This unique book by Stewart Shapiro looks at a range of philosophical issues and positions concerning mathematics in four comprehensive sections. Part I describes questions and issues about mathematics that have motivated philosophers since the beginning of intellectual history. Part II is an historical survey, discussing the role of mathematics in the thought of such philosophers as Plato, Aristotle, Kant, and Mill. Part III covers the three major positions held throughout the twentieth century: the idea that mathematics is logic (logicism), (...) the view that the essence of mathematics is the rule-governed manipulation of characters (formalism), and a revisionist philosophy that focuses on the mental activity of mathematics (intuitionism). Finally, Part IV brings the reader up-to-date with a look at contemporary developments within the discipline. This sweeping introductory guide to the philosophy of mathematics makes these fascinating concepts accessible to those with little background in either mathematics or philosophy. (shrink)

What is Functionalism? Functionalism is one of the major proposals that have been offered as solutions to the mind/body problem. Solutions to the mind/body problem usually try to answer questions such as: What is the ultimate nature of the mental? At the most general level, what makes a mental state mental? Or more specifically, What do thoughts have in common in virtue of which they are thoughts? That is, what makes a thought a thought? What makes a pain a pain? (...) Cartesian Dualism said the ultimate nature of the mental was to be found in a special mental substance. Behaviorism identified mental states with behavioral dispositions; physicalism in its most influential version identifies mental states with brain states. Functionalism says that mental states are constituted by their causal relations to one another and to sensory inputs and behavioral outputs. Functionalism is one of the major theoretical developments of Twentieth Century analytic philosophy, and provides the conceptual underpinnings of much work in cognitive science. (shrink)

Disjunctivism about perceptual appearances, as I conceive of it, is a theory which seeks to preserve a naïve realist conception of veridical perception in the light of the challenge from the argument from hallucination. The naïve realist claims that some sensory experiences are relations to mind-independent objects. That is to say, taking experiences to be episodes or events, the naïve realist supposes that some such episodes have as constituents mind-independent objects. In turn, the disjunctivist claims that in a case of (...) veridical perception like this very kind of experience that you now have, the experiential episode you enjoy is of a kind which could not be occurring were you having an hallucination. The common strategy of arguments from hallucination set out to show that certain things are true of hallucinations, and hence must be true of perceptions. For example, it is argued that hallucinations must have non-physical objects of awareness, or that such states are not relations to anything at all, but are at best seeming relations to objects. In insisting that veridical perceptual experience is of a distinct kind from hallucination, the disjunctivist denies that any of these conceptions of hallucination challenges our conception of veridical perceptions as relations to mind-independent objects. More specifically, I assume that the disjunctivist advocates naïve realism because they think that this position best articulates how sensory experience seems to us to be just through reflection. If the disjunctivist is correct in this contention, then anyone who accepts the conclusion of the argument from hallucination must also accept that the nature of sensory experience is other than it seems to us to be. In turn, one may complain that any such error theory is liable to lead to sceptical consequences. A Humean scepticism about the senses launches a challenge about our knowledge of the world through questioning the conception we have of what sense experience is, and how it can provide knowledge of the world. (shrink)

The functionalist view of the nature of the mind is now widely accepted. Like behaviorism and physicalism, functionalism seeks to answer the question "What are mental states?" I shall be concerned with identity thesis formulations of functionalism. They say, for example, that pain is a functional state, just as identity thesis formulations of physicalism say that pain is a physical state.

This entirely new translation of Critique of Pure Reason is the most accurate and informative English translation ever produced of this epochal philosophical text. Though its simple and direct style will make it suitable for all new readers of Kant, the translation displays an unprecedented philosophical and textual sophistication that will enlighten Kant scholars as well. This translation recreates as far as possible a text with the same interpretative nuances and richness as the original. The extensive editorial apparatus includes informative (...) annotation, detailed glossaries, an index, and a large-scale general introduction in which two of the world's preeminent Kant scholars provide both a succinct summary of the structure and argument of the Critique and a detailed account of its long and complex genesis. (shrink)

Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a 'nature' than that confers on them. Parsons also analyzes the concept of intuition (...) and presents a conception of it distantly inspired by that of Kant, which describes a basic kind of access to abstract objects and an element of a first conception of the infinite. (shrink)

