This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse (...) Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis--finding the "right axioms" to prove fundamental theorems--and giving a novel approach to logic. Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the "right axiom" to prove it. By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics. (shrink)

I present a new interpretation of Wittgenstein's later philosophy of logic and mathematics. This interpretation, like others, emphasizes Wittgenstein's attempt to reconcile platonistic and constructivistic approaches. But, unlike other interpretations, mine explains that attempt in terms of Wittgenstein's position about the relations between our concepts of necessity and provability. If what I say here is correct, then we can rescue Wittgenstein from the charge of naive relativism. For his relativism extends only to provability, and not to necessity.

Big Ideas for Little Kids includes everything a teacher, a parent, or a college student needs to teach philosophy to elementary school children from picture books. Written in a clear and accessible style, the book explains why it is important to allow young children access to philosophy during primary-school education. Wartenberg also gives advice on how to construct a "learner-centered" classroom, in which children discuss philosophical issues with one another as they respond to open-ended questions by saying whether they agree (...) or disagree with what others have said. (shrink)

This paper explores different interpretations of the word ‘deep’ as it is used by mathematicians, with a large number of examples illustrating various criteria for depth. Most of the examples are theorems with ‘historical depth’, in the sense that many generations of mathematicians contributed to their proof. Some also have ‘foundational depth’, in the sense that they support large mathematical theories. Finally, concepts from mathematical logic suggest that it may be possible to order certain theorems or problems according to ‘logical (...) depth’. (shrink)

This Element aims to present an outline of mathematics and its history, with particular emphasis on events that shook up its philosophy. It ranges from the discovery of irrational numbers in ancient Greece to the nineteenth- and twentieth-century discoveries on the nature of infinity and proof. Recurring themes are intuition and logic, meaning and existence, and the discrete and the continuous. These themes have evolved under the influence of new mathematical discoveries and the story of their evolution is, to a (...) large extent, the story of philosophy of mathematics. (shrink)

Hilbert took to using ideal elements in the 1890's, in both algebraic number theory and geometry. His Zahlbericht of 1897 popularized the concept of the ideal introduced by Dedekind in 1871 (which in turn formalized the concept of "ideal number" introduced by Kummer in the 1840's). His geometric work likewise followed a long history of ideal elements, some that originated in geometry and others that originated elsewhere and were applied to geometry. Important examples were:Piero della Francesca's Ideal City.Points at infinity (...) in projective geometry, originating in the 1430s from the "vanishing points" in perspective drawing.Imaginary points in algebraic geometry, originating from the imaginary numbers used by .. (shrink)

The following three related contributions jointly serve to lift up elements of the thought of the anthropolo­gist Gregory Bateson that can be fruitfully compared with elements of Michael Polanyi’s thought. In a brief introduction, William Stillwell reviews Bateson’s life and developing interests. Stillwell also provides, in a creative dialog form akin to Bateson’s own dialogs, a short review article on Noel Charlton’s Understanding Gregory Bateson: Mind, Beauty and the Sacred Earth. The third piece is Jere Moorman’s short 1991 (...) essay (now out of print) discussing Polanyi’s ideas about tacit knowing and their connection with Bateson’s ideas about the double bind. (shrink)

How the concept of proof has enabled the creation of mathematical knowledge. The Story of Proof investigates the evolution of the concept of proof--one of the most significant and defining features of mathematical thought--through critical episodes in its history. From the Pythagorean theorem to modern times, and across all major mathematical disciplines, John Stillwell demonstrates that proof is a mathematically vital concept, inspiring innovation and playing a critical role in generating knowledge. Stillwell begins with Euclid and his influence (...) on the development of geometry and its methods of proof, followed by algebra, which began as a self-contained discipline but later came to rival geometry in its mathematical impact. In particular, the infinite processes of calculus were at first viewed as "infinitesimal algebra," and calculus became an arena for algebraic, computational proofs rather than axiomatic proofs in the style of Euclid. Stillwell proceeds to the areas of number theory, non-Euclidean geometry, topology, and logic, and peers into the deep chasm between natural number arithmetic and the real numbers. In its depths, Cantor, Gödel, Turing, and others found that the concept of proof is ultimately part of arithmetic. This startling fact imposes fundamental limits on what theorems can be proved and what problems can be solved. Shedding light on the workings of mathematics at its most fundamental levels, The Story of Proof offers a compelling new perspective on the field's power and progress. (shrink)

