become increasingly important in areas such as computer science and artificial intelligence. Fuzzy logic, in particular, has motivated major technological developments in recent years. Susan Haack's Deviant Logic provided the first extended examination of the philosophical consequences of alternative logics. In this new volume, Haack includes the complete text of Deviant Logic , as well as five additional papers that expand and update it. Two of these essays critique fuzzy logic, (...) while three augment Deviant Logic 's treatment of deduction and logical truth. Haack also provides an extensive new foreword, brief introductions to the new essays, and an updated bibliography of recent work in these areas. Deviant Logic, Fuzzy Logic will be indispensable to students of philosophy, philosophy of science, linguistics, mathematics, and computer science, and will also prove invaluable to experienced scholars working in these fields. (shrink)

Bringing together evolutionary biology, psychology, and philosophy, Henry Plotkin presents a new science of knowledge, one that traces an unbreakable link between instinct and our ability to know.

Kant's views on logic and logical theory play an important role in his critical writings, especially the Critique of Pure Reason. However, since he published only one short essay on the subject, we must turn to the texts derived from his logic lectures to understand his views. The present volume includes three previously untranslated transcripts of Kant's logic lectures: the Blumberg Logic from the 1770s; the Vienna Logic (supplemented by the recently discovered Hechsel Logic) from the early 1780s; and the (...) Dohna-Wundlacken Logic from the early 1790s. Also included is a new translation of the Jasche Logic, compiled at Kant's request and published in 1800 but which also appears to stem in part from a transcript of his lectures. Together these texts provide a rich source of evidence for Kant's evolving views on logic, on the relations between logic and other disciplines, and on a variety of topics (e.g. analysis and synthesis) central to Kant's mature philosophy. They also provide a portrait of Kant as lecturer, a role in which he was both popular and influential. This volume contains substantial editorial apparatus: a general introduction, linguistic and factual notes, glossaries of key terms (both German/English and English/German) and concordances relating Kant's lectures to Georg Frederich Meier's Excerpts from the Doctrine of Reason, the book on which Kant lectured throughout his life and in which he left extensive notes. (shrink)

Here the author of How to Solve It explains how to become a "good guesser." Marked by G. Polya's simple, energetic prose and use of clever examples from a wide range of human activities, this two-volume work explores techniques of guessing, inductive reasoning, and reasoning by analogy, and the role they play in the most rigorous of deductive disciplines.

If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space and time, it is not at all obvious that this is also the case (...) with respect to the objects that are studied in mathematics. In addition to that, the methods of investigation of mathematics differ markedly from the methods of investigation in the natural sciences. Whereas the latter acquire general knowledge using inductive methods, mathematical knowledge appears to be acquired in a different way, namely, by deduction from basic principles. The status of mathematical knowledge also appears to differ from the status of knowledge in the natural sciences. The theories of the natural sciences appear to be less certain and more open to revision than mathematical theories. For these reasons mathematics poses problems of a quite distinctive kind for philosophy. Therefore philosophers have accorded special attention to ontological and epistemological questions concerning mathematics. (shrink)

One of the most intriguing features of mathematics is its applicability to empirical science. Every branch of science draws upon large and often diverse portions of mathematics, from the use of Hilbert spaces in quantum mechanics to the use of differential geometry in general relativity. It's not just the physical sciences that avail themselves of the services of mathematics either. Biology, for instance, makes extensive use of difference equations and statistics. The roles mathematics plays in these theories is also varied. (...) Not only does mathematics help with empirical predictions, it allows elegant and economical statement of many theories. Indeed, so important is the language of mathematics to science, that it is hard to imagine how theories such as quantum mechanics and general relativity could even be stated without employing a substantial amount of mathematics. (shrink)

The problem of the criterion says that we can know a proposition only if we first know a criterion of truth and vice versa, hence, we cannot know any proposition or any criterion of truth. The epistemic levels response says that since knowledge does not require knowledge about knowledge, we can know a proposition without knowing a criterion of truth. This response (advocated by Chisholm and Van Cleve) presupposes that criteria of truth are epistemic principles. In general, however, criteria of (...) truth are not epistemic principles, so the epistemic levels response to the problem of the criterion fails. (shrink)

