Bartlomiej Skowron

In this article, we propose a hermeneutic analysis and approach to mathematical intuition, briefly highlighting differences from strictly phenomenological approaches (Husserl, Ingarden). We analyse various ways in which mathematical intuition manifests itself, including the use of intuitive concepts—which are informal yet active in the creation of mathematical knowledge—and background knowledge. Furthermore, we contrast intuitive actions with those strictly determined by the rules of formalised theories, and we point to the use of intuitive models. Mathematicians operate within a horizon of implicit and unspoken assumptions that form a hermeneutic horizon, even when they attempt to codify mathematical theories in a strictly formalised manner, and hermeneutics methodically reconstructs this background knowledge. An analysis of historical case studies reveals that such horizontal implicit components of mathematical knowledge are always present in the development of mathematics. They usually take the form of ontological and structural presuppositions that influence the shape of the formalisation employed.

This book is a collection of articles authored by renowed Polish ontologists living and working in the early part of the 21st century. Harking back to the well-known Polish Lvov-Warsaw School, founded by Kazimierz Twardowski, we try to make our ontological considerations as systematically rigorous and clear as possible – i.e. to the greatest extent feasible, but also no more than the subject under consideration itself allows for. Hence, the papers presented here do not seek to steer clear of methods of inquiry typical of either the formal or the natural sciences: on the contrary, they use such methods wherever possible. At the same time, despite their adherence to rigorous methods, the Polish ontologists included here do not avoid traditional ontological issues, being inspired as they most certainly are by the great masters of Western philosophy – from Plato and Aristotle, through St. Thomas and Leibniz, to Husserl, to name arguably just the most important.

The contributions gathered here demonstrate how categorical ontology can provide a basis for linking three important basic sciences: mathematics, physics, and philosophy. Category theory is a new formal ontology that shifts the main focus from objects to processes. The book approaches formal ontology in the original sense put forward by the philosopher Edmund Husserl, namely as a science that deals with entities that can be exemplified in all spheres and domains of reality. It is a dynamic, processual, and non-substantial ontology in which all entities can be treated as transformations, and in which objects are merely the sources and aims of these transformations. Thus, in a rather surprising way, when employed as a formal ontology, category theory can unite seemingly disparate disciplines in contemporary science and the humanities, such as physics, mathematics and philosophy, but also computer and complex systems science.

Braddon-Mitchell and Miller put forward the claim that the relation of being-the-same-person is gradable: a person can be the same person tomorrow as today, but only half the same. To justify their thesis, they propose a model of persons that is intended to be metaphysically neutral. This article sets out to show that such a model implicitly contains strong metaphysical assumptions that run contrary to the authors’ own statements. Using Roman Ingarden’s phenomenological ontology, we aim to demonstrate that within the model in question persons exist intentionally, where this leads to a caricature rather than an adequate account of persons. In opposition to Braddon-Mitchell and Miller, we argue that the justification presented for the thesis of the gradability of the relation of being-the-same-person in fact makes it reasonable to demand that an adequate model of persons be posited. We present Kurt Lewin’s holistic topological-dynamic model, and show that in this context it is not being-the-same-person that is gradable, but rather the characteristic (of persons) that is dimension. Thus, we reject the thesis of the gradability of the relation of being-the-same-person involved in personal identity.

This article examines the use of mathematical concepts in philosophy, focusing on topology, which may be viewed as a modern supplement to geometry. We show that Plato and Parmenides were already employing geometric ideas in their research, and discuss three examples of the application of topology to philosophical problems: the first concerns the analysis of the Cartesian distinction between res extensa and res cogitans, the second the ontology of possible worlds of Wittgenstein's Tractatus, and the third Leibniz's monadology. We also consider the role of topology in mathematical explanations of the sort found in science, arguing that it can perform a role in philosophy that is of comparable importance.

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.

Virtual objects, such as online shops, the elements that go to make up virtual life in computer games, virtual maps, e-books, avatars, cryptocurrencies, chatbots, holograms, etc., are a phenomenon we now encounter at every turn: they have become a part of our life and our world. Philosophers—and ontologists in particular—have sought to answer the question of what, exactly, they are. They fall into two camps: some, pointing to the chimerical character of virtuality, hold that virtual objects are like dreams, illusions and fictions, while others, citing the real impact of virtuality on our world, take them to be real—an actual part of the real world, just like other real objects. In this article, we defend the thesis that both sides are wrong. Using Roman Ingarden’s phenomenological ontology, we advocate a position according to which a virtual object is a computationally grounded intentional object that has its existential foundation in computational processes, which are compliant with a certain model of computation. We point out that virtuality is framed by some kind of ideal mathematical objects: i.e., mathematical models of computation, which in turn fall, each of them, under their respective ideas. We also refer to the idea of natural computation, which in conjunction with the ontological analysis carried out leads to the thesis that an object can be more or less virtual.

It is a well-known fact that mathematics plays a crucial role in physics; in fact, it is virtually impossible to imagine contemporary physics without it. But it is questionable whether mathematical concepts could ever play such a role in psychology or philosophy. In this paper, we set out to examine a rather unobvious example of the application of topology, in the form of the theory of persons proposed by Kurt Lewin in his Principles of Topological Psychology. Our aim is to show that this branch of mathematics can furnish a natural conceptual system for Gestalt psychology, in that it provides effective tools for describing global qualitative aspects of the latter’s object of investigation. We distinguish three possible ways in which mathematics can contribute to this: explanation, explication and metaphor. We hold that all three of these can be usefully characterized as throwing light on their subject matter, and argue that in each case this contrasts with the role of explanations in physics. Mathematics itself, we argue, provides something different from such explanations when applied in the field of psychology, and this is nevertheless still cognitively fruitful.

Mac Lane identified and meticulously described dozens of mathematical ideas in his book Mathematics: Form and Function. At the same time, he acknowledged the difficulty of precisely defining the nature of these ideas, while firmly rejecting any association with platonism. In this paper, I reconstruct Mac Lane’s ontology of ideas by drawing upon Roman Ingarden’s phenomenological ontology. I argue that, contrary to his own assertions, Mac Lane’s ideas belong to the realm of ideal beings, and that his mathematical forms constitute intentional objects in Ingarden’s sense. Furthermore, I present Mac Lane as one of the foremost theorists of ideas.