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Ancient Mathematics (Sciences of Antiquity Series) by Serafina Cuomo

By Serafina Cuomo

The theory of Pythagoras, Euclid's "Elements", Archimedes' solution to locate the amount of a sphere: all elements of the important legacy of old arithmetic. yet old arithmetic was once additionally approximately counting and measuring, surveying land and attributing mystical value to the quantity six.
This quantity bargains the 1st obtainable survey of the self-discipline in all its style and variety of practices. The interval coated levels from the 5th century BC to the 6th century advert, with the focal point at the Mediterranean area. issues include:
* arithmetic and politics in classical Greece
* the formation of mathematical traditions
* the self-image of mathematicians within the Graeco-Roman period
* arithmetic and Christianity
* and using the mathematical earlier in overdue antiquity.

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Although nowhere did Aristotle explicitly state that the mathematics of his day had reached the ideal described in his works, or that it was his discipline of choice, many of the examples deployed in the Posterior Analytics, which dealt with scientific demonstration particularly, were taken from mathematics. 57 An element of necessity is also required: one has to be aware that what is in a such-and-such way could not be otherwise, for instance the diagonal cannot be commensurable. In fact, the proof that the diagonal cannot be commensurable (of which more below) is used by Aristotle to exemplify the so-called privative demonstration, also known as reductio ad absurdum or proof per impossibile: if the consequences of a certain statement are absurd, the statement itself is false.

We may ask the crucial question again: to what extent does Aristotle reflect the actual mathematical practice of his day? The reader will probably know the answer by now: we simply cannot tell for sure. His direct involvement with mathematics may have been negligible and his admiration for the subject remarkable, yet, most of the times Aristotle names mathematicians it is in order to criticize them, either because of fallacious arguments, or on even more exquisitely philosophical grounds. Thus, he attacks the Platonists for believing that mathematical objects have an existence separate from the things in the world.

Consequently, what Aristotle says about mathematics, even in his capacity as a philosopher, would seem to reflect historical circumstances to a greater degree than Plato. But we will be cautious anyway. Like Plato, Aristotle was very interested in the mathematicians’ method, and it is on mathematical procedures that some of his philosophical discussions concentrate. He provides the earliest extant account of what 31 E A R LY G R E E K M AT H E M AT I C S : T H E E V I D E N C E criteria should be followed to obtain demonstrative validity, and several of his works are devoted to an analysis of demonstration, argumentation, forms of discourse – not just how knowledge is to be gained, but how it is to be organized, expressed and defended against objections.

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