To understand Galileo we have to appreciate that he was before all else a fervent neo-Platonic Renaissance Platonist. The Medieval doctrine of the harmonious universe reached its most sophisticated exposition in the works of Pico della Mirandola and Marsilio Ficino in late fifteenth century Florence. Lorenzo the Magnificent’s country villa at Carregi, just outside Florence, became a kind of monastery or ashram to which the artists and poets of the Florentine renaissance flocked, and it was from these beginnings that the Academy of Florence developed. It was here that Ficino held court and from here that he disseminated his doctrines. Plato’s works in particular were studied with rapt enthusiasm and at Carregi a votive lamp was kept constantly burning before his bust. Why did Plato inspire the aesthetes and philosophers of fifteenth century Florence so much? It was because of his teaching that the cosmos is moved by love of beauty.
A primary inspiration for Pico and Ficino was the Roman poet Seneca’s commentary on the ancient icon of the three graces. In classical portrayals these are always three female figures, the outer two of whom face the viewer while the middle one has her back turned, with interlinked arms and hands. According to Seneca the three figures compose an allegory concerned with generosity. ‘Why the Graces are three, why they are sisters, why they interlace their hands’ is explained by Seneca through the triple rhythm of generosity, which consists of giving, accepting and returning. Why one has her back to us, while the other two face us, is explained because from one benefit that we give two will return. Pico and Ficino radically re-interpreted this figure. Now the graces are portrayed with one facing us, one turned half away from us and looking to the side, and one with her back to us. Now the icon is a triple allegory of the divine love coming to earth in the form of beauty, its reception by mankind which it transforms through the rapture that it bestows, and its return to heaven in the enraptured soul. This doctrine of revelation, reception and return is expressed as emanatio, raptio and remeatio. Alternatively the figure is expressed as Pulchritudo, Amor and Voluptas. Beauty inspires love which leads to delight. One grace is looking towards us, one is looking sideways because she is already seeking heaven from whence the beauty that transfixes her came, and another has her back to us because she is looking directly towards it. The divine presence on earth is revealed in beauty and beauty is loved by mankind, which leads in turn to a pleasure of the soul so exquisite it cannot be accounted for in purely earthly terms.
Another aspect of Plato’s teaching that was most important to his latter-day disciples at Carregi is that beauty is mathematical, for beauty is a product of proportion and number. This was a doctrine that Plato had inherited from Pythagoras. According to legend Pythagoras had been passing a brazier’s shop and was struck by the harmony of the noises caused by the beating of the metal. Intrigued, he entered the shop and realised that the pitch of the sounds was in proportion to the weight of the metal. When he got home he hung similar weights from different lengths of string and, conversely, different weights from equal lengths, and discovered that the three sounds most harmonious to the human ear were proportionate to the different weights and lengths. A proportion of 1:2 such as 2 lbs to 4 lbs or 6 units of length of string to 12, was the equivalent of a musical octave. That of 2:3 produced a fifth. And that of 3:4 produced a fourth. This was the beginning of the whole Platonic myth. Pythagoras took his number mysticism further. He constructed a table of ten opposites, in terms of which everything in the world can be understood. Of these dichotomies the most important was the first, that between the Limited and the Unlimited. Pythagoras had inherited this idea of fundamental opposites from previous philosophers, but what was original about him was the claim that underneath the dichotomies lie numbers which have a mystical meaning. One is not simply the first unit used in counting but an expression, not merely a symbol, of unity. Two is an expression of the dichotomies into which in a sublunary earth unity has become divided. Three is time, which is made up of a beginning, a middle and an end. Four was the number of points required to construct a pyramid, the simplest of the perfect solids, and therefore the first example of solid material things. These first four numbers were of crucial importance because their order forms a pyramid, the so-called tetractys, thus:
X X X
X X X X
Contained within the tectratys is the mystery of how diversity comes out of unity, and how differentiated material solids come out of the immaterial . Together the four numbers add up to ten, which was the basis of Pythagoras’s mathematics. Mathematics was not simply a useful mental exercise, it was a mystical way of life. The disciples of Pythagoras took an oath: ‘I swear by the discoverer of the tectratys, which is the spring of all our wisdom, the perennial fount and root of nature’. In the Pythagorean view reality is literally made of music which is indistinguishable from mathematics. According to Aristotle:
The Pythagoreans, as they are called, devoted themselves to mathematics: they were the first to advance this study, and having been brought up in it, they thought that its principles were the principles of all things. Since of these principles numbers are by nature the first, and in numbers they seemed to see many resemblances to the things that exist and come into being…..since again they say that the attributes and ratios of the musical scales were expressible in numbers; since, then, all things seemed in their whole nature to be modelled after numbers, and numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number’
 See Edgar Wind 1980 Pagan Mysteries in the Renaissance Oxford Univ Press pp26-53
 Jamie James 1993 The Music of the Spheres London chapters 2& 3
 Aristotle Metaphysics.