The mathematics of logics or Logic of mathematics. Critical problems in the History of Science.

Raffaele Pisano
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Università di Roma, “La Sapienza”

In the past Logic had glorious years, even in medieval times, during which it was part of the famous Quadrivium of teaching.

Abstract

In the past Logic had glorious years, even in medieval times, during which it was part of the famous Quadrivium of teaching. It historically developed its solidity and identity by shouldering the development of those Sciences such as Mathematics, Physics and Chemistry; even though its methodological process has been above all influenced by the first of these. However the strong historical and cultural reasons which confirm the importance of this Science are contrasted by the difficulty of its fundaments, above all, Mathematics of Logic. How can we overcome something that seems to be an aporia? In this paper, which begins with a reflection on the ambiguity found both in natural and scientific language, I investigate a particular aspect of Logics: negative and the double negative sentences in ancient scientific books.  Such an aspect would produce in fundaments of Science, still more than the validity of the Tertium non datur law, the valid distinction between classical Logic and the non-classical forms. Henceforth, I shall attempt to show that non-classical Logic may assume, besides its intellectually based role on which a theory is built, also a sharp category of historical interpretation (i. e., Sadi Carnot’s Thermodynamics), from which it is possible to understand what the author thought even implicitly, so it is possible to elevate his fundamental hypotheses beyond his written work.