On a Revision of Logicism

Abstract

In Mathematical Knowledge, Mark Steiner discusses several schools in the philosophy of mathematics; in the first two chapters he attempts to criticize what he believes is a correct reconstruction of several epistemic aspects of logicism and what he assumes are serious difficulties of Quine’s philosophy of mathematics.

Our main claim is that his critical revision of logicism in incorrect On the one hand, he gave an equivocal characterization of its central problems and theses; as a result, he actuality has criticized claims which were never supported by the logicist philosophers. On the other hand, he wrongly evaluated Quine’s reduction of Arithmetic into Theory of Sets. In fact, his criticism is based in an inadequate conception of some features of Quine’s philosophy of mathematics. Thus, Steiner wrongly understood Quine’s view about the reduction of theories by assuming that such a reduction should satisfy such requirements as clarity, certainty and an improvement in operativity. Finally, he blamed Quine’s reduction for raising problems that, truly, do not emerge.

As a consequence, logicism seems to stay beyond Steiner’s criticism.
[R.J.G.]