The term ‘naturalism’ has no very precise meaning in contemporary philosophy. Its current usage derives from debates in America in the first half of the last century. The self-proclaimed ‘naturalists’ from that period included John Dewey, Ernest Nagel, Sidney Hook and Roy Wood Sellars. These philosophers aimed to ally philosophy more closely with science. They urged that reality is exhausted by nature, containing nothing ‘supernatural’, and that the scientific method should be used to investigate all areas of reality, including the (...) ‘human spirit’ (Krikorian 1944, Kim 2003). (shrink)

This book provides an accessible, critical introduction to the three main approaches that dominated work in the philosophy of mathematics during the twentieth century: logicism, intuitionism and formalism.

v. 1. The methodology of scientific research programmes.--v. 2. Mathematics, science, and epistemology.

What is the nature of number systems and arithmetic that we use in science for quantification, analysis, and modeling? I argue that number concepts and arithmetic are neither hardwired in the brain, nor do they exist out there in the universe. Innate subitizing and early cognitive preconditions for number— which we share with many other species—cannot provide the foundations for the precision, richness, and range of number concepts and simple arithmetic, let alone that of more complex mathematical concepts. Numbers and (...) arithmetic, and mathematics in general, have unique features—precision, objectivity, rigor, generalizability, stability, symbolizability, and applicability to the real world—that must be accounted for. They are sophisticated concepts that developed culturally only in recent human history. I suggest that numbers and arithmetic are realized through precise combinations of non-mathematical everyday cognitive mechanisms that make human imagination and abstraction possible. One such mechanism, conceptual metaphor, is a neurally instantiated inference-preserving cross-domain mapping that allows the conceptualization of abstract entities in terms of grounded bodily experience. I analyze how the inferential organization of the properties and “laws” of arithmetic emerge metaphorically from everyday meaningful actions. Numbers and arithmetic are thus—outside of natural selection—the product of the biologically constrained interaction of individuals with the appropriate cultural and historical phenotypic variation supported by language, writing systems, and education. (shrink)

Mathematics poses a difficult problem for methodological naturalists, those who embrace scientific method, and also for ontological naturalists who eschew non-physical entities such as Cartesian souls. Mathematics seems both essential to science but also committed to abstract non-physical entities while methodologically it seems to have no place for experiment or empirical confirmation. The chapter critically reviews a number of responses naturalists have made including logicism, Quinean radical empiricism, and Penelope Maddy’s variant thereof and suggests some further problems both for ontological (...) and for methodological naturalists. (shrink)

It is often taken for granted by writers who propose--and, for that matter, by writers who oppose--'justifications' of inductions, that deduction either does not need, or can readily be provided with, justification. The purpose of this paper is to argue that, contrary to this common opinion, problems analogous to those which, notoriously, arise in the attempt to justify induction, also arise in the attempt to justify deduction.

One way to solve the epistemic regress problem would be to show that we can acquire justification by means of an infinite regress. This is infinitism. This view has not been popular, but Peter Klein has developed a sophisticated version of infinitism according to which all justified beliefs depend upon an infinite regress of reasons. Klein's argument for infinitism is unpersuasive, but he successfully responds to the most compelling extant objections to the view. A key component of his position is (...) his claim that an infinite regress is necessary, but not sufficient, for justified belief. This enables infinitism to avoid a number of otherwise compelling objections. However, it commits infinitism to the existence of an additional feature of reasons that is necessary and, together with the regress condition, sufficient for justified belief. The trouble with infinitism is that any such condition could account for the connection between justification and truth only by undermining the rationale for the regress condition itself. (shrink)

The question that is the subject of this article is not intended to be a sociological or statistical question about the practice of today’s mathematicians, but a philosophical question about the nature of mathematics, and specifically the method of mathematics. Since antiquity, saying that mathematics is problem solving has been an expression of the view that the method of mathematics is the analytic method, while saying that mathematics is theorem proving has been an expression of the view that the method (...) of mathematics is the axiomatic method. In this article it is argued that these two views of the mathematical method are really opposed. In order to answer the question whether mathematics is problem solving or theorem proving, the article retraces the Greek origins of the question and Hilbert’s answer. Then it argues that, by Gödel’s incompleteness results and other reasons, only the view that mathematics is problem solving is tenable. (shrink)