This paper explores how to explain the applicability of classical mathematics to the physical world in a radically naturalistic and nominalistic philosophy of mathematics. The applicability claim is first formulated as an ordinary scientific assertion about natural regularity in a class of natural phenomena and then turned into a logical problem by some scientific simplification and abstraction. I argue that there are some genuine logical puzzles regarding applicability and no current philosophy of mathematics has resolved these puzzles. Then I introduce (...) a plan for resolving the logical puzzles of applicability. (shrink)

This book intends to show that radical naturalism, nominalism and strict finitism account for the applications of classical mathematics in current scientific theories. The applied mathematical theories developed in the book include the basics of calculus, metric space theory, complex analysis, Lebesgue integration, Hilbert spaces, and semi-Riemann geometry. The fact that so much applied mathematics can be developed within such a weak, strictly finitistic system, is surprising in itself. It also shows that the applications of those classical theories to the (...) finite physical world can be translated into the applications of strict finitism, which demonstrates the applicability of those classical theories without assuming the literal truth of those theories or the reality of infinity. Both professional researchers and students of philosophy of mathematics will benefit greatly from reading this book. (shrink)

The dissertation studies the mathematical strength of strict constructivism, a finitistic fragment of Bishop's constructivism, and explores its implications in the philosophy of mathematics. ;It consists of two chapters and four appendixes. Chapter 1 presents strict constructivism, shows that it is within the spirit of finitism, and explains how to represent sets, functions and elementary calculus in strict constructivism. Appendix A proves that the essentials of Bishop and Bridges' book Constructive Analysis can be developed within strict constructivism. Appendix B further (...) develops, within strict constructivism, the essentials of the functional analysis applied in quantum mechanics, including the spectral theorem, Stone's theorem, and the self-adjointness of some common quantum mechanical operators. Some comparisons with other related work, in particular, a comparison with S. Simpson's partial realization of Hilbert's program, and a discussion of the relevance of M. B. Pour-El and J. I. Richards' negative results in recursive analysis are given in Appendix C. ;Chapter 2 explores the possible philosophical implications of these technical results. It first suggests a fictionalistic account for the ontology of pure mathematics. This leaves a puzzle about how truths about fictional mathematical entities are applicable to science. The chapter then explains that for those applications of mathematics that can be reduced to applications of strict constructivism, fictional entities can be eliminated in the applications and the puzzle of applicability can be resolved. Therefore, if strict constructivism were essentially sufficient for all scientific applications, the applicability of mathematics of mathematics in science would be accountable. The chapter then argues that the reduction of mathematics to strict constructivism also reduces the epistemological question about mathematics to that about elementary arithmetic. The dissertation ends with a suggestion that a proper epistemological basis for arithmetic is perhaps a mixture of Mill's empiricism and the Kantian views. (shrink)

This article attempts to motivate a new approach to anti-realism (or nominalism) in the philosophy of mathematics. I will explore the strongest challenges to anti-realism, based on sympathetic interpretations of our intuitions that appear to support realism. I will argue that the current anti-realistic philosophies have not yet met these challenges, and that is why they cannot convince realists. Then, I will introduce a research project for a new, truly naturalistic, and completely scientific approach to philosophy of mathematics. It belongs (...) to anti-realism, but can meet those challenges and can perhaps convince some realists, at least those who are also naturalists. (shrink)

I argue that the most popular versions of naturalism imply nominalism in philosophy of mathematics. In particular, there is a conflict in Quine's philosophy between naturalism and realism in mathematics. The argument starts from a consequence of naturalism on the nature of human cognitive subjects, physicalism about cognitive subjects, and concludes that this implies a version of nominalism, which I will carefully characterize. The indispensability of classical mathematics for the sciences and semantic/confirmation holism does not affect the argument. The disquotational (...) theory of reference and truth is discussed but rejected. This argument differs from the Benacerrafian arguments against realism, because it does not rely on any specific assumption about the nature of knowledge or reference. It differs from the popular objections to the indispensability argument for realism as well, because it can admit both indispensability and holism. This argument motivates a new, radically naturalistic and nominalistic approach to philosophy of mathematics. (shrink)

We show that the following results in the classical theory of unbounded linear operators on Hilbert spaces can be proved within the framework of Bishop's constructive mathematics: the Kato-Rellich theorem, the spectral theorem, Stone's theorem, and the self-adjointness of the most common quantum mechanical operators, including the Hamiltonians of electro-magnetic fields with some general forms of potentials.

