The important point is that these concepts in the physical world can be experimentally verified in the ordinal sense without recourse to a numerical scale. Examples are: entropy (from the ordering "adiabatically accessible" on the states of a thermodynamical substance), temperature (hotter than), price in economics from the relationship "worth more than" (under the generic name of "utility function") and of course in the synthetic approach to axiomatic euclidean geometry (using the ordering induced by the "in between" axiom). This is one of the central problems of the theory of measurement and explains how one can pull the reals out of a hat from a system of axioms which do not contain the concept of number explicitly. This is the basis of many results which show that natural orderings which arise there are induced by a real- valued function. There is a characterisation of the reals purely in terms of its order structure which explains its ubiquity as a model in mathematics and physics: a totally ordered, Dedekind complete set which has a countable order dense subset but no largest or smallest element. As suggested by some of the comments, I would like to rephrase my question: How can I deduce from the mathematical properties of the real numbers that they are mathematically interesting, and thus that they are the optimal formalization of our intuitions of geometry and infinitesimal operations? The familiar number sets $\mathbb$ and other concepts based on these notions (like almost all of Analysis, Differential topology, etc.), so I am well aware of the fact, that I should not refuse the real numbers as old-fashioned even in case, I don't receive many answers.Įdit: Of course, my question is implicitly based on my strong belief that mathematical interestingness and categorical interestingness are equivalent concepts.
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