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The Foundations of Geometry by David Hilbert

The book Grundlagen der Geometrie (The Foundations of Geometry) published by Hilbert in 1899 substitutes a formal set, comprised of 21 axioms, for the traditional axioms of Euclid. They avoid weaknesses identified in those of Euclid, whose works at the time were still used textbook-fashion. Hilbert's work is the cornerstone of modern geometry and its simple, elegant, rigor has had a profound impact in many other areas of modern science. He contributed substantially to the establishment of the formalistic foundations of mathematics.

David Hilbert (1862-1943) was a German mathematician, recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries.

See also: Anschauliche Geometrie, translated into English as Geometry and the Imagination.

 

Chapters:

Introduction

1. The Five Groups of Axioms

2. The Compatibility and Mutual Independence of the Axioms

3. The Theory of Proportion

4. The Theory of Plane Areas

5. Desarguesís Theorem

6. Pascalís Theorem

7. Geometrical Constructions Based Upon the Axioms IĖV

Conclusion

Source: Project Gutenberg http://www.gutenberg.org

 

Hilbert's famous address Mathematical Problems was delivered to the Second International Congress of Mathematicians in Paris in 1900.

Following an extract from the address, in which Hilbert speaks of his views on mathematics:

 

"To new concepts correspond, necessarily, new signs. These we choose in such a way that they remind us of the phenomena which were the occasion for the formation of the new concepts. So the geometrical figures are signs or mnemonic symbols of space intuition and are used as such by all mathematicians. Who does not always use along with the double inequality a > b > c the picture of three points following one another on a straight line as the geometrical picture of the idea "between"?

 

Who does not make use of drawings of segments and rectangles enclosed in one another, when it is required to prove with perfect rigour a difficult theorem on the continuity of functions or the existence of points of condensation?

 

Who could dispense with the figure of the triangle, the circle with its centre, or with the cross of three perpendicular axes? Or who would give up the representation of the vector field, or the picture of a family of curves or surfaces with its envelope which plays so important a part in differential geometry, in the theory of differential equations, in the foundation of the calculus of variations and in other purely mathematical sciences?

 

The arithmetical symbols are written diagrams and the geometrical figures are graphic formulas; and no mathematician could spare these graphic formulas, any more than in calculation the insertion and removal of parentheses or the use of other analytical signs. "

 

 

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Last updated: January 5, 2007.
 

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