Extracts from Scientific Method in Ptolemy’s Harmonics by Andrew Barker (Cambridge University Press 2004):
Mathematics is not the study of all quantities and all quantitative relations indiscriminately. It is the science of beauty. Its task, at the theoretical level, is to interpret, in terms of ‘rationally’ or mathematically intelligible form, the features, movements or states which, when they are present in perceptible phenomena, constitute their aesthetic excellence. p.264
Those of our senses through which we are able to perceive some things as beautiful are therefore involved in an intimate collaboration with mathematical reason. p.264
Since beauty is the manifestation to the senses of that which reason understands as perfect in form, the senses to which beauty is undetectable lack sensitivity, which sight and hearing possess, to those distinctions which, from a rational point of view, are the most significant. p.265
the mathematical sciences have a single objective, the analysis and understanding of the formal basis of beauty p.266
The conception of mathematical science which Ptolemy has presented is that of a capacity that does not merely analyse sets of quantitative relations, but homes in on those that are of special significance, and discovers the principles on which their significance rests. p.268