Borges & Leibniz's two labyrinths

There is a famous sentence in Leibniz's Platonic dialogue Pacidius Philalethi-

Duo sunt nimirum labyrinthi humanae mentis, unus circa compositionem continui, alter circa naturam libertatis, qui ex eodem infiniti fonte oriuntur" which translates in to English as:

"There are, of course, two labyrinths of the human mind, one concerning the composition of the continuum, the other concerning the nature of liberty, which arise from the same source of the infinite".

Borges's "The Two Kings and the Two Labyrinths" features a King of Babylon who constructs an intricate labyrinth. He tells a visiting Arab monarch to enter it. The Arab manages to find his way out. He tells the Babylonian that he too has a labyrinth in his Kingdom which, if God so wills, the Babylonian will experience for himself. Then the Arabs conquer Babylon. Its King is taken, by camel, into the middle of the  desert. There he is abandoned to die of hunger and thirst. 

One might say, the Babylonian labyrinth is a test of ingenuity. There is some sequence of choices which enables one to survive it. Free Will can triumph if it is coupled with certain admirable qualities of heart and head. The Arab King is intelligent. He has fortitude. He survives the Babylonian labyrinth. By contrast, Allah's labyrinth appears to be an implacable destiny. But is this really so?

 I suppose, if the Babylonian is physically fit and wise in the ways of the Bedouin, he too has a chance to escape. As the Prophet said, the camel freely wanders the desert and can survive without an owner. It is not inconceivable that such a camel chances upon the Babylonian and rescues him. Alternatively, if some Arab clan sees an advantage in keeping the Babylonian King alive, they might search him out. But, we suspect, God has not willed any such outcome. Equally, one might say, the Babylonian monarch had the chance to acquire qualities of heart and head such that he might effect his own rescue or motivate others to do so. 

The continuum is a useful hypothesis just as the skill of finding water in the desert or of befriending passing camels is a useful skill. But it is not a necessary hypothesis for the same reason that one can survive quite well in Babylon without learning the ways of the Bedouin. Moreover, what is useful pertains to the finite and the contingent not the infinite or the essential. Thus Leibniz's two labyrinths are mere sketches- dessins d'enfants- of the idea of the labyrinth. But what is that idea? On the one hand, there is a shortest path- an Ariadne's thread- and on the other a longest path, which may be infinite if one goes round in circles. But space too is is defined by a 'least action' metric. It may be finite but unbounded.

In a paper titled 'Brouwer meets Husserl' Mark Van Atten writes-

There is an age-old problem in mathematics how to analyse the straight line (‘the continuum’). Traditional mathematics thinks of the straight line as a large number of isolated points lying next to each other, like grains of sand. As Aristotle already pointed out, the problem is that this isolation breaks the line’s continuity. A line is continuous through and through; a continuum is not made up from grains of sand but rather from strings of melted cheese. The mathematician L.E.J. Brouwer was the first to show how to rectify the situation mathematically: his choice sequences provide a means to give a mathematical form to the strings of cheese.

The question is whether the sequence is law-like or 'lawless'. If choices are being made to maximize or minimize an objective function, I suppose one could say there is a 'natural' (non-arbitrary) sequence which, in some sense, is 'law-like', provided the objective function concerns something genuinely useful.  

Mark Van Atten seeks a 'phenomenological' (in the style of Husserl) account of choice sequences. I think, Utility supplies it. The problem is that co-evolved processes are difficult to represent mathematically. It is likely that the thing will be 'multiply realizable' and anti-fragile. In other words, there will be lots of different phenomenologies or, indeed, none at all. People just imitate what smart people are doing. 

Borges, in conversation with Herbert Simon, confessed to being a 'compatibilist'. Freedom was a necessary illusion compatible with belief in a deterministic universe. But, the opposite could be equally true. One might think all one's actions were determined by some occult force while also believing that the universe is wholly random if not illusory simply. The question is, is it useful to have such beliefs? For some people- sure. Borges did very well out of Leibniz's two labyrinths. Did he, like Socrates, discover that he himself was the Minotaur? If so, he was too well bred to tell the tale. That was his charm.

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