Grant H., Kleiner I. Turning Points in the History of Mathematics

Basel: Birkhäuser, 2016. — 110 p. — (Compact Textbooks in Mathematics) — ISBN: 978-1-4939-3263-4.Provides a comprehensive overview of the major turning points in the history of mathematics, from Ancient Greece to the present
Substantial reference lists offer suggestions for resources to learn more about the topics discussed
Problems and projects are included in each chapter to extend and increase understanding of the material for students
Ideal resource for students and teachers of the history of mathematics
This book explores some of the major turning points in the history of mathematics, ranging from ancient Greece to the present, demonstrating the drama that has often been a part of its evolution. Studying these breakthroughs, transitions, and revolutions, their stumbling-blocks and their triumphs, can help illuminate the importance of the history of mathematics for its teaching, learning, and appreciation.
Some of the turning points considered are the rise of the axiomatic method (most famously in Euclid), and the subsequent major changes in it (for example, by David Hilbert); the “wedding,” via analytic geometry, of algebra and geometry; the “taming” of the infinitely small and the infinitely large; the passages from algebra to algebras, from geometry to geometries, and from arithmetic to arithmetics; and the revolutions in the late nineteenth and early twentieth centuries that resulted from Georg Cantor’s creation of transfinite set theory. The origin of each turning point is discussed, along with the mathematicians involved and some of the mathematics that resulted. Problems and projects are included in each chapter to extend and increase understanding of the material. Substantial reference lists are also provided.
Turning Points in the History of Mathematics will be a valuable resource for teachers of, and students in, courses in mathematics or its history. The book should also be of interest to anyone with a background in mathematics who wishes to
learn more about the important moments in its development.Axiomatics-Euclid’s and Hilbert’s: From Material to FormalEuclid’s Elements
Hilbert’s Foundations of Geometry
The Modern Axiomatic Method
Ancient vs. Modern AxiomaticsFurther ReadingSolution by Radicals of the Cubic: From Equations to Groups and from Real to Complex NumbersCubic and Quartic Equations
Beyond the Quartic: Lagrange
Ruffini, Abel, Galois
Complex Numbers: Birth
Growth
MaturityFurther ReadingAnalytic Geometry: From the Marriage of Two Fields to the Birth of a ThirdDescartes
Fermat
Descartes’ and Fermat’s Works from a Modern Perspective
The Significance of Analytic GeometryFurther ReadingProbability: From Games of Chance to an Abstract TheoryThe Pascal–Fermat Correspondence
Huygens: The First Book on Probability
Jakob Bernoulli’s Ars Conjectandi (The Art of Conjecturing)
De Moivre’s The Doctrine of Chances
Laplace’s Théorie Analytique des Probabilités
Philosophy of Probability
Probability as an Axiomatic TheoryFurther ReadingCalculus: From Tangents and Areas to Derivatives and IntegralsSeventeenth-Century Predecessors of Newton and Leibniz
Newton and Leibniz: The Inventors of Calculus
The Eighteenth Century: Euler
A Look Ahead: FoundationsFurther ReadingGaussian Integers: From Arithmetic to ArithmeticsAncient Times
Fermat
Euler and the Bachet Equation x2.+.2.=.y3
Reciprocity Laws, Fermat’s Last Theorem, Factorization of IdealsFurther ReadingNoneuclidean Geometry: From One Geometry to ManyEuclidean Geometry
Attempts to Prove the Fifth Postulate
The Discovery (Invention) of Noneuclidean Geometry
Some Implications of the Creation of Noneuclidean GeometryFurther ReadingHypercomplex Numbers: From Algebra to AlgebrasHamilton and Complex Numbers
The Quaternions
Beyond the QuaternionsFurther ReadingThe Infinite: From Potential to ActualThe Greeks
Before Cantor
Cantor
Paradoxes Lost
Denumerable (Countable) Infinity
Paradoxes Regained
Arithmetic
Two Major ProblemsFurther ReadingPhilosophy of Mathematics: From Hilbert to GödelLogicism
Formalism
Gödel’s Incompleteness Theorems
Mathematics and Faith
Intuitionism
Nonconstructive ProofsFurther ReadingSome Further Turning PointsNotation: From Rhetorical to Symbolic
Space Dimensions: From 3 to n (n..3)
Pathological Functions: From Calculus to Analysis
The Nature of Proof: From Axiom-Based to Computer-Assisted
Experimental Mathematics: From Humans to MachinesFurther Reading