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    $\begingroup$ I don't understand your point, sorry. Could you make it clearer? I am saying that in the end, to fit experimental data, you will need "numbers" that can be compared to each other. From an experimental point of view, we are pretty sure that a $Z_0$ flying at $0.5 c$ has a far bigger energy than an electron flying at the same speed. Whatever the model I assume, the final theoretical prediction has to be comparable i.e has to take values on a field of numbers which has an order relation. However, sadly, all field extensions of $\mathbb{R}$ (the complex, quaternions, octonions) do not have it. $\endgroup$ Commented Oct 28, 2013 at 23:46
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    $\begingroup$ Well, the point is as follows: Say, you have a field $\vec E = \{0,1,2\}$. Foundamentally it is a vector. I guess, there is no problem in comparing it to some other field, say $\vec E'=\{2,1,0\}$. So, on foundamental level we are sometimes dealing with quantities, which are not ordered, hence not comparable in your sense. Why then do we need to be always able to compare observed, measured quantities to each other? What is different between observed and foundamental quantities, that one need to be ordered, while others don't? $\endgroup$ Commented Oct 28, 2013 at 23:52
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    $\begingroup$ This is all true. But then I read it as: sometimes we need to compare quantities, hence some measurements should be numbers. Then, why can't some other measurements be, say, vectors? I guess then your thesis statement is "Among all posible measurements we need to have those which produce reals, in order to be able to compare them with each other". $\endgroup$ Commented Oct 29, 2013 at 1:36
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    $\begingroup$ Oh wow only now I fully understand what you say! Yes, exactly. In principles the measure results could be reals and "other things". But we must have at least the reals, and we shall never compare the "other things". Now I see. Thanks for sharing your opinions! $\endgroup$ Commented Oct 29, 2013 at 1:39
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    $\begingroup$ A small remark: I think you are confusing finite fields $\mathbb F_p = \mathbb Z/p\mathbb Z$, which are cyclic as groups, but don't have a norm compatible with the algebraic structure, and the p-adic fields $\mathbb Q_p$ (and their ring of integers $\mathbb Z_p$) which are harder to define, but which are complete normed fields in which calculus can be done. $\endgroup$ Commented Oct 29, 2013 at 2:07