Reverse formalism 16

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