The paper presents an interpretation of Platonism, the seeds of which can be found in the writings of Gödel and Wittgenstein. Although it is widely accepted that Wittgenstein is an anti-Platonist the author points to some striking affinities between Gödel’s and Wittgenstein’s accounts of mathematical concepts and the role of feeling and intuition in mathematics. A version of Platonism emerging from these considerations combines realism with respect to concepts with a view of concepts as accessible to feeling and able to (...) guide our behavior through feeling. (shrink)

In a series of papers Ladyman and Presnell raise an interesting challenge of providing a pre-mathematical justification for homotopy type theory. In response, they propose what they claim to be an informal semantics for homotopy type theory where types and terms are regarded as mathematical concepts. The aim of this paper is to raise some issues which need to be resolved for the successful development of their types-as-concepts interpretation.

An Aristotelian Philosophy of Mathematics breaks the impasse between Platonist and nominalist views of mathematics. Neither a study of abstract objects nor a mere language or logic, mathematics is a science of real aspects of the world as much as biology is. For the first time, a philosophy of mathematics puts applied mathematics at the centre. Quantitative aspects of the world such as ratios of heights, and structural ones such as symmetry and continuity, are parts of the physical world and (...) are objects of mathematics. Though some mathematical structures such as infinities may be too big to be realized in fact, all of them are capable of being realized. Informed by the author's background in both philosophy and mathematics, but keeping to simple examples, the book shows how infant perception of patterns is extended by visualization and proof to the vast edifice of modern pure and applied mathematical knowledge. (shrink)

Homotopy Type Theory is a putative new foundation for mathematics grounded in constructive intensional type theory that offers an alternative to the foundations provided by ZFC set theory and category theory. This article explains and motivates an account of how to define, justify, and think about HoTT in a way that is self-contained, and argues that, so construed, it is a candidate for being an autonomous foundation for mathematics. We first consider various questions that a foundation for mathematics might be (...) expected to answer, and find that many of them are not answered by the standard formulation of HoTT as presented in the ‘HoTT Book’. More importantly, the presentation of HoTT given in the HoTT Book is not autonomous since it explicitly depends upon other fields of mathematics, in particular homotopy theory. We give an alternative presentation of HoTT that does not depend upon ideas from other parts of mathematics, and in particular makes no reference to homotopy theory, and argue that it is a candidate autonomous foundation for mathematics. Our elaboration of HoTT is based on a new interpretation of types as mathematical concepts, which accords with the intensional nature of the type theory. 1 Introduction2 What Is a Foundation for Mathematics?2.1 A characterization of a foundation for mathematics2.2 Autonomy3 The Basic Features of Homotopy Type Theory3.1 The rules3.2 The basic ways to construct types3.3 Types as propositions and propositions as types3.4 Identity3.5 The homotopy interpretation4 Autonomy of the Standard Presentation?5 The Interpretation of Tokens and Types5.1 Tokens as mathematical objects?5.2 Tokens and types as concepts6 Justifying the Elimination Rule for Identity7 The Foundations of Homotopy Type Theory without Homotopy7.1 Framework7.2 Semantics7.3 Metaphysics7.4 Epistemology7.5 Methodology8 Possible Objections to this Account8.1 A constructive foundation for mathematics?8.2 What are concepts?8.3 Isn’t this just Brouwerian intuitionism?8.4 Duplicated objects8.5 Intensionality and substitution salva veritate9 Conclusion9.1 Advantages of this foundation. (shrink)

Thought experiments have played a pivotal role in many debates within ethics—and in particular within applied ethics—over the past 30 years. Nonetheless, despite their having become a commonly used philosophical tool, there is something odd about the extensive reliance upon thought experiments in areas of philosophy, such as applied ethics, that are so obviously oriented towards practical life. Herein I provide a moderate defence of their use in applied philosophy against those three objections. I do not defend all possible uses (...) of thought experiments but suggest that we should distinguish between legitimate and illegitimate uses. Their legitimate uses are determined not so much by the modal content of any actual thought experiment itself, but by the extent to which the argument in which it is nested follows basic tenets of informal logic and respects the fundamental contingency of applied ethical problems. In pursuing these ideas, I do not so much provide a set of criteria for their legitimate use, but more modestly present two significant ways in which their use can go awry. (shrink)

The definitive version of this article is published in Philosophical Books 48.3 July 2007 pp. 233-240 by Blackwell Publishing, and is available at www.blackwell-synergy.com.

