The author cites work on visual perception which indicates that in order to study perception it is necessary to replace such classical geometrical notions as betweeness, straightness, perpendicularity, and parallelism with more general concepts. The term tolerance geometry is used for any geometry when primitive notions are obtained from the classical ones by substituting closeness for identify. The axioms for tolerance geometry on the line are stated and some basic theorems are proved. In this geometry the traditional ternary relation of betweeness is replaced by the tolerance relation of epsilon-betweeness. In particular, the axioms are stated in terms of a pair (A,B) where A is a finite set and B is a ternary relation on A. (Author/FL)