La freccia e il cerchio
anno 6, numero 6, 2015
pp. 48-50

Giuseppe Giordano
From “Ideal” Mathematics to Mathematics of Nature

   The discourse about the passage from “Ideal” Mathematics to Mathematics of Nature entails following a path that, in Western culture, – for reasons that we will see afterward – essentially concerns Geometry. The period involved in this passage corresponds to the history of our civilization, from the Greeks up to the present time. So it stands to reason my decision to propose a fast itinerary through almost three thousand years of history; an itinerary that does not pretend to solve the problem, but rather to offer a point of view.
   I would like to talk about the modalities according to which the relation between mathematics and reality is developed; a relation that takes part in an epochal change: from an approach according to which the world finds its objectivity in mathematics and it is bounded to it, to a point of view according to which mathematics becomes “concrete” and does not claim to describe objectively a reality through limitation and adjustment, but that instead renders it in its imperfection.
   It is widely known that mathematics has Oriental origins, well away from Western culture, that has its origins in Ancient Greece. But it’s exactly in the Greece of the first philosophers that something new and remarkable takes place. Immanuel Kant identifies this innovation in 1787 in the preface of the second edition of the Critique of Pure Reason: «A new light must have flashed on the mind of the first man (Thales, or whatever may have been his name) who demonstrated the properties of the isosceles triangles. For he found that it was not sufficient to meditate on the figure, as it lay before his eyes, or the conception of it, as it existed in his mind, and thus endeavour to get at the knowledge of its properties, but that it was necessary to produce these properties, as it were, by a positive a priori construction; and that, in order to arrive with certainty at a priori cognition, he must not attribute to the object any other properties than those which necessarily followed from that which he had himself, in accordance with his conception, placed in the object».
   With these words – that describe concisely why mathematics is a science, since it is “synthetic a priori” knowledge – Kant not only recognises the need for the consequences that, in the mathematical discussion, derive from assumptions, but either reveals an identification that takes its origin in Greece, in the Greece of Talete, the identification of Mathematics with rationality. The turning point of Talete (or someone else on his behalf) consists just of this: the transformation of mathematics into the key of interpretation of reality. This is the role of “mathematics in Western culture”- quoting the title of a book by Morris Kline of 1950.
   In order to support what has been said until now, just think about the description of those philosophers that consider mathematics and geometry as the cornerstone of their thoughts, the Pythagoreans; more concretely, I was referring to the description made by Aristotle in the Metaphysics, when, after reminding that «Plato does not consider Numbers as perceivable», asserts that «according to Pythagoreans, Numbers are the proper things and do not consider mathematical Entities as intermediate point between those and these»; after this, Aristotle asserts critically that: «[Pythagoreans] are not seeking for theories and causes to account for observed facts, but rather forcing their observations and trying to accommodate them to certain theories and opinions of their own, and were thus setting up, one might say, as joint organizers of the universe».
   Therefore, Pythagoreans are the first that identify the reduction to mathematics as the rational essence of reality. They have been rapidly followed by Plato, although with the differences that Aristotle synthetically pointed out. The Athenian philosopher, in fact, in the Timaeus asserts that behind the experienced reality there is the work of a Demiurge that “molded with shapes and numbers” the matter. Everything can be reduced to shapes; and the basic shape is the triangle. However, according to Plato the basis of all that does not stand in the real world, but rather in the ideal world, in contrast with what sustained by the Pythagoreans. The ideal world is the one that the “divine creator” took in consideration to mould the material world.
   According to Plato, mathematics is important since it functions – above all for the philosopher – as the mean to get the immutable essence of the true being in the multitude of the existence. In this case, we are talking about the themes expressed in the Republic (where the author, in addition, talks about arithmetic as a mean of mind elevation, through the use of abstract numbers) that lay the foundations for the attribution of a main role to mathematics, or better, to the mathematical reason in Western culture. Therefore, it’s not by chance that a great mathematician – incidentally – Alfred North Whitehead, affirms that the whole history of the Western philosophy it is none other than a commentary on Plato.
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