Stelios Negrepontis, born 22 February 1939, Ph.D. in Mathematics at University of Rochester, N.Y,1965. Assistant Professor at Indiana University, Bloomington, 1965, Assistant & Associate Professor at McGill University, Montreal, 1966-1976, Professor, Athens University, 1973-2006, Professor Emeritus, 2006-. Chairman, Mathematics, 1983-1989, 1992-1994 and Vice Rector 1986-1988 of Athens University. Specializes in ultrafilters, topology, Banach spaces, infinitary combinatorics, history and philosophy of Mathematics, especially Plato and Pythagoreans.
TITLE: The Birth of Number Theory from Pythagorean Music
by Stelios Negrepontis
This is joint work with Vassiliki Farmaki.
Euclid’s Elements is the most famous book on Mathematics ever written. For many centuries, certainly till the 17th century, it was the Bible of Mathematics. One of the reasons for its fame is its axiomatic approach of Geometry, the first in the History of Mathematics.
But the Elements is not only about Geometry. Three of its 13 Books are pure Number Theory. Book VII is known to be Pythagorean and it forms the foundation of modern Number Theory, since it contains all the concepts and tools needed, notably the principle of the Least [equivalent to the principle of mathematical induction] and the “Euclidean”, but in fact Pythagorean, algorithm (“anthyphairesis”, as it was called), for the rigorous proof of the Fundamental Theorem of Arithmetic [every number is the product in a unique way of prime numbers].
But it comes as a great (negative) surprise that Book VII has no Axioms whatsoever. No real explanation has been given so far and the reason for the absence of Axioms has been a mystery. We will try to provide an explanation.
Looking more closely into Book VII we are struck by several peculiarities if we compare it with the modern theory of rational numbers, such as (i) in the definition of proportion only unequal numbers are allowed,, (ii) the representation of numbers is by means of lines, (iii) there is no operation of addition of fractions, (iv) the multiplication of fractions is possible only in the very special case a/b.b/c=a/c,
(v) there is a most surprising proof of the commutativity of the multiplication of numbers,
Again no satisfactory explanation has ever been given.
Initially our main intent was an attempt to explain the absence of axioms. The principal idea was to apply to Book VII Aristotle’s Topics principle of dynamic interaction between Axioms and Definitions. Book VII possesses a good definition of proportion but no axioms, and is thus, according to Aristotle’s principle (considered too optimistic in modern times, axiomatization is now considered necessary) in a most perfect stage. So there must have been an earlier more primitive stage of Book VII, when the good definition was not discovered yet; in this stage the propositions using directly this definition in their proof in Book VII, and these are four in number, cannot be proved without the definition, and thus necessarily are turned into axioms.
This approach provides a satisfying and perhaps convincing explanation for the absence of axioms. But going back to the definition-less form of Book VII turns out to be very fruitful. We are surprised to find that the four mathematical axioms that result when the basic definition of Book VII is taken away, in terms of numbers, ratios of numbers/fractions, and rational numbers (equivalence classes of fractions),
have their exact counterpart in the four empirical properties needed in Nicomachus’ memorable, even if partly fabricated, account, in terms of
length of chords, bichords, and musical intervals (equivalence classes of bichords), of Pythagoras’ mathematization of the purely musical Babylonian 4chord.
This correspondence strongly suggests that the number-theoretic Book VII evolved from the Pythagorean mathematization of Music, and took its final form with the discovery of the Euclidean algorithm. This is confirmed by the fact that all the unexplained peculiarities we have noted above become clear witnesses of its musical origin. For example, for (i) a bichord with equal chords makes no sense in music, for (iv) the composition of musical intervals makes sense only in a trichord
In conclusion, while it is generally thought that questions set by the Pythagoreans on Music “looked for answers in Mathematics”, as Andrew Barker puts it, we have argued that on the contrary the arithmetical Book VII evolved from early Pythagorean Music.
