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Elements is mostly formal, but it's also concrete and visual. Euclid developed arithmetic and algebra through constructive geometry, which relies on our visual
by gowld 2y ago
Elements is mostly formal, but it's also concrete and visual.
Euclid developed arithmetic and algebra through constructive geometry, which relies on our visual intuition to solve problems. Non-concrete problems were totally out of scope. Even curved surfaces (denying the parallel postulate) were byond Euclid.
Notably, Elements didn't have imaginary or transcendental numbers. Euclid made no attempt to unify line lengths and arc lengths, and had nothing to say about what fills the gaps between the algebraically (geometrically!) constructible numbers.
- paulddraper 2y ago> Elements didn't have imaginary or transcendental numbers Elements has pi.* *It proves the ratio of a circle's area to the square of its diameter is constant.
- gowld 2y agoBut doesn't say how large that constant value is.
- Someone 2y ago> Euclid […] had nothing to say about what fills the gaps between the algebraically (geometrically!) constructible numbers Did he even know those gaps existed? Euclid lived around 300BC. The problem of squaring the circle had been proposed around two centuries before that (https://en.wikipedia.org/wiki/Anaxagoras#Mathematics https://en.wikipedia.org/wiki/Anaxagoras#Mathematics), but I don’t think people even considered it to be impossible by that time.