4. Euclidean Space Kant’s reconciliation of the views of both the empiricists and rationalists enables Mathematics to be used as a tool for understanding space. Mathematics can be regarded as a synthetic a priori truth, providing an insight of the world even though our knowledge of it derived is independently of observation. As such, Euclidean geometry, the axioms on which the structure of Euclidean space is defined, may be considered to be both universal and necessary – a fundamental precondition for our understanding and experience of space. By way of the first requirement of the axiomatic method, an axiom is a premise for reasoning that is typically so obvious that is accepted without controversy (Singh, 2015a, p. 3). There are five key postulates or axioms which Euclidean geometry is based on. Postulate 1: For every point P and every point Q not equal to P, there …show more content…
A topology on a set X consists of a collection of subsets called open sets associated with the topology (Singh, 2015b, p. 1). The topology of a space allows for the discussion of notions of continuity. To illustrate this, suppose that in making a Euclidean diagram, we periodically lift our pencil and begin again at another point while drawing what was supposed to be a straight line (Maudlin, 2012, p. 6). Thus rather than a single, continuous line, two disconnected lines will be produced. To differentiate between these two outcomes, the space must be endowed with a topological structure or a topology on a set X, which is given by a set D of subsets of x such that (1) every union of sets of D is a set in D, and (2) every finite intersection of sets of D is a set in D (Singh, 2015b, p. 1). Given this topology on a set X, the notions of interior, exterior and boundary points relative to any subset of x can be