Ikujiro Nonaka, whose formative experience is Japanese, is an established scholar who has written about large business organizations. He sees knowledge at the heart of the organization and its products and aims to develop Michael Polanyi’s conception of tacit knowledge in a practical direction to enhance organizational “knowledge creation.” For Nonaka, what matters is the practice, the doing, the embodiment of knowledge. An organization can amplify and crystallize individuals’ tacit knowledge in a process that allows them to experience deeper understanding. (...) Nonaka holds that it is iimportant to explore the potential that knowledge holds. His spiral process describes disciplined practices that make tacit knowledge independent and available to restructure the organizational knowledge context. (shrink)

This paper is an exposition and evaluation of the Agustín Rayo's views about the epistemology and metaphysics of mathematics, as they are presented in his book The Construction of Logical Space.

This volume is a study of the influence of Timaeus on the development of Western cosmology in three axial periods of European culture: Late Antiquity, Middle Ages and Renaissance.

Glymour and Quine propose two different formal criteria for theoretical equivalence. In this paper we examine the relationships between these criteria.

This book is a fiftieth anniversary republication of Thomas Nagel's "What Is It Like to Be a Bat?", a classic in the philosophy of mind. Through its argument for the irreducible subjectivity of consciousness, it played an essential role in making the study of consciousness a central part of philosophy, psychology, and neuroscience. It also spurred the now flourishing scientific attention to the consciousness of non-human creatures: mammals, birds, fish, mollusks, and insects. The book also includes a second essay (...) offering Nagel's more recent thoughts on the most promising positive response to the mind-body problem, as posed in the original essay. (shrink)

The purported fact that geometric theories formulated in terms of points and geometric theories formulated in terms of lines are “equally correct” is often invoked in arguments for conceptual relativity, in particular by Putnam and Goodman. We discuss a few notions of equivalence between first-order theories, and we then demonstrate a precise sense in which this purported fact is true. We argue, however, that this fact does not undermine metaphysical realism.

This paper presents a simple pair of first-order theories that are not definitionally (nor Morita) equivalent, yet are mutually conservatively translatable and mutually 'surjectively' translatable. We use these results to clarify the overall geography of standards of equivalence and to show that the structural commitments that theories make behave in a more subtle manner than has been recognized.

Thomas Reid saw the three subjects of logic, rhetoric, and the fine arts as closely cohering aspects of one endeavor that he called the culture of the mind. This was a topic on which Reid lectured for many years in Glasgow, and this volume presents as near a reconstruction of these lectures as is now possible. Though virtually unknown today, this material in fact relates closely to Reid's published works and in particular to the late Essays on the Intellectual (...) Powers of Man and Essays on the Active Powers of Man. When composing these works, Reid drew primarily on his lectures on "pneumatology," which presented a theory of the mental powers, broadly conceived. These lectures were basic to the course on the culture of the mind that explained the cultivation of the mental powers. Although the Essays also included some elements from the material on the culture of the mind, the bulk of the latter was left in manuscript form, and Alexander Broadie's edition restores this important extension of Reid's overall work. In addition, this volume continues the attractive combination of manuscript material and published work, in this case Reid's important and well-known essay on Aristotle's logic. This text was corrupted in earlier editions of Reid's works and is now restored to the state in which Reid left it. This volume underscores Reid's great and growing significance, viewed both as a historical figure and as a philosopher. At the same time, it is of great interdisciplinary importance. While the material emerges directly from the core of Reid's philosophy, as now understood, it will appeal widely to people in literary, cultural, historical, and communications studies. In this regard, the present volume is a true fruit of the Scottish Enlightenment. (shrink)

Wilhelm (Forthcom Synth 199:6357–6369, 2021) has recently defended a criterion for comparing structure of mathematical objects, which he calls Subgroup. He argues that Subgroup is better than SYM \(^*\), another widely adopted criterion. We argue that this is mistaken; Subgroup is strictly worse than SYM \(^*\). We then formulate a new criterion that improves on both SYM \(^*\) and Subgroup, answering Wilhelm’s criticisms of SYM \(^*\) along the way. We conclude by arguing that no criterion that looks only to the (...) automorphisms of mathematical objects to compare their structure can be fully satisfactory. (shrink)