Dermot Moran provides a lucid, engaging, and critical introduction to Edmund Husserl's philosophy, with specific emphasis on his development of phenomenology. This book is a comprehensive guide to Husserl's thought from its origins in nineteenth-century concerns with the nature of scientific knowledge and with psychologism, through his breakthrough discovery of phenomenology and his elucidation of the phenomenological method, to the late analyses of culture and the life-world. Husserl's complex ideas are presented in a clear and expert manner. Individual chapters explore (...) Husserl's key texts including _Philosophy of Arithmetic_, _Logical Investigations_, _Ideas_ I, _Cartesian Meditations_ and _Crisis of the European Sciences_. In addition, Moran offers penetrating criticisms and evaluations of Husserl's achievement, including the contribution of his phenomenology to current philosophical debates concerning consciousness and the mind. _Edmund Husserl_ is an invaluable guide to understanding the thought of one of the seminal thinkers of the twentieth century. It will be helpful to students of contemporary philosophy, and to those interested in scientific, literary and cultural studies on the European continent. (shrink)

David Hume's Treatise of Human Nature, composed before the author was twenty-eight years old, was published in 1739 and 1740. In revising the late L.A. Selby-Bigge's edition of Hume's Treatise Professor Nidditch corrected verbal errors and took account of Hume's manuscript amendments. He also supplied the text of theof the Treatise following the original 1740 edition and provided an apparatus of variant readings.

The thesis is a study of the notion of intuition in the foundations of mathematics which focuses on the case of natural numbers and hereditarily finite sets. Phenomenological considerations are brought to bear on some of the main objections that have been raised to this notion. ;Suppose that a person P knows that S only if S is true, P believes that S, and P's belief that S is produced by a process that gives evidence for it. On a phenomenological (...) view the relevant evidence is provided by intuition , and this should be the case in either ordinary perceptual knowledge or in mathematical knowledge. Intuition is to be understood in terms of fulfillments of intentions. Knowledge is a product of intuition and intention. In the case of mathematical knowledge it is said that there is a construction for a mathematical statement S if and only if the intention expressed by S is fulfilled . Constructions are thus viewed as intuition processes that could actually or possibly be carried out. In elementary parts of mathematics they might be characterized in terms of certain classes of recursive functions. This view is discussed in the case where S is taken to be a singular statement about natural numbers or finite sets, and also where S is taken to be a general statement about such objects. The distinction between intuition of and intuition that is also investigated in this context. ;It is pointed out how on a phenomenological view a number of central problems about mathematical intuition can be avoided: problems about the analogousness of perceptual and mathematical intuition, about causal accounts of knowledge in mathematics, and about structuralism in mathematics. The bearing of the account on issues concerning constructivism and platonism is also discussed. (shrink)

Excerpt from A History of Formal Logic In this edition of the most considerable history Of formal logic yet published, the Opportunity has Of course been taken to make some adjustments seen to be necessary in the original, with the author's full concurrence. Only in 36, however, has the numeration of cited passages been altered owing to the introduction of new matter. Those changes are as follows. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. (...) Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works. (shrink)

Science Without Numbers caused a stir in 1980, with its bold nominalist approach to the philosophy of mathematics and science. It has been unavailable for twenty years and is now reissued in a revised edition with a substantial new preface presenting the author's current views and responses to the issues raised in subsequent debate.

This volume will be an essential resource for anyone working in the central areas of philosophy, and the starting point for future research in this fascinating ...

Robert Hanna presents a fresh view of the Kantian and analytic traditions that have dominated continental European and Anglo-American philosophy over the last two centuries, and of the connections between them. But this is not just a study in the history of philosophy, for out of this emerges Hanna's original approach to two much-contested theories that remain at the heart of contemporary philosophy. Hanna puts forward a new 'cognitive-semantic' interpretation of transcendental idealism, and a vigorous defense of Kant's theory of (...) analytic and synthetic necessary truth, making this compelling reading for all who are interested in these fundamental philosophical issues. (shrink)

