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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
See also: Anschauliche
Geometrie, translated into English as
Geometry and the Imagination.
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. Desargues’s Theorem
6. Pascal’s Theorem
7. Geometrical Constructions Based
Upon the Axioms I–V
Source: Project Gutenberg
Hilbert's famous address Mathematical Problems was
delivered to the Second International Congress of Mathematicians in Paris in
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
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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