This paper serves as a kind of field guide to certain passages in the literature which bear upon the foundational theory of abstract objects. The foundational theory assimilates ideas from key philosophers in both the analytical and phenomenological traditions. I explain how my foundational theory of objects serves as a common ground where analytic and phenomenological concerns meet. I try to establish how the theory offers a logic that systematizes a well-known phenomenological kind of entity, and I try to show (...) the various ways the theory systematizes the ideas of many analytic philosophers. The ideas of Plato, Leibniz, Frege, Russell, Goedel and even Kripke become connected through those of Brentano, Meinong, Husserl, and Mally. -/- The field guide will not only document the passages in which the distinction between two kinds of predication originates, but also document the other surprising, and often unrelated, contexts where the distinction reappears in the work of others. It will also document ways in which the theory can be used to represent precisely the ideas of the philosophers mentioned above. The resulting guide will bring together the works of many different authors, including some clearly within the analytic tradition, some clearly within the phenomenological tradition, and some who straddle the divide. (shrink)

This book is intended for those who have no previous acquaintance with the topics of which it treats, and no more knowledge of mathematics than can be acquired at a primary school or even at Eton. It sets forth in elementary form the logical definition of number, the analysis of the notion of order, the modern doctrine of the infinite, and the theory of descriptions and classes as symbolic fictions. The more controversial and uncertain aspects of the subject are subordinated (...) to those which can by now be regarded as acquired scientific knowledge. These are explained without the use of symbols, but in such a way as to give readers a general understanding of the methods and purposes of mathematical logic, which, it is hoped, will be of interest not only to those who wish to proceed to a more serious study of the subject, but also to that wider circle who feel a desire to know the bearings of this important modern science. (shrink)

This thesis draws on Jacques Derrida's idea of 'differance' to argue that, within post-Darwinian humanist culture, the category of 'the human' can only be defined by differentiating it from 'the animal'. Following Derrida, this project seeks to 'determine the number, form, sense, or structure' of the 'plural and repeatedly folded frontier' between these two categories. Chapter 1, 'Becoming Human', examines the chronological boundaries between the human and extinct hominids such as 'Homo erectus' and Neanderthals. It reads contemporary scientific accounts of (...) 'how we became human' to demonstrate that they preserve the conceptual framework of earlier creation myths. However, despite their humanism, these accounts inevitably unsettle the boundaries between human and animal by revealing the play of traces across them. Chapter 2, 'Acting Human', reads Descartes' 'Discourse on the Method' with Judith Butler's theory of gender performance to argue that the human can only be identified by its behaviour. Those who do not behave 'correctly', such as people with autism, threaten humanism and are consequently punished. Conversely, when animals are seen to 'act human' this is dismissed as anthropomorphism. However, these possibilities demonstrate that human behaviour is not tied to an internal essence. Chapter 3, 'Talking Human', deconstructs the opposition between human language and animal communication which underpins contemporary humanist discourse. The human voice is identified with presence, truth and subjectivity, while animals are mute, inarticulate objects. However, the human subject is never fully in control of its communication, as demonstrated by blushing and involuntary nonverbal 'leakage'. I conclude that 'language' and 'the human' are constituted only by referring to each other. This thesis critiques the mythology of humanism in order to challenge the unethical acts that are committed in the name of the human. (shrink)