The Kripkean metaphysical modality (i.e. possibility and necessity) is one of the most important concepts in contemporary analytic philosophy and is the basis of many metaphysical speculations. These metaphysical speculations frequently commit to entities that do not belong to this physical universe, such as merely possible entities, abstract entities, mental entities or qualities not realizable by the physical, which seems to contradict naturalism or physicalism. This paper proposes a naturalistic interpretation of the Kripkean modality, as a naturalist’s response to these (...) metaphysical speculations. It will show that naturalism can accommodate the Kripkean metaphysical modality. In particular, it will show that naturalism can help to resolve the puzzles surrounding Kripke’s a posteriori necessary propositions and a priori contingent propositions. (shrink)

There are three major theses in Plantinga’s latest version of his evolutionary argument against naturalism. (1) Given materialism, the conditional probability of the reliability of human cognitive mechanisms produced by evolution is low; (2) the same conditional probability given reductive or non-reductive materialism is still low; (3) the most popular naturalistic theories of content and truth are not admissible for naturalism. I argue that Plantinga’s argument for (1) presupposes an anti-materialistic conception of content, and it therefore begs the question against (...) materialism. To argue for (2), Plantinga claims that the adaptiveness of a belief is indifferent to its truth. I argue that this claim is unsupported unless it again assumes an anti-materialistic conception of content and truth. I further argue that Plantinga’s argument for (3) is not successful either, because an improved version of teleosemantics can meet his criticisms. Moreover, this version of teleosemantics implies that the truth of a belief is (probabilistically) positively related to its adaptiveness, at least for simple beliefs about physical objects in human environments. This directly challenges Plantinga’s claim that adaptiveness is indifferent to truth. (shrink)

This book intends to show that, in philosophy of mathematics, radical naturalism (or physicalism), nominalism and strict finitism (which does not assume the reality of infinity in any format, not even potential infinity) can account for the applications of classical mathematics in current scientific theories about the finite physical world above the Planck scale. For that purpose, the book develops some significant applied mathematics in strict finitism, which is essentially quantifier-free elementary recursive arithmetic (with real numbers encoded as elementary recursive (...) Cauchy sequences of rational numbers). Applied mathematical theories developed in the book include the basics of calculus, metric space theory, complex analysis, Lebesgue integration, Hilbert spaces, and semi-Riemann geometry (sufficient for the basic applications in classical quantum mechanics and general relativity). The fact that so much applied mathematics can be developed within such a weak, strictly finitisitc system is perhaps surprising in itself. It also shows that the applications of those classical theories to the finite physical world can be translated into the applications of strict finitism, which demonstrates the applicability of those classical theories without assuming the literal truth of those theories or the reality of infinity. The first chapter of the book contains an informal introduction to its philosophical motivation and technical strategy. The book is intended for students and researchers in philosophy of mathematics. -/- Contents -/- Preface 1 Introduction 2 Strict Finitism 3 Calculus 4 Metric Space 5 Complex Analysis 6 Integration 7 Hilbert Space 8 Semi-Riemann Geometry References Index. (shrink)

The indispensability argument for abstract mathematical entities has been an important issue in the philosophy of mathematics. The argument relies on several assumptions. Some objections have been made against these assumptions, but there are several serious defects in these objections. Ameliorating these defects leads to a new anti-realistic philosophy of mathematics, mainly: first, in mathematical applications, what really exist and can be used as tools are not abstract mathematical entities, but our inner representations that we create in imagining abstract mathematical (...) entities; second, the thoughts that we create in imagining infinite mathematical entities are bounded by external conditions. (shrink)

In a recent debate, Rosenberg claims that only the methods of natural science can deliver genuine knowledge, while Williamson rejects Rosenberg’s extreme methodological naturalism and insists that we have genuine philosophical and humanistic knowledge not achievable by hard-scientific methods alone. This paper responds to the debate. I will argue that physicalism, together with contemporary neurocognitive and evolutionary knowledge, implies that some of our intuitions and mental simulations used in the humanities and philosophy are justified methods for achieving knowledge but are (...) practically irreplaceable with hard-scientific methods. That is, extreme methodological naturalism is in conflict with physicalism. The argument also shows that some moderate version of methodological naturalism can be consistent with physicalism. Physicalism is the strong version of ontological naturalism and is supposed to be accepted by strongly committed naturalists like Rosenberg. Therefore, to be self-consistent, these naturalists should adopt physicalism plus a moderate version of methodological naturalism, rather than Rosenberg’s extreme version. (shrink)

This book demonstrates how a radical version of physicalism (‘No-Self Physicalism’) can offer an internally coherent and comprehensive philosophical worldview. It first argues that a coherent physicalist should explicitly treat a cognitive subject merely as a physical thing and should not vaguely assume an amorphous or even soul-like subject or self. This approach forces the physicalist to re-examine traditional core philosophical notions such as truth, analyticity, modality, apriority because our traditional understandings of them appear to be predicated on a cognitive (...) subject that is not literally just a physical thing. In turn, working on the assumption that a cognitive subject is itself completely physical, namely a neural network-based robot programmed by evolution (hence the term ‘No-Self’), the book proposes physicalistic theories on conceptual representation, truth, analyticity, modality, the nature of mathematics, epistemic justification, knowledge, apriority and intuition, as well as a physicalistic ontology. These are meant to show that this No-Self Physicalism, perhaps the most minimalistic and radical version of physicalism proposed to date, can accommodate many aspects that have traditionally interested philosophers. Given its refreshingly radical approach and painstakingly developed content, the book is of interest to anyone who is seeking a coherent philosophical worldview in this age of science. (shrink)