Turning Points in the History of Mathematics

Chapter 1  Axiomatics-Euclid’s and Hilbert’s: From Material to Formal
     1.1 Euclid’s Elements
     1.2 Hilbert’s Foundations of Geometry
     1.3 The Modern Axiomatic Method
     1.4 Ancient vs. Modern Axiomatics
     References
     Further Reading

Chapter 2  Solution by Radicals of the Cubic: From Equations to Groups and from Real to Complex Numbers
     2.1 Introduction
     2.2 Cubic and Quartic Equations
     2.3 Beyond the Quartic: Lagrange
     2.4 Ruffini, Abel, Galois
     2.5 Complex Numbers: Birth
     2.6 Growth
     2.7 Maturity
     References
     Further Reading

Chapter 3  Analytic Geometry: From the Marriage of Two Fields to the Birth of a Third
     3.1 Introduction
     3.2 Descartes
     3.3 Fermat
     3.4 Descartes’ and Fermat’s Works from a Modern Perspective
     3.5 The Significance of Analytic Geometry
     References
     Further Reading

Chapter 4  Probability: From Games of Chance to an Abstract Theory
     4.1 The Pascal–Fermat Correspondence
     4.2 Huygens: The First Book on Probability
     4.3 Jakob Bernoulli’s Ars Conjectandi (The Art of Conjecturing)
     4.4 De Moivre’s The Doctrine of Chances
     4.5 Laplace’s Théorie Analytique des Probabilités
     4.6 Philosophy of Probability
     4.7 Probability as an Axiomatic Theory
     4.8 Conclusion
     References
     Further Reading

Chapter 5  Calculus: From Tangents and Areas to Derivatives and Integrals
     5.1 Introduction
     5.2 Seventeenth-Century Predecessors of Newton and Leibniz
     5.3 Newton and Leibniz: The Inventors of Calculus
     5.4 The Eighteenth Century: Euler
     5.5 A Look Ahead: Foundations
     References
     Further Reading

Chapter 6  Gaussian Integers: From Arithmetic to Arithmetics
     6.1 Introduction
     6.2 Ancient Times
     6.3 Fermat
     6.4 Euler and the Bachet Equation x2.+.2.=.y3
     6.5 Reciprocity Laws, Fermat’s Last Theorem, Factorization of Ideals
     6.6 Conclusion
     References
     Further Reading

Chapter 7  Noneuclidean Geometry: From One Geometry to Many
     7.1 Introduction
     7.2 Euclidean Geometry
     7.3 Attempts to Prove the Fifth Postulate
     7.4 The Discovery (Invention) of Noneuclidean Geometry
     7.5 Some Implications of the Creation of Noneuclidean Geometry
     References
     Further Reading

Chapter 8  Hypercomplex Numbers: From Algebra to Algebras
     8.1 Introduction
     8.2 Hamilton and Complex Numbers
     8.3 The Quaternions
     8.4 Beyond the Quaternions
     References
     Further Reading

Chapter 9  The Infinite: From Potential to Actual
     9.1 The Greeks
     9.2 Before Cantor
     9.3 Cantor
     9.4 Paradoxes Lost
     9.5 Denumerable (Countable) Infinity
     9.6 Paradoxes Regained
     9.7 Arithmetic
     9.8 Two Major Problems
     9.9 Conclusion
     References
     Further Reading

Chapter 10  Philosophy of Mathematics:  From Hilbert to Gödel
     10.1 Introduction
     10.2 Logicism
     10.3 Formalism
     10.4 Gödel’s Incompleteness Theorems
     10.5 Mathematics and Faith
     10.6 Intuitionism
     10.7 Nonconstructive Proofs
     10.8 Conclusion
     References
     Further Reading

Chapter 11  Some Further Turning Points
     11.1 Notation: From Rhetorical to Symbolic
     11.2 Space Dimensions: From 3 to n (n..3)
     11.3 Pathological Functions: From Calculus to Analysis
     11.4 The Nature of Proof: From Axiom-Based to Computer-Assisted
     11.5 Experimental Mathematics: From Humans to Machines
     References
     Further Reading

Index