ACADEMIC WORKS

TOWARDS A COGNITIVE FOUNDATION OF MATHEMATICS

In this dissertation I approach the philosophy of mathematics and its foundations from the point of view of modern cognitive science. The first, introductory part, considers a cognitive foundations program, in general philosophical terms. The second part, which brings in the cognitive-mathematical substance, demonstrates how a cognitive approach can empower the exploration and provide insight on classical issues in the philosophy of mathematics.

THE LOGIC OF LOVE

This philosophical work lays the groundwork for a game-theoretic account of (romantic) love, substantiating the folk-psychological conception of love as 'a unification of souls'. It does so by setting up an appropriate universal framework of cognitive agency, that accommodates such unifications and motivates them. This framework applies the gene’s eye view of evolution to the evolution of cognition, integrating it with a distributed, dynamic theory of selfhood – and the game-theoretic principles of agent-unification that govern these dynamics. The application of this framework to particular biological settings produces love as a theoretical evolutionary prediction (unveiling its rationality). Through this, the connection of the strategic normativity to love's real-life behavioral and phenomenological expressions is systematically explored.

RANDOM SEQUENCES

Different and many times provably-incompatible technical definitions, all claiming to formally characterize the vague notion of randomness, have appeared in the past century. This work is a personal attempt to put a few of these related mathematical developments into a general philosophical perspective. It is not at all a conclusive summary of the modern notion of randomness and its underlying philosophy. Quite the contrary - it is a mere sketch. Nonetheless, one hopefully demonstrating not only the rich technical diversity pertaining to the notion, but the delicate interplay between the technical developments and the evolving philosophical intuitions behind the word, and even more importantly – the possibility of revealing the order underlying that diversity, the structure determining the possibilities for notions of randomness and how they are constrained, as well as the relations among them.

EQUIVALENCE RELATIONS & TOPOLOGICAL AUTOMORPHISM GROUPS IN SIMPLE THEORIES

While writing [Hru97], Hrushovski tried to construct a simple theory whose Lascar group isn't totally-disconnected, but failed. The question of whether such a thing exists is still open. Since then, following related developments in the field of simple theories and the intense development of the field itself in general, this question has now come to be seen as one of uttermost importance. 

This work is meant to serve, first of all, as an introduction to the aforementioned problem. Not in the sense of merely giving the definitions that are needed for the problem to be stated, but rather of supplying the background necessary in order to truly get "a feel", an understanding of what's going on. Necessary whether one wants to actually tackle the problem, or just effectively get into these deep territories despite beginning at a general model-theoretic starting-point (only basic knowledge of types, saturation and the likes is assumed; an acquaintance with stability or simplicity theory does help, though). 

More generally, this work is aimed at giving a more unified view of the profound relations between groups of automorphisms, kinds of definability & types, and topological properties, that appear in this abstract area of mathematics. 

On-going research (by Hrushovski & Pillay) in dependent theories and their relations to some of the issues explored in this work is sketched in the appendix. 

Beyond clarifications & elaborations, mostly-minor fixes to existing proofs, a reworking of small portion of [HP] in the appendix, and the fact that [Hru97] was never published (just circulated around), none of the results here are probably new (except for proposition 4.2.3).