Axiomatics

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

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:

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

Hilbert just states

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

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 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.

References

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

• Perfect numbers are complex, complex numbers might be perfect
• Fundamental Theorem of Algebra: Statement and Significance
• What's in a proof?
• More about proofs
• Axiomatics
• Intuition and Rigor
• How to Prove Bolzano's Theorem
• Early attempts
• Proofs of the Fundamental Theorem of Algebra
  • Remarks on Proving The Fundamental Theorem of Algebra
    • Sketch of a Proof by Birkhoff and MacLane
    • Sketch of Second Proof (after Cauchy)
    • Details of the Cauchy's Proof
    • Note on the Extreme Value Theorem
    • Sketch of Proof by the methods of the theory of Complex Variables (after Liouville)
    • Maximum Modulus Theorem
  • A Proof of the Fundamental Theorem of Algebra: Standing on the shoulders of giants
  • Yet Another Proof of the Fundamental Theorem of Algebra
  • Fundamental Theorem of Algebra - Yet Another Proof
  • A topological proof, going in circles and counting
  • A Simple Complex Analysis Proof
  • An Advanced Calculus Proof

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