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Notes towards path hierarchy Euclidean geometry is taken as fundamental to most physics, and to most so-called "exact" science generally. It is, however, a special case, that relies largely on intuition, which is another name for the self-evident— that is, for things that may be 'taken for granted', that are taken, most often tacitly, to be the case without need of proof, such as equality, right angles and parallelity. The Euclidean view has problems, such as with "ir-ratio-nality", where two lengths cannot both be expressed as whole numbers of the same unit, or as a ratio of such whole numbers, as with, for example, the square root of two, and, for another, the ratio of the circumference of a circle to its diameter. Resort is made to approximation by a rational number in such cases. I take the view that there is no such number. When a number is said to be irrational, the intuition is that that number exists, has a value, and pertains—to a place, say. But it does not have a value, so cannot exist. The geometric object – such as a diagonal of a square, for example – exists, but no number of equal units corresponds with or to it. To me, this brings the whole notion of "length" into question, as lengths are numbers. There should be no unspecifiable lengths. Projective geometry does not suffer from this limitation, as it does not specify or require numerical equality of lengths. In fact, it does not specify length at all. PG can and does specify and count intervals, however, according to the manners of incidence, or non-incidence, of their elements, without regard to size or length. ------------------------------------------------------------------------------- (a) The so-called "projective transformation of space into itself" provides the fundamental and most general basis of measure-as-such, and with it metrication, which is a systematisation of measurement. (b) The concept of an absolute, replicable and additive "unit" (one-off) of size, associating number with geometric elements, needs clarification, and proper definition for the projective context. (c) Correction and redefinition of tacit, spatial intuitions concerning geometric elements and intervals— for example, Euclid's "First Common Notion", concerning distributed, absolute equality and its "self-evidence". Self-evidence is just intuition, i.e., that which is "known" tacitly, without proof, as in (b), above. (d) If a line is composed of points, i.e., of places that are com - posed, posed together, it is a composite, a plurality, not an element, which is a singularity. If points do not have size, they cannot be com-ponents ("placed-togethers"), and so cannot assemble by com-position - i.e., by "putting together", com - posing. So neither lines nor curves can be composed of points. This bears directly on the notion of continuity. (e) It follows from (d) above that, if two lines are incident in a point, that point is not a component of either line. It is just the place at which they meet. |