About the type of numbers allowed by axioms and Nature

Question

I have a question which has 4 different subcases or "avatars":

1) Has every "interesting" class of number been invented? 2) Has every "possible" class of number been invented? 3) Does Nature use every interesting class of number for something in the physical realm? 4) Does Nature use every possible class of number for something in the physical world?

Prime numbers, integer numbers, fractional numbers, irrational numbers, real numbers, complex numbers, quaternionic numbers, octonionic numbers, Cayley-Graves numbers, p-adic numbers, tropical numbers, surreal numbers, transfinite numbers, adelic numbers, grassmannian numbers, clifford numbers, Cayley-Dickson numbers, ternary numbers, ideal numbers, p-ary numbers, ... What else?

Nature has no saying in mathematics: if something fits it, good. Otherwise, who cares? — DonAntonio, May 17 '13 at 16:04
@DonAntonio Clearly, some mathematical concepts are better represented in nature than others. — William Stagner, May 17 '13 at 16:06
That, @WilliamStagner, could be true, yet I think it is not what the OP asked. — DonAntonio, May 17 '13 at 16:47
It is quite possible that nature "uses" only a finite subset of natural numbers, and all the rest of what we do is approximations for dealing with huge sets of values. — Thomas Andrews, May 17 '13 at 16:51
Voting to close as nonconstructive. What makes a class of numbers interesting? Highly subjective. — Gerry Myerson, May 19 '13 at 06:37

score 0 · Answer 1 · answered May 19 '13 at 05:22

Nature has everything to do with mathematics. Mathematics is the natural science, inherent in the physical world, of quantity, with its amazing implications from set theory to category theory and far beyond. There are endless classes of integers alone. Just glance at the Bell Labs catalog of integer sequences. There is no reason to think that mathematical invention and discovery will ever reach an end as long as we exist. By the way, WE invent and discover what can exist in nature, not only in mathematics but in every area of our enterprise from agriculture and airplanes to Zeppelins and zoos. The idea that mathematics is not of the natural world was proposed by loafing weavers of invisible cloth long ago and is long long long overdue to be abandoned.

About the type of numbers allowed by axioms and Nature

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