I think the trick is that there’s an intermediate layer of map, in between arithmetic and physical objects, which you aren’t seeing: finite set cardinality. You aren’t “adding” sheep, you’re taking the union of two disjoint sets of sheep. Everything between arithmetic and cardinality is provable mathematics, but cardinality maps much more closely to the operation you’re actually performing on sheep. In this example, you can’t get from addition on number of sheep to addition on economic value, because economic-value(set-of-sheep) is a function which does not have sigma-additivity (which is a fancy way of saying that the value of a set of sheep is not always equal to the sum of the values of the subsets).
The intermediate map resolves the map-territory problems with arithmetic, but doesn’t really address the problem here, which is more than that the map of arithmetic doesn’t correspond perfectly with the territory, but that there are -other valid maps- of that same territory which can evaluate the same real-world operation (adding a sheep to a field) on the basis of a different abstract operation.
“What are you counting” is the operative question here; arithmetic only makes sense if you count additive properties.
I think the trick is that there’s an intermediate layer of map, in between arithmetic and physical objects, which you aren’t seeing: finite set cardinality. You aren’t “adding” sheep, you’re taking the union of two disjoint sets of sheep. Everything between arithmetic and cardinality is provable mathematics, but cardinality maps much more closely to the operation you’re actually performing on sheep. In this example, you can’t get from addition on number of sheep to addition on economic value, because economic-value(set-of-sheep) is a function which does not have sigma-additivity (which is a fancy way of saying that the value of a set of sheep is not always equal to the sum of the values of the subsets).
The intermediate map resolves the map-territory problems with arithmetic, but doesn’t really address the problem here, which is more than that the map of arithmetic doesn’t correspond perfectly with the territory, but that there are -other valid maps- of that same territory which can evaluate the same real-world operation (adding a sheep to a field) on the basis of a different abstract operation.
“What are you counting” is the operative question here; arithmetic only makes sense if you count additive properties.