It is intended to confute the opinion of a Platonic Archimedes, through the study of the fundamental theses of his Sandreckoner (Psammites) and of its particular logical-linguistic aspects, but especially of an Aristotelic Archimedes, as Delsedine (1970) maintains in his article "L'infini numérique dans l'Arénaire d'Archimède". He writes: The Sandreckonerrépond à la nécessitè d'adapter la notation numérique à l'idée de l'infinité potentielle de l'ensamble des nombres naturales" 1 . First, it is

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... focused on the general aspects of the work, which highlight its Enlightenment and Pythagorean-Democritean character, then it is passed to the analysis of its particular linguistic and logical aspects and of its fundamental theses, translated into symbolic form, in which it is still giving prominence to its Pythagorean-Democritean or Italic character (Boscarino, 1999 (Boscarino, , 2010 (Boscarino, , 2011 (Boscarino, , 2012 ). Keywords Apeiron, Meghethos, Aritmos, Monas, Pletos, Pythagorean-Democritean or Italic Tradition of Though, Platonic-Aristotelian or Ionic Tradition of Though 1 See Delsedine, 1970. G. Boscarino 9 for this reason, I made it the object of my greatest scientific and philosophic attention, in the reading as also in the commentary and interpretation, a complex work therefore, both from the scientific and philosophical point of view and from the linguistic one. The work meanwhile denies the cliché of an aseptic Greek science, like the one that appears in the text of Euclid, made up of definitions, axioms and theorems, without cultural background, since Archimedes' work is written in the form of a letter to a king, Gelon of Syracuse, who is credited with a certain mathematical competence and a certain astronomical knowledge 2 and which is intended to engage him in a fight against a prejudice or more critical and fideistic prejudices, common in the religious beliefs of many, as of the poets and philosophers, namely that there cannot be any numbers which are capable of counting large multiplicities, as much as the grains of the sea sand can be, but also those ones forming the known universe. We think of the poet Pindar, of the fifth century BC, lived for a time in Syracuse, which will be then Archimedes homeland, who had already written "the sand escapes the number" (Olympian Ode, II, 98) and all the contemporary or nearly so, to Archimedes, religious biblical literature, cheering the innumerability of the sand of the sea, the number of which can be known only by a wisdom superior to man's one, which can be only God's one (All wisdom comes from the Lord and it is with him for all centuries. The sand of the sea and the drops of rain and the days of the centuries, who can never count them? Only one is wise and very terrible sitting upon his throne: God) 3 . Therefore deeply enlightened, ante litteram, work, so in the wake of the thought of Democritus, the only philosopher, who is mentioned in the surviving works of Archimedes, who dared to imagine a homogeneous universe, from the physical and metaphysical point of you, a world that consists of only physical atoms, as the grains of sand are, which Archimedes imagine forming his hypothetical physical universe, of which he wants to count the number, which is then neither inhomogeneous nor hierarchical, such as that of Plato and Aristotle, as in the wake of the thought of Parmenides, according to which "the not being neither you can think it nor you can express it, because thinking is the same as being" 4 , as well as in the wake of sophistic thought, of criticism of mythical and religious uncritical prejudice , not on a human scale. Only those who are able to think in a mathematical way, that is as to say for Archimedes, in a rational way, this concludes is work, know how to become free of prejudice, or know how to think and nominate numbers as large as they want and assign them to multiplicities as well great as they want, they are too big as the entire universe that they know I think finally, King Gelon, that all these things will seem unbelievable to most people, who are inexperienced in mathematical things, but those that will cultivate those ones and will apply to know the distances and sizes of the earth, the sun, the moon and the whole world, will admit the result of my demonstration. And that is why I have deemed it appropriate that even you took knowledge 5 . The work denies then the other stereotype of a Greek mathematics, aimed at founding the science of geometry alone and take it forward, because it is directed to the foundation of a science of numbers, as is clear from the same testimony of Archimedes, in this work, in which it is said that Archimedes has already sent arithmetic, unfortunately lost, written to a scientist of his time, Zeuxippus. But, I shall try to show to you, by means of geometrical proofs, which you will be able to follow, that, among the numbers named by me and exhibited in the writings sent to Zeuxippus, some exceed not only the number of the mass of sand equal in volume to the earth filled in the manner described by me, but also that of a mass equal in volume to the cosmos 6 . We should ask ourselves why his writings of arithmetic have been lost and not those of geometry. We believe that the reasons were not random and accidental, but much deeper, external to the progress of science. The work denies yet another prejudice about the progress of the Greek science in an unique sense, with no conflict on the field, only of Platonic-Aristotelian approach, especially in the field of astronomy, when it bears the testimony of a heliocentric system, such as that of Aristarchus, III century BC, of an entirely different approach, the Pythagorean-Democriteanone, that is not only of a not geocentric universe, such as that of Pythago-