As I read about Mandelbrot, I kept thinking of Thales (of Miletus), who also used geometry to solve problems - like of navigation, and also connected "philosophy" to economics. According to Aristotle's Politics, he bought olive presses when demand was low, to rent them at a higher price when demand was high, to prove the value of "philosophy" - i.e. observation of the heavens.
Thales is known mostly as the first to use experience and evidence, not myths, as support - which is important in connection with Mandelbrot in the sense that logos is valued more than mythos. And this brings us back to Aristotle - to his Metaphysics - where philosophers are distinguished from poets in that they seek to escape their ignorant wonder. Another way to talk about logos vs. mythos is abstract thought vs. mythical stories often about gods. While Thales saw gods in all things, Cicero wrote that for him, "god was the mind that moulded all things out of water" (e.a.), which was later cited by monists. I'm not sure whether non-mythological accounts of meteorology were considered a threat to Olympian religion, like later science was thought to be by some.
Thales had been educated by an Egyptian priest - and because he also considered the electricity in amber (ηλεκτρον), I can't help but think of Plato's Timaeus, in which a myth about amber is recounted. The myth is retold as an Egyptian priest tells a visiting Greek why "you Hellenes are never anything but children, and there is ... no old opinion handed down among you by ancient tradition, nor any science which is hoary with age ... Whereas just when you and other nations are beginning to be provided with letters and the other requisites of civilized life, after the usual interval, the stream from heaven ... comes pouring down, and leaves only those of you who are destitute of letters and education; and so you have to begin all over again like children, and know nothing of what happened in ancient times".
Speaking of ancient times, we might wish to consider King Solomon and whether he was a good king as Dante does in his Divine Comedy's Paradiso. In those poems, poetry seems to become logos, as arguments are set out as to why King Solomon may be considered good. Not only that, but in that same canto xiii, he mentions science, and even Thales' eponymous theorem. That canto is primarily about the problem of appearing right vs. the truth ("fit in the truth as centre in a circle"). "Let not the people be too swift to judge" the canto ends.
As if Dante were writing Heidegger's lecture on Things, he states: "kings, of whom are many, and the good are rare". Dante sees contingencies in the movement of things from heaven, which "nature gives it evermore deficient". (Similar to Thales, Dante believes "things generated, which the heaven produces by its own motion.") Thus it may happen, in such an imperfect world - tampering with the ideal stamp: the multitude not necessarily being a picture of the truer, and better, "essential nature" (Heidegger) - that men may "render distorted their straight faces" and allow current opinion to blind them or their pride to blind their intellect. Truth may be distorted. "Let not the people be too swift to judge."
That line reads like a Delphic maxim, "short, memorable phrases" - the kind that Socrates attributed to the "sages," who included Thales. In Plato's Protagoras, Socrates praises such maxims for their wisdom, particularly if uttered at the right time, like a child may be made upright by the force of admonition. He also says man may be too young to determine the "wares of knowledge" which - unlike material wares - cannot be carried in a vessel but must be carried in the soul, where they either do harm or good. Socrates uses the question of goodness to show how hard it is to live out the maxims that are the sign of the educated man: it isn't even possible to prove whether or not virtue can be taught.
That the good life is so hard to achieve is presented in Socrates' portrayal of Thales in the Theaetetus: so concerned with looking at the stars, he falls into a pit. But Thales, like Socrates, is a philosopher, so it doesn't make sense that we take this image literally. Indeed, the passage continues to describe the goal of the philosopher to escape ... not wonder, which is the beginning point of philosophy ... but the earth, to reach the heavens. And part of this formula is essentially to know oneself: "just because they do not think they are such as they are, they are so all the more truly; for they do not know the penalty of unrighteousness, which is the thing they most ought to know." This passage is a digression in a discussion of flux, in the larger discussion of the nature of knowledge.
Socrates later goes on to show how a man can be ignorant of what he knows by reason of his knowledge (199). This is not unlike the warning in Dante, but more dramatic, because it appears Socrates is condemning himself, when really he is warning us about judgement and what we think is knowledge.
He also talks about ideas as birds - some flocked together, some on their own, and explains that to possess an idea is like to cage a bird. He seems to be addressing how it is that we organise ideas once we're old enough to cultivate them: "When we are young, the aviary is empty; after a time the birds are put in; for under this figure we may describe different forms of knowledge;—there are some of them in groups, and some single, which are flying about everywhere; and let us suppose a hunt after the science of odd and even, or some other science." The challenge is to see the world as it is, not how it appears.
We have mention of man's higher interests, the idea that the good in the world has been achieved by idealists. And we know that mythos was used in the presentation of ideals to uphold the ideal when there was no logical support.
Thales thought water was the first principle, the permanent entity. He was concerned with the cause of things, not just what they did. It is easy to imagine his conclusion when imagining the importance of maritime trade and fish for sustenance: water could unite all things. Within this framework, his ideas are "essential" - to use Heidegger''s word.
And then we have Mandelbrot, in exile, a war refugee, a maverick scientist at the outskirts of academia. He found repetitions of self-sameness in broken shapes. There is a pattern that - like Socrates' birds - can apparently be used. As I read the article about him, though, I kept thinking that he was a Socratic-type philosopher for at least trying to reach some sort of ideal; even if he never fully reached it. He was not a fake philosopher, and tried to see things for what they were. The essential idea seems to be there is repetition in chaos.
How to apply this knowledge is difficult. It is possible to cage ourselves into ideas, as if we ourselves were but bird-ideas. But if we don't let such explanations take the wonder out of the world, they could inspire us to ever more beginnings. The principle cause remains hidden and is unproved by theory. Therefore, we have an empty, or blank, atrium. How we organise is up to us, as is how final we take knowledge to be. But that was pretty well covered in the Theaetetus and the stakes of right vs. true.