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Full Idea
Euclid's fifth 'parallel' postulate says if there is an infinite straight line and a point, then there is only one straight line through the point which won't intersect the first line. This axiom is independent of Euclid's first four (agreed) axioms.
Gist of Idea
Euclid's parallel postulate defines unique non-intersecting parallel lines
Source
report of Euclid (Elements of Geometry [c.290 BCE]) by Michèle Friend - Introducing the Philosophy of Mathematics 2.2
Book Ref
Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.24
A Reaction
This postulate was challenged in the nineteenth century, which was a major landmark in the development of modern relativist views of knowledge.
| 8623 | Proof reveals the interdependence of truths, as well as showing their certainty [Euclid, by Frege] |
| 13907 | If you pick an arbitrary triangle, things proved of it are true of all triangles [Euclid, by Lemmon] |
| 6297 | Euclid's geometry is synthetic, but Descartes produced an analytic version of it [Euclid, by Resnik] |
| 9603 | An assumption that there is a largest prime leads to a contradiction [Euclid, by Brown,JR] |
| 8738 | Postulate 2 says a line can be extended continuously [Euclid, by Shapiro] |
| 22278 | Euclid relied on obvious properties in diagrams, as well as on his axioms [Potter on Euclid] |
| 8673 | Euclid's parallel postulate defines unique non-intersecting parallel lines [Euclid, by Friend] |
| 10250 | Euclid needs a principle of continuity, saying some lines must intersect [Shapiro on Euclid] |
| 10302 | Euclid says we can 'join' two points, but Hilbert says the straight line 'exists' [Euclid, by Bernays] |
| 14157 | Modern geometries only accept various parts of the Euclid propositions [Russell on Euclid] |
| 9894 | A unit is that according to which each existing thing is said to be one [Euclid] |
| 1600 | Euclid's common notions or axioms are what we must have if we are to learn anything at all [Euclid, by Roochnik] |