Tuesday, July 24, 2007

Why Take the Fifth?

" No person shall be held to answer for a capital, or otherwise infamous crime, unless on a presentment or indictment of a Grand Jury, except in cases arising in the land or naval forces, or in the Militia, when in actual service in time of War or public danger; nor shall any person be subject for the same offense to be twice put in jeopardy of life or limb; nor shall be compelled in any criminal case to be a witness against himself, nor be deprived of life, liberty, or property, without due process of law; nor shall private property be taken for public use, without just compensation."

... "'defence' of Lie's behaviour by referring to the close relationship between genius and madness really created a generally accepted explanation which has survived up to the present. By this act of 'defence' Klein did his old friend an incredible injustice."1

Fred Vaughan headshot by Fred Vaughan

We all know what it means to "take the fifth". It ain't good!

There have been many attempts to reduce the number, modify the structure, and alter the phraseology of Euclid's postulates, but it has been found that for plane projective geometry they are by and large very sound as initially presented. However, there seems to have been little effort to determine whether there might be a different postulate more appropriate that the fifth for modification to provide compatibility with the formalism of relativity and our current view of the universe.

That one of Euclid's postulates upon which he based The Elements of his geometry might be flawed, or worse yet, unnecessary is, of course, an integral part of present day establishmentarian mathematics and physics. The Fifth Postulate, that through any point only one line can be drawn parallel to any other has been unanimously selected as the culpable postulate invalidated by the current understanding of relativity and cosmology at larger scales of our universe.

Long before that mathematicians began exploring alternative geometrical possibilities deriving from the elimination of this assumption after repeatedly failing to reduce it to a provable theorem. This was before there was any inkling that we might actually live in such an alternative universe.2 Gauss actually attempted measurements employing light signals to determine based on such empirical evidence whether that might be the case, however. But with the advent of Einstein's relativity, bold conjectures of a combined spacetime exhibiting strange geometrical properties have been totally accepted by the scientific community, so that alternative-fifth-postulate-geometries thrive; notwithstanding this feeding frenzy on the Fifth, Postulate convincing evidence that another of Euclid's postulates is invalid continues to be denied.

Relativity provides the analytic work of pioneering mathematicians a context of immediate relevance and it should not be surprising that their work would have been re-evaluated with renewed interest. These former discoveries concerning viable geometries not requiring Euclid's Fifth Postulate revitalized mathematical physics.

One must note that even in the general theory of relativity, physical experiments are always considered as being conducted within locally-Lorentz reference frames. What this means is that even though an observer may experience wild gyrations of acceleration due to gravitation or his own rocket engines, at each moment in time it is only his instantaneous velocity relative to what is being observed that is pertinent to the geometry of his current observations. This is where one must begin if the objective is to map observations between oneself and other observers in relative motion. So the Lorentz geometry of special relativity would seem to be the local geometry of choice. This has been thought to involve a flat spacetime, but it is hardly without distortion as the author has discussed elsewhere. In particular relativistic aberration distorts the directions of objects in one frame of reference relative to where those objects are to be seen in the other. The coordinate axes of the other observer are not immune to this distortion

Let us look at Euclid's five postulates and attempt to determine for ourselves which one seems most likely to be at odds with such observational inferences made from Lorentz reference frames. Here are all five postulates3:

  1. Only one straight line can be drawn between any two points.
  2. A finite straight line can be extended indefinitely.
  3. Only one circle of a given radius can be centered at a given point.
  4. Through a point at a distance from a given line there is only one line that can be drawn through the point that is perpendicular to the given line.
  5. Through a point at a distance from a given line there is only one line that can be drawn that is parallel to the given line. 4

In lieu of the apparent directional distortions of the three perpendiculars that constitute the spatial axes of Lorentz reference frames of various observers in relative motion, one can but wonder why there has been this preoccupation with the Fifth Postulate anyway? What we have found is that each of all possible coincident observers with unique relative velocities would witness all other observers' perpendicular directions to be misaligned with regard to their own. Parallel lines of sight in one frame of reference would remain parallel for the others although they would in concert be pointing off in other directions.

So it seems self-evident that to make sense of the coordination of the geometrical observations and constructions between relatively moving observers, we must reject the Fourth Postulate! It seems to the author that we may even need a new theory of perpendiculars. But his elder sister did nickname him "Perpendicular" — Perpy for short — so maybe such stigmata warps ones sense of geometrical rectitude.

On that charge I think I will claim my Fifth Amendment right.

1 Written by Marius Lie's friend and collaborator Friedrich Engel at his death. The quote is provided gratuitously as being of possible interest to this audience.

2 Robert Bonola, Non-Euclidean Geometry, Dover, New York (1955), originally published 1914. Supplements within this book contain "The Theory of Parallels" by Nicholas Lobachevski, and "The Science of Absolute Space" by John Bolyai. The book also provides a context for the pioneering efforts of such names as Gerolamo Saccheri (1667-1733), Johann Lambert (1728-1777), Adrien Legendre (1752-1833), Wolfgang Bolyai (1775-1856), Friedrich Wachter (1792-1817), Bernhard Thibaut (1776-1832), Karl Gauss (1777-1855, Ferdinand Schweikart (1780-1859), Franz Taurinus (1794-1874), Nicholas Lobachevski (1793-1856), John Bolyai (1802-1860), B. Riemann (1826-1866), Ludwig Helmholtz (1821-1894), and Marius Lie (1842-1899).

3 This version involves only a slight rephrasing of those given by Sir Thomas Heath in The Elements of Euclid. Changes parallel Playfair's rephrasing of the Fifth Postulate.

4 In 1795, John Playfair (1748-1819) offered an alternative version of the originally translated postulate involving interior angles, which was: That if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the straight lines, if produced indefinitely, will meet on that side on which the angles are less that two right angles. This alternative version, of course, gives rise to the identical geometry of Euclid. It is Playfair's version of the Fifth Postulate that most often appears in discussions

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