return to homepage Squaring the Circle
It is commonly claimed that it has been proved, first by Lindemann, that the circle cannot be squared. I will show that this is false. If all this is true, then why have we been taught that the circle cannot be squared? It is because the problem has been used not as a real problem of geometry or construction, but as an illustration of new theories. Contemporary mathematicians are most proud of their theory of transcendental numbers, a theory that is not very old. It is not surprising that living mathematicians would be more interested in advertising modern achievements than in fairly presenting a problem that comes out of the ancient past. Mathematicians enjoy giving mystical qualities to numbers, and transcendentals bend to this desire more easily than anything else except complex numbers or transfinite numbers. By stressing that they are unconstructible, mathematicians want you to understand that they are far more esoteric, and therefore worthy of esteem, than ordinary numbers such as normal people can understand. Contemporary math has arrived at the furthest reaches of abstraction, where physicality and constructibility are considered to be plebeian matters, nearly beneath notice. What a person can or cannot do with a real tool is mechanics, not mathematics. What are of interest now are only the equations, and the equations tell us that π cannot be expressed algebraicly. That is the fact that must be learned, and if an ancient problem is misrepresented and much logic bludgeoned to teach it, what does it matter? *Most books admit that the circle can be squared if one is allowed to roll the given circle. This creates a line of length π and saves us from having to construct that length with the compass. This is strictly disallowed, however, since a line created by rolling is not “constructed.” But the fact that the length π can be constructed or achieved by any means confirms my arguments above. Once rolling is allowed and the resulting line is drawn, that line is understood (and admitted to be, by most modern mathematicians) exactly π. It is not an approximation of π, or π with a margin of error. It is π, here and now and on our diagram, perfect and complete. The reason its existence in this case confirms my arguments above is that it is proof of π being determinate. The end of that drawn or imagined line in the diagram created by rolling is a mathematical point, which is a precise limit. That is to say, the line ends abruptly, without any sort of fuzziness or smearing, quantum or otherwise. This is even clearer when we are given or achieve a triangle with one side of length π. Both ends of the line are constrained by the figure, by definition. Each side of the triangle ends in a discrete point, not in some bog of an infinite regression. A drawn triangle, which is an abstract geometic figure, is understood to be a figure that exists at infinity, where all lines are perfect—with zero width. If the lines and figures of abstract geometry are at infinity in this way, then all transcendental lengths must also be at infinity—that is to say, exhausted. This means that we don’t need to construct them with compasses and other real tools, no matter what problem we are working on. We simply define all transcendentals as complete, in the same way that we define all our other abstractions as complete or perfect.
If this paper was useful to you in any way, please consider donating a dollar (or more) to the SAVE THE ARTISTS FOUNDATION. This will allow me to continue writing these "unpublishable" things. Don't be confused by paying Melisa Smith--that is just one of my many |