CEFISES OLOFOS Seminar - Putting the dialectics of the confrontation between the scientific and the manifest image at the very core of mathematics

Title: Putting the dialectics of the confrontation between the scientific and the manifest image at the very core of mathematics.

Abstract: In this talk I apply the ideas of my current book project on interpretation in mathematics to this year’s topic of OLOFOS: the scientific vs. manifest image, here within mathematics. In that book I defend a view on mathematics according to which mathematical objects are seen as created by mathematicians by means of their formal definition, but have from the moment of creation on an existence independent of their creator’s mind. Those objects can be studied, but nobody has direct access to them. So, despite the apparent certainty that is associated with mathematical results, this study is in essence of a fallible nature, just like results obtained in the empirical sciences. I will argue that the discovery of mathematical truth can best be approached through a dialectical process. This approach has several advantages, among which: it proposes a non-platonistic way in which we can understand Gödel’s distinction between formal proof and truth, it allows for (possibly) non-trivial paradoxical objects, and it is closer to the nature of mathematical practice in which debates and some amount of uncertainty are indispensable.

It seems sufficiently faithful to Sellars’ conceptual distinctions to take the scientific image of mathematics to be the formal apparatus of axioms and definitions on the one hand and the manifest image as the content of these theories for human beings doing. Arithmetic is a good example. Its scientific image would be constituted by formal theories for arithmetic (Peano arithmetics, the finite ordinals in ZFC set theory, etc.) and the manifest image would then be what the natural numbers are for a human epistemic agent and the inferential role they play inside her web of beliefs and the actions she engages in (physical actions, speech acts, as well as mental actions). The dialectical process I aim to explicate in my book project could then correspond to the process that brings a human agent from a intuitive number concept to a formalism, to the intended model of the formalism (natural numbers), to e.g. a refined formalism inside set theory, to a refined interpretation, etc.

Besides making these philosophical conceptualizations, I will also offer a brief introduction to the formal work done in the book, which is an attempt to offer a logical foundation for this dialectical approach to mathematics. The basis of it is a dynamic collection theory that allows for incomplete and inconsistent collections, the definition of which is in terms of criteria for membership and (possibly independent) criteria for non-membership. By means of so called adaptive logics (first devised by Diderik Batens), one reasons about such collections under the assumption that the collection is consistent and complete, unless and until it is proven that they are not. The instability of the subsequent estimations of the consistency and the completeness of the collections allows for a dialectical dynamics of the results that can be obtained about the members of the collection.  Finally, as an illustration, if time permits, I will informally present one such collection, the collection of the sets in the Von Neumann hierarchy, as the result of a process bringing us from the scientific to the manifest image (and back).