Reverse formalism 16. - Sanders, Sam (Synthese, 2020)
In his remarkable paper Formalism 64, Robinson defends his eponymous position concerning the foundations of mathematics, as follows: Any mention of infinite totalities is literally meaningless. We should act as if infinite totalities really existed. Being the originator of Nonstandard Analysis, it stands to reason that Robinson would have often been faced with the opposing position that 'some infinite totalities are more meaningful than others', the textbook example being that of infinitesimals (versus less controversial infinite totalities). For instance, Bishop and Connes have made such claims regarding infinitesimals, and Nonstandard Analysis in general, going as far as calling the latter respectively a debasement of meaning and virtual, while accepting as meaningful other infinite totalities and the associated mathematical framework. We shall study the critique of Nonstandard Analysis by Bishop and Connes, and observe that these authors equate 'meaning' and 'computational content', though their interpretations of said content vary. As we will see, Bishop and Connes claim that the presence of ideal objects (in particular infinitesimals) in Nonstandard Analysis yields the absence of meaning (i.e. computational content). We will debunk the Bishop–Connes critique by establishing the contrary, namely that the presence of ideal objects (in particular infinitesimals) in Nonstandard Analysis yields the ubiquitous presence of computational content. In particular, infinitesimals provide an elegant shorthand for expressing computational content. To this end, we introduce a direct translation between a large class of theorems of Nonstandard Analysis and theorems rich in computational content (not involving Nonstandard Analysis), similar to the 'reversals' from the foundational program Reverse Mathematics. The latter also plays an important role in gauging the scope of this translation.