Chapter 12: The Principle of Computational Equivalence

Section 9: Implications for Mathematics and Its Foundations

[Axioms for] geometry

Euclid gave axioms for basic geometry around 300 BC which were used with fairly little modification for more than 2000 years. In the 1830s, however, it was realized that the system would remain consistent even if the so-called parallel postulate was modified to allow space to be curved. Noting the vagueness of Euclid's original axioms there was then increasing interest in setting up more formal axiom systems for geometry. The best-known system was given by David Hilbert in 1899—and by describing geometrical figures using algebraic equations he showed that it was as consistent as the underlying axioms for numbers.

The axioms given here are illustrated below. They were developed by Alfred Tarski and others in the 1940s and 1950s. (Unlike Hilbert's axioms they require only first-order predicate logic.) The first six give basic properties of betweenness of points and congruence of line segments. The second- and third-to-last axioms specify that space has two dimensions; they can be modified for other dimensions. The last axiom is a schema that asserts the continuity of space. (The system is not finitely axiomatizable.)

The axioms given can prove most of the results in an elementary geometry textbook—indeed all results that are about geometrical figures such as triangles and circles specified by a fixed finite number of points, but which do not involve concepts like area. The axioms are complete and consistent—and thus not universal. They can however be made universal if axioms from set theory are added.

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From Stephen Wolfram: A New Kind of Science [citation]