The Annual European Meeting of the Association for Symbolic Logic, also known as the Logic Colloquium, is among the most prestigious annual meetings in the field. The current volume, Logic Colloquium 2007, with contributions from plenary speakers and selected special session speakers, contains both expository and research papers by some of the best logicians in the world. This volume covers many areas of contemporary logic: model theory, proof theory, set theory, and computer science, as well as philosophical logic, including tutorials on cardinal arithmetic, on Pillay's conjecture, and on automatic structures. This volume will be invaluable for experts as well as those interested in an overview of central contemporary themes in mathematical logic.

The Logical Must is an examination of Ludwig Wittgenstein's philosophy of logic, early and late, undertaken from an austere naturalistic perspective Penelope Maddy has called "Second Philosophy." The Second Philosopher is a humble but tireless inquirer who begins her investigation of the world with ordinary perceptual beliefs, moves from there to empirical generalizations, then to deliberate experimentation, and eventually to theory formation and confirmation. She takes this same approach to logical truth, locating its ground in simple worldly structures and our knowledge of it in our basic cognitive machinery, tuned by evolutionary pressures to detect those structures where they occur.
In his early work Tractatus Logico-Philosophicus, Wittgenstein also links the logical structure of representation with the structure of the world, but he includes one key unnaturalistic assumption: that the sense of our representations must be given prior to-independently of-facts about how the world is. When that assumption is removed, the general outlines of the resulting position come surprisingly close to the Second Philosopher's roughly empirical account. In his later discussions of logic in Philosophical Investigations and Remarks on the Foundations of Mathematics, Wittgenstein also rejects this earlier assumption in favor of a picture that arises in the wake of the famous rule-following considerations. Here Wittgenstein and the Second Philosopher operate in even closer harmony-locating the ground of our logical practices in our interests, our natural inclinations and abilities, and very general features of the world-until the Second Philosopher moves to fill in the account with her empirical investigations of the world and cognition. At this point, Wittgenstein balks, but as a matter of personal animosity rather than philosophical principle.

The philosopher Penelope Maddy is well-known for her pursuit of 'Second Philosophy', a form of naturalism that sees the methods of philosophy as indistinguishable from those of the empirical sciences. This volume collects eleven of her recent essays (five new and six reprinted), exploring a wide range of philosophical topics—from methodology, epistemology, and the philosophy of science, to the philosophies of logic, arithmetic, and higher mathematics. Though the topics vary widely, each essay bears in one way or another on the description, exploration, or application of Second Philosophy, revealing the underlying systematic character of Maddy's thought. The title essay traces the source of second-philosophical thinking to the 'natural philosophy' of the early modern period, when 'science' and 'philosophy' weren't separate disciplines; a companion essay, drawing second-philosophical morals for the realism/instrumentalism debate in the philosophy of science rounds out the opening section on philosophical method. The second section, on external world skepticism, is largely historical: an essay comparing the naturalistic credentials of Hume and Reid, then one each on Moore and Wittgenstein. A second-philosophical examination of debates over truth and reference, starring J. L. Austin, opens the section on language and logic, followed by a broad-brush description of historical landmarks in the philosophy of logic and an executive summary of the Second Philosopher's view. The concluding section on mathematics begins with an essay addressed to undergraduates on the ontology of number and another assessing the bearing of contemporary developmental psychology on the philosophies of logic and arithmetic. The concluding essay is an attempt to revive the often-ridiculed if-thenist position in the philosophy of mathematics. Maddy's second-philosophical essays offer new insight into long-standing questions in the philosophy of science, epistemology, the philosophies of language, logic, mathematics-all with an eye to the methodological themes that connect them.

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.

The Annual European Meeting of the Association for Symbolic Logic, also known as the Logic Colloquium, is among the most prestigious annual meetings in the field. The current volume, Logic Colloquium 2007, with contributions from plenary speakers and selected special session speakers, contains both expository and research papers by some of the best logicians in the world. This volume covers many areas of contemporary logic: model theory, proof theory, set theory, and computer science, as well as philosophical logic, including tutorials on cardinal arithmetic, on Pillay's conjecture, and on automatic structures. This volume will be invaluable for experts as well as those interested in an overview of central contemporary themes in mathematical logic.

Many philosophers these days consider themselves naturalists, but it's doubtful any two of them intend the same position by the term. In this book, Penelope Maddy describes and practises a particularly austere form of naturalism called 'Second Philosophy'. Without a definitive criterion for what counts as 'science' and what doesn't, Second Philosophy can't be specified directly - 'trust only the methods of science!' or some such thing - so Maddy proceeds instead by illustrating the behaviours of an idealized inquirer she calls the 'Second Philosopher'. This Second Philosopher begins from perceptual common sense and progresses from there to systematic observation, active experimentation, theory formation and testing, working all the while to assess, correct and improve her methods as she goes. Second Philosophy is then the result of the Second Philosopher's investigations. Maddy delineates the Second Philosopher's approach by tracing her reactions to various familiar skeptical and transcendental views (Descartes, Kant, Carnap, late Putnam, van Fraassen), comparing her methods to those of other self-described naturalists (especially Quine), and examining a prominent contemporary debate (between disquotationalists and correspondence theorists in the theory of truth) to extract a properly second-philosophical line of thought. She then undertakes to practise Second Philosophy in her reflections on the ground of logical truth, the methodology, ontology and epistemology of mathematics, and the general prospects for metaphysics naturalized.

