Foundations of Geometry

by David Hilbert

Along with the writings of Hilbert's friend and correspondent Frege, Hilbert's Foundations of Geometry set the stage for Russell and Whitehead's Principia Mathematica. Hilbert presents a new axiomatization of geometry to algebra, and introduces the distinction between mathematics and metamathematics, with a new theory of proof.

Genres: Mathematics, Science, Geometry, Nonfiction

226 pages, Paperback

First published January 1, 1922

About the author

David Hilbert (23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory, the calculus of variations, commutative algebra, algebraic number theory, the foundations of geometry, spectral theory of operators and its application to integral equations, mathematical physics, and the foundations of mathematics (particularly proof theory).

Hilbert adopted and defended Georg Cantor's set theory and transfinite numbers. In 1900, he presented a collection of problems that set the course for much of the mathematical research of the 20th century.

Hilbert and his students contributed significantly to establishing rigor and developed important tools used in modern mathematical physics. Hilbert is known as one of the founders of proof theory and mathematical logic.

Reviews

[David Olmsted · Rating 4 out of 5]

This book is based upon lectures given in German during the 1898-1899 school year by the renowned mathematician David Hilbert. This is the English translation published in 1902. The purpose of this book was to clarify the geometry of Euclid for the modern era when multiple consistent geometries were discovered based upon differing approaches to parallel lines. Towards this end Hilbert separates the problematic parallel line axioms from other axioms which he grouped together as axioms of connection, order, congruence, and continuity. What is significant is that Hilbert did not explicitly include the concept of distance avoiding the inexactness of the length of the triangular hypotenuse or the ratio of PI. Instead, he used the axioms of congruence to get around this difficulty.

Yet all is not perfect. Hilbert defines continuity twice, first as a theorem (theorem 3) as a consequence of the axioms of connection and order and later as an axiom stating that any point could exist between any other pair of points. As such he falls short of his goal of creating a simple and complete set of independent axioms. Apparently a comprehensive geometry needs to consider in more rigor and interconnectedness the ideas involving limits, distance, equality, and continuity.

[Robert · Rating 5 out of 5]

A must read classic for any geometer or and instructor of geometry. If you have not read this book, then you do not have a solid grounding in the modern development of geometry.

[Forked Radish]

Epigraph:
“All human knowledge begins with intuitions, thence passes to concepts and ends with ideas.”
Immanuel Kant, Critique of Pure Reason, Part 2, Sec. 2. [unattested]
What it should say:
All human folly, error, and misconception begins with intuitions, thence passes to concepts and ends with ideas!
Extensive observations and experiments are ALWAYS an a priori necessity for all TRUE knowledge (an unfortunate, though necessary, tautology). Stick that in your pipe and smoke it Manny! (if you really did write such a stupid thing).

[Michael Greer · Rating 5 out of 5]

My favorite work in geometry. Orderly, concise, clear and innovative.

[Jovany Agathe · Rating 4 out of 5]

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