Timeline for How do Euclid’s postulates imply that a line has more than two points? [closed]
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| Aug 19, 2021 at 23:33 | history | closed |
Sam Hopkins Roland Bacher David Roberts♦ Konstantinos Kanakoglou Lee Mosher |
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| Aug 19, 2021 at 23:33 | comment | added | Lee Mosher | There are various good books which trace the mathematical history of filling in various gaps in Euclid's axioms (the one you suggest is not the only gap). Take a look, for example at Geometry: Euclid and Beyond by Hartshorne. | |
| Aug 19, 2021 at 20:56 | review | Close votes | |||
| Aug 19, 2021 at 23:36 | |||||
| Aug 19, 2021 at 20:52 | comment | added | Buzz | My high school geometry book included axioms that a line contained two points; a plane contained three non-collinear points; and space contained four non-coplanar points. Axioms like these are needed for a rigorous description of Euclidean geometry, but after introducing them, the textbook never mentioned them again. | |
| Aug 19, 2021 at 20:41 | comment | added | JoshuaZ | It turns out that there are a lot of different implicit assumptions in Euclid's axioms which are not stated explicitly. It wasn't until around the end of the 19th century that actually rigorous axiomatic treatments of geometry were presented. See for example Hilbert's axioms en.wikipedia.org/wiki/Hilbert%27s_axioms . | |
| Aug 19, 2021 at 20:40 | comment | added | paul garrett | These are the sorts of things that Hilbert's reconsideration of Euclid's axioms addressed/added... | |
| Aug 19, 2021 at 20:37 | review | First posts | |||
| Aug 19, 2021 at 21:10 | |||||
| Aug 19, 2021 at 20:36 | history | asked | Shillington | CC BY-SA 4.0 |