The Greek miracle of mathematics and the birth of philosophy occurred jointly. The well-known epistemologist M. Serres writes in this regard: 

 “Now, to reply to the problem of the Greek beginning of geometry compels one, exactly, to ask how it is passed from one language to another, from one language considered natural and from its alphabetical connotation to the natural rigorous language and from its alphabetical connotation to the rigorous and systematic language for numbers, measures, axioms and rigorous reasoning, written and annotated in equations and figures” (Serres: 1993, 122).

The language of the esprit of geometry was born by means of pure figures and demonstrations. But who was the geometer-philosopher? He was the man who was able to transfer his mind from the practical, the finite, the particular and the surrounding world to the vision and theoretic knowledge of the known and unknown, of spaces and times seen inside the horizon of one open infinite. On historical revolutions, continuity and discontinuity, pre-historical path and progressive expansion of mathematics I refer again to M. Serres who, by means of his fascinating narrative, has written “the quantitative purification of its concepts, the always strengthened power of its methods, the movement in ahead towards a mathematical conceived as horizon allows to think a shape evolutionary connected, but articulated from stages, stages or crisis, total reorganizations of a knowledge transmitted without losses, therefore incessantly accumulated”(Serres: 1993, 20). We have not mentioned the archaic geometry in the time by the Greeks, the concrete empirical science that began in Miletus with a kind of investigation called historiê; it’s a sure thing that the Greek geometricians were the first to use the demonstration proof in order to obtain greater rigor for their sciences.  

During the Hellenistic age, in a certain way, science and philosophy separated, Athens remained the centre of philosophical research, Alexandria the centre of scientific studies and discoveries, but the speculative spirit of the first Greeks continued and the sciences improved substantially.

It is in this epoch that Euclid came to the fore.

It is most probable that he received his mathematical training in Athens, and he may himself have been a Platonist. Proclus says that Euclid was of the school of Plato and in close touch with that philosophy. One thing is however certain, namely that Euclid taught and founded a school at Alexandria. 

As known his principal work is the mathematical and geometric treatise Elements composed originally by 13 books, and written in Egypt near 300 BC. Euclid's books are in the fields of Euclidean geometry, as well as the ancient Greek version of number theory, the oldest extant axiomatic deductive treatment of geometry, and has proven instrumental in the development of logic and modern science. On this basis, a large number of propositions are proved with a high degree of deductive rigour including a collection of definitions, postulates (axioms), propositions (theorems), and proofs. The Elements were one of the very first books to go to press, and second only to the Bible in number of published editions. Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework. For centuries, when the medieval quadrivium was included in the curriculum of all university students, the knowledge of at least part of Euclid's Elements was required all students. 

During the 17^th century, a very important century for the European history, after the recovery and the publication of the Greek text in the editio princeps on September 1533 at Basel, what was the mathematicians’s attitude towards the Elements? First of all there was an increasing, detached interest for the text, both from a philological and epistemological point of view, followed by the Latin versions. In between we have the first one realized by F. Commandino in 1572 at Pisa, then translated in Italian by the same at Urbino in 1575, followed by P.C. Clavius’ always of XV Books, printed at Rome in 1574, and many others printed in national European languages.

For decades the European mathematicians of 17^th and 18^th century were engaged in the revisiting and restating the work of Euclide in canonical form, but they tried also to publish it at a popular level. The more pressing academic requirement in 17^th century was to restore the Elements, as happened with the publication of Euclides restitutus 1658 by G. Borrelli in Latin, the V Book of the Elements by Galileo, published in 1674 by V. Viviani in Italian, the Euclide restituto 1680 and 1686 by V. Giordano in Italian, the Nouveaux elements de geometries 1667 by F. Arnauld in France, Euclidis elementorum books XV breviter demonstrati the 1655 by I. Barrow and Geometric Euclidis elementa novo order ac methodo to fere demonstrata 1678 by N. Mercator in Latin, just to mention some. There were no scarsity of adaptations such as reduced editions for the requirements of the general readers. Currently the most printed and remarkable were the first four books, those of plane geometry. It remained, nevertheless, acquired that the thirteen books included the so-called arithmetical books. 

But in order to realize the Greek miracle of geometry let me remember the suggestive words written in full 18^th century by the philosopher I. Kant in the Preface to Kritik der reinen Vernunft 1787 2^nd edn:

“A new light broke upon the first person who demonstrated the isosceles triangle (whether he was called “Thales” or had some other name). For he found that what he had to do was not to trace what he saw in this figure, or even trace its mere concept, and read off, as it were, from the properties of the figure; but rather that he had to produce the latter from what he himself thought into the object and presented (through construction) according to apriori concepts and that in order to know something securely a priori he had to ascribe to the thing nothing except what followed necessarily from what he himself had put into it in accordance with its concept”. (Kant: 1787, B Xiip).

And after referring to the physics through scientists as Galileo and Torricelli “They comprehended that reason has insights only into what it itself produces according to its own design; that it must take the lead with principles for its judgments according to constant laws and compel nature to answer its questions, rather than letting nature guide its movements by keeping reason, as it were, in treading-strings; for otherwise accidental observation, made according to no previously designed plan, can never connect up into a necessary law, which is yet what reason seeks and requires. Reason, in order to be thought by nature, must approach nature with its principles in one hand, according to which alone the agreement among appearances can count as law, and, in the other hand, the experiments thought out in accordance with these principles yet in order to be instructed by nature not like a pupil, who has recited to him whatever the teacher wants to say, but like an appointed judge who compels witnesses to answer the questions he puts to them”. (Ibid. B xiv).

Jumping until the 20^th century, right after World War I among philosophers and literati who took part in a wide-ranging forum on the theme of Europe such as M. Heidegger, E. Husserl, Ortega y Gasset, Nikolaj Berdjaev, I must remember P. Valery, quoted recently on this Magazine by Tzvetan Todorov, who treated the European cultural identity by means of famous pages. Valery as poet and writer presents an imaginary European Hamlet leaning from a terrace that overlooks the whole of Europe and looking at million of ghosts. "But he is an intellectual Hamlet who reflects on the life and on the death of the truth. His ghosts are the objects of our controversies; his regrets are the titles of our glory. He is oppressed by the weight of his discoveries, of his knowledge, and is incapable of any action. He reflects on the boredom of a past that needs to be re-discovered, at the folly of continual innovation. He is thorn between two crevasses, since two are the dangers confronting the world: order and disorder" (Valery: 1994, 32).  

After descending from the terrace, the European Hamlet picks up a skull and recognizes it for the trace of the development of European history: here is the skull of Leonardo da Vinci, and then that of Leibniz, who, between the 17^th and the 18^th century, was looking for a universal European peace, and that of Kant who was also looking for “perpetual peace” at the antipodes of Heraclitus, the founder of European philosophy. From Kant’s skull issue that of Hegel, of Marx "Kant qui genuit Hegel, qui genuit Marx, qui genuit… " And so on.   And if were the European intellectual to abandon all the skulls of the past and throw them in a ditch, would he still be himself? What would the European intellectual become? It is time to say goodby to ghosts since it has no need for them any longer? Surely Valery would have add the names of Pythagoras, Euclid and Archimedes, Galileo, Descartes and Leibniz, as mathematician, Newton etc... in the list, given the fact that within the same essay he doesn’t forget geometry. 

Finally he closes with a statement that is not properly coherent and scientific, but a plausible one, the same remembered by Todorov defining European man as a man with three fundamental characteristic: “As far as I am concerned any people who have been influenced throughout history by Greece, Rome and Christianity are Europeans”. But later on he gives us one of the best praising of mathematics and geometry “Greek geometry has been that incorruptible model, and not only model, proposed for every type of knowledge that aims at the state of perfection: an incomparable model of the more typical qualities of the European reasoning. I never think of classic art without necessarily taking as an example that monument to reason that is Greek geometry. The construction of such a monument has demanded rare and more naturally incomparable gifts. The men who have constructed it were solid and sagacious workers, deep thinkers, but also talented artists with the fineness and the exquisite sense of the perfection…” (Valery 1994, 53).

So in company with Serres, in a short synthesis of the philosophical and scientific value of mathematics, we can affirm: “Mathematics is therefore: so objective that it is the only one that is truly collective; so collective that it is the only one that it is truly objective; so useless that it is the only one that is truly useful; so outer that it is the only one that is truly inner; so inner that it is the only one that is truly outer; so in the being that is outstanding at the knowledge; so in being that it excels in knowledge; so abstract that it is the only one that is truly concrete, so concrete that it has been believed, sometimes, that its space was the shape of the external sense” (Serres: 1993, 267-8).

In conclusion, it seems impossible to deny that from the new spirit of world-openness, the spirit of re-birth, which appeared on the stage of human history with the Greek “discovery of the world,” European culture and science achieved its peculiar form of “identity” through the epochal stages of its history.

As far as I am personally concerned I would only apologize me for the apparent eurocentrism of this paper. It wasn’t my intention.

Francesco Tampoia
Bitonto (Italy)

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References:

. Martin Heidegger, Was ist das- Die Philosophie?, Neske Pfulligen 1956. See: Francesco Tampoia, Actos VII Congreso “Cultura  Europea” Pamplona 2002-2005 - Paper “Philosophers and Europe: M. Heidegger, G. Gadamer, J. Derrida”
. Klaus Held, The Origin of Europe with the Greek Discovery of the World, Epoché, Volume 7,   Issue 1 (Fall 2002) ISSN 1085-1968.      
. Immanuel Kant, Critique of Pure Reason, 1787, translated and edited by Paul Guyer-Allen  W. Wood , Cambridge University Press 1998
. Michel Serres, Les origines de geometries, Flammarion 1993.
. Paul Valery, Essais quasi politique(Variété) in Oeuvres, vol. 1, Paris, Gallimard 1957.
. J. P. Vernant, Mythe et pensée chez les Grecs. Etudes de psychologie historìque, Libraire Francois Maspero, Paris 1965.

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