Speech at the event of the Academy of Athens, dedicated to the Day of the Greek Language, on the topic: “Remarks on the contribution of the Greek language to the development of mathematical science“, former President of the Republic, Academician and Professor Emeritus of the EKPA Faculty of Law Mr. Prokopios Pavlopoulos moved on to a series of highlights. In particular, the former President of the Republic, among other things, noted the following:

Prologue

The example of the development of mathematical science, in the broad sense of the word, in ancient Greece through the Greek language is highly indicative of the contribution of the ancient Greek spirit – starting with the thought of the pre-Socratics – to the discovery of the scientific method. The beginning dates back to the 5th century BC. century, mainly with Leucippus and Democritus, in the field of atomic theory and Pythagoras in the field of mathematics. The sequel is quite revealing, with Thales and Euclid in the first place in geometry and Diophantus in algebra, of course in keeping with the scientific name it later adopted. It is also necessary to clarify what has already been indicated in the smallest detail regarding the connection of mathematical science in Ancient Greece with the Greek language. And this is because, from the point of view of scientific creativity, the “meeting” of the Greek language with mathematical science was, of course, among others. And this “meeting” was the cause, which explains why other, of course, the peoples of antiquity had remarkable knowledge in the wider field of mathematics. However, the Greeks were the first to lay the foundation of mathematical science with Euclid’s Elements, the works of Eudoxus and Archimedes as colophon, referring to the type of mathematics known from the 17th century onwards as “infinite mathematics”. and the general mathematical thinking of Diophantus, deriving, among other things, from the following:

I. The texture of the Greek language

Obviously, there was something special in the linguistic structure of the Greek language that contributed to the establishment of relations between it and mathematics.

A. And this is revealed mainly in the following two “features”: first of all, in the special grammatical state of the Greek language, precisely in the presence of a definite article, something non-existent, for example. in Latin. The definite article developed in metomeric Greek from the pronominal use of “o, h, to”. The article, based on specific grammatical constructions of the language, creates an “abstract expression”. And this, in temporal continuity, leads to the “abstract noun.” And the “abstract” by definition and by its nature is the basis for the formation of logical judgments and reasoning, an element inherent in the development of Mathematical Thinking, especially in its early phase.

B. And, secondly, the aforementioned favorable grammatical condition was accompanied, in terms of the relationship between Greek language and mathematical thought, by a more general condition that governed Greek thought from the very beginning: an “unlimited” and “universal” tendency towards finding an “atomic units” in each field of activity of the scientist. This is the moment when “phthongos”, “isolated” in oral and audible speech, as the last individual and indivisible acoustic unit, formed the basis for the transformation of the Phoenician alphabet into Greek. And it was this impulse that “isolated”, in particular, by the method of Democritus, the atom, separating it as a primary, vaporous and indivisible unit of matter.

II Mathematics in Ancient Greece

As J. L. Hardy noted (L’organisation des mathématiques grecques de Théetète à Archimède, Paris, ed. Vrin, 1997, p. 270 and especially 276 ff.), the contribution of the ancient Greeks to the field of mathematics was unique in so far as it concerned the synthesis of the problem and evidence.

A. And to be precise, then, on the one hand, the formulation of the problem, by formulating a theorem with the mathematical property that gives it the most general form. And, on the other hand, adding an appropriate proof in the process of solving the problem. Moreover, here one should not underestimate the fact that production and induction developed very early in the Greek intellect, with the appearance of the empirical language of the Ionians in philosophy and already in archaic poetry.

B. Thus, by virtue of this symbolic “path”, the Greek language made it possible to create language formulations of such “abstractness” that they could easily “reconcile” even with extremely abstract mathematical concepts. That is why the Greek language turned out to be the most suitable “metal-language” from the point of view of mathematics, causing an “explosion” of mathematical knowledge in general.

III Greek language and “symbols”

However, the aforementioned – emphatically unique – contribution of the ancient Greeks to the creation of mathematical science was made possible thanks to the equally unique originality of the Greek language. Consequently, peoples who did not have the appropriate “language tools” did not take appropriate scientific steps.

A. In essence, the above feature lies, first of all, in the fact that the Greek language is almost “ideal” in the field of the production of “symbols”, is able to adequately illustrate the Thought, so as to structure the “ark of knowledge” necessary for universal scientific creation. Mainly the “ark of knowledge” that paved the way for the founding and development of mathematics (see instead of another reference Chr. Fili “The Ancient Greek Contributions of Modern Mathematics”, ed. Papasotiriou, Athens, 2010, p. 877).

