Dear Sirs,
Heisenberg inequality states that the measurement values of the particle position coordinates (x,y,z) have necessary errors. So Archimedean axiom is not valid in quantum mechanics.
1. Could we develop a new geometry where the notion "congruence" contains the notion "error"? Anybody explored it?
2. If the item 1is developed then Archimedean axiom could be valid in quantum mechanics? Could anybody comment it or show my mistake?
I am appreciate of any references.
I afraid to say that it is not possible unless you can define the notion of error there. Euclidean system is a sound logical system not on approximation. So not possible to include error.
This is a very good question.
In addition to the very helpful answers already given, you may want to explore the notion of error in congruence in Euclidean geometry in terms of the proof of theorems. To start this line of reasoning, see
http://www.math.wustl.edu/~sk/eolss.pdf
See page 26 and consider erroneous theorems about congruence.
Another plausible path for a consideration of error and conguence is in terms of mistaken notation. See
https://www.jstor.org/stable/27964580?seq=1#page_scan_tab_contents
https://www.researchgate.net/publication/259860516_Proof-Related_Reasoning_in_High_School_Textbooks
https://www.researchgate.net/publication/234764010_How_Well_Do_Students_Write_Geometry_Proofs
Dr. Sharon L. Senk, Michigan State University, is on RG and many of her papers are downloadable.
Article Proof-Related Reasoning in High School Textbooks
Article How Well Do Students Write Geometry Proofs?
Dear Prof. James F Peters,
I thank you very much for your articles and information.
Not "error" but rather "precision", "uncertainty" or "accuracy". While this notion is very natural (and in fact necessary) in physics, it is not applicable to classical geometry which is, in some sense, "perfect". Think about the length of square diagonal, not expressible exactly in any computer.
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