In cognitive growth, the mental objects we think about are constructed in several different ways, each having a different status. The visual objects we see are direct perceptions of the outside world, or rather, our own personal constructions of what we think we see in the outside world. Later in geometry, objects such as a “point” or a “line” take on a more abstract meaning. A point is no longer a pencil mark with finite size (so that a child may imagine a finite number of points on a line segment (e.g. Tall, 1980)), but an abstract concept that has “position but no size”. A straight line is no longer a physical mark made using pencil and ruler, but an imagined, perfectly straight line, with no thickness which can be continued as far as required in either direction. In Euclid, a line is defined as “breadthless length” and a straight line “lies evenly with the points on itself”. These words do not define a straight line in any absolute sense, but they help to convey the meaning of the perfect Platonic object which we may “see” lying behind any inadequate physical picture. As Hardy observed: