Many philosophers these days consider themselves naturalists, but it's doubtful any two of them intend the same position by the term. In Second Philosophy, 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" for example), so Maddy proceeds instead by illustrating the behaviors of an idealized inquirer she calls the "Second Philosopher". (...) mhis Second Philosopher begins from perceptual common sense experimentation, theory formation and testing, working all the while to asses, 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 practice 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. (shrink)

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 favor of another approach--naturalism. Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be successfully applied in set theory. Her 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.

Husserl and Mathematics explains the development of Husserl's phenomenological method in the context of his engagement in modern mathematics and its foundations. Drawing on his correspondence and other written sources, Mirja Hartimo details Husserl's knowledge of a wide range of perspectives on the foundations of mathematics, including those of Hilbert, Brouwer and Weyl, as well as his awareness of the new developments in the subject during the 1930s. Hartimo examines how Husserl's philosophical views responded to these changes, and offers a (...) pluralistic and open-ended picture of Husserl's phenomenology of mathematics. Her study shows Husserl's phenomenology to be a method capable of both shedding light on and internally criticizing scientific practices and concepts. (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.

Normativity concerns what we ought to think or do and the evaluations we make. For example, we say that we ought to think consistently, we ought to keep our promises, or that Mozart is a better composer than Salieri. Yet what philosophical moral can we draw from the apparent absence of normativity in the scientific image of the world? For scientific naturalists, the moral is that the normative must be reduced to the nonnormative, while for nonnaturalists, the moral is that (...) there must be a transcendent realm of norms. _Naturalism and Normativity_ engages with both sides of this debate. Essays explore philosophical options for understanding normativity in the space between scientific naturalism and Platonic supernaturalism. They articulate a liberal conception of philosophy that is neither reducible to the sciences nor completely independent of them—yet one that maintains the right to call itself naturalism. Contributors think in new ways about the relations among the scientific worldview, our experience of norms and values, and our movements in the space of reason. Detailed discussions include the relationship between philosophy and science, physicalism and ontological pluralism, the realm of the ordinary, objectivity and subjectivity, truth and justification, and the liberal naturalisms of Donald Davidson, John Dewey, John McDowell, and Ludwig Wittgenstein. (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)

The Logical Must is an examination of Wittgenstein's philosophy of logic, early and late, from an austere naturalistic perspective called "Second Philosophy.".

Can the straight line be analysed mathematically such that it does not fall apart into a set of discrete points, as is usually done but through which its fundamental continuity is lost? And are there objects of pure mathematics that can change through time? Mathematician and philosopher L.E.J. Brouwer argued that the two questions are closely related and that the answer to both is "yes''. To this end he introduced a new kind of object into mathematics, the choice sequence. But (...) other mathematicians and philosophers have been voicing objections to choice sequences from the start. This book aims to provide a sound philosophical basis for Brouwer's choice sequences by subjecting them to a phenomenological critique in the style of the later Husserl. (shrink)

In this paper I will give a systematic account of Husserl's notion of the natural attitude in the development from its first presentation in Ideas I (1913) until Husserl's last years. The problem of the natural attitude has to be dealt with on two levels. On the thematic level, it is constituted by the correlation of attitude and horizon, both stemming from Husserl's theory of intentionality. On the methodic level, the natural attitude is constituted by three factors: naturalness, naivety and (...) normality. I shall conclude by sketching out a possible motivation for leaving the natural attitude and thus for entering the sphere of phenomenology. (shrink)

Can the straight line be analysed mathematically such that it does not fall apart into a set of discrete points, as is usually done but through which its fundamental continuity is lost? And are there objects of pure mathematics that can change through time? The mathematician and philosopher L.E.J. Brouwer argued that the two questions are closely related and that the answer to both is "yes''. To this end he introduced a new kind of object into mathematics, the choice sequence. (...) But other mathematicians and philosophers have been voicing objections to choice sequences from the start. This book aims to provide a sound philosophical basis for Brouwer's choice sequences by subjecting them to a phenomenological critique in the style of the later Husserl. "It is almost as if one could hear the two rebels arguing their case in a European café or on a terrace, and coming to a common understanding, with both men taking their hat off to the other, in admiration and gratitude. Dr. van Atten has convincingly applied Husserl's method to Brouwer's program, and has equally convincingly applied Brouwer's intuition to Husserl's program. Both programs have come out the better." Piet Hut, professor of Interdisciplinary Studies, Institute for Advanced Study, Princeton, U.S.A. (shrink)

Husserl repeatedly has claimed that (1) mathematics without a philosophical foundation is not a science but a mere technique; (2) philosophical considerations may lead to the rejection of parts of mathematical practice; but (3) they cannot lead to mathematical innovations. My thesis is that Husserl's third claim is wrong, by his own standards.

In seinen ALogischen UntersuchungenA (1900/01) hatte Husserl die EigenstAndigkeit und IdealitAt logischer Gebilde gegenA1/4ber einem falschen Subjektivismus und Psychologismus geltend gemacht. Es ging ihm dabei nicht um die formale Logik selbst als solche, sondern um die BegrA1/4ndungsfunktion der Logik fA1/4r eine apriorische Wissenschaftslehre. Diese in den AIdeenA (1913) weitergefA1/4hrte Konstitutionsproblematik wird nun in dem Werk A1/4ber AFormale und transzendentale LogikA im Sinne einer AKritik der logischen VernunftA vertieft und durch den Aoebergang von der formalen zur transzendentalen Logik begrifflich entschiedener gefaAt.