A metaphysical foundation for mathematical philosophy

Abstract

Although mathematical philosophy is flourishing today, it remains subject to criticism, especially from non-analytical philosophers. The main concern is that even if formal tools serve to clarify reasoning, they themselves contribute nothing new or relevant to philosophy. We defend mathematical philosophy against such concerns here by appealing to its metaphysical foundations. Our thesis is that mathematical philosophy can be founded on the phenomenological theory of ideas as developed by Roman Ingarden. From this platonist perspective, the "unreasonable effectiveness of mathematics in philosophy"-to adapt Wigner's phrase-is analogous to that of mathematical explanations in science. As success-criteria for mathematical philosophy, we propose that it should be correct, responsive, illuminating, promising, relevant, and adequate.

Towards a topological philosophy

“…(Metaphorically speaking, such structures are, we think, something from which philosophical thinking can be struck or shaped, much as model figures are from stone or clay.) Mathematics, then, on our account not only provides a language for use in science and philosophy but also constitutes an invaluable source , thanks to its models, structures, and so forth (Wójtowicz and Skowron 2022).…”

“…There are various ways in which one might argue that it is hardly surprising that mathematics works so well in philosophy. Most notably, given a strong metaphysical assumption to the effect that there are ideas similar to those introduced by Plato, it can be argued that there is some overlap between mathematical and philosophical ideas(Wójtowicz and Skowron 2022), and that it is because of this substantial similarity at the level of ideas that mathematics fits so well into philosophical considerations.…”

Introduction to topo-philosophy

“…What the reader may wish to see, however, is more of the author's synthesis for the presented ideas, expanded beyond "Towards general topology of object and its parts" in section 7. One may also follow up on Skowron's ideas in his recently published paper 'A metaphysical foundation for mathematical philosophy" (Wójtowicz and Skowron, 2022).…”