A Concise History of Mathematics for Philosophers

Title page
Imprints page
Abstract page
Contents
A Concise History of Mathematics for Philosophers
	Preface
	1 Irrational Numbers and Geometry
		PREVIEW
		1.1 The Pythagorean Theorem
		1.2 Irrationality
		1.3 Operations on Lengths and Numbers
		1.4 Axiomatics
		1.5 Philosophical Issues
			Intuition and logic.
			Meaning and existence.
			Discrete and continuous.
	2 Infinity in Greek Mathematics
		PREVIEW
		2.1 Irrationality and Non-termination
		2.2 Areas and Volumes
		2.3 The Method of Exhaustion
		2.4 The Theory of Proportions
		2.5 Archimedes and Actual Infinity
		2.6 Philosophical Issues
			Intuition and logic.
			Meaning and existence.
			Continuous and discrete.
	3 Imaginary Numbers
		PREVIEW
		3.1 Quadratic and Cubic Equations
		3.2 Bombelli’s Algebra of Imaginary Numbers
		3.3 The Convenience of Imaginary Numbers
		3.4 Realizing the Imaginary
		3.5 Philosophical Issues
			Intuition and logic.
			Meaning and existence.
			Discrete and continuous.
	4 Calculus and Infinitesimals
		Preview
		4.1 Infinite Series
		4.2 Algebraic Geometry
		4.3 Infinitesimal Calculus
		4.4 Infinitesimals: Criticism and Avoidance
		4.5 Complex Analysis
		4.6 Philosophical Issues
			Intuition and logic.
			Meaning and existence.
			Discrete and continuous.
	5 Continuous Functions and Real Numbers
		PREVIEW
		5.1 The Fundamental Theorem of Algebra
		5.2 The Intermediate Value Theorem
			Intermediate Value Theorem.
		5.3 Definition of Real Numbers
		5.4 Counter-intuitive Curves
		5.5 Philosophical Issues
			Intuition and logic.
			Meaning and existence.
			Discrete and continuous.
	6 From Non-Euclidean Geometry to Arithmetic
		PREVIEW
		6.1 The Parallel Axiom
		6.2 Non-Euclidean Geometry
		6.3 The Impact of Non-Euclidean Geometry
		6.4 Arithmetization of Geometry
		6.5 Vector Geometry
		6.6 Philosophical Issues
			Intuition and logic.
			Meaning and existence.
			Discrete and continuous.
	7 Set Theory and Its Paradoxes
		PREVIEW
		7.1 Before Cantor
		7.2 Cantor’s Diagonal Argument
		7.3 Higher Infinities
		7.4 Aftermath of the Diagonal Argument
		7.5 Philosophical Issues
			Intuition and logic.
			Meaning and existence.
			Discrete and continuous.
	8 Formal Systems
		PREVIEW
		8.1 Hilbert
		8.2 The Systems of Peano and Zermelo
		8.3 Frege’s System for Logic
		8.4 Completeness and Incompleteness
		8.5 Philosophical Issues
			Intuition and logic.
			Meaning and existence.
			Discrete and continuous.
	9 Unsolvability and Incompleteness
		PREVIEW
		9.1 Computability
		9.2 Unsolvability
		9.3 Incompleteness
		9.4 The Incompleteness of Set Theory
		9.5 Philosophical Issues
Notes
Bibliography
The Philosophy of Mathematics
The Philosophy of Mathematics
	Elements in the Series