Knowledge and the philosophy of number : what numbers are and how they are known /
"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: th...
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| Main Author: | |
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| Format: | Electronic eBook |
| Language: | English |
| Published: |
London :
Bloomsbury Publishing Plc,
2020.
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| Series: | Mind, Meaning and Metaphysics
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| Subjects: | |
| Online Access: | CONNECT |
MARC
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| 100 | 1 | |a Hossack, Keith, |e author. | |
| 245 | 1 | 0 | |a Knowledge and the philosophy of number : |b what numbers are and how they are known / |c Keith Hossack. |
| 264 | 1 | |a London : |b Bloomsbury Publishing Plc, |c 2020. | |
| 300 | |a 1 online resource (217 pages) | ||
| 336 | |a text |b txt |2 rdacontent | ||
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| 490 | 0 | |a Mind, Meaning and Metaphysics | |
| 588 | 0 | |a Print version record. | |
| 505 | 0 | |a Intro; Title Page; Copyright Page; Contents; Preface; Introduction; 1 Mathematical Knowledge; 2 The Sceptical Consequence; 3 The Logic of Quantity; 4 Equality; 5 The Homomorphism Theorem; Chapter 1: Properties; 1.1 Predicables; 1.2 Different Accounts of Predication; 1.3 Criticism of Davidson; 1.4 Property Realism; 1.5 Kinds of Property; 1.6 Magnitudes; 1.7 Ratios; 1.8 Numbers; Chapter 2: Frege's Theory of Concepts; 2.1 No Explanation of Naturalness; 2.2 Second-Order Logic; 2.3 Non-standard Models of Arithmetic; 2.4 Frege's Theorem; 2.5 The Incompleteness of Plural Logic | |
| 505 | 8 | |a Chapter 3: The Logic of Quantity3.1 Taxonomizing Logical Subjects; 3.2 Ontological Parts; 3.3 The Logic of 'and'; 3.4 Comparison with the Magnitudes Axioms; 3.5 The Least Upper Bound Property; Chapter 4: Mereology; 4.1 Mereology; 4.2 Virtual Classes; 4.3 Mereology Interpreted as about Individuals; 4.4 The Category of Quantity; 4.5 The Axioms of the Mereology of Pluralities; 4.6 The Axioms of the Mereology of Continua; 4.7 Equivalence of the Various Axiomatizations; From Tarski's Axioms to the Axioms of Simons; From the Axioms of Simons to the Common Axioms | |
| 505 | 8 | |a From the eight Common Axioms to the Axioms of TarskiProof of Axiom A8 for Continua; Chapter 5: The Homomorphism Theorem; 5.1 The Equality Axioms; 5.2 Common Structure; 5.3 The Common Structure of a Mereology and Its System of Magnitudes; 5.4 Congruence Relations on Semigroups; 5.5 Congruences on Groups; 5.6 Congruences on Positive Semigroups; 5.7 The Homomorphism Theorem; 5.8 Sizes of Quantities; Chapter 6: The Natural Numbers; 6.1 Numerical Equality; 6.2 Tallying; 6.3 Is Tallying an Equality?; 6.4 Is It a priori that Tallying Is an Equality?; 6.5 Are the Axioms of Peano Arithmetic True? | |
| 505 | 8 | |a 6.6 Zero Is Not a Number6.7 The Natural Number 1; 6.8 Every Number Has a Successor; Chapter 7: Multiplication; 7.1 What Is an 'Axiom'?; 7.2 Set-theoretic Constructions; 7.3 Mysterious Multiplication; 7.4 Euclid's Definition of Multiplication; 7.5 The Multiplication Axioms of Peano Arithmetic; Chapter 8: Ratio; 8.1 Relative Size; 8.2 Eudoxus's Definition of Proportion; 8.3 Ratios of Magnitudes; 8.4 Proportionality as an Equivalence Relation; 8.5 Ratios of Natural Numbers; 8.6 The Positive Real Numbers; Chapter 9: Geometry; 9.1 Geometrical Equality; 9.2 Congruence Is an Equality | |
| 505 | 8 | |a 9.3 The Lengths Are a Complete System of Magnitudes9.4 Multiplication and Division of Lengths; 9.5 Transcendental Real Numbers; 9.6 Doubts about Euclidean Geometry; 9.7 Euclid Presupposed in Non-Euclidean Geometry; 9.8 What Is a priori in Euclid?; 9.9 Should We Base the Reals on Set Theory?; Chapter 10: The Ordinals; 10.1 The Discovery of the Ordinals; 10.2 The Set-theoretic Account of Order; 10.3 Are Relations the Source of Order?; 10.4 Serial Reference; 10.5 Longer Series; 10.6 Equality of Series; 10.7 The Ordinals Are a System of Magnitudes; 10.8 How Many Ordinal Numbers Are There? | |
| 500 | |a 10.9 Stopping at the Constructive Ordinals | ||
| 520 | |a "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."-- |c Provided by publisher | ||
| 500 | |a EBSCO eBook Academic Comprehensive Collection North America |5 TMurS | ||
| 650 | 0 | |a Number theory. | |
| 650 | 0 | |a Mathematics |x Philosophy. | |
| 730 | 0 | |a WORLDSHARE SUB RECORDS | |
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| 776 | 0 | 8 | |i Print version: |a Hossack, Keith. |t Knowledge and the Philosophy of Number : What Numbers Are and How They Are Known. |d London : Bloomsbury Publishing Plc, ©2020 |z 9781350102903 |
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