How Not to look at a cube
Bertrand Russell defined an external object as a circular system radiating possible impressions. The esoteric remark reminds me of a simple analogy that often comes to mind when I consider the difficulty (or impossibility) of obtaining a comprehensive and definitive understanding of something.
Place an ordinary cube on the table before you and ask yourself: How many angles can it be viewed from? A tabulation of possible perspectives begins with the six front-on views of its six faces which, transposed to a two-dimensional surface, like a sheet of paper, could be represented by six squares. |
Next you balance the cube on each of its eight corners and view it from above, so that three faces are exposed to view and two corners diagonally opposite are aligned, as if you were going to spin it like a top. These eight different views could be represented by eight different hexagons.
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After that you set the cube on each of its twelve edges and view it from above, so that two faces are exposed and diagonally opposite edges are aligned, for a further twelve hexagons. |
Having now tabulated all of the symmetrical perspectives (twenty six, in total) your task becomes more problematic. Take any one of the above views and tilt the cube or alter your position relative to it. Doing so creates another, slightly different, asymmetric perspective, since a transposition of it cannot be superimposed exactly on any of the polygons already tabulated... |
To explain it all differently, imagine tracing the outline of a cube onto a piece of glass. How many polygons would it be possible to create, however minute the differences between them, by altering the position of the cube and treating perspectives that yield identical polygons but expose different faces as distinct? To say that none of the above perspectives is the correct or definitive way of viewing a cube (which, by the way, was one of the motivating assumptions of analytical Cubism) is hardly controversial. But it follows from this that it is impossible to see a cube from a correct or definitive perspective. From every possible angle, most of the surface area of the cube is hidden from view, for to apprehend a cube from every possible perspective simultaneously, that is, to see it definitively, our eye would need to be a hollow sphere with the cube suspended at its center. We never actually perceive but only intuit its complete shape after we have turned it over, seen it from every angle, and mentally conflated the different perspectives. Now consider the difference in complexity between a simple cube and a real-world phenomenon (the industrial revolution, modern art, drug addiction) and let the possible perspectives of the cube symbolize the possible viewpoints of that phenomenon. As with the cube, it is impossible to see every aspect of a phenomenon at once. But unlike the cube, it will be extremely difficult if not impossible to, firstly, perceive every possible viewpoint in sequence because there are almost infinitely many, and secondly, to mentally conflate these innumerable possible viewpoints to arrive at an intuitive understanding of the phenomenon in question. Most people will, in fact, settle on a single viewpoint, often one that has the rejection of alternative viewpoints built into it, thereby foreclosing the possibility of obtaining a better understanding. Keeping with the analogy to geometry, the universe, or even people and things, appear to us not as cubes but as polyhedra with infinite sides that are continuously changing and re-tessellating. The truth about each of them is cognitively closed, something to be ever approached but never arrived at. The rules of the game should therefore be as follows: See as many viewpoints as you can; consider every possible viewpoint of every phenomenon componential and provisional and never accede finally and inflexibly to any single viewpoint—even if that viewpoint happens to be a conflation of multiple viewpoints, for it should also be obvious that we must regard combinations of two or more viewpoints as distinct and to our already-vast list add every possible permutation of the base set of possible viewpoints. And finally, however impossible our task, we should never give up in despair, never let our reason "fust in us unus'd," opening ourselves up to every kind of error or cowering into oblivion. As the Talmud instructs its students, “It is not given to you to complete the law. Neither are you free to desist from it.” |