Part iii: The geometry of reality

cdr 03-30


'The geometry of reality' -- I first wondered about this when I was reading Lobachevsky and learning that those chalkboard parallel lines I remembered from grade school did very different things when they were let alone. Though this is out of character, I wanted to recurrect some former, more polished and focused writing on the subject of geometry and uncertainty.

In the uncertainty whether the perpendicular AE is the only line which does not meet DC, we will assume that it may be possible that there are still other lines, for example AG, which do not cut DC, howsoever they be prolonged.

Lobachevsky discards Euclid's troublesome fifth postulate and proceeds to develop a system of geometry that not only omits an equivalent to Euclid’s fifth postulate, but actually denies the fifth postulate as a necessary rule for parallel lines. [As a reminder, Euclid's fifth postulate states: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles]. After acknowledging the uncertainty that comes with any line said to be produced indefinitely, Lobachevsky asserts a new way of determining parallels that can allow for greater accuracy and a more functional system of geometry.

Euclid’s elements of geometry are awe-inspiring for their elegance, accessibility, and structure. Each definition serves a purpose, each postulate and common notion is required for the definitions to work within the proofs, each proof relies necessarily on what has come before it; the pieces fit, the proofs are lucid, there is no equivocation. But even in the midst of this clarity there are some uncertainties amongst the otherwise self-evident proofs. His axioms begin with elements that seem to be defined by their metaphysical qualities, not by their function or cause in the physical world:

A point is that which has no parts

A line is a breadthless length

It takes some mental stretching to conceive of what is meant by these definitions, especially when one realizes that the point and the line are not objects of contemplation, but must be used in the construction of physical proofs and in other “real-world” applications.

In proposition 1.4 we are asked to imagine one given triangle being “applied” to a second given triangle, with the assumption that doing so will reveal that the triangles coincide and are thus congruent. But how does this application occur? Again we must stretch our mind to move a fixed geometrical element from its apparently static location to a different location.

This requirement in the proposition is difficult to accept; motion is something we reluctantly attribute to the figures composed of lines, not easily finding where motion is implied in Euclid’s definitions. His definitions seem to describe elements which remain fixed and unchanging; their qualities and nature are reliable, and thus there is no uncertainty in their application within a proof. There is no surprise maneuver of a line that we can imagine after reading only the definitions. Lines will be breadthless lengths, always and indefinitely, just as points will never have any parts. Nevertheless, the triangle is applied.
Proof 1.4 introduces a second uncertain element: the proof is only accomplished by introducing the reductio method and proving the impossibility of one claim to show the truth of its opposite. And while this is still a satisfying proof, it is significantly less satisfying than the preceding three proofs which progressed from definitions and common notions in a direct manner.

Why are these parts of Euclid’s proofs more difficult to accept than the clear definitions and directly constructed propositions? Geometry is supposed to provide us with a grammar by which we can speak of the natural world and all of the motions and bodies we find in it. The uncomfortable elements of proposition 1.4 comes from the need to manipulate in a functional way elements that seem to be too precise for this usage. I do not mean to say that Euclid’s elements are not useful in the natural world; there is, however, an initial surprise in the mind when the idea of applying one triangle to another is introduced, or when, in setting out to study geometry, one finds oneself face to face with a definition that seems more metaphysical than physical.

The elegance, reliability, and precision that are the greatest characteristics of Euclid’s geometry do not admit easily of anything uncertain. Since the greater part of his definitions and propositions strike us as self-evident and unarguable, the few places where we feel surprise at how the elements are used (and a consequent surprise at feeling surprised) stick out in sharper relief as conspicuously less certain. But any surprise the mind experiences in proposition 1.4 is quickly assuaged by a return to direct proof and clear constructions.

A parallel line, by Euclid’s definition, is the one, unique, non-cutting line which, when taken with its pair, will have its interior angles add up exactly to the sum of two right angles. These lines can be produced indefinitely and will remain always parallel; any other pair of lines that does not conform to this definition must meet at some point in their production. His definition of parallel lines is precise and admits no exception.

Lobachevsky’s geometry does not rush past uncertainty or surprise so quickly. Instead of accepting Euclid’s parallels, Lobachevsky takes his definition and finds not precision, but uncertainty. He asks whether we are entirely certain that there can be only one non-intersecting line. Could there not be another species of line, the non-intersecting, but non-parallel line? Instead of one unique, non-cutting line, we propose instead, one first, non-cutting line, the line which is the boundary or parallel line.

In theorem 16, Lobachevsky challenges Euclid’s fifth postulate, which defines parallel lines by the sum of their interior angles. He shows in this theorem that there are, in fact, three sorts of lines that can pass through a given point with respect to a given line, and that there are actually two first non-cutting lines, one on either side of the perpendicular to a given line. Thus, there are sides of parallelism; the angle of parallelism on one side of a perpendicular can be created on the other side to form two first non-cutting lines (two boundary lines which are at an angle less than two right angles to one another). This first non-cutting line on either side is the parallel, but there is an infinite amount of other non-cutting, non-parallel lines which go through the given point in the space outside the areas created by the sum angle of parallelism (“sum angle” being meant as the angles of parallelism taken together on the side of the boundary lines facing the given line, and again on the side of the construction corresponding to the prolongations of the two boundary lines).

In order to accommodate this potential plenum of non-cutting, non-parallel lines, Lobachevsky must assert that the angle of parallelism is not necessarily the exact sum of two right angles, but is instead a function of the length of the perpendicular between the two parallel lines. In the realm of “uncertainty,” we have hit upon something that is at least manageable, if not certain. So long as we can determine the distance of the perpendicular to a given line, we can determine the angle of parallelism, and thus find our parallel on each side, as well as the region of non-parallel, non-cutting lines.

This sort of ‘certainty’ will not result in one precise parallel line, as would be the case with Euclid’s definition, but in providing us with the possibility of other non-cutting, non-parallel lines, we actually have a smaller margin of error in our result [“error” being used as a line incorrectly identified as a parallel line, or an incorrect supposition that a certain line, if produced indefinitely, will never intersect its “parallel”]. Euclid’s definition allows us to be absolutely certain regarding one specific line, but leaves us susceptible to an enormous margin of error once we’re dealing with extreme magnitudes of production [we could be absolutely wrong about “parallel” lines which seem to contain interior angles that add up to two right angles, and since we cannot be there at the place where they would meet, we accept an incorrect supposition].

There are two certainties then; Euclid’s certainty is consistently developed, but Lobachevsky presents a different set of starting premises, with the promise to develop from them a breed of geometry as consistent as Euclid’s, though entirely different in nature. Lobachevsky doesn’t provide a proof-based system that is as palatable as Euclid’s, but rather a functional system, and one that is revealed to be a more accurate “grammar” for use in analyzing the natural world.