For more than 2000 years the notion persisted that Euclid has developed Geometry on the basis of 5 simple facts known as Postulates. The five were indeed simple (it's not my intention at this point to talk of the famous fifth postulate and its history) but were not the only facts Euclid used to construct Geometry. He gave definitions to the terms of Geometry and introduced a variety of Common Notions (e.g., If equals be added to equals, the wholes are equal.) that would count nowadays among axioms along with the Postulates.

Each of the 13 books of the Elements starts with a new set of Definitions. For example, Book III one starts with

Equal circles are those the diameters of which are equal, or the radii of which are equal.

This says that wherever two shapes are drawn according to the definition of a circle (a line of points equidistant from a fixed point) with equal radii, the shapes are equal. Compare this to the Fourth Postulate:

All right angles are equal to one another.

By definition (#10), an angle is right if it's obtained at the intersection of two straight lines where two adjacent angles are equal. Again it says that, if one follows the rule of construction (drawing intersecting lines with equal adjacent angles), the result will be one and the same right angle. The distinction between a Definition in Book III and a combination of a Definition and a Postulate in Book I is obscure. The lesson we learn is that Euclid, although a source of the axiomatic method, frequently in his proofs relied on intuition in addition to his axioms.

D.Hilbert around the turn of the century offered a modern day axiomatization of the plane geometry based on 16 axioms. What's more interesting and edifying is that Hilbert does not define the terms used in his axioms. Euclid says

Definition 1. A point is that which has no part.
Definition 2. A line id breadthless length.
Postulate 1. Through any two points there is a straight line.

Hilbert just states

Axiom I-1. Through any two distinct points there is always a straight line.

There is no definition of what is a point or a straight line. By the end of the 19th century it became an acceptable fact that the terms of a theory are defined by its axioms. All that may be said of objects of a theory is conveyed by its axioms. Realization that any definition beyond that implied by the axioms depends on human intuition and has no place in a formal theory was quite formative in the development of modern mathematics starting with the 19th century.

Another example of an early axiomatization is supplied by Newton's fundamental Philosophiae naturalis principia mathematica. The Second Law of Motion reads

... the change of motion is proportional to the impressed force ...

Hence, if no impressed force is present, the motion remains unchanged (which is also given separately as the Law of Inertia.) One may compare this with the preceding definition (IV):

... the impressed force is the action on a body that changes its state of rest or of uniform rectilinear motion

The definition and the axiom state exactly the same thing; for, if the force is defined as that which changes the state of motion, in the absence of force the latter remains unchanged. Ernest Mach in his 1883's Mechanics reformulated Newton's theory by defining its terms only through their usage in axioms. Subsequently and independent of the theory, he also explained the terms' intuitive meaning.

Another example will illustrate how hard is it to entirely ban intuition from a mathematical proof.


  1. R. von Mises, in J.Newman, The World of Mathematics, Simon and Schuster, 1956

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