II.6 The Development of the Idea of Proof, Considerations, Greek Mathematics: Pages 129 -132.

The development of the idea of proof seemed to occur around 500 B.C. with deductive argument.  Thales of Miletus, mathematician, philosopher and scientist, is the first mathematician known by name.  Euclid’s books the “Elements” were written around 300 B.C. and there were one of the most successful attempts to organize mathematical concepts.  Mathematics stands out in the sciences in the unique way in which t relies on “proof”.  There are other attempts of this kind including induction which are not dealt with in this present account.

There was an early reliance on geometric justifications.  These later, with the development of non-Euclidean geometry and problems in the foundations of analysis, led to a fundamental change in mathematical orientation.  Arithmetic and set theory eventually provided the certainty and clarity from which other disciplines drew their legitimacy and clarity.

The proofs in Euclid’s Elements all have six parts and an associated diagram.  Some terms in the elements now have different meanings, for example, two figures are said to be equal if their areas are equal.  The six parts of the proofs are: Protasis (enunciation), Ekthesis (setting out), Diorismos (construction), Kataskeue (construction), Apodeixis (proof) and Sumperasma (conclusion).

An example from the elements is the following IX.35 which reads, after translation by Sir Thomas Heath, as follows: “If as many numbers as we please be in continued proportion, and there be subtracted from the second and the last numbers equal to the first , then, as the excess of the second is to the first, so will the excess of the  last be to all those before it.”  In today’s mathematics this may be written: (an+1 – a1) : (a1 + a2 + … + an) = (a2 – a1) : a1.

There was an accompanying figure for this proposition IX.35 which included four line segments of different lengths with various points included on these segments.  The proof refers to this diagram.

In Greek mathematical texts proofs sometimes relied on the “exhaustion method” which was based on a double contradiction.  An example of this method is provided in Euclid’s proposition XII.2 of Euclid’s Elements.  Other methods were sometimes used in proof including: synchronized motion of two lines and mechanical devices.  However, the Euclid style of proof remained a model to be followed whenever possible.

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