## Neo-Logicism? An Ontological Reduction of Mathematics to Metaphysics

### Author

Edward N. Zalta
### Reference

*Erkenntnis*, **53**/1-2 (2000): 219-265
### Abstract

In this paper, we describe "metaphysical reductions", in which the
well-defined terms and predicates of *arbitrary* mathematical
theories are uniquely interpreted within an axiomatic, metaphysical
theory of abstract objects. Once certain (constitutive) facts about a
mathematical theory T have been added to the metaphysical theory of
objects, theorems of the metaphysical theory yield both an analysis of
the *reference* of the terms and predicates of T and an analysis of
the *truth* of the sentences of T. The well-defined terms and
predicates of T are analyzed as denoting abstract objects and abstract
relations, respectively, in the background metaphysics, and the
sentences of T have a reading on which they are true.
After the technical details are sketched, the paper concludes with
some observations about the approach. One important observation
concerns the fact that the proper axioms of the background theory
abstract objects can be reformulated in a way that makes them sound
more like logical axioms. Though I shall not claim that they are
logical axioms, some philosophers have argued that we should accept
(something like) them as being logical.

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