Taking stock

"Deconstruction insists not that truth is illusory
but that it is institutional." ~ Terry Eagleton


On its face, "parasimplicity" looks a lot like an excuse to make things that are really simple a lot more complex. But, as I hope you've picked up by now, what parasimplicity - or anything else - looks like isn't close to being what it really is. For instance - and this is just a 'for instance' - it also serves as a handy tool for making things that are really complex a lot more simple. The Law of Fives is extremely simple, and as good a shorthand for the very complex things it actually references (which this collection of blogs to date has similarly referenced but at more length, including the obligatory self-similar discussions of the Law of Fives itself). The reduction of all paradoxes to the twin paradigms of All-in-One and One-in-All is another parasimplistic operation (actually the final operation in a chain that begins with the appreciation of all statements as interactions of paradoxes, but we'll get to that).

So, how do we apply parasimplicity? How do mountains become not-mountains, and then not-rivers become rivers? If you've been paying attention, you may already know. What you know may even be what I was trying to say; equally, what you know may be more than what I know. We can't know what we don't know, but we'll get to that, too.

Let's tie in the koan to our five-layer reality cake.

Mountains are mountains and rivers are rivers - this we can consider to be referring to the objective mountains and rivers. Bear in mind that these are not mountains and rivers we can ever directly know: we form subjective impressions of the relations of properties we intuit as belonging to the objective realm, and that is as close as we can get to knowing them.

Mountains are not-mountains and rivers are not-rivers - these not-mountains and not-rivers are not, as you might be forgiven for thinking, equivalent in our model to the subjective mountain- and river-impressions. The not-mountain is the intersubjective mountain: the mountain that emerges from discourse, from interrogation of our subjective mountain-impressions. IF the objective mountain is real, and IF our impressions of it are accurate, and IF we share our impressions truthfully, and IF we don't later edit or filter our consensus to fit some concept of 'truth' - and those are all very big 'ifs' indeed, which we'll review when we turn to Baconian Idols in the near future - then the not-mountain may be apparently identical with the mountain (this is one of the cruder approaches to paradox resolution, in fact - the rejection of the paradox as presented on the grounds that the presentation is corrupted by one or more of these factors). But not-mountains are, well, not mountains..

Mountains are again mountains and rivers are again rivers - these 'again' mountains and rivers are the transcendental mountains and rivers, which we have said are immanent upon the subjective (we might, with a sly wink at Dali, say they are immanent upon the objective as well; in fact Dali's paranoiac-critical method is another rewarding subject for study). We have already said that we can't directly know the objectively real mountain - so how can we possibly hope to know the transcendental mountain beyond? We cannot cross the same river twice (so claims Heraclitus), so how can we know the transcendental river that is beyond all those once-crossed iterations?

The answer is that we need to work against our brilliant knowledge-building engines, our glorious rational Big Brains. We are hardwired to recognize patterns, and we are hardwired to filter and sort the data our brains receive to make it a coherent conscious experience (a trivial example with which you're probably familiar: optics being what it is, the visual data we receive on our retinae is inverted; our brains flip the image over during processing to restore it to its putative objective orientation). However, if we seek the transcendental, that which is equally remote from all things, that in which mountains are again mountains is equivalent to mountains are not mountains or even mountains are rivers - we won't find it after our brains are through processing the data.

Recall that objective entities are entirely separate from subjective ones. Yet our brains, in processing the data from the objective, produce impressions that are subjective and are qualitatively the same as Humean ideas that have no relation to empirical data whatsoever. What this tells us is that the process of rational cognition is capable of bridging the divide that separates objective and subjective - and what that is really telling us is that, somewhere in there, we are working in the transcendental. Each and every one of us, it turns out, is also a Gateless Gate.

The transcendental Universe is self-similar, not only with the objective, but with the myriad subjective Universe-impressions. We could, perhaps, approach some rational understanding of the transcendental if we could somehow simultaneously apperceive all of those possible subjective information-states; but the self-similarity of the Universe, embodied in the Gateless Gate of each self-aware consciousness, makes this unnecessary.

How do you pass through a Gateless Gate? Begin with a gate, and take the gate away so it becomes gateless. Then pass through. 'Get OUT,' as Crowley had it.

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.