The womb of truth

“Every experience is a paradox in that it means to be absolute,
and yet is relative; in that it somehow always goes beyond itself
and yet never escapes itself.” ~ T. S. Eliot

 

Over the last couple of posts, we’ve looked at paradoxes – statements that seem to be understandable in our representative system, but produce confounding results. Mandelbrot’s statement that a coastline gets longer the shorter the scale of measurement becomes is an example of a paradox of infinite recursion; Godel’s statement that any logical system must be inconsistent or incomplete exemplifies a paradox of self-referentiality. I will hereby gift you another of my unsubstantiated assertions: all paradoxes are either paradoxes of infinite recursion, and so statements about the One-in-All; or they are paradoxes of self-referentiality, and so statements about the All-in-One.

These concepts of One-in-All and All-in-One, to which we briefly alluded some time ago in a discussion of the Phoenix, are important in theology, where they provide analogies for the Divine. Within the pentapartite model of reality outlined early in the life of this blog, these concepts are transcendental – they derive meaning only as relations with ideals, or metarelations.

Systems constructed by our rational faculty cannot grasp these metarelations, because they are bound to the objective and subjective realms. Even though I am providing you the raw material for a scheme that describes metarelations, it necessarily falls short of being properly descriptive – my assertion that there exists something beyond our understanding is not at all the same thing as an assertion that this specific entity here is understandable as being beyond our understanding (in fact, you might be able to recognize this second construction as a restatement of the paradox of self-referentiality). Nevertheless, an examination of paradoxes has value – not only as an intellectual exercise, but also as a spiritual one.

Nicolas of Cusa, known also as Cusanus, elaborated a sophisticated philosophy around this notion of paradox as a womb of truth in the transcendental sense. He posited a cosmology in which God was both within and beyond the All of Creation and the Nothingness of Void; he described God as the non aliud, the ‘not-other,’ that is, the thing which is neither One nor the Other (this can be seen as a challenge to the Aristotelian Law of the Excluded Middle, an anticipation of Godel’s Incompleteness Theorem, or yet another restatement of the paradox of self-referentiality). For Cusanus, God was the unimaginable union of All and Nothing in One.

(A quick aside: note that Cusanus here introduces a third element to our earlier picture of One-in-All and All-in-One. In fact, we can now talk of One-in-Nothing, All-in-Nothing, One-in-All, Nothing-in-All, and All-in-One. We could talk of Nothing-in-One, but that would actually be Two, harking back to our earlier discussion of essential numerology. The cosmology of One, Nothing, and All is another restatement of the Law of Fives.)

Cusanus accepted that God was unknowable, in accordance with Church teaching (he was a Bishop of Rome in the Catholic Church). He nevertheless felt that we could understand something of the Divine, seeing perhaps “as through a glass darkly” but seeing nonetheless. Cusanus believed this could be accomplished by meditation upon the coincidentia oppositorum, the “marriage of opposites” – in the sense that paradoxes simultaneously defy and unify the opposites of True and False in a bivalent logic, they are ripe for Cusanian study.

Wheels within wheels

“Clouds are not spheres, mountains are not cones, coastlines are not circles,
and bark is not smooth, nor does lightning travel in a straight line.”
~ Benoit Mandelbrot

 

Of course, there’s more to mathematics than numbers. Mathematics is a way of modeling reality, and there are a range of approaches to that modeling. One of these is to simply interrogate some aspect of reality, and attempt to devise a representation that models the result.

The French mathematician Benoit Mandelbrot took on the formidable challenge of modeling the physical contours of things in the objective realm. He pursued a method of modelling the shapes of clouds and mountains and coastlines, which were not immediately apparent as obeying any coherent mathematical principle. In 1967, he published a paper that asked the innocuous question “How Long is the Coast of Britain?” – it was to prove revolutionary, and would give rise to the concept of fractals. This concept is very important to my own parasimplistic worldview, but of course it has far more important implications than that.

Coastlines, it turns out, are tricky things to measure. If one attempted to measure the coastline in units of, say, 10-meter lengths – approximating the actual contours of the coastline – one would find a shorter result than if one used 1-meter lengths. The 1-meter lengths would give a better approximation, and would be longer as a result. In fact, as Mandelbrot’s paper illustrated, the shorter the unit of measurement becomes, the longer the overall measurement becomes. In the limit of an infinitesimally small unit of measure, the coastline of Britain becomes infinite.

This might seem at first blush to be absurd, but Mandelbrot expanded on this result to show its consistency with a whole family of known mathematical relationships that exhibit the property of self-similarity – that is, the curve viewed at a large scale resembles the same curve at successively smaller scales. With a self-similar curve modeling variable x against variable y, the appearance of the curve between, say, 1 and 2 will be the same as the appearance of the curve between 1.0 and 1.1, or between 1.00 and 1.01, or at any smaller scale of measure. Such curves are said to have a Hausdorff dimension between 1 and 2 – the upper bound, curves of Hausdorff dimension 2, are known as Peano curves and have the property on successive iteration of completely filling the space over which they are measured, after the fashion of a ‘Greek key’ motif. Moreover, Mandelbrot listed several examples of naturally occuring self-similar relations – famously including the leaf fronds of ferns.

Mandelbrot was, like most mathematicians, building on the work of predecessors (including in this case Lewis Fry Richardson, who had tackled the coastline paradox himself and posited a mathematical law governing coastlines that foreshadowed Mandelbrot’s result that coastlines were self-similar). The focus of his 1967 paper, which was to form the basis of his work until his death of pancreatic cancer in 2010, was actually a modern application of a very ancient paradox proposed by the Greek philosopher Zeno of Elea – the dichotomy paradox, famously elaborated in his paradigm of a race between Achilles and the tortoise. Zeno’s paradoxes give us a useful structure for considering parasimplicity and being-in-time (what Heidegger refers to as Sein-in-der-Welt), and we will be returning to them.