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.

Our much-valued mathematical knowledge rests on two supports: the logic of proof and the axioms from which those proofs begin. Naturalism in Mathematics investigates the status of the latter, the fundamental assumptions of mathematics. These were once held to be self-evident, but progress in work on the foundations of mathematics, especially in set theory, has rendered that comforting notion obsolete. Given that candidates for axiomatic status cannot be proved, what sorts of considerations can be offered for or against them? That is the central question addressed in this book. One answer is that mathematics aims to describe an objective world of mathematical objects, and that axiom candidates should be judged by their truth or falsity in that world. This promising view-realism-is assessed and finally rejected in favour of another-naturalism-which attends less to metaphysical considerations of objective truth and falsity, and more to practical considerations drawn from within mathematics itself. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be helpfully applied in the assessment of candidates for axiomatic status in set theory. Maddy's clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.

When engaged in mathematics, most people 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? Penelope Maddy delineates and defends a novel version of mathematical realism that answers the traditional questions and refocuses philosophical attention on the pressing foundational issues of contemporary mathematics.

The Logical Must is an examination of Ludwig Wittgenstein's philosophy of logic, early and late, undertaken from an austere naturalistic perspective Penelope Maddy has called "Second Philosophy." The Second Philosopher is a humble but tireless inquirer who begins her investigation of the world with ordinary perceptual beliefs, moves from there to empirical generalizations, then to deliberate experimentation, and eventually to theory formation and confirmation. She takes this same approach to logical truth, locating its ground in simple worldly structures and our knowledge of it in our basic cognitive machinery, tuned by evolutionary pressures to detect those structures where they occur. In his early work Tractatus Logico-Philosophicus, Wittgenstein also links the logical structure of representation with the structure of the world, but he includes one key unnaturalistic assumption: that the sense of our representations must be given prior to-independently of-facts about how the world is. When that assumption is removed, the general outlines of the resulting position come surprisingly close to the Second Philosopher's roughly empirical account. In his later discussions of logic in Philosophical Investigations and Remarks on the Foundations of Mathematics, Wittgenstein also rejects this earlier assumption in favor of a picture that arises in the wake of the famous rule-following considerations. Here Wittgenstein and the Second Philosopher operate in even closer harmony-locating the ground of our logical practices in our interests, our natural inclinations and abilities, and very general features of the world-until the Second Philosopher moves to fill in the account with her empirical investigations of the world and cognition. At this point, Wittgenstein balks, but as a matter of personal animosity rather than philosophical principle.

Many philosophers these days consider themselves naturalists, but it's doubtful any two of them intend the same position by the term. In this book, Penelope Maddy describes and practices a particularly austere form of naturalism called "Second Philosophy." Without a definitive criterion for what counts as "science" and what doesn't, Second Philosophy can't be specified directly - "trust only the methods of science " or some such thing - so Maddy proceeds instead by illustrating the behaviors of an idealized inquirer she calls the "Second Philosopher." This Second Philosopher begins from perceptual common sense and progresses from there to systematic observation, active experimentation, theory formation and testing, working all the while to assess, correct and improve her methods as she goes. Second Philosophy is then the result of the Second Philosopher's investigations.
Maddy delineates the Second Philosopher's approach by tracing her reactions to various familiar skeptical and transcendental views (Descartes, Kant, Carnap, late Putnam, van Fraassen), comparing her methods to those of other self-described naturalists (especially Quine), and examining a prominent contemporary debate (between disquotationalists and correspondence theorists in the theory of truth) to extract a properly second-philosophical line of thought. She then undertakes to practise Second Philosophy in her reflections on the ground of logical truth, the methodology, ontology and epistemology of mathematics, and the general prospects for metaphysics naturalized.

How do you know the world around you isn't just an elaborate dream, or the creation of an evil neuroscientist? If all you have to go on are various lights, sounds, smells, tastes and tickles, how can you know what the world is really like, or even whether there is a world beyond your own mind? Questions like these - familiar from science fiction and dorm room debates - lie at the core of venerable philosophical arguments for radical skepticism: the stark contention that we in fact know nothing at all about the world, that we have no more reason to believe any claim - that there are trees, that we have hands - than we have to disbelieve it. Like non-philosophers in their sober moments, philosophers, too, find this skeptical conclusion preposterous, but they're faced with those famous arguments: the Dream Argument, the Argument from Illusion, the Infinite Regress of Justification, the more recent Closure Argument. If these can't be met, they raise a serious challenge not just to philosophers, but to anyone responsible enough to expect her beliefs to square with her evidence. What Do Philosophers Do? takes up the skeptical arguments from this everyday point of view, and ultimately concludes that they don't undermine our ordinary beliefs or our ordinary ways of finding out about the world. In the process, Maddy examines and evaluates a range of philosophical methods - common sense, scientific naturalism, ordinary language, conceptual analysis, therapeutic approaches - as employed by such philosophers as Thomas Reid, G. E. Moore, Ludwig Wittgenstein, and J. L. Austin. The result is a revealing portrait of what philosophers do, and perhaps a quiet suggestion for what they should do, for what they do best.

Naturalism in Mathematics investigates how the most fundamental assumptions of mathematics can be justified. One prevalent philosophical approach to the problem--realism--is examined and rejected in favour of another approach--naturalism--which attends more closely to practical considerations drawn from within mathematics itself. Penelope Maddy defines naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her clear, original discussion is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines.