The idea that a mathematical theory must rest on a few postulates accepted without proof has a history more than two thousand years long, and over these two millennia the very meaning of the word “postulate” changed profoundly.
Euclid’s model
Around 300 BC Euclid gathered in his Elements all the geometry of his time, organising it for the first time deductively: five postulates, five common notions and a handful of definitions, and from there, one theorem after another, 465 propositions. For Euclid the postulates were self-evident truths: statements so obvious as to need no proof, such as “through two points passes one and only one straight line”.
One of the five, however, did not seem obvious at all: the fifth postulate, the one about parallels. In the equivalent formulation due to Playfair it states that, given a line and a point outside it, there is exactly one parallel to the line through that point. For centuries mathematicians suspected it was a hidden theorem and tried to deduce it from the other four. Everyone failed.
The discovery of non-Euclidean geometries
In the nineteenth century it was understood why all those attempts had failed: the fifth postulate is independent of the others. Denying it leads to no contradiction, but to new, perfectly consistent geometries. Carl Friedrich Gauss reached this conclusion first, but published nothing; it was Nikolai Lobachevsky and, independently, János Bolyai who published, around 1830, a geometry in which through a point pass infinitely many parallels and the sum of the interior angles of a triangle is less than .
The consequence was a conceptual revolution: a postulate is no longer a “truth” about the world, but a free assumption. The only requirement asked of it is not to be true, but to be consistent (to generate no contradictions).
Hilbert’s Grundlagen
Seen with this more critical eye, even Euclid’s work showed cracks: in several proofs he used, without stating them, facts “read off the figure” — for instance that a point lies between two others, or that two circles intersect. In 1882 Moritz Pasch was the first to make these order assumptions explicit (the “Pasch axiom” about triangles).
The crowning achievement came in 1899 with the Grundlagen der Geometrie (“Foundations of Geometry”) of David Hilbert. Hilbert reorganised all of elementary geometry into five groups of axioms — of incidence, of order, of congruence, of parallels and of continuity — filling every gap left by Euclid. His most radical idea concerns the primitive terms: point, line and plane are no longer defined or drawn, but take on meaning only from what the axioms say about them. Hilbert put it in a remark that has become famous: one must be able to say, at any moment, “instead of points, lines and planes: tables, chairs and beer mugs”. What matters is not what those objects are, but which relations they satisfy.
This is how the modern axiomatic method was born: axioms do not describe self-evident truths, but implicitly define the objects they speak of, and are judged by consistency, independence and completeness. It is the same method by which, still today, every mathematical theory is founded (Boyer; Kline; Katz).
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Topics: Euclidean geometry Concepts: Postulate · Theory People: Euclid · Gauss · Nikolai Lobachevsky · János Bolyai · Moritz Pasch · David Hilbert