Euclid's Fifth Axiom
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Back to Basics PageTurning. Instructional concepts for a saner approach to the global precarious. Is spacetime curved? Oh yeah, Einstein's General Relativity scores and, oh yeah, it keeps on scoring, and so, oh yeah, spacetime must be curved. Not so fast please, for it might be that the constructivists' paradigm is colouring what we take from General Relativity. If the world is sets ... but who says the world is sets? Ernst Zermelo says? John Von Neumann says? General Relativity enables one to dissolve out the force of gravity as a vector and replace it with the energymomentum tensor. This is difficult because the tensor approach relies completely on a geodesic description. Now because it scores points in evidence, e.g. the Eddington experiment, we may be inclined to treat gravity merely as one of the inputs to a geodesic equation. It is a subtle point however, that gravity cannot be broadly dissolved out as a force. In the definition of energy, when we talk of a force we mean to include gravity. The gravitational potential energy has been on the canvas since the very beginning of energy science. General Relativity cannot wipe out the energy science. Therefore gravity as a force, indeed as a vector, lives on. Readers who have seen some light in Stanley's position (see my discussion for teachers here), may still be wondering a little. Cases and facts to be distinguished? How so? By using the phrase "where it is the case that" in front of an equation (e.g. "where it is the case that x = 5.5 lbs") we obtain a reference. More broadly, instead of just one equation, there may be equations plural, inequations and other statements of the case, joined with connectives from the simple propositional calculus. What goes after the phrase "where it is the case that" in general just completes a way of refering to space, without yet classifying the space as a set. It is just somewhere. So let there be three coordinate variables to be evaluated in metres or in feet or in whatever other length unit is appropriate. For the sake of argument, let them be x, y and z, to be evaluated in feet. Euclid's first axiom states: a straight line can be drawn from any point to any point. This makes so much sense for our threedimensional space. We can surely grant it. There are points in order leading away more or less adinfinitum in any direction. We may then turn to the theory of relations and let our first two points be found lying on an axis: we must choose one of the three coordinate variables for this axis. Let us choose x. Then the chosen axis runs where it is the case that y = 0 ft and z = 0 ft. By saying it runs where these equations in y and z hold we are intimating that points on the line are distinguished by x value. The distance between the selected points on the axis is then the algebraic difference between the two x values. The line required by Euclid's first axiom is then the part of the axis lying between the points. It transpires that a line can be considered straight for the purposes of Euclid's geometry if it runs where it is the case that K, for some linear relation K of the three coordinate variables. With a little thought, one can verify that this program for refering to space relies on drawing a distinction between a variable and a value. While the constructivist's paradigm may disparage such a distinction, yet it can be very useful and surely it must have a pedigree of some kind. The constructivists' paradigm may have already been in bed when Russell's Paradox, associated with the philosopher Bertrand Russell, produced a profound headscratching and then a movement of the literati (gaining steam by the mid1930s). For if one thinks like Stanley from Taupo Primary School (see the discussion for teachers here), then one can readily define what an indeterminate quantity is. It may follow that we should treat Bertrand Russell's paradoxical set as indeterminate; the equation sign used in set theory can have its transitivity property put subject to a condition. Examples from mechanics showing that indeterminates are already with us, even before any quantum mechanics, even before any set theory, can be given. Further to the geometry wherein a line can be just where a relation holds, a point typically will need three equations to pin it down. For example, one point is at the place where it holds that x = 0 ft, y = 0 ft and z = 0 ft. Now take the values that correspond to a point and put them together as an ordered triple, e.g. (0 ft, 0 ft, 0 ft). Due to the impress of the constructivists' paradigm, this ordered triple is also a point. The term relation graph is usually referring to a set populated with these latter abstract points and this is all right; set theory does have some uses. (The constructivists' fault was just to take it too far.) It is perhaps a slight drawback from plaintiff's memorandum RZC3 (see the reference here) that geometric lines and the corresponding relation graphs are joined as one in the definition of a linear relation. The following distillation from the memorandum has then been corrected accordingly, not without awondering how much reflex constructivism remains in the head of one who is trying to be quit of it. If the relation graphs are pictured just so, there will indeed be lines that are graphs; but they need not be pictured; and lines can be merely places where relations hold. An Introduction to the Barycenter Including Underlying Definitions from Elementary Calculus With a sufficiently powerful derivative on board, it is possible to explicate the kind of relation that holds definitively on a straight line. It also becomes apparent that angles between lines may be defined from coordinate derivatives, with the aid of a series expansion for the inverse tangent function. Hence indeed one may become interested in the question of a proof for Euclid's fifth axiom. The following proof is confined to a plane and also suffers from the abovementioned lingering graphs affection. It is yet good enough as an indication, especially as a demonstration plane may be translated and rotated into any position within our threedimensional space. Generalising the finding of this proof, it appears that all of Euclid's five axioms are given if we agree to count points as places, in the logical universe, where relations hold, concerning three coordinate variables. A Proof of Euclid's Fifth Axiom using Concepts of Elementary Calculus Note: This is best viewed in a specialised PDF reader, e.g. Foxit Reader. When opening it, agree to the fullscreen view and use your keyboard's Page Up and Page Down keys for navigating. If because of energy science, we cannot dissolve out gravity as a vector field, how do we reconcile the energymomentum tensor? Rather than a riff on spacetime curvature, we may postulate that Einstein has cornered proper time. The first clue that Einstein has cornered proper time is that the null geodesic is defined as the worldline along which precisely zero proper time passes. There is no gravitational cycle in a photon! The second clue is that an abnormality in the motion of a planet's perihelion can be viewed as an expression of a year's relativity. This is so because to know whether a motion is abnormal or not, one needs to have a model of normality on board. It is easy enough to rustle up a coordinate system for space that handily has Euclid's five axioms on board. Therefore one must be careful. What does one mean: spacetime is curved! Maybe there are category mistakes in some of the romanticising and glamourising.
