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Hopefully I'm not wrong to suspect that the various formal axiomatic systems, which mathematicians develop, have varying amounts of empirical support (not that I generally know such systems, except by name, admittedly). Similarly, I suspect that some of them have no empirical support. Therefore, to begin sorting this out, I request the following.

Of the formal axiomatic systems, used in math, which have names and which you happen to know: please name any formal axiomatic system(s) which has/have: (A.) produced at least one equation or model that agrees with physical observation under some physical interpretation; or (B.) never produced at least one equation or model that agrees with physical observation under some physical interpretation.

Thank you.

user50489
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  • For (A) see Peano'axioms; they can prove : $1+1=2$ and this equation agrees with physical observation regarding apples. – Mauro ALLEGRANZA Nov 08 '14 at 13:02
  • Speaking of apples, Mauro ALLEGRANZA, you sure picked "the low-hanging fruit" in an effort to answer this, didn't you? Anyway, thank you for your comment; and I'm still curious to see how large our (A.) and (B.) lists can become, once we get past the obvious responses such as the Peano axioms. – user50489 Nov 08 '14 at 13:36
  • Mainly, I think you're missing the point. Formal axiomatic systems have nothing to do with "empirical support" or "physical observation" or anything like that: formal axiomatic systems are simply tools for reasoning about things that have prescribed properties using only those properties. –  Nov 08 '14 at 14:25
  • I guess my notion is that an equation or model, which can be derived from a formal axiomatic system (but possibly not derived from another formal axiomatic system) in combination with a creative physical interpretation, actually can be evaluated with respect to empirical observation. That is, the various formal axiomatic systems may yield different equations or models, which are not all evaluated identically with respect to empirical observation following physical interpretation. In that sense, the various formal axiomatic systems have "varying amounts of empirical support" in my view. – user50489 Nov 08 '14 at 14:53

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If I take Peano arithmetic and interpret $1$ to mean "the sky", $2$ to mean "color", every other number to mean "blue", and $x+y$ to mean "the $y$ of $x$", then we see that our interpretation of "$1+2=3$" holds: the color of the sky is indeed blue.

Of course, most arithmetic statements would be nonsensical under this interpretation....

  • I could be wrong...but doesn't Peano arithmetic require that its mathematical objects be interpreted at a bare minimum as certain types of numbers? I.e. natural numbers, which at a bare minimum have particular cardinal and ordinal meanings--which "the sky", "color", and "blue" don't have? – user50489 Nov 08 '14 at 18:01
  • If my comment above is right...and I realize it might not be...then I would think we aren't allowed by Peano arithmetic to interpret addition however we want. Rather, we are restricted to interpreting addition at a bare minimum as somehow affecting cardinality, ordinality, or another characteristic of natural numbers given by Peano's system. – user50489 Nov 08 '14 at 18:25