Feng Ye

The thesis Physicalism about Cognitive Subjects (PCS) says that human cognitive subjects and processes are themselves physical things and processes, namely, human brain neural networks, neural processes, and their physical interactions with human environments. This paper will first introduce reasons supporting PCS and introduce my previous researches exploring implications of PCS, which include a nominalist, fictionalist and finitist account of human practices of classical mathematics. After these introductions, this paper will argue that PCS implies liberalism for mathematical practices, which can even legitimize logically inconsistent practices in some situations. Several philosophical questions about ultrafinitism, experimental mathematics and the set-theoretic multiverse view will then be answered in this context. Finally, since PCS has answered the ontological and epistemological questions about infinity in mathematics, this paper will offer some analyses on two other questions concerning infinity that are still open.

This chapter will extend the model developed in Chaps. 3 and 4 to model some abstract concepts and abstract thoughts composed of abstract concepts, which do not represent any physical things, properties or states directly. It will focus on mathematical concepts and thoughts and the result is a radically physicalistic, nominalistic and strictly finitistic philosophy of mathematics. It will first argue that, in terms of their semantic, ontological and epistemological aspects, mathematical concepts are very similar to concepts ‘representing fictional entities’ in fictions and a mathematical theory is similar to a fiction. The real difference between them consists in their pragmatic and aesthetic aspects. A physicalistic account of fictions will be illustrated in the course of this argument. Then, the rest of this chapter will illustrate cognitive functions of mathematical concepts and thoughts and offer physicalistic, nominalistic and strictly finitistic accounts for understanding, knowledge, intuition, objectivity, truth, and alleged infinity in mathematical practices. The chapter ends with an introduction to a logical explanation of the applicability of classical mathematics in the sciences under physicalism, nominalism and strict finitism. The details of this logical explanation of applicability have been published elsewhere.

This chapter first argues that there is a serious problem in contemporary philosophy: given contemporary cognitive science on human cognitive (vs. phenomenal experiential and spiritual) activities, we should admit that human cognitive subjects are human brains, not any non-physical, amorphous, soul-like Subject hiding inside a brain, but many traditional philosophical concepts and theories exactly presuppose a non-physical Subject of cognition. This presupposition threatens the internal coherence of many philosophers’ philosophical positions, irrespective of whether they endorse physicalism or naturalism. For sympathizers of physicalism, recognizing this problem should motivate No-Self Physicalism, a version of physicalism that calls for renewing the entire traditional philosophy to get rid of all presuppositions of non-physical Subjects and redefine philosophical concepts and reconstruct philosophical theories by explicitly treating human brains as the subjects of cognition and action. No-Self Physicalism is a framework for doing philosophy in all philosophical areas, not just a metaphysical position in philosophy of mind. After motivating No-self Physicalism this way, this chapter will elaborate the major ideas of No-Self Physicalism and summarize reasons supporting them. These together constitute an introduction to No-Self Physicalism, while Chaps. 3–8 of this book belong to doing real philosophy under the framework of No-Self Physicalism.

This chapter will develop a model for concepts and conceptual representation under the framework of No-Self Physicalism. Concepts are assumed to be concrete neural structures in brains. It will be shown that this model correctly predicts the representation relation for a variety of concepts and that it successfully resolves the Problem of Indeterminacy, Problem of Circularity, and several well-known philosophical puzzles about conceptual representation. This model will serve as the basis for developing other physicalistic theories on truth, analyticity, modality, apriority etc. in later chapters of this book. The model will take many ideas from contemporary theories of concepts in cognitive psychology. The major new contribution here is to design a way to combine ingredients from these psychological theories into a single model, so that it can resolve the philosophical problems and puzzles that current theories of conceptual representation in the philosophical literature cannot resolve. The model will consist of several layers. Each concept is assumed to be a structured entity consisting of layered components, with perceptual images as the most basic components. Concepts in a brain are linked into a conceptual network. This chapter will start with an overview of the model before diving into the details.