Here, Bob Hale and Crispin Wright assemble the key writings that lead to their distinctive neo-Fregean approach to the philosophy of mathematics. In addition to fourteen previously published papers, the volume features a new paper on the Julius Caesar problem; a substantial new introduction mapping out the program and the contributions made to it by the various papers; a section explaining which issues most require further attention; and bibliographies of references and further useful sources. It will be recognized as the (...) most powerful presentation yet of a neo-Fregean program. (shrink)

In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He does this by establishing that both platonism and anti-platonism are defensible views. Introducing a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, he proceeds to defend anti-platonism (in particular, mathematical fictionalism) against various attacks, most notably the Quine-Putnam indispensability attack. He concludes by arguing that it is not simply that we do not currently have any good argument (...) for or against platonism, but that we could never have such an argument and, indeed, that there is no fact of the matter as to whether platonism is correct. (shrink)

Edmund Husserl has argued that we can intuit essences and, moreover, that it is possible to formulate a method for intuiting essences. Husserl calls this method 'ideation'. In this paper I bring a fresh perspective to bear on these claims by illustrating them in connection with some examples from modern pure geometry. I follow Husserl in describing geometric essences as invariants through different types of free variations and I then link this to the mapping out of geometric invariants in modern (...) mathematics. This view leads naturally to different types of spatial ontologies and it can be used to shed light on Husserl's general claim that there are different ontologies in the eidetic sciences that can be systematically related to one another. The paper is rounded out with a consideration of the role of ideation in the origins of modern geometry, and with a brief discussion of the use of ideation outside of pure geometry. (shrink)

As Louisiana and Cuba emerged from slavery in the late nineteenth century, each faced the question of what rights former slaves could claim. Degrees of Freedom compares and contrasts these two societies in which slavery was destroyed by war, and citizenship was redefined through social and political upheaval. Both Louisiana and Cuba were rich in sugar plantations that depended on an enslaved labor force. After abolition, on both sides of the Gulf of Mexico, ordinary people-cane cutters and cigar workers, laundresses (...) and labor organizers-forged alliances to protect and expand the freedoms they had won. But by the beginning of the twentieth century, Louisiana and Cuba diverged sharply in the meanings attributed to race and color in public life, and in the boundaries placed on citizenship. Louisiana had taken the path of disenfranchisement and state-mandated racial segregation; Cuba had enacted universal manhood suffrage and had seen the emergence of a transracial conception of the nation. What might explain these differences? Moving through the cane fields, small farms, and cities of Louisiana and Cuba, Rebecca Scott skillfully observes the people, places, legislation, and leadership that shaped how these societies adjusted to the abolition of slavery. The two distinctive worlds also come together, as Cuban exiles take refuge in New Orleans in the 1880s, and black soldiers from Louisiana garrison small towns in eastern Cuba during the 1899 U.S. military occupation. Crafting her narrative from the words and deeds of the actors themselves, Scott brings to life the historical drama of race and citizenship in postemancipation societies. (shrink)

Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

Wittgenstein's work remains, undeniably, now, that off one of those few philosophers who will be read by all future generations.

This book does three main things. First, it defends mathematical platonism against the main objections to that view (most notably, the epistemological objection and the multiple-reductions objection). Second, it defends anti-platonism (in particular, fictionalism) against the main objections to that view (most notably, the Quine-Putnam indispensability objection and the objection from objectivity). Third, it argues that there is no fact of the matter whether abstract mathematical objects exist and, hence, no fact of the matter whether platonism or anti-platonism is true.

Over the last few decades Michael Dummett developed a rich program for assessing logic and the meaning of the terms of a language. He is also a major exponent of Frege's version of logicism in the philosophy of mathematics. Over the last decade, Neil Tennant developed an extensive version of logicism in Dummettian terms, and Dummett influenced other contemporary logicists such as Crispin Wright and Bob Hale. The purpose of this paper is to explore the prospects for Fregean logicism within (...) a broadly Dummettian framework. The conclusions are mostly negative: Dummett's views on analyticity and the logical/non-logical boundary leave little room for logicism. Dummett's considerations concerning manifestation and separability lead to a conservative extension requirement: if a sentence S is logically true, then there is a proof of S which uses only the introduction and elimination rules of the logical terms that occur in S. If basic arithmetic propositions are logically true-as the logicist contends-then there is tension between this conservation requirement and the ontological commitments of arithmetic. It follows from Dummett's manifestation requirements that if a sentence S is composed entirely of logical terminology, then there is a formal deductive system D such that S is analytic, or logically true, if and only if S is a theorem of D. There is a deep conflict between this result and the essential incompleteness, or as Dummett puts it, the indefinite extensibility, of arithmetic truth. (shrink)

