Formalism 25

TL;DR

This paper reexamines Abraham Robinson's philosophical stance, providing evidence that he was a formalist rather than a finitist, and clarifies the distinction between his views on mathematics and metamathematics.

Contribution

It offers a corrected interpretation of Robinson's philosophy, challenging previous claims of finitism and emphasizing his formalist approach.

Findings

Robinson's writings support a formalist philosophical stance.

Robinson explicitly opposed finitist positions in his philosophical writings.

Robinson distinguished between mathematics and metamathematics, advocating finitism only for the latter.

Abstract

Abraham Robinson's philosophical stance has been the subject of several recent studies. Erhardt following Gaifman claims that Robinson was a finitist, and that there is a tension between his philosophical position and his actual mathematical output. We present evidence in Robinson's writing that he is more accurately described as adhering to the philosophical approach of Formalism. Furthermore, we show that Robinson explicitly argued {against} certain finitist positions in his philosophical writings. There is no tension between Robinson's mathematical work and his philosophy because mathematics and metamathematics are distinct fields: Robinson advocates finitism for metamathematics but no such restriction for mathematics. We show that Erhardt's analysis is marred by historical errors, by routine conflation of the generic and the technical meaning of several key terms, and by a philosophical {parti pris}. Robinson's Formalism remains a viable alternative to mathematical Platonism.