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Foundations and Fundamental Concepts of Mathematics

Howard Eves
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This is a specialized math history book that looks at the growth of axiomatics. It starts out even before there were axioms, with some approximate geometric formulas developed by the ancient Egyptians and Babylonians, and follows how things got gradually more formal and rigorous up through the foundational crises and the development of mathematical logic in the early twentieth century. The intended use is probably as a textbook for an elective course for upper-division undergraduate math majors, or possibly for pre-service teachers. It’s probably too difficult for non-majors. The book includes a lot of detailed technical information, but does not develop much mathematics per se and does not require previous knowledge. The present volume is a 1997 Dover reprint of the 1990 third edition from PWS-Kent; the earlier editions were in 1958 and 1965.

The book is interesting because it looks both at the forces that led to more axiomatization and at the increased abstraction that an axiomatic approach led to. The coverage is slanted towards geometry, but there is also a good bit on logic and set theory. I thought the coverage of real analysis was weak. It does cover the construction of the real numbers, but does not give a good idea of the struggles that went into the arithmetization of analysis, the changing definition of function, and Weierstrass’s program to make analysis more rigorous.

The book was written as a companion to Eves’s earlier book An Introduction to the History of Mathematics (first edition 1953, sixth edition 1990). I have not seen the companion book, but it appears to be a comprehensive and straightforward history of math.

The book has an enormous number of exercises, of various sorts. Many are drill and some ask for proofs of simple theorems stated in the body; none seem very difficult. The most interesting ones give a fallacious argument and ask the student to debunk it.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.


1. Mathematics Before Euclid
  1.1 The Empirical Nature of pre-Hellenic Mathematics
  1.2 Induction Versus Deduction
  1.3 Early Greek Mathematics and the Introduction of Deductive Procedures
  1.4 Material Axiomatics
  1.5 The Origin of the Axiomatic Method
2. Euclid's Elements
  2.1 The Importance and Formal Nature of Euclid's Elements
  2.2 Aristotle and Proclus on the Axiomatic Method
  2.3 Euclid's Definitions, Axioms, and Postulates
  2.4 Some Logical Shortcomings of Euclid's Elements
  2.5 The End of the Greek Period and the Transition to Modern Times
3. Non-Euclidean Geometry
  3.1 Euclid's Fifth Postulate
  3.2 Saccheri and the Reductio ad Absurdum Method
  3.3 The Work of Lambert and Legendre
  3.4 The Discovery of Non-Euclidean Geometry
  3.5 The Consistency and the Significance of Non-Euclidean Geometry
4. Hilbert's Grundlagen
  4.1 The Work of Pasch, Peano, and Pieri
  4.2 Hilbert's Grundlagen der Geometrie
  4.3 Poincaré's Model and the Consistency of Lobachevskian Geometry
  4.4 Analytic Geometry
  4.5 Projective Geometry and the Principle of Duality
5. Algebraic Structure
  5.1 Emergence of Algebraic Structure
  5.2 The Liberation of Algebra
  5.3 Groups
  5.4 The Significance of Groups in Algebra and Geometry
  5.5 Relations
6. Formal Axiomatics
  6.1 Statement of the Modern Axiomatic Method
  6.2 A Simple Example of a Branch of Pure Mathematics
  6.3 Properties of Postulate Sets--Equivalence and Consistency
  6.4 Properties of Postulate Sets--Independence, Completeness, and Categoricalness
  6.5 Miscellaneous Comments
7. The Real Number System
  7.1 Significance of the Real Number System for the Foundations of Analysis
  7.2 The Postulational Approach to the Real Number System
  7.3 The Natural Numbers and the Principle of Mathematical Induction
  7.4 The Integers and the Rational Numbers
  7.5 The Real Numbers and the Complex Numbers
8. Sets
  8.1 Sets and Their Basic Relations and Operations
  8.2 Boolean Algebra
  8.3 Sets and the Foundations of Mathematics
  8.4 Infinite Sets and Transfinite Numbers
  8.5 Sets and the Fundamental Concepts of Mathematics
9. Logic and Philosophy
  9.1 Symbolic Logic
  9.2 The Calculus of Propositions
  9.3 Other Logics
  9.4 Crises in the Foundations of Mathematics
  9.5 Philosophies of Mathematics
Appendix 1. The First Twenty-Eight Propositions of Euclid
Appendix 2. Euclidean Constructions
Appendix 3. Removal of Some Redundancies
Appendix 4. Membership Tables
Appendix 5. A Constructive Proof of the Existence of Transcendental Numbers
Appendix 6. The Eudoxian Resolution of the First Crisis in the Foundations of Mathematics
Appendix 7. Nonstandard Analysis
Appendix 8. The Axiom of Choice
Appendix 9. A Note on Gödel's Incompleteness Theorem
Bibliography; Solution Suggestions for Selected Problems; Index