This is a textbook on the history, philosophy and foundations of mathematics. One of its aims is to present some interesting mathematics, not normally taught in other courses, in a historical and philosophical setting. The book is intended mainly for undergraduate mathematics students, but also for students in the sciences, humanities and education with a strong interest in mathematics. It proceeds in historical order from about 1800 BC to 1800 AD and then presents some selected topics of foundational interest from the 19th and 20th centuries. Among other material in the first part, the authors discuss the renaissance method for solving cubic and quartic equations and give rigorous elementary proofs that certain geometrical problems posed by the ancient Greeks (e.g. the problem of trisecting an arbitrary angle) cannot be solved by ruler and compass constructions. Among the topics in the second part, they sketch a proof of Godel's incompleteness theorem and discuss some of its implications, and also present the elements of category theory. The authors' approach to most of these matters is new.
0 Introduction.- 0 Introduction.- I: History and Philosophy of Mathematics.- 1 Egyptian Mathematics.- 2 Scales of Notation.- 3 Prime Numbers.- 4 Sumerian-Babylonian Mathematics.- 5 More about Mesopotamian Mathematics.- 6 The Dawn of Greek Mathematics.- 7 Pythagoras and His School.- 8 Perfect Numbers.- 9 Regular Polyhedra.- 10 The Crisis of Incommensurables.- 11 From Heraclitus to Democritus.- 12 Mathematics in Athens.- 13 Plato and Aristotle on Mathematics.- 14 Constructions with Ruler and Compass.- 15 The Impossibility of Solving the Classical Problems.- 16 Euclid.- 17 Non-Euclidean Geometry and Hilbert’s Axioms.- 18 Alexandria from 300 BC to 200 BC.- 19 Archimedes.- 20 Alexandria from 200 BC to 500 AD.- 21 Mathematics in China and India.- 22 Mathematics in Islamic Countries.- 23 New Beginnings in Europe.- 24 Mathematics in the Renaissance.- 25 The Cubic and Quartic Equations.- 26 Renaissance Mathematics Continued.- 27 The Seventeenth Century in France.- 28 The Seventeenth Century Continued.- 29 Leibniz.- 30 The Eighteenth Century.- 31 The Law of Quadratic Reciprocity.- II: Foundations of Mathematics.- 1 The Number System.- 2 Natural Numbers (Peano’s Approach).- 3 The Integers.- 4 The Rationals.- 5 The Real Numbers.- 6 Complex Numbers.- 7 The Fundamental Theorem of Algebra.- 8 Quaternions.- 9 Quaternions Applied to Number Theory.- 10 Quaternions Applied to Physics.- 11 Quaternions in Quantum Mechanics.- 12 Cardinal Numbers.- 13 Cardinal Arithmetic.- 14 Continued Fractions.- 15 The Fundamental Theorem of Arithmetic.- 16 Linear Diophantine Equations.- 17 Quadratic Surds.- 18 Pythagorean Triangles and Fermat’s Last Theorem.- 19 What Is a Calculation?.- 20 Recursive and Recursively Enumerable Sets.- 21 Hilbert’s Tenth Problem.- 22 Lambda Calculus.- 23 Logic fromAristotle to Russell.- 24 Intuitionistic Propositional Calculus.- 25 How to Interpret Intuitionistic Logic.- 26 Intuitionistic Predicate Calculus.- 27 Intuitionistic Type Theory.- 28 Gödel’s Theorems.- 29 Proof of Gödel’s Incompleteness Theorem.- 30 More about Gödel’s Theorems.- 31 Concrete Categories.- 32 Graphs and Categories.- 33 Functors.- 34 Natural Transformations.- 35 A Natural Transformation between Vector Spaces.- References.
