Tuesday, November 20, 2012

Ontology of the Natural Numbers

From time to time WB puts up a short post with two, three or four fascinating issues/questions/problems. Besides triggering considerable envy of his concision and breadth, this usually gets me thinking, usually about things I haven't thought about for years.

In particular, I wondered what a machine intelligence/semantic web version of an ontology for the natural numbers would look like. Recall that such a specification of a conceptualization should have some concepts or categories, some rules relating them, and perhaps some instances as well as an inference scheme for deducing implicit consequences. As it happens, just such a conceptualization of the natural numbers has been developed - the Peano Axioms. We have a concept, the natural number, an instance '0', the notion of the successor of a natural number, some relations among numbers, and first or second order logic for an inference engine.

Notice that this conceptualization doesn't say anything about the "ultimate" meaning of a number, about alternate conceptualizations that might or might not be equivalent, or the relation to our intuitions or physical models of numbers.

This eschewing of ultimate meaning evades some difficult or perhaps impossible questions, but it turns out to be amazingly powerful.