This is a good question with many possible answers.

A good place to start in answering this question is

https://www.researchgate.net/publication/2207709_Formal_and_rigid_geometry_an_intuitive_introduction_and_some_applications#pag:19:mrect:(398.75,471.69,14.60,9.96)

Article Formal and rigid geometry: an intuitive introduction, and so...

Hilbert regarded the Archimedean Axiom practically as an experimental principle. This is discussed in “The Philosophy of Mathematical Practice” by Paolo Mancosu, for example. The relationship between mathematics and physics is open to dispute. Yet, if it is admitted that physics is grounded on measurement, non-Archimedean geometry should be considered non-physical.

In my opinion, the problem with that axiom doesn’t lie as much in the uncertainty principle, but rather in the infinitesimals. The problem of associating measures with infinitesimals is bypassed when the derivative is calculated, but local geometries may come across it. For this reasons I would suggest to have a look at “Two Essays on the Archimedean versus Non-Archimedean Debate” (Elemer Rosinger) https://arxiv.org/abs/0809.4509.

    In Physics it is paradigmatic the Einstein's General Theory of Relativity, where the Riemann geometry is employed for the space-time. But I suppose that for calculating optimal trajectories for aircrafts on the Earth atmosphere the logic is to employ also the Riemann geometry.

Dear Mrs. Sara Liyuba Vesely,

You said "Yet, if it is admitted that physics is grounded on measurement, non-Archimedean geometry should be considered non-physical." 

But I see the opposite conclusion from the physical measurement.  Archimedean axiom is as follows. You have two arbitrary straight line segments of length a and b. Then there is a such natural number n that na>b. Or we can create the segment having the length na which is longer than the length b of the other segment. In any physical measurement the one segment, say a can not have arbitrary length due to the measurement error, say of the length d. In other words a>d. Therefore in any physical measurements Archimedean axiom is not true.

Please clarify this issue.

Dear dr. Anatoly A. Khripov,

in my opinion, except for Heisenberg’s uncertainty principle, which is embodied in quantum mechanics, the experimental error is not part of a theory. For example, Newton’s laws don’t state what error is prescribed in doing measurements. Thus, it is not excluded that an improved apparatus can improve the accuracy of measurements.

In quantum mechanics there is a built in uncertainty, but the way of handling it is different from the non-Archimedean one implied by infinitesimals of first, second, third… order. Hilbert states the Axiom of Archimedes among the continuity axioms. Saying that it is an “experimental” axiom he means that physics deals with magnitudes that can be measured as precisely as required. (In this respect, please note dr. Baldomir's answer). The other axiom of continuity that Hilbert takes into account is Dedekind’s.

Finally, my comment didn't mean that I myself think that the Archimedean Axiom is necessary for the development of physics.

Applications of non-Archimedean analysis are considered by scientific school of V.S.Vladimirov and some other groups. The key words are "p-adical mathematical physics".

There is active community on p-adical math physics, with conferences and journals.

Some active researchers are

• I.V. Volovich (p-adical quantum mechanics, p-adical string theory, etc. Steklov Mathematical Institute),
• E.I. Zelenov (p-adical quantum mechanics, Steklov Mathematical Institute),
• A.Kh. Bikulov (p-adical diffusion and protein dynamics, Semenov Institute of Chemical Physics),
• V.A. Arutyunov (p-adical diffusion and protein dynamics, self-organisation, Semenov Institute of Chemical Physics),
• A.Yu.Khrennikov (p-adical quantum mechanics, probability, cognitive science, etc. International Center for Mathematical, Modelling in Physics and Cognitive Sciences, Linnaeus University, Sweden).

P-adical models are applicable to different systems with hierarchy: tertiary protein structure, spin-glass, cognitive science etc.

P-adical diffusion is successfully used to describe kinetics of protein folding.

Application of p-adical geometry to space-time was considered at speculative level by I.V. Volovich at (approximately) 1970s.

Dear Mikhail G. Ivanov,

I would be very grateful if you give internet addresses of that groups or emails in St. Petersburg, Russia? ·

Thank you for an advance

Dear Mikhail G. Ivanov,

What I wrote above was classical Archimedean axiom of Euclid geometry. But the notion of non-Archimedean geometry is not the absence of classic Archimedean axiom! I read from the article "Non-Archimedean Geometry and Physics on Adelic Spaces" by Branko DRAGOVICH (downloaded from arxiv site) that the non-Archimedean geometry is a geometry which has the strong triangle inequality:

d(x,y)

To find p-adic people in St. Petersburg it would be useful to request somebody inside the p-adic community (I am near, but not inside), E.g. you could e-mail to S.V. Kozyrev from Steklov Mathematical Institute (Moscow) (see the link).

http://www.mathnet.ru/php/person.phtml?option_lang=rus&personid=11785

Strong triangle inequality means in linear space

|a+b|

Dear Mikhail G. Ivanov,

Thank you very much for all your messages and clarifications!!

Dear Mr. Mikhail G Ivanov,

I did not see before. But now I ask you. The below inequality is impossible:

|a+b|

Inequality

|a+b|