Aggregate Theory Versus Set Theory

Abstract

Maddy’s 1990 arguments against Aggregate Theory were undermined by the shift in her position in 1997. The present paper considers Aggregate Theory in the light this, and the recent search for ‘New Axioms for Mathematics’. If Set Theory is the part-whole theory of singletons, then identifying singletons with their single members collapses Set Theory into Aggregate Theory. But if singletons are not identical to their single members, then they are not extensional objects and so are not a basis for Science. Either way, the Continuum Hypothesis has no physical interest. I first show that, because there are non-sortal predicates, Frege’s attempt to derive Arithmetic from Logic stumbles at its very first step. For there are properties without a number, and the contingency of that condition means Frege’s definition of zero is not obtainable from Logic. This result then points to the need to consider more fully properties without a number, and so to generate a theory of continua based on mereological aggregates rather than sets containing numbers of things.