Something similar may be said of the concepts straight line, plane, of three-dimensionality and of the validity of Pythagoras' theorem.
Now, Euclidean geometry, in the form in which it has been handed down to us from Euclid, uses the fundamental concepts "straight line" and "plane" which do not appear to correspond, or at any rate, not so directly, with experiences concerning the position of rigid bodies.
For example, one defines the mathematical concept by arbitrary combination of other concepts ("four straight lines bounding a plane surface so that the opposite sides are not parallel to each other"[1]), accompanied by a "sensible sign" that displays the relations among the parts of all objects so defined.
In the axiomatizations that they developed, the basic concepts, called primitives, are implicitly defined by the postulates: thus, such concepts as point, straight line, intersection, betweenness, and plane are related to each other in a merely formal manner.
Lindeboom and others (2005) state that 'we departed from the concept that straight lines are the best approach for future policy and maintenance' (Lindeboom and others 2005, p. 18).
Start with the concept of a straight line, and there is no alternative to accepting it as the shortest.
On this it must be remarked that the concept of the straight line may be reduced to that of the distance.1 Moreover, geometricians were less concerned with bringing out the relation of their fundamental concepts to experience than with deducing logically the geometrical propositions from a few axioms enunciated at the outset.
They are generalization of the concept of a straight line in the plane.
The first four Books of Euclid's Elements are about straight lines and circles, but it is well known that the concept of a straight line receives only a most unsatisfactory definition.
He can analyze [zergliedern] and make distinct the concept of a straight line, or of an angle, or of the number three, but he will not come upon any other properties that do not already lie in these concepts.
All this contributed to the central importance attributed to a non-metrical geometry based on little more than the concept of the straight line and on the incidence properties of lines and planes.
