On mathematical truth: Wittgenstein and the dissolution of Benacerraf’s dilemma

Abstract

According to Benacerraf, our most influential theory of meaning leads us to believe that we use mathematical expressions to refer to abstract objects, while our most influential theory of knowledge cannot fathom the idea of a subject knowing about an object that is not causally related to her. So mathematics is either a useful formal linguistic system that we invented, or we can discover causally impassive and inactive objects through intuition. In my view, we only arrive at such a dilemma if we misconceive something along the way.
Thus, the goal of this thesis is to dissolve Benacerraf’s dilemma. First, I argue for the inadequacy of the dilemma’s premises, then I demonstrate the possibility of a nonproblematic third alternative. As the argument goes, the dilemma rests on premises that presuppose the truth of the representationalist metasemantics and the causal theory of knowledge. However, these presuppositions promote confusions regarding the expressive function of mathematical language and raise the problematic necessity for postulating abstract elements as regress-stoppers. There are two insights from Wittgenstein at the basis of my criticism: (i) that use is explanatory of meaning, and (ii) that the way we use mathematical statements serves a normative instead of a descriptive function. Then I proceed to develop these insights into an expressivist and pragmatic account of mathematical truth, explained in terms of social practices, lifeforms and their evolutionary history. I argue this view can satisfy Benacerraf’s conditions while avoiding his presuppositions and the problems generated by them, thus effectively dissolving this dilemma.

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