In Embodied Minds in Action, Robert Hanna and Michelle Maiese work out a unified treatment of three fundamental philosophical problems: the mind-body problem, the problem of mental causation, and the problem of action. This unified treatment rests on two basic claims. The first is that conscious, intentional minds like ours are essentially embodied. This entails that our minds are necessarily spread throughout our living, organismic bodies and belong to their complete neurobiological constitution. So minds like ours are necessarily alive. The (...) second claim is that essentially embodied minds are self-organizing thermodynamic systems. This entails that our mental lives consist in the possibility and actuality of moving our own living organismic bodies through space and time, by means of our conscious desires. The upshot is that we are essentially minded animals who help to create the natural world through our own agency. This doctrine--the Essential Embodiment Theory--is a truly radical idea which subverts the traditionally opposed and seemingly exhaustive categories of Dualism and Materialism, and offers a new paradigm for contemporary mainstream research in the philosophy of mind and cognitive neuroscience. (shrink)

This book makes available to the English reader nearly all of the shorter philosophical works, published or unpublished, that Husserl produced on the way to the phenomenological breakthrough recorded in his Logical Investigations of 1900-1901. Here one sees Husserl's method emerging step by step, and such crucial substantive conclusions as that concerning the nature of Ideal entities and the status the intentional `relation' and its `objects'. Husserl's literary encounters with many of the leading thinkers of his day illuminates both the (...) context and the content of his thought. Many of the groundbreaking analyses provided in these texts were never again to be given the thorough expositions found in these early writings. Early Writings in the Philosophy of Logic and Mathematics is essential reading for students of Husserl and all those who enquire into the nature of mathematical and logical knowledge. (shrink)

In Rationality and Logic, Robert Hanna argues that logic is intrinsically psychological and that human psychology is intrinsically logical. He claims that logic is cognitively constructed by rational animals and that rational animals are essentially logical animals. In order to do so, he defends the broadly Kantian thesis that all rational animals possess an innate cognitive "logic faculty." Hanna 's claims challenge the conventional philosophical wisdom that sees logic as a fully formal or "topic-neutral" science irreconcilably separate from the species- (...) or individual-specific focus of empirical psychology.Logic and psychology went their separate ways after attacks by Frege and Husserl on logical psychologism--the explanatory reduction of logic to empirical psychology. Hanna argues, however, that--despite the fact that logical psychologism is false--there is an essential link between logic and psychology. Rational human animals constitute the basic class of cognizers or thinkers studied by cognitive psychology; given the connection between rationality and logic that Hanna claims, it follows that the nature of logic is significantly revealed to us by cognitive psychology. Hanna 's proposed "logical cognitivism" has two important consequences: the recognition by logically oriented philosophers that psychologists are their colleagues in the metadiscipline of cognitive science; and radical changes in cognitive science itself. Cognitive science, Hanna argues, is not at bottom a natural science; it is both an objective or truth-oriented science and a normative human science, as is logic itself. (shrink)

In this paper I study the variants of the notion of completeness Husserl pre-sented in “Ideen I” and two lectures he gave in Göttingen in 1901. Introduced primarily in connection with the problem of imaginary numbers, this notion found eventually a place in the answer Husserl provided for the philosophically more im-portant problem of the logico-epistemological foundation of formal knowledge in sci-ence. I also try to explain why Husserl said that there was an evident correlation between his and Hilbert’s notion (...) of completeness introduced in connection with the axiomatisation of geometry and the theory of real numbers when, as many commen-tators have already observed, these two notions are independent. I show in this paper that if a system of axioms is complete in Husserl’s sense, then its formal domain, the manifold of formal objects it determines, does not admit any extension. This is precisely the idea behind Hilbert’s notion of completeness in question. Therefore, the correlation Husserl noted indeed exists. But, in order to see it, we must consider the formal domain determined by a formal theory, not its models. (shrink)