Traditionally there were two schools of Pythagoreans, the acousmatics and the mathematici. Philosophers, following Aristotle in this, have mainly focused on the first. Yet the acousmatics are arguably the old school, closer to the original thought of Pythagoras. I am not interested here in being the more authentic Pythagorean. However I am interested in this other direction and in how Pythagoras himself would have approached aesthetics. This material is based mainly on my understanding of the Waterfield translation in The First Philosophers. Pythagoras looked at the world and found a preponderance of harmony. If he had a central aesthetic concept it would be harmony. The earliest Pythagoreans were interested in numbers for two reasons, because they have meaning in proportions, for example in harmonious chords in music, and because they can operate as symbols. The two are related since a number is often meaningful as a symbol of a proportion. According to most scholars, and contrary to what we were taught in school, mathematical theorems and formulae were not of much interest to the first Pythagoreans.
The acousmatics were also concerned about akousmata. These were sayings, or rather simple but obscure answers to questions, ones that I suspect had a more literal or practical interpretation hiding behind. This can be inferred from the fact that in Diogenes Laertius their sayings have standard interpretations. For example the injunction "not to pick up things that have fallen" is explained as "not to eat in immoderate quantities." So when we turn to the accousmata we are looking at layers of interpretation. The question could be seen as a riddle. However the standard answer could be seen as another riddle. There was also, as in the injunctions, probably an explanation of the answer, although even this might be obscure. In this respect the accousmata and the injunctions are like Zen Koans. Two examples are: "What are the Isles of the Blessed? The Sun and moon." and "What is the Delphic oracle? The tetraktys, which is the harmony in which the Sirens sing." After mentioning these, Iamblicus also mentions "What is most moral? To sacrifice?" and "What is wisest? Number." One suspects that the worst way to take these is as literally true. For example the one on sacrifice is intended to be a riddle itself. We are supposed to interpret what is meant by saying "to sacrifice" or by suggesting that to sacrifice is the most moral. Sacrifice is a practice already under suspicion by philosophers, as we see in Xenophanes and Heraclitus. And numbers are not literally wise, although the explanation might be that knowing the secrets of numbers is.
My hypothesis is that the early Pythagoreans saw riddles and harmonies everywhere and that this is a way of seeing the world around us as deeply aesthetic, as charged with an aura of meaning. Harmonies, riddles and numbers all have hidden structures. Another puzzle might be their most characteristic oath: "No, by him who handed down to our company the tetraktys, the fount which holds the roots of every-flowing nature." The tetraktys is a numerical structure made up of proportions and shaped in the form of a triangle. Yes, the tetractys is made up of 1+2+3+4=10, and ten is considered the most perfect number, but also this is a basis for an aesthetic ordering of nature and of the processes of natural change. As Sextus Empiricus says, "it is their view that the whole universe is organized on harmonic principles" and this involves three concords based on the proportions 4;3, 3:2, and 2:1. As an experiment, Hippasus made four bronze discs with thicknesses that had exactly these three proportions, and which, when struck "produced a concord." Aristotle clarifies this when he says that the Pythagoreans believed that "numbers are the primary mathematical principles" and that there are "many analogues to things that are and that come into being" in numbers. Note that the claim so far is not precisely that all things are numbers but that numbers are symbols and like that they are like the other things they symbolize. The idea is that a certain attribute of numbers is justice and another is soul and mind, and these are associated with ratios and harmonies. Indeed, "the whole natural world seemed basically to be an analogue of numbers" from which it was concluded that "the elements of numbers are the elements of all things, and that the whole universe is harmony and number." This is ambiguous: the element of numbers might not actually be numbers, and the whole universe might be understandable only because we can understand it by way of applying to it numbers or numbers to it in the symbolic (and not measuring or mathematical) way we are discussing. We think today that this is a matter of seeing it in mathematical terms, and this is partly because Galileo and the other Renaissance philosophers saw Pythagoras in these terms. But it is more like astrology, where numbers make up a symbol system that makes sense of the universe in terms of ratios and harmonies. Aristotle sees this in terms of the ordering of the planets and posits that the Pythagoreans needed, for example, a counter-earth to make their system complete numerically. But let's come back down to earth, back even to home and everyday life, where the idea might be related to the building of a harmonious community, perhaps a Pythagorean one, one in which numbers play an explanatory role by providing a kind of symbolic overlay on experience. One example of this is the system of opposites that Aristotle mentions in the same section, as in such pairs as limit and unlimited, odd and even, unity and multiplicity, right and left, male and female, still and moving, straight and bent, light and darkness, good and bad, male and female. Deconstructionism made a business of seeing interpretation as a matter of deconstructing this table of opposites, to which Derrida and others added even more opposites. If deconstruction is an aesthetic move, since interpretive in a non science-like way, then that upon which it is based, the very idea of opposites to be deconstructed, is another aesthetic move.
Now consider the following passage: "The Pythagoreans, as a result of observing that many properties of numbers exists in perceptible bodies, came up with the idea that existing things are numbers" Focus on the first part of the passage. They observed in perceptible things properties of numbers: What can this mean? We will see that the properties of numbers are their symbolic associations particularly in relation to certain ratios. So seeing a property of a number in a perceptible thing is seeing a thing as a symbol, as having heightened meaning because of its being a riddle with a name, the name being a number. Looking at the world in terms of numbers where numbers refer to different harmonies is looking at the world, not only the cosmos but everyday life, as music, as sensuous experience endowed with meaning. Why do the Pythagoreans believe that existing things consist of numbers? They do because, as Aristotle tells us, "the properties of numbers exist in musical harmony, in the heavens, and in many other cases." Again, Aristotle insists that perceptible things are made up of number. So, for example, "reciprocity or equality is a property of justice" equality being a property of numbers, justice being understood then as the first square number, usually considered the number 4. Their approach to marriage as a concept is useful in understanding how this works, as marriage is taken as 5 because it is the union of the odd and the even, the odd being male and the even female. Marriage, on this view, is a harmony, but one based on two opposites, a point similar to that of Heraclitus, that harmony is based on war, on a special kind of tension (perhaps he protested too much in attacking Pythagoras). Marriage oddly is a deconstruction of the opposites avant la lettre. A harmonious marriage is a deconstruction of opposites. Another way that Pythagoreans negotiated life by way of riddles was through what were called tokens: so as Aristotle's fragment in Porphyry puts it "do not walk on the highways" is a recommendation not to follow the opinions of the many.