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As is well known "God made the natural numbers; all else is the work of man" (Leopold Kronecker). However, "what would correspond more to the spirit of physics would be a mathematical theory of the integers in which numbers, when they became very large, would acquire, in some sense, a "blurred" form and would not be strictly defined members of the sequence of natural numbers as we consider it. The existing theory is, so to speak, over-accurate: adding unity changes the number, but what does the addition of one molecule to the gas in a container change for the physicist? If we agree to accept these considerations even as a remote hint of the possibility of a new type of mathematical theory, then first and foremost, in this theory one would have to give up the idea that any term of the sequence of natural numbers is obtained by the successive addition of unity - an idea which is not, of course, formulated literally in the existing theory, but which is provoked indirectly by the principle of mathematical induction. It is probable that for "very large" numbers, the addition of unity should not, in general, change them" http://iopscience.iop.org/0036-0279/28/4/L06/ (P.K. Rashevskii, On the dogma of the natural numbers).

Was Rashevskii's this idea (generalization of the notion of natural numbers more "consonant" to the spirit of physics) ever realized in some form? I'm aware of trigonometric and elliptic numbers of Frenkel and Turaev http://link.springer.com/chapter/10.1007%2F978-1-4612-4122-5_9 (Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions), but are they the generalizations in the spirit of Rashevskii? More likely they are just another example of "the work of man" based ultimately on the God given natural numbers.

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  • $\begingroup$ Isn't that very un-sciency? What "large number" means is very dependent on the scale you're looking at but reality doesn't depend on scaling. $\endgroup$
    – Johannes Hahn
    May 20, 2014 at 9:01
  • $\begingroup$ Of course, if realized, this will mean an introduction of some new physical constant to define the meaning of "large number". For example velocities much less than the light velocity add in the usual way. But if one velocity is the light velocity it will not change under the (relativistic) addition of velocities. By the way I agree that Rashevskii's idea is somewhat vague but intriguing. $\endgroup$
    – Zurab Silagadze
    May 20, 2014 at 9:16
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    $\begingroup$ Sounds like floating point numbers to me. :-) $\endgroup$
    – Michael
    May 20, 2014 at 9:32
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    $\begingroup$ parity effects are everywhere in physics, so an integer may be large, but whether it's even or odd makes a crucial difference; the statement that "the addition of unity should have no effect" is definitely not true. $\endgroup$
    – Carlo Beenakker
    May 20, 2014 at 10:25
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    $\begingroup$ Precisely how large integers should be blurred seems to be domain-specific. In the case of passing to large scale behavior (e.g., gas molecules and the masses of heavy objects), the intentional ignorance of fine detail seems to fall into the domain of renormalization. The discrete nature of molecules is by itself a low-energy approximation. $\endgroup$
    – S. Carnahan ♦
    May 21, 2014 at 0:54

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