LATE ANCIENT MATHEMATICS: THE QUESTIONS
Clement of Alexandria is a bit early for our purposes, because he died in AD 216, but he offers some interesting insights into the dialectic between traditional and new Christian values. One of his works, for instance, is devoted to arguing that the evangelic pronunciation against wealthy people (it would be easier for a camel to go through a needle’s eye than for a rich man to enter heaven) is actually not to be taken literally, and that the wealthy need not worry too much about their salvation, as long as they redistribute some of their possessions.3 An example of how the new beliefs could be accommodated to extant and definitely unchanged social and economic hierarchies. Mathematics comes onto Clement’s horizon for several reasons. First, as part of the pagan educational curriculum and thus as a Greek form of knowledge. In this guise, the potential dangers of some branches of mathematics are made amply clear: in the course of interpreting the prophet Enoch, Clement even concludes that humans were taught astronomy by the evil angels.4 On a less dramatic note, astronomy and geometry (and dialectic) are seen as futile – they do not teach the real truth; or they (in this case geometry and music, and grammar and rhetoric) are represented as the
handmaids of philosophy, which itself, while limited, at least is a quest for truth.5 But Clement cuts the ‘Greekness’ of traditional knowledge down to size. He underlines that Greek philosophy is heavily indebted to non-Greek one, and that all technai, philosophy, geometry, astrology and the division of the year into months were invented by non-Greeks.6 On the other hand, he appreciates the value of some philosophical authors, especially Plato and Pythagoras, and also of some mathematics. Moses, he tells us, studied arithmetic and geometry, and mathematics (following a common Platonist tenet) is useful to train the soul towards non-material, higher, realities. Moreover, mathematics represents order, in particular the harmony and regularity of God’s creation: in line with what is stated in the Bible, He is the measure, weight and number of the universe, the one who has counted the depth of the oceans and the number of hairs on everyone’s head. Clement states explicitly that God possesses mathematical knowledge; he draws a parallel between faith and science, which both start from undemonstrated first principles, and engages in quite a lot of numerology. The size of the ark of the covenant in the tabernacle in Jerusalem is discussed from four different mathematical points of view: arithmetical, geometrical, astronomical and musical.7 Here is part of the arithmetical explanation:
‘And the days of the men will be’ it says ‘120 days’ [...] the hundredand-twenty is a triangular number and has been formed from the equality of the 64, of which the composition part by part gives birth to squares, 1 3 5 7 9 11 13 15, on the other hand from the unequality of the 56, seven of the even numbers starting from two, which give birth to the rectangles, 2 4 6 8 10 12 14. According to yet another interpretation the number hundred 20 has been formed from four <numbers>, one, the triangular <number> fifteen, another, the square <number> 25, the third the pentagonal <number> 35, the fourth the hexagonal <number> 45.8
For the definition of triangular, pentagonal and hexagonal numbers, as well as for an explanation of the ‘birth’ of squares and rectangles, the reader has to be referred to Nicomachus or Iamblichus – it is material that does not appear prominently in Euclid’s Elements, where we only find definitions of square and rectangular numbers.