Towards psychological foundations of mathematics.

Abstract

In the algebraic world, addition and multiplication are special. In the geometric world, lines, circles, right angles, parallel lines, etc. are special. Attempts to lay foundations for arithmetic (such as Peano’s axioms or the ZFC axioms) are evidence that addition and multiplication are special. Likewise, the fact that Euclid’s axioms for plane geometry include lines, circles, right angles, parallel lines, etc. are evidence that these objects are special. While these foundations are successful at doing what is required of them - facilitating the proving of theorems about addition, multiplication, lines, circles, etc. - they do not address the question of why these objects are special; indeed, these foundations rest on the presupposition of this fact. This thesis attempts to answer this question. When we say that addition, multiplication, and the fundamental geometric objects are special, we mean primarily psychologically so, but also mathematically. In order to explain the special status (psychologically speaking) of addition and multiplication, we use four foundational mathematical concepts - monotonicity, convexity, continuity, and isomorphism - to uniquely identify natural, rational, and real addition, and (positive) real multiplication. To explain the special status (psychologically speaking) of lines, circles, right angles, and parallel lines, we use three foundational mathematical concepts - distance, symmetry, and betwixity (betweenness) - and reconstruct Euclidean plane geometry from these concepts. We then show that each of these concepts is - prior to being a formal mathematical concept - a fundamental and intuitive psychological concept; a preverbal principle of perceptual organization that is biologically based and shapes how humans and non-humans alike perceive the world. Essentially, we are laying down psychological foundations for arithmetic and geometry. In the course of laying down these foundations, we ask questions about familiar mathematical objects that mathematicians do not usually ask; our goals are somewhat distinct from those of the mathematician. Consequently, in our efforts to answer such questions we sometimes find a new mathematical perspective from which to view these familiar objects; in this way, we also help to (partially) explain why these familiar objects are mathematically special, and not merely psychologically so. In particular, we lean heavily on the order-theoretic structure of the natural numbers; we also view natural addition through topological, metric-theoretic, and lattice-theoretic lenses