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. 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 Nonstandard Analysis yields the absence of meaning. We will debunk the Bishop–Connes critique by establishing the contrary, namely that the presence of ideal objects 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, similar to the ‘reversals’ from the foundational program Reverse Mathematics. The latter also plays an important role in gauging the scope of this translation. (shrink)

Interest in thought experiments (TEs) derives from the paradox: “How can findings that carry conviction result from a new experiment conducted entirely within the head?” Historical studies have established the importance of TEs in science but have proposed disparate hypotheses concerning the source of knowledge in TEs, ranging from empiricist to rationalist accounts. This article analyzes TEs in think‐aloud protocols of scientifically trained experts to examine more fine‐grained information about their use. Some TEs appear powerful enough to discredit an existing (...) theory—a disconfirmatory purpose. In addition, confirmatory and generative purposes were identified for other TEs. One can also use details in transcript data, including imagery reports and gestures, to provide evidence for a central role played by imagistic simulations in many TEs, and to suggest that these simulations can generate new knowledge using several sources, including the “extended application” of perceptual motor schemas, implicit prior knowledge, and spatial reasoning operations, in contrast to formal arguments. These sources suggest what it means for TEs to be grounded in embodied processes that can begin to explain the paradox above. This leads to a rationalistic view of TEs as using productive internal reasoning, but the view also acknowledges the historical role that experience with the world can play in forming certain schemas used in TEs. Understanding such processes could help provide a foundation for developing a larger model of scientific investigation processes grounded on imagistic simulation (Clement, 2008). (shrink)