The parallel postulate was found to be optional when non-Euclidean geometries came under earnest exploration. This was a paradigm-shift, and a breakthrough in mathematics, yet historically people often found it to be a failing in mathematics to the point where, too late in the game, great mathematicians such as Lagrange were still trying to prove the postulate. The reason we wanted Euclidean geometry to be true is it reduces space to a quantifiable reality, so that at the moment of breakthrough, the favorite non-Euclidean geometries could be translated into Euclidean geometry using a metric. Now, we can have very strange kinds of vague metrics for gauging distance. The basic difference in these early non-Euclidean geometries was that there could be more than one parallel line going through the same point, with respect to another line. In other words, the parallel postulate was false for these other geometries.
Euclid himself was so exceedingly careful in intellect that he kept the parallel postulate out of his exposition of geometry, out of his innovative axiomatic form, until he felt it was absolutely necessary. This means that the beginning of Euclid’s first textbook is true in all geometries, and is called “Neutral Geometry.” These precious first theorems are universally true, but this was just not enough for Euclid or his line of mathematicians. They wanted more to be true, so, perhaps under the weight of his contemporaries and ancestors, Euclid abandoned his misgivings about the postulate and asserted Euclidean Geometry as universally true.
How did he make this assertion? You could say his assertion was existential. He asserted that there is only one line through a given point with respect to another line in the same plane. He made this choice out of what we now know to be many options. I may be talking about free will, but not quite. I am sure Euclid believed his books to be about something true, and only his work failed, not reality. He was somehow persuaded or convinced that the parallel postulate was the right choice, and not without examples to the contrary. Geometry on a sphere is not Euclidean: you can have a triangle with three right angles using the line at the equator and two lines going through the north pole. We didn’t know the world was round back then you say? Some scholars back then did, since its circumference was being estimated by scholars as early as 2,000 years ago in Alexandria. Aristotle’s model for the Earth is inconsequentially different from a sphere, though the model appeared flat. Aristotle instructed Alexander the Great to conquer to the edge of the world and, by continuing, he’d end up back in his home eventually. Now, scientists seem to agree that gravity bends space out of shape, and is not Euclidean. The debates of brilliant minds back then, as well as now, were varied, of course, and the over-simplifications from our compulsory education about flat-Earth vs round-Earth, Euclidean vs Non-Euclidean are stark, foolish and in service of the illusion of progress.
The thing to notice is that the alternatives between Neutral Geometry and Euclidean Geometry was a kind of pluralism of parallel lines through the given point, and with respect to another line. This gave rise to a pluralism of quantifying or qualifying distance. Similar to asserting that the only alternative to Being is Not-Being, and the vagaries of cloud-gazing was out of the question. Now we have whole specialized languages to describe the gap between Being and Not-Being: such as psychological hallucination. Leaving out the modern hallucinations of distance, and the looming history of the field of number theory (eventually abandoned by the Greeks, and later connected to geometry in Descartes), how do we choose between these options? Enter Rhetoric, left stage.
Aristotle’s definition of rhetoric was that it was an Art of persuasion, not a Science, even though the above shows how deeply dependent Euclid was on Rhetoric.
So, when I say that Rhetoric is the stuff that connects our planet with the “universe” (another overly-ambitious grab-word from Science), I mean it scientifically, in the Art of choosing how to define distance, because the most important practice of Science, that gives it its importance, is this territory-grabbing from Rhetoric. And Rhetoric is merely the guard of its master: Poetry. In this essay, I have shown how Rhetoric orients people in understanding the gaps between disciplines. Rhetoric is a way of understanding things: the Sciences, Philosophy, and even translation and other parts of Poetic discourse.
Questions Are Power 