The philosophy and the mysticism of Pythagoras may be argued to have an appeal for just a limited number of scholars and persons interested in these particular areas. His discovery of the role of numbers in music, on the other hand, is of greatest importance to the musicians, although most of them may not even remember his name. But his greatest “popular” achievement, in geometry, has made him a household name, as everyone who has ever attended middle school, must surely remember the so-called Pythagoras Theorem, which proposes that in a right-angled triangle the sum of the squares on the sides adjoining the right angle is equal to the square on the side facing it. (The Russians call this construction “Pythagoras’ Pants.”)
There has been a lot of sheer anguish, however, expressed on account of the fact that this theorem highlights the necessitated existence of “incommensurables,” that is, of irrational numbers. Alongside with the famous “golden triangle,” characterized by the well-behaved numbers three, four, and five, there is an even simpler triangle, with the legs one and one, which accounts for the hypotenuse that cannot be expressed as a fraction of any rational numbers. (As proved in Euclid’s Book X.)
This defeat of rationality at the hands of the Master himself has caused a lot of concern, originally among the Greeks themselves, who saw it as the defeat of the founding principle of Pythagoreanism that numbers are the first principles of everything. At the heart of the confusion was the presumption of rationality of God.
What, then, can we say in defense of irrationality? It is true that the quality of rationality cannot be denied to God, as such a denial would constitute a demonstrable deficiency, which is contrary to God’s nature. But, by the very same token, irrationality cannot be denied to Him either, on exactly the same grounds. Therefore the existence of irrational numbers should by no means constitute an inconsistency, as, in conformity with my assertion in the related entry Reason And Passion (to be posted next), God’s nature necessarily includes and embraces both rationality and irrationality. Consequently, Pythagoreanism gets an honorable pass here, with a special distinction for encouraging this very promising discussion, which will be most productively continued into the future.
There has been a lot of sheer anguish, however, expressed on account of the fact that this theorem highlights the necessitated existence of “incommensurables,” that is, of irrational numbers. Alongside with the famous “golden triangle,” characterized by the well-behaved numbers three, four, and five, there is an even simpler triangle, with the legs one and one, which accounts for the hypotenuse that cannot be expressed as a fraction of any rational numbers. (As proved in Euclid’s Book X.)
This defeat of rationality at the hands of the Master himself has caused a lot of concern, originally among the Greeks themselves, who saw it as the defeat of the founding principle of Pythagoreanism that numbers are the first principles of everything. At the heart of the confusion was the presumption of rationality of God.
What, then, can we say in defense of irrationality? It is true that the quality of rationality cannot be denied to God, as such a denial would constitute a demonstrable deficiency, which is contrary to God’s nature. But, by the very same token, irrationality cannot be denied to Him either, on exactly the same grounds. Therefore the existence of irrational numbers should by no means constitute an inconsistency, as, in conformity with my assertion in the related entry Reason And Passion (to be posted next), God’s nature necessarily includes and embraces both rationality and irrationality. Consequently, Pythagoreanism gets an honorable pass here, with a special distinction for encouraging this very promising discussion, which will be most productively continued into the future.
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