Knowledge and the Philosophy of Number

What Numbers Are and How They Are Known

Dr Keith Hossack (Birkbeck College, University of London, UK)

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English

Bloomsbury Academic
26 August 2021

Philosophy: metaphysics & ontology; Philosophy: logic; Philosophy of mathematics; Number theory

Series: Mind, Meaning and Metaphysics

If numbers were objects, how could there be human knowledge of number? Numbers are not physical objects: must we conclude that we have a mysterious power of perceiving the abstract realm? Or should we instead conclude that numbers are fictions?

This book argues that numbers are not objects: they are magnitude properties. Properties are not fictions and we certainly have scientific knowledge of them. Much is already known about magnitude properties such as inertial mass and electric charge, and much continues to be discovered. The book says the same is true of numbers.

In the theory of magnitudes, the categorial distinction between quantity and individual is of central importance, for magnitudes are properties of quantities, not properties of individuals. Quantity entails divisibility, so the logic of quantity needs mereology, the a priori logic of part and whole. The three species of quantity are pluralities, continua and series, and the book presents three variants of mereology, one for each species of quantity.

Given Euclid’s axioms of equality, it is possible without the use of set theory to deduce the axioms of the natural, real and ordinal numbers from the respective mereologies of pluralities, continua and series. Knowledge and the Philosophy of Number carries out these deductions, arriving at a metaphysics of number that makes room for our a priori knowledge of mathematical reality.

By: Dr Keith Hossack (Birkbeck College University of London UK)
Imprint: Bloomsbury Academic
Country of Publication: United Kingdom
Dimensions: Height: 234mm, Width: 156mm,
Weight: 313g
ISBN: 9781350277960
ISBN 10: 1350277967
Series: Mind, Meaning and Metaphysics
Pages: 216
Publication Date: 26 August 2021
Audience: College/higher education , Primary
Format: Paperback
Publisher's Status: Active

Preface Introduction 1. Properties 2. Frege's Theory of Concepts 3. The Logic of Quantity 4. Mereology 5. The Homomorphism Theorem 6. The Natural Numbers 7. Multiplication 8. Ratio 9. Geometry 10. The Ordinals Notes References Index

Keith Hossack is Reader in Philosophy, Birkbeck College, University of London, UK.

Reviews for Knowledge and the Philosophy of Number: What Numbers Are and How They Are Known

Recommended. Upper-division undergraduates, graduate students, and faculty. * CHOICE * Hossack's book revives the ancient view that numbers are magnitudes and thus a special kind of property, which are instantiated in the physical world. This view receives a powerful mathematical and philosophical development and defense. The book is highly recommended to anyone with an interest in metaphysics or the philosophy of mathematics. * Oystein Linnebo, Professor of Philosophy, University of Oslo, Norway *