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Wonky Circles and Infinity

30th of October, 2022

If one Googles the question: “how many sides does a circle have?”, the automated response given by the search engine is “zero”. The first result for an organic web page for, searching from the United Kingdom, is for an educational site called, which states that:

“Here at Twinkl, we have decided that the answer to this question depends on the age of the children being taught. In our primary education resources, we say that a circle has one curved side. However, it is also possible to challenge older children with the concept that a circle has an infinite amount of straight sides. Whichever answer you choose to give to this question, Twinkl have lots of fantastic resources describing the properties of 2D shapes. Why not check out this fantastic video, which describes the properties of common 2D shapes using real-life examples?”

There are no perfect circles in the material world, which is surprising because there are also no perfect circles inside thinking minds either. What we have—what we have created for ourselves—are descriptions of what such shapes might be like if they were to exist, and instructions on how to construct them if we have infinite time and precision. We know that if we ever come across a shape where all points along its perimeter are exactly equidistant from its exactly known central point, then we have a circle. We will of course never find a shape like this, and we do not have the resources to build one. We also do not have the cognitive resources to imagine one.

In our earliest ancestors the pattern of electric signals that informed the simple operation of single celled organisms became more and more complex, and once multicellular life forms needed to navigate a competitive space they had to develop brains. The more complex the organism, the more complex the syntactic patterns of electrical signals that crackled around the machinery it uses to model the world. Eventually, and no one knows when this happened, the signal resembled mentalese: the internal language that we think with. The once purely practical necessity of electrical process was able to now arrange itself in the complexity of an animal brain, which interprets sense data. An ape sees lightning strike a tree, and feels not an instinct towards self preservation, but fear.

Planning for the future means that we need to produce small scale schematic models of world states. We can make moves in these imagined worlds, and base our behaviour in the full scale material one according to outcomes in the constructed ones. Most of the time this is not done intentionally: we simply generate these dioramas automatically and in various levels of detail all the time. The consciousness peruses experience in the form of a multi-media language of sense data: sights and smells and textures, but behind this surface detail which can be replicated crudely by cameras and microphones are a set of matryoshka like representations of this tapestry of sense which is constantly reacting to every whimsical and involuntary suggestion of possibility.

We might see a face in profile and model the rest, the unseen. Without this, we would probably be startled constantly at seeing people with half a mouth and one eye. The face that is modelled is done so automatically and not rendered by the mind’s eye: rendering it takes mental power. Looking at the face of someone in our present company is not like looking at a photograph of any old face, because we understand that the impression we have of the living face is part of a continuity of its faceness: that what it is depends on how it behaves and where it is in relation to other things in both space and in time. The mouth might move if the face begins to speak. The eyebrows might furrow if the mind behind them has decided that your staring is making it uncomfortable. We don’t render these actively in the mind’s eye, but we constantly model ghostly afterimages of all these possible variations which might have existed instead, and each afterimage can be toyed with in the same way. All that is sacrificed is clarity, or a sense of the concrete reality of the image in question.

In fact all these images, whatever level of detail they happen to be in, are equally unreal or real insofar as they are ideas which have as much reality as one is ideologically inclined to give them. What separates them in the end isn’t how real they are, or how well they resemble something material (nothing we could ever imagine resembles anything that happens in the physical world) but the degree to which their level of detail allows us to iterate upon them further. We might think of something very far away (like Australia) and then think of something further (the crab nebula) and then go further (the edge of the observable universe). Being in Australia is something we understand well enough: we know some things about how it might feel to be there and the sort of things that tend to happen there. We may be wrong in subtle ways about this but the impression is going to be more or less accurate.

How Australia is, rough approximation.

Trying to do the same for the crab nebula reveals the stark difference in these mental pictures. The crab nebula looks like an iris when photographed from a great distance, but being there means being inside a multiple lightyear-wide cloud of gas: something we have no frame of reference for. Trying to imagine being at the edge of the observable universe reveals that this picture is in fact not even of a place, but of a conceptual limit to the possibility of space in relation to our current position. Distance, time, and causality are collapsed by the successive production of pocket universes with just enough detail to satisfy some intellectual utility, but falling short of the full infinity of the real universe. A succession of concentric wonky circles.

Where cognition is concerned—and this applies to both human and machine cognition—there aren’t really any possible representations of infinite values. The values instead sometimes get so large that we intuit the limit beyond which further increase becomes irrelevant, so that very large numbers instead begin to behave as though they are infinite. In a practical sense, a regular polygon with a googol sides is indistinguishable from a circle. If such a thing could be constructed in the physical universe, it would be indistinguishable from a regular polygon with a googolplex sides. If an immortal man were left drifting alone in space for a googol years, and then brought home for a year before being left adrift again for a googolplex years, would the ostensibly unfathomable difference between these two lengths of time even be detectable to him? Both timescales defy the touch of consciousness—they are two eternities.

The circle, like pi or Euler’s constant e, is not a thing but the hypothetical result of an impossible process. No engineer will ever need to know pi or e beyond a few tens of decimal places, which is fortunate for the enterprise. The imaginary flat space that the perfect circle describes is infinitely populated with points, and the hypothetical attempt to model this space beyond simply settling (as the engineer does) for an imperfect circle becomes then not an act of understanding and parsing data, but an artistic representation. It is a mapping of the internal infinity that the prism of consciousness offers us in the multiplicities of our images. The internal infinity of consciousness is the spiritual infinity of multiplicity rather than the (impossible) numerical infinity of magnitude. We use it to think about God, to imagine civilisations that will never exist, to play chess, and to remain unperturbed by images of cloven faces. I have decided to ignore the possibility that a circle has zero sides, because it is inelegant for my present purposes. You can imagine a different piece in which I take that possibility seriously. It goes to different places.

Further Reading:

Vision, Attention, Consciousness
The Shining Ice
Ballet for Inanimate Dancers