Dear Sirs,
All sciences uses Euclid geometry in many cases. For that purpose one should choose what is straight line, point, plane, etc. For example light raw can be chosen as a straight line. But straight line as mathematical object should satisfy all Euclid axioms.
Does any object in nature satisfy Archimedes axiom?
Please see Hilbert axiomatic system for the Archimedes axiom definition at
https://en.wikipedia.org/wiki/Hilbert%27s_axioms
I enclosed the quote:
"Axiom of Archimedes. If AB and CD are any segments then there exists a number n such that n segments CD constructed contiguously from A, along the ray from A through B, will pass beyond the point B."
This is a good question with many possible answers.
In addition to the Wikipedia page cited by @Anatoly A Khripov, see
https://en.wikipedia.org/wiki/Archimedean_property
For the history of geometry up 1870, see
http://arxiv.org/pdf/1409.1140.pdf
The Archimedian axiom is discussed on page 6. Dehn showed that it is impossible to prove Legendre’s first theorem with the incidence, the order, the congruence axioms and without Archimedes’ axiom:
http://www1.mat.uniroma1.it/ricerca/rendiconti/ARCHIVIO/2010(1)/47-66.pdf
More to the point, the Archimedes' axiom does apply to such things as geographic surveys and ordinary architectural measurements, but it is not possible to apply Archimedes' axiom in the measurement of all distances. In the following thesis, it is pointed out that a measurement of distances smaller than Planck's length is impossible. See
https://dea.lib.unideb.hu/dea/bitstream/handle/2437/174121/disszertacio7_titkositott.pdf;jsessionid=0665A2F074997BC58EB50246D7205012?sequence=1
Dear Prof. James F Peters,
Thank you very much for the important information for me.
It well known that in quantum mechanics one can not precisely measure a coordinate x and the x-projection of momentum px due to Heisenberg`s inequality dx*dpx>h/(4pi), where dx -rms value of x-coordinate, h-Plank constant. Therefore we can not precisely measure the trajectory of moving quantum object, say electron. In this sense we can not precisely measure the x, y, z coordinates of this trajectory or the electron position point.
Based on the above I suggest that maybe some mathematicians or physicists invented non-Euclidean geometry with the modified Archimedes axiom? I mean the following or the like modification of Archimedes axiom (sorry for mistakes, I am physicist):
"If AB and CD are segments such that the lengths of AB and CD >PL, where PL is some real number, then there exists a number n such that n segments CD constructed contiguously from A, along the ray from A through B, will pass beyond the point B."
Could you advise me some articles which are close to my above suggestions?
I thank you for an advance.
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