On the role of infinity in physics: comparing continuous to discrete and hyperfinite models

Continuous models and infinite limits are pervasive in physics. It is common to use real numbers and calculus, motivated only by mathematical convenience, which leaves open philosophical questions. As for any mathematical formalism used in physics, one may ask which properties of these standard models are representative (corresponding to something physical) and which are artefacts (superfluous structure). To differentiate between the two, in this project, we compare the standard way of doing physics with two alternative approaches: discrete and finite models (without infinities and without continuity; see, e.g., Lee 1983) and hyperfinite models in physics (based on non-standard analysis, which provides an alternative formalization of infinity and continuity; see, e.g., Albeverio et al. 1986). We will focus on case studies in classical mechanics and statistical physics that can be treated with standard physics as well as with both alternative approaches (e.g., Helms & Loeb 1979, Marsden & West 2001, Bagarello 1999, and Borrmann et al. 2000). To the best of our knowledge, there has been no earlier attempt to combine these parts of the literature on the foundations of physics in this way and to address them together.

Keywords:foundations of physics, infinity, infinitesimals

Disciplines:Applied mathematics in specific fields, Astronomy and space sciences, Classical physics, Materials physics, Mathematical physics, Quantum physics, General pedagogical and educational sciences, Communication sciences, Philosophy