By Sebastien Gandon
Sebastien Gandon goals at reversing the viewpoint. His target is to provide an image of Russell's logicism according to a close studying of the advancements facing complex arithmetic - specifically projective geometry and the speculation of volume. This ebook isn't just the 1st learn ever made from the 'later' parts of PoM and PM, it additionally exhibits how this shift of standpoint compels us to alter our view of the logicist application taken as a whole.
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Extra info for Russell's Unknown Logicism: A Study in the History and Philosophy of Mathematics
Nevertheless, some properties remain the same. In particular, incidence relations between the elements of a figure are usually (see below) not affected: lines are transformed into lines, pencils are transformed into pencils. Let’s call ‘projective’ the properties that are invariant by homography, and ‘metrical’ those that are not. This distinction led Poncelet to differentiate between two kinds of theorem. 2 ABCD and A¢B¢C¢D¢ are in perspective from centre O Pythagoras’s theorem is clearly a metrical theorem, because the property of being a right-angled triangle is not projective.
Of course, he recognized that they were points on the line that did not belong to the net. But he seemed to have believed that the density of the Möbius net was enough to guarantee that, as soon as the images of all the points of the net were defined, the images of the extra points would be fixed. In a modern (anachronistic) language, Von Staudt confused two distinct ordinal properties: density and density in a set. 26 For deriving the fundamental theorem, this is the latter ordinal property which is needed.
35 It would seem at first that there is an insuperable intuitive and conceptual gap between ordinal and incidence notions. The way the points are located on a line seems indeed to have nothing to do with the way points and lines intersect on a plane. Pieri’s beautiful result proves that this is not the case – it shows that, for ordering the points on a line, one is only required to set appropriate constraints on the way lines intersect in the plane. This idea is breathtaking: what, indeed, seems to be more ultimate than order?