Hossack K. Knowledge and the Philosophy of Number: What Numbers Are and How They Are Known

London: Bloomsbury Academic, 2020. — 217 p.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.Title Page
Copyright PageMathematical Knowledge
The Sceptical Consequence
The Logic of Quantity
Equality
The Homomorphism Theorem
Properties
Predicables
Different Accounts of Predication
Criticism of Davidson
Property Realism
Kinds of Property
Magnitudes
Ratios
Numbers
Frege’s Theory of Concepts
No Explanation of Naturalness
Second-Order Logic
Non-standard Models of Arithmetic
Frege’s Theorem
The Incompleteness of Plural Logic
The Logic of Quantity
Taxonomizing Logical Subjects
Ontological Parts
The Logic of ‘and’
Comparison with the Magnitudes Axioms
The Least Upper Bound Property
Mereology
Mereology
Virtual Classes
Mereology Interpreted as about Individuals
The Category of Quantity
The Axioms of the Mereology of Pluralities
The Axioms of the Mereology of Continua
Equivalence of the Various Axiomatizations
From Tarski’s Axioms to the Axioms of Simons
From the Axioms of Simons to the Common Axioms
From the eight Common Axioms to the Axioms of Tarski
Proof of Axiom A for Continua
The Homomorphism Theorem
The Equality Axioms
Common Structure
The Common Structure of a Mereology and Its System of Magnitudes
Congruence Relations on Semigroups
Congruences on Groups
Congruences on Positive Semigroups
The Homomorphism Theorem
Sizes of Quantities
The Natural Numbers
Numerical Equality
Tallying
Is Tallying an Equality?
Is It a priori that Tallying Is an Equality?
Are the Axioms of Peano Arithmetic True?
Zero Is Not a Number
The Natural Number
Every Number Has a Successor
Multiplication
What Is an ‘Axiom’?
Set-theoretic Constructions
Mysterious Multiplication
Euclid’s Definition of Multiplication
The Multiplication Axioms of Peano Arithmetic
Ratio
Relative Size
Eudoxus’s Definition of Proportion
Ratios of Magnitudes
Proportionality as an Equivalence Relation
Ratios of Natural Numbers
The Positive Real Numbers
Geometry
Geometrical Equality
Congruence Is an Equality
The Lengths Are a Complete System of Magnitudes
Multiplication and Division of Lengths
Transcendental Real Numbers
Doubts about Euclidean Geometry
Euclid Presupposed in Non-Euclidean Geometry
What Is a priori in Euclid?
Should We Base the Reals on Set Theory?
The Ordinals
The Discovery of the Ordinals
The Set-theoretic Account of Order
Are Relations the Source of Order?
Serial Reference
Longer Series
Equality of Series
The Ordinals Are a System of Magnitudes
How Many Ordinal Numbers Are There?
Stopping at the Constructive Ordinals
The Existence of Sets
Notes