Constructing numbers through moments in time: Kant's philosophy of mathematics

Abstract

Among the various theses in the philosophy of mathematics, intuitionism is the thesis that numbers are constructs of the human mind. In this thesis, a historical account of intuitionism will be exposited- - from its beginnings in Kant's classic work, Critique of Pure Reason, to contemporary treatments by Brouwer and other intuitionists who have developed his position further. In chapter II, I examine the ontology of Kant's philosophy of arithmetic. The issue at hand is to explore how Kant, using intuition and time, argues for numbers as mental constructs. In chapter III, I examine how mathematics for Kant yields synthetic a priori truth, which is to say an informative statement about the world whose truth can be known independently of observation. In chapter IV, I examine how intuitionism developed under the care of Brouwer and others (e.g. Dummett) and how Hilbert sought to address issues in Kantian philosophy of mathematics with his finitist approach. In conclusion, I examine briefly what intuitionism resolves and what it leaves to be desired.