This chapter gives a detailed study of diagram-based reasoning in Euclidean plane geometry (Books I, III), as well as an exploration how to characterise a geometric practice. First, an account is given of diagram attribution: basic geometrical claims are classified as exact (equalities, proportionalities) or co-exact (containments, contiguities); exact claims may only be inferred from prior entries in the demonstration text, but co-exact claims may be asserted based on what is seen in the diagram. Diagram control by constructions is necessary (...) for this to work. Case-branching occurs when a construction renders a diagram un-representative. The roles of diagrams in reductio arguments, and of objection in probing a demonstration, are discussed. (shrink)

Oxford Philosophical Texts Series Editor: John Cottingham The Oxford Philosophical Texts series consists of authoritative teaching editions of canonical texts in the history of philosophy from the ancient world down to modern times. Each volume provides a clear, well laid out text together with a comprehensive introduction by a leading specialist, giving the student detailed critical guidance on the intellectual context of the work and the structure and philosophical importance of the main arguments. Endnotes are supplied which provide further commentary (...) on the arguments and explain unfamiliar references and terminology, and a full bibliography and index are also included. The series aims to build up a definitive corpus of key texts in the Western philosophical tradition, which will form a reliable and enduring resource for students and teachers alike. David Hume's aim in writing An Enquiry concerning Human Understanding was to introduce his philosophy to a European culture in which many educated people read original works of philosophy. He gives an elegant and accessible presentation of strikingly original and challenging views about the limited powers of human understanding, the attractions of scepticism, the compatibility of free will and determinism, and weaknesses in the foundations of religion. Hume's philosophy was highly controversial in the eighteenth century and remains so today. The text printed in this edition is that of the Clarendon critical edition of Hume's works. A substantial introduction by the editor explains the intellectual background to the work and surveys its main themes. The volume also includes detailed explanatory notes on the text, a glossary of terms, a full list of references, and a section of supplementary readings. (shrink)

In this new book, Michael Friedman argues that Kant's continuing efforts to find a metaphysics that could provide a foundation for the sciences is of the utmost ...

Officially, for Kant, judgments are analytic iff the predicate is "contained in" the subject. I defend the containment definition against the common charge of obscurity, and argue that arithmetic cannot be analytic, in the resulting sense. My account deploys two traditional logical notions: logical division and concept hierarchies. Division separates a genus concept into exclusive, exhaustive species. Repeated divisions generate a hierarchy, in which lower species are derived from their genus, by adding differentia(e). Hierarchies afford a straightforward sense of containment: (...) genera are contained in the species formed from them. Kant's thesis then amounts to the claim that no concept hierarchy conforming to division rules can express truths like '7+5=12.' Kant is correct. Operation concepts ( ) bear two relations to number concepts: and are inputs, is output. To capture both relations, hierarchies must posit overlaps between concepts that violate the exclusion rule. Thus, such truths are synthetic. (shrink)

The seventeenth century saw dramatic advances in mathematical theory and practice. With the recovery of many of the classical Greek mathematical texts, new techniques were introduced, and within 100 years, the rules of analytic geometry, geometry of indivisibles, arithmatic of infinites, and calculus were developed. Although many technical studies have been devoted to these innovations, Mancosu provides the first comprehensive account of the relationship between mathematical advances of the seventeenth century and the philosophy of mathematics of the period. Starting with (...) the Renaissance debates on the certainty of mathematics, Mancosu leads the reader through the foundational issues raised by the emergence of these new mathematical techniques, including the influence of the Aristotelian conception of science in Cavalieri and Guldin, the foundational relevance of Descartes' Geometrie, the relation between geometrical and epistemological theories of the infinite, and the Leibnizian calculus and the opposition to infinitesimalist procedures. In the process Mancosu draws a sophisticated picture of the subtle dependencies between technical development and philosophical reflection in seventeenth century mathematics. (shrink)

In this paper, I investigate an important aspect of Kant’s theory of pure sensible intuition. I argue that, according to Kant, a pure concept of space warrants and constrains intuitions of finite regions of space. That is, an a priori conceptual representation of space provides a governing principle for all spatial construction, which is necessary for mathematical demonstration as Kant understood it.Author Keywords: Kant; Space; Pure sensible intuition; Philosophy of mathematics.

