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The correlation of the facts of sense-experience effected by the alternative account of nature extends beyond the physical properties of motion and the properties of congruence. It gives an account of the meaning of the geometrical entities such as points, straight lines, and volumes, and connects the kindred ideas of extension in time and extension in space.
The permanence is itself meaningless apart from some immediate judgment of self-congruence. Otherwise how is an elastic string differentiated from a rigid measuring rod? Each remains the same self-identical object. Why is one a possible measuring rod and the other not so? The meaning of congruence lies beyond the self-identity of the object.
On the other hand Russell is in an equally strong position when he asserts that, as a fact of observation, we do find it, and what is more agree in finding the same congruence relation.
Finally, it must be remarked that, as Luther puts it, “Faith always goes against appearances.” The religious conception of the world not only never grows directly out of a scientific and general study of things, but it can never be brought into absolute congruence with it.
We are now in possession of a theory of parallels and a theory of perpendiculars and a theory of motion, and from these theories the theory of congruence can be constructed. It will be remembered that a family of parallel levels in any moment is the family of levels in which that moment is intersected by the family of moments of some other time-system.
Russell in effect pointed out that apart from minor inexactitudes a determinate congruence relation is among the factors in nature which our sense-awareness posits for us. Poincaré asks for information as to the factor in nature which might lead any particular congruence relation to play a preeminent rôle among the factors posited in sense-awareness.
Some objects can palpably alter as they move for example, an elastic thread; but a yard measure does not alter if made of the proper material. What is this but a judgment of congruence applied to the train of successive positions of the yard measure? We know that it does not alter because we judge it to be congruent to itself in various positions.
The first of these axioms, forming the fifth axiom of congruence, will be called the axiom of 'kinetic symmetry. It expresses the symmetry of the quantitative relations between two time-systems when the times and lengths in the two systems are measured in congruent units. The axiom can be explained as follows: Let α and β be the names of two time-systems.
Space and time are now interconnected; and this peculiar factor of time which is so immediately distinguished among the deliverances of our sense-awareness, relates itself to one particular congruence relation in space. Congruence is a particular example of the fundamental fact of recognition. In perception we recognise.
So that this Northeasterne currant must at the length abruptly bow towards vs South on the West side of Finmarke and Norway: or else strike downe Southwest aboue Groneland and Iseland, into the Northwest straight we speake of, as of congruence it doeth, if you marke the situation of that Region, and by the report of M. Frobisher experience teacheth vs.
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