# Democracy gives birth to mathematical proof!

Especially in cases where the scientific research and documentation is not taken into account, that tendency can result in false opinions in the general context of the popular greek saying ‘everything can be traced back in the ancient greek civilization’.

Off course, the reliable tool of scientific research is always available to help us make fair judgments and what follows is an example of the use of research to justify the birth of mathematical proof which, yes (!), has taken place in Greece, around 430 BC, with a possible error of ± 20 years.

The research for justifying the birth of mathematical proof was being conducted for the last ten years by the writer Apostolos Doxiadis (who, among other subjects, has studied mathematics as well!).

Today, it is known that Egyptians, influenced by Babylonians, developed strong calculative tools that consecutively influenced Greeks around 6th century BC, though mathematics developed by Greeks during the 4th century BC and especially Euclid’s Elements don’t have anything in common with mathematics of previous civilizations.

The first officially recorded theorem is the ‘Squaring of Menisci’, written by Greek mathematician Hippocrates of Chios (not to be confused with the father of Western medicine Hippocrates of Kos!).

Hippocrates of Chios (470-400 BC)

That specific theorem includes some unique characteristics when compared to the mathematical tools developed previously, the following:

-It is written in a generic way

-It is based in well known truths

-It uses logical syllogisms

-It concludes to indisputable results

On the other hand, neither Hippocrates or Euclid’s mathematical proofs are complete and above all, they don’t include detailed syllogisms!

When examining the proofs written by Euclid, we can state that they, mostly, consist of ‘incomplete syllogisms’ (syllogismus truncatus) or ‘enthymemes’, as incomplete syllogisms are wider known in literature.

Aristotle defines enthymemes as rhetorical syllogisms, specifically, his opinion was that ‘in Rhetoric there is no need to say obvious things, moreover neither to say every single thing, otherwise this becomes a boring procedure’. In other words, Euclid’s mathematical proofs are Rhetoric!!

Euclid

This is a first indication that while Democracy and Culture were flourishing in Athens and the citizens in Agora were trying to convince the public for the correctness of their opinion, the rhetorical persuasion (‘pistis’) became the basis of mathematical proof. In that way, Greeks used the rhetorical speech to develop a new logical and substantiative way of thinking that, respectively, influenced the development of the modern civilization.

In addition, the relationship between Rhetoric and mathematical proof is justified when comparing the ancient texts with the first mathematical theorems, where it is easily concluded that exist common patterns of speech and thought like ‘chiasms’ and ‘cyclic combinations’, facts that amplify the concept of literary ‘genealogy’ for mathematical proof. According to Doxiadis, poetic narration was the basis for rhetorical persuasion which then led to the development of logical and mathematical proof.

It can also be noticed that the evolvement of poetic narration to mathematical proof doesn’t takes place in a ‘revolutionary’ way, instead it exhibits a step-approach and adaptive procedure. Thus, a standard rule of the ‘ideas evolution theory’ is applied, the rule of ‘exaptation’ which states that ‘to face new problems, old tools can be used in new ways’ (Lev Vygotsky).

To my mind, a significant lesson learned comes out from this new discovery. For another time it is verified that, contrary to the modern trend of over-specialization and unilateralism where the scientific society is divided in ‘positive’ and ‘theoretical’ parties, both ‘positive’ and ‘theoretical’ mental competences are interconnected and interoperable.

I strongly believe that a modern person must strive to develop both kinds of competences in a symmetrical way, a procedure that can help him understand the today’s challenges in a clear and precise manner.

Bearing that in mind, the iniciative of Apostolos Doxiadis and his colleagues of the team ‘Thales and Friends’, is considered very important for the education of new generation in Greece and has to be officially recognized and supported by the Greek educational authorities.

Apostolos Doxiadis delivered a presentation on his research for the birth of mathematical proof in Museum of Cycladic Art, Athens, in which a new lecture series was launched, titled ‘Up to date information on the study of ancient greek mathematics’.

Detailed information here