Book 1, Proposition 1[düzenle]
- The very first geometric proof in the Elements, shown in the figure above, is that any line segment is part of a triangle; Euclid constructs this in the usual way, by drawing circles around both endpoints and taking their intersection as the third vertex. His axioms, however, do not guarantee that the circles actually intersect, because they do not assert the geometrical property of continuity, which in Cartesian terms is equivalent to the completeness property of the real numbers.
Starting with Moritz Pasch in 1882, many improved axiomatic systems for geometry have been proposed, the best known being those of Hilbert, George Birkhoff, and Tarski.
- The full historical story behind this opinion is of course a complicated one. Three interrelated factors, however, are consistently tied to its emergence, and it is these I will discuss. They are: the generality problem, the modern mathematical understanding of continuity, and the modern axiomatic method. The first is a puzzle that has surrounded
Euclid’s proofs from the time they were conceived. The second is a 19th century conceptual development which seemed to expose diagrammatic methods as hopelessly imprecise. And the third is a methodological development, also occurring in the 19th century, which provided a clear and exact way to understand both geometric generality and continuity.
- The general observation here, if accepted, is fatal to the legitimacy of Euclid’s diagrammatic proofs. Again and again, Euclid takes intersection points to exist because they appear in his diagrams. Indeed, he does this right away in proposition I,1 when he introduces the intersection point of two constructed circles. The only thing Euclid offers as justification is a diagram where two circles cross. Nowhere do we find the articulation of a continuity principle which by modern standards seems necessary to secure the point’s existence.
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