Vittorio G Hosle

University of Notre Dame, German and Russian, Faculty Member

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I am a philosopher whose studies cover a broad range of topics, both systematic and historical. My main interest is the systematic reconstruction of the tradition of objective idealism in a way that is compatible with the knowledge that the empirical sciences have delivered in the last centuries.
Phone: 574-631-5121
Address: University of Notre Dame
318 O'Shaughnessy Hall
Notre Dame, IN 46556

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MISZELLEN A CONFUSION OF ἘΛΑΣΣΟΝΑ AND ΜΕΙΖΟΝΑ IN OUR TEXT OF PROCLUS' COMMENTARY ON THE FIRST BOOK OF EUCLID'S ELEMENTS

Rheinisches Museum fur Philologie 160, 2017

Proclus' commentary on the first book of Euclid's Elements is doubtlessly the longest and most in... more Proclus' commentary on the first book of Euclid's Elements is doubtlessly the longest and most intellectually complex ancient text in the philosophy of mathematics that has been preserved. For readers inclined more toward mathematics than philosophy, the most exciting passages are probably to be found in Proclus' comments on prop. 1,27-29 of Euclid. 1,29 is the converse of 1,27 f. While many propositions in the Elements are converses of the ones immediately preceding them, 1,29 is the only converse whose demonstration requires a new postulate-the fifth one, which is not used by Euclid in the earlier part of the work (1,1-28). The fifth postulate states that, if a straight line, falling on two other straight lines, makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on the side on which the angles less than the two right angles are situated. The lack of evidence of this axiom-given that the Greeks were sufficiently familiar with asymptotes for curves-as well as the anomaly of the relation between 1,27 f. and 1,29 explain why mathematicians, already in antiquity, tried to prove the fifth postulate on the basis of the other postulates and axioms. Proclus refers to Ptolemy's attempt and aptly demonstrates its failure. He himself then proposes a new demonstration-which, however, would later suffer the same fate as that of Ptolemy suffered at Proclus' own hands (which will be the case for all such 'proofs' until the 19 th century, when mathematicians developed hyperbolic geometry as an equally consistent alternative to Euclidean geometry). Proclus' demonstration is circular, for he appeals to an axiom of Aristotle, which cannot be assumed unless one already presupposes the fifth postulate. However, immediately before his own proposal Proclus discusses an objection that claimed to show that lines produced from angles less than two right angles could n e v e r meet-the contrary proposition to the fifth postulate. The argument, which is not relevant for my purpose, is analogous to Zeno's paradox about Achilles and the tortoise. As the latter paradox can be refuted by showing that both the space and the time required to pass through the space decrease continuously, 1 so the for-1) See Aristotle, Physics 6,2 and 6,9.

Vittorio G Hosle

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