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Mathematics is usually described as a deductive science. The set of axioms, with which we start, should be as economical as possible, hopefully consistent, and deductively strong with as many as possible “desirable” consequences. How do we achieve sufficient for an axiomatic system?
In this work we present the unorthodox thesis that the deductive strength in Mathematics comes, perhaps exclusively, from its paradoxical nature, namely from its proximity to the contradictory, a proximity that almost always takes the form of a Finitization of the Infinite. We support our thesis by examining Euclidean Geometry; Number Theory; Incommensurability and periodic anthyphairesis/continued fractions in the Mathematics and the Philosophy of the Pythagoreans, Zeno, Theaetetus, Plato; ratios of magnitudes and method of exhaustion in weakly finitized form by Eudoxus, and Real numbers and Calculus in the strongly finitized form by Dedekind completeness; Set theory axioms such as the axiom of choice, with special reference to compactness and ultrafilters, and Gödel’s program with axioms of large cardinals. In the last two sections we argue that Beauty in Mathematics and we suggest that the “Unreasonable Effectiveness of Mathematics in the Natural Sciences” are both manifestations of the Paradoxical Nature of Mathematics.