Quine often argued for a simple, untyped system of logic rather than the typed systems that were championed by Russell and Carnap, among others. He claimed that nothing important would be lost by eliminating sorts, and the result would be additional simplicity and elegance. In support of this claim, Quine conjectured that every many-sorted theory is equivalent to a single-sorted theory. We make this conjecture precise, and prove that it is true, at least according to one reasonable notion of theoretical (...) equivalence. Our clarification of Quine’s conjecture, however, exposes the shortcomings of his argument against many-sorted logic. (shrink)

This paper makes an observation about the ``amount of structure'' that different classical and relativistic spacetimes posit. The observation substantiates a suggestion made by Earman and yields a cautionary remark concerning the scope and applicability of structural parsimony principles.

The Essays on the Active Powers of Man was Thomas Reid's last major work. It was conceived as part of one large work, intended as a final synoptic statement of his philosophy. The first and larger part was published three years earlier as Essays on the Intellectual Powers of Man. These two works are united by Reid's basic philosophy of common sense, which sets out native principles by which the mind operates in both its intellectual and active aspects. The (...) Active Powers shows how these principles are involved in volition, action, and the ability to judge morally. Reid gives an original twist to a libertarian and realist tradition that was prominently represented in eighteenth-century British thought by such thinkers as Samuel Clarke and Richard Price. (shrink)

Explanations contribute to our understanding of the world byembedding phenomena into general nomic patterns that we recognize in the world. Manyof these patterns are, of course, causal ones, but the declaration as ``causal'' often fails to determinethe explanatory power of the pattern. More important is the systematization capacity and the empiricalcontent of the pattern or theory with respect to explanations. We can specify these parameters moreprecisely within the framework of the structuralist view of theories.

The concept of an ecological niche (econiche) has been used in a variety of ways, some of which are incompatible with a relational or functional interpretation of the term. This essay seeks to standardize usage by limiting the concept to functional relations between organisms and their surroundings, and to revise the concept to include epistemic relations. For most organisms, epistemics are a vital aspect of their functional relationships to their surroundings and, hence, a major determinant of their econiche. Rejecting the (...) traditional dualism of organism and environment, an econiche is defined as the reciprocal (dual) of a functionally specified class of organisms (FSTU). From this perspective, an econiche necessarily implies a certain type of organism, and a class of functionally similar organisms implies a special econiche.The econiche concept is also discussed in relation to other ecological terms that reflect the distributional patterns of organisms, such as habitat, and the concept of an empty niche is criticized. (shrink)

The definition of art is controversial in contemporary philosophy. Whether art can be defined has also been a matter of controversy. The philosophical usefulness of a definition of art has also been debated. -/- Contemporary definitions can be classified with respect to the dimensions of art they emphasize. One distinctively modern, conventionalist, sort of definition focuses on art’s institutional features, emphasizing the way art changes over time, modern works that appear to break radically with all traditional art, the relational properties (...) of artworks that depend on works’ relations to art history, art genres, etc. – more broadly, on the undeniable heterogeneity of the class of artworks. The more traditional, less conventionalist sort of definition defended in contemporary philosophy makes use of a broader, more traditional concept of aesthetic properties that includes more than art-relational ones, and puts more emphasis on art’s pan-cultural and trans-historical characteristics – in sum, on commonalities across the class of artworks. Hybrid definitions aim to do justice to both the traditional aesthetic dimension as well as to the institutional and art-historical dimensions of art, while privileging neither. (shrink)

It has been suggested that we can tell whether two theories are equivalent by comparing the structure that they ascribe to the world. If two theories posit different structures, then they must be i...

" Our various cultures are symbolic environments or "spiritual ecologies" within which the Human Eros can thrive. This is how we inhabit the earth. Encircling and sustaining our cultural existence is nature.

In his Treatise on the Virtues, Aquinas discusses the character and function of habit; the essence, subject, cause, and meaning of virtue; and the separate intellectual, moral, cardinal, and theological virtues. His work constitutes one of the most thorough and incisive accounts of virtue in the history of Christian philosophy. John Oesterle's accurate and elegant translation makes this enduring work readily accessible to the modern reader.

Thomas Alexander shows that the primary, guiding concern of Dewey's philosophy is his theory of aesthetic experience.