It's well-known that Kant believed that intuition was central to an account of mathematical knowledge. What that role is and how Kant argues for it are, however, still open to debate. There are, broadly speaking, two tendencies in interpreting Kant's account of intuition in mathematics, each emphasizing different aspects of Kant's general doctrine of intuition. On one view, most recently put forward by Michael Friedman, this central role for intuition is a direct result of the limitations of the syllogistic logic (...) available to Kant. On this view, Kant's reasons for introducing intuition are taken to be logical or mathematical, rather than philosophical. The other tendency, which I shall try to develop here, emphasizes an epistemological or phenomenological role for intuition in mathematics arising out of what may loosely be called Kant's ‘antiformalism.’This paper, which focuses specifically on the case of geometry, falls into two parts. First, I consider Kant's discussion of intuition in the Metaphysical Exposition of the concept of space. (shrink)

David Hume's Enquiry concerning Human Understanding is the definitive statement of the greatest philosopher in the English language. His arguments in support of reasoning from experience, and against the "sophistry and illusion"of religiously inspired philosophical fantasies, caused controversy in the eighteenth century and are strikingly relevant today, when faith and science continue to clash. The Enquiry considers the origin and processes of human thought, reaching the stark conclusion that we can have no ultimate understanding of the physical world, or indeed (...) our own minds. In either sphere we must depend on instinctive learning from experience, recognizing our animal nature and the limits of reason. Hume's calm and open-minded skepticism thus aims to provide a new basis for science, liberating us from the "superstition" of false metaphysics and religion. His Enquiry remains one of the best introductions to the study of philosophy, and his edition places it in its historical and philosophical context. (shrink)

According to Alberto Coffa in The Semantic Tradition from Kant to Carnap, Kant’s account of mathematical judgment is built on a ‘semantic swamp’. Kant’s primitive semantics led him to appeal to pure intuition in an attempt to explain mathematical necessity. The appeal to pure intuition was, on Coffa’s line, a blunder from which philosophy was forced to spend the next 150 years trying to recover. This dismal assessment of Kant’s contributions to the evolution of accounts of mathematical necessity is fundamentally (...) backward-looking. Coffa’s account of how semantic theories of the a priori evolved out of Kant’s doctrine of pure intuition rightly emphasizes those developments, both scientific and philosophical, that collectively served to undermine the plausibility of Kant’s account. What is missing from Coffa’s story, apart from any recognition of Kant’s semantic innovations, is an attempt to appreciate Kant’s philosophical context and the distinctive perspective from which Kant viewed issues in the philosophy of mathematics. When Kant’s perspective and context are brought out, he can not only be seen to have made a genuinely progressive contribution to the development of accounts of mathematical necessity, but also to be relevant to contemporary issues in the philosophy of mathematics in underappreciated ways. (shrink)

According to Alberto Coffa in The Semantic Tradition from Kant to Carnap, Kant’s account of mathematical judgment is built on a ‘semantic swamp’. Kant’s primitive semantics led him to appeal to pure intuition in an attempt to explain mathematical necessity. The appeal to pure intuition was, on Coffa’s line, a blunder from which philosophy was forced to spend the next 150 years trying to recover. This dismal assessment of Kant’s contributions to the evolution of accounts of mathematical necessity is fundamentally (...) backward-looking. Coffa’s account of how semantic theories of the a priori evolved out of Kant’s doctrine of pure intuition rightly emphasizes those developments, both scientific and philosophical, that collectively served to undermine the plausibility of Kant’s account. What is missing from Coffa’s story, apart from any recognition of Kant’s semantic innovations, is an attempt to appreciate Kant’s philosophical context and the distinctive perspective from which Kant viewed issues in the philosophy of mathematics. When Kant’s perspective and context are brought out, he can not only be seen to have made a genuinely progressive contribution to the development of accounts of mathematical necessity, but also to be relevant to contemporary issues in the philosophy of mathematics in underappreciated ways. (shrink)

This book provides a reading of Kant's theory of the construction of mathematical concepts through a fully contextualised analysis. In this work the author argues that it is only through an understanding of the relevant eighteenth century mathematics textbooks, and the related mathematical practice, that the material and context necessary for a successful interpretation of Kant's philosophy can be provided.