This chapter will extend the model developed in the last chapter to cover thoughts and truth and then use the model to give an account of belief ascription, analyticity, modality, and some other topics under the framework of No-Self Physicalism. Section 4.1 will describe the structures and functions of thoughts in individual brains. Section 4.2 first characterizes truth for thoughts with respect to their environments and then demonstrates how truth and usefulness are two distinct properties of thoughts but are probabilistically positively correlated in normal environments. Section 4.3 then discusses some social phenomena about thoughts, including thought individuation, the public meaning of a sentence, and propositions in the traditional sense. Sections 4.4 and 4.5 apply this model to characterize belief ascription and analyticity respectively. Finally, Sect. 4.6 will use the model to give an account of modality. It will include an interpretation of Kripke’s metaphysical modality and show that metaphysically necessary but a posteriori truths are of the same nature as physically necessary but a posteriori truths (and are therefore nothing surprising). All these show that No-Self Physicalism can offer many new insights into issues in philosophy of language.

The ontology of physicalism says that all that really exist are physical. Then, three major ontological questions remain. (1) Is ‘physical’ referring to those recognized by our current physics or whatever recognized by a future, ideal physics theory? (2) Assuming that the physical are those recognized by our current physics, what exactly are they? (3) What are mental properties and how are they related to the physical? This chapter will try to answer these questions. I will argue that we should use ‘physical’ to mean whatever recognized by our current physics. I will reject the traditional object-property ontological framework and propose a new ontological framework with physical systems as the only existent entities. Most properties in the traditional framework will be treated as fictions, similar to abstract mathematical entities treated in Chap. 5. This answers the question (2) above. It also implies that alleged multiply realizable mental properties are fictions. Further, I will use the ontology of physical systems to give an account of the semantics of mental predicates and discuss several different senses in which the mental is reducible or irreducible to the physical. These will answer the question (3) above.

This chapter will round up our studies in No-Self Physicalism by giving a general characterization of No-Self Physicalism as a global philosophical worldview and responding to some well-known arguments against physicalism. Section 8.1 will summarize the basic tenets of No-Self Physicalism by using the physicalistic notions of concept, truth, analyticity, necessity and apriority characterized in previous chapters. Section 8.2 will respond to the knowledge argument, conceivability argument and explanatory gap argument against physicalism. Resources from previous chapters will also be utilized.

This chapter studies epistemology and methodology under No-Self Physicalism. There are three objectives. The first is to propose a physicalistic framework for undertaking various types of epistemic evaluations, with epistemic justification and knowledge as two special types. The framework belongs to process reliabilism and is based on the model of concepts and thoughts developed in Chaps. 3 and 4, as well as some new hypotheses about cognitive processes introduced here. The framework can successfully respond to well-known objections against process reliabilism in the literature. The second objective is to offer a physicalistic definition of apriority. This definition implies that a hypothetical complete physical description of the world would entail, a priori, all truths about the world, including truths stated with phenomenal concepts. This is so without assuming analytical reducibility of phenomenal concepts to physical concepts. This means that No-Self Physicalism is a priori and synthetic physicalism. The third objective is to argue that several kinds of intuitions that we use in the humanities and philosophy are justified methods for achieving knowledge, while at the same time they are not practically reducible to hard-scientific methods. It means that No-Self Physicalism supports moderate, not extreme, methodological naturalism.

This chapter is a supplementary demonstration of the problem of subject in contemporary philosophy raised in the last chapter. I will examine more examples in more detail, to show that philosophers who apparently favor physicalism frequently presuppose a non-physical Subject of cognition in their philosophical concepts and views. Two case-studies on Carnap and Quine will be presented in Sects. 2.1 and 2.2 respectively. Both Carnap and Quine apparently favor physicalism. However, I will show how some of their assertions implicitly but indispensably presuppose the image of a Subject versus the External World and therefore presuppose a non-physical Subject. I will argue that this fact threatens the internal coherence of their philosophical positions. These analyses apply to many other contemporary philosophers as well, because they adopt the Carnapian and Quinean philosophical locution and thus share the image of a Subject versus the External World. Section 2.3 will continue to examine a few more philosophical concepts and views, showing that traditional understandings of them presuppose the image of a Subject versus the External World. I hope these detailed analyses can convince philosophers that there is a serious problem of subject in contemporary philosophy.

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.

This book intends to show that radical naturalism (or physicalism), 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 (sufficient for the applications in classical quantum mechanics and general relativity). 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.

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

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.

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

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.

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

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

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

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.