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The best conceptualization of the concept of number that I've come across is Von Staudt's construction of the rationals using the concept of harmonic tetrads. A

The best conceptualization of the concept of number that I've come across is Von Staudt's construction of the rationals using the concept of harmonic tetrads. At first it seems weirdly over-complicated, but there's a moment where it clicks and then a shocking beauty. It's like a version of special relativity that's even more relative, like, what could observers agree on if they didn't even agree on the speed of light? Turns out they can still calculate cross ratios, but then even the notion of cross ratio can be turned inside out, and you get your fundamental notion of number out of a deeper concept of harmony.

phoe-krk 4y ago
> The best conceptualization of the concept of number that I've come across is Von Staudt's construction of the rationals using the concept of harmonic tetrads. Do you have any more material for this? Google seems to give me no good results.
- jacobolus 4y ago
  From what I understand, Von Staudt and some other geometers were trying to push back against algebra/arithmetic taking over geometry, and against projective geometry invariants like the cross ratio being defined in terms of algebraic relations between Euclidean distances. So he flipped things around and developed arithmetic in terms of basic projective geometry relations instead. (Projective geometry is geometry without circles, angle measures, distances, or parallelism, only straight lines.) See Coolidge (1934) "The Rise and Fall of Projective Geometry". AMM 41(4), pp. 217-228 https://www.jstor.org/stable/2302023 https://www.jstor.org/stable/2302023 Google scholar search: https://scholar.google.com/scholar?q=staudt+throws https://scholar.google.com/scholar?q=staudt+throws https://en.wikipedia.org/wiki/Karl_Georg_Christian_von_Staudt#Algebra_of_throws https://en.wikipedia.org/wiki/Karl_Georg_Christian_von_Staud... https://en.wikipedia.org/wiki/Cross-ratio https://en.wikipedia.org/wiki/Cross-ratio
  - phoe-krk 4y ago
    Thanks!

phoe-krk 4y ago

> The best conceptualization of the concept of number that I've come across is Von Staudt's construction of the rationals using the concept of harmonic tetrads. Do you have any more material for this? Google seems to give me no good results.

jacobolus 4y ago
From what I understand, Von Staudt and some other geometers were trying to push back against algebra/arithmetic taking over geometry, and against projective geometry invariants like the cross ratio being defined in terms of algebraic relations between Euclidean distances. So he flipped things around and developed arithmetic in terms of basic projective geometry relations instead. (Projective geometry is geometry without circles, angle measures, distances, or parallelism, only straight lines.) See Coolidge (1934) "The Rise and Fall of Projective Geometry". AMM 41(4), pp. 217-228 https://www.jstor.org/stable/2302023 https://www.jstor.org/stable/2302023 Google scholar search: https://scholar.google.com/scholar?q=staudt+throws https://scholar.google.com/scholar?q=staudt+throws https://en.wikipedia.org/wiki/Karl_Georg_Christian_von_Staudt#Algebra_of_throws https://en.wikipedia.org/wiki/Karl_Georg_Christian_von_Staud... https://en.wikipedia.org/wiki/Cross-ratio https://en.wikipedia.org/wiki/Cross-ratio
- phoe-krk 4y ago
  Thanks!