B. In this regard, it is especially significant that symbols also contributed to the use of numbers both in their nature and the purpose of the basis of mathematical thinking, which can be deduced from the following: Etymologically, the word “number” comes from the compound from the verb “ararisko” – a doubled form from “aro” – which means, among other things, the harmonic connection of data among themselves, and the noun “itma”, which means “step”. In other words, a “number” is a means of putting something in its proper place in relation to related data, and in this light, numerical reasoning can take the next steps, especially when formulating a problem. It is precisely this function of numbers that, as explained above, was decisively facilitated by symbols, mainly from the point of view of setting the problem. Especially since, as documented by the History of Sciences itself, prior to the use of symbols, with Greek as the “vehicle” for their formation, the scientific approach and cultivation of mathematics had not “appeared” sufficiently.

C. Through the “tools” of the Greek language, the mathematical thinking of Diophantus of Alexandria also developed.

1. And this is because the peculiarity of the Greek language allowed Diophantus to open the way to the “foreshadowing” state of the formulation of a mathematical problem, under conditions leading to the so-called “polynomial algebraic equations”, such as the “binomial equation” αxn =bxm or “three-term equation” It is also accepted that the search for the so-called “Pythagorean triads” – that is, integral solutions to the equation “x2 + ψ2 = γ2” – is part of the classical tasks of the entire analysis of Diophantus.That’s right, Diophantus’ Arithmetic – in thirteen books, six of which have survived in Greek and four in Arabic translation – and today are considered the “foundation” of the creation of a tradition that led to the creation of algebraic thinking and analysis of the modern era. For, as J. Klein wrote, “modern algebra and modern formalism arose as a result of Vieta’s interaction with Diophantus ” (see J. Klein, “The World of Physics and the “Physical” World”, translated by D. Lappas, M. Mytilineos, G. Sayas, T. Spyrou, G. Christianid is, Neysis 7 (1998), 41–74 English text published in 1981 in The St. Johns Review” and reprinted in a collection, R.B. Williamson. and E. Zucherman (eds.) “Jacob Klein: Lectures and Essays,” Annapolis, MD, St. John’s College Press, 1985, 1-34).

2. It should be noted that the term “algebra” belongs to the Persian mathematician, astronomer and geographer Abu Musa al-Khwarizmi. Which much later, in the 9th century – almost 500 years after Diophantus – presented, however, for the first time a systematic solution of a quadratic polynomial equation. So, it is worth asking oneself: would al-Khwarizmi, even at his age, have been in his algebraic analysis without Diophantus’s pioneering mathematical “legacy” regarding the creation of “predecessors” of the above algebraic methods of problem solving, thanks to the “subtractive” power of the Greek language?

IV. Formulation of the production method

In support of what has been said in this case, it is worth making a special reference to the book by Reviel Netz, professor of Greek mathematics and astronomy at Stanford University in the USA, one of the leading researchers in the work of Archimedes, entitled: “Formation of the production method in Greek mathematics – a study in the light of cognitive science” ( translated into Greek by V. Spiropoulou, edited by G. Christianidis and M. Sialaru, University Press of Crete, Heraklion, 2019).

A. This book explores a topic of great importance to the entire history of Western civilization. That is, the formulation of a productive proof in classical Greek mathematics. The impressive, literally, originality of the aforementioned work by Reviel Netz lies in the fact that the author focuses his analysis on two fundamental characteristics of the practice of Greek proofs: the written diagram and the technical, logo language. Without losing sight in the slightest of the material and social conditions and practices within which the above-mentioned characteristics “arose” in the course of the evolution of Greek mathematics.

B. In particular, Reviel Netz pointed out that the methods then developed by the Greek mathematicians—focusing on Euclid, Archimedes, and Apollonius—for constructing the letters in their diagrams, and hence the constant interaction between text and diagram in their proofs, were the key to ” birth” of productive proof. Thus, Reviel Netz was able to sufficiently clarify the underlying cognitive processes.

Epilogue

This fact, however, has a second, wider meaning, concerning the course of all science in the West, that is, the course of this Western civilization. What is characteristic is what, in conclusion, Reviel Netz himself emphasizes in the introduction he wrote for the Greek edition of his aforementioned work: “Greek mathematicians discovered a certain practice and a certain set of tools that made a particular project possible. : consistent implementation of productive evidence. This will play a significant role in the emergence of Western science. The prospect of proof led to the mathematization of all science and, finally, to the Newtonian program, which, with its success, opened the way for the industrial revolution and the rise of the West” (ibid., pp. XV-XVI).”