By Eisenhart L. P.
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Additional resources for Affine Geometries of Paths Possessing an Invariant Integral
W e shall see that the data of mathematics are not without their premises; they are not, as the Germans say, voraussetzungslos; and though math ematics is built up from nothing, the mathematician does not start w i t h nothing. H e uses mental i m plements, and i t is they that give character to his science. Obviously the theorem of parallel lines is one instance only of a difficulty that betrays itself every where i n various forms; i t is not the disease of geometry, but a symptom of the disease.
53 kind of argument, and its statements are practically tautologies. The case is different w i t h causation. The class of abstract notions of which causation is an instance is much more complicated. keenest thinkers was in deep earnest when he doubted the possibility of p r o v i n g this obvious state ment. A n d K a n t , seeing its kinship w i t h geometry and algebra, accepted i t as a priori and treated i t as being on equal terms w i t h mathematical axioms. Yet there is an additional element i n the formula of causation which somehow disguises its a priori origin, and the reason is that it is not as r i g i d l y a priori as are the norms of pure logic.
M a x i m u m intensity. Each of these rays, thus ide ally constructed, is a representation of the straight line which being the shortest path between the start ing-point A and any other point, is the climax of directness: i t is the upper limit of effectiveness and its final boundary, a 11011 plus ultra. I t is a m a x i m u m because there is no loss of efficacy. , the greatest intensity on the shortest path that is reached among infinite possibilities of progression by u n i f o r m l y following up all.