Pythagoreanism postulates that number in some sense invisibly underlies all things. Today this position is called 'structural realism'. This can be formulated in a number of different ways, and is in any formulation controversial. The basic intuition, though, is that, now that we are able to model and simulate nature so well, it seems that in principle our models are fundamentally isomorphic with nature itself.
Another postulate of Pythagoreanism is that, in a different sense, all is music. This is in a way an epistemological claim in addition to being an ontological claim - because it means that by listening to music we are able to directly access nature. Music is pedagogy. Philosophy isn't even entirely possible without the experience of music. The classic example is the ratios that generate pitch. The truth of nature qua music goes very deep. It is well known that atoms bond with one another to form molecules according to harmonic laws that are literally exactly the same as those that govern tuning.
Further, these two aspects of Pythagoreanism (the world as digital and the world as harmonic) can be seen as two sides of the same coin. Think again of the atom - the term "quantum" refers to the fact that electron shells are basically digital. An electron's distance from its nucleus must be a whole number - 1, 2 etc. And this digital character is required precisely because of the physics of waves. The electron shell can fit one wavelength or two, but does not admit degrees in between. In other words, atoms are digital because they are harmonic.
But what exactly is to be gained from these speculations? They support the attractive idea, proposed by Schopenhauer, Emerson and others, that we have a direct window to the absolute in our own experience. We can grasp absolute reality in a way through our very emotions.
Nevertheless it is important to be careful with this - which is why the discussion of Pythagoreanism belongs to metaphysics rather than cosmogony. It is not clear that even atoms and mathematics, much less subjective aesthetic experience, really participate in the absolute. The absolute resists all hypostatization, reification and objectification. The absolute does not need humanity, nor does need mathematics or physical laws - or at least we are able to imagine it as such.
Then again, if we really posit such an utterly transcendent (OIOIONIC) absolute, we have sidestepped an important question: why is there a world at all, rather than no world? Is it really for no reason at all? Doesn't the intuition persists that somehow the absolute needs the world, that the articulations of nature and thought are filling a void in God's own heart? In any case, with this question we enter the properly Hegelian realm of the speculative