Preaffine Spaces with Visible Points

From old times one presumes that in geometry points have no extent of any sort or kind. And nowadays one is told that the elements of a certain set designated as a figure are the points. On the other hand the length, width, area, or volume of a figure is often thought to be yielded by some extent of points lying closely in a row or packed to the full. Especially concerning what is called continua these things are incomprehensible to us. Really each point in school-mathematics is pretty large, not exactly a pin-point, and various figures there will be covered with a finite number of points. And yet, since the noted David HILBERT, none asserts that points have no extent. Of course this has no immediate connection with the fact that the measure of a point should be equal to zero. In any case geometric points are ordinarily given as ”figures” by describing, and they consequently have some fair extent in space. But yet from such ”visible points” an affine geometry is unquestionably constructed and soon after an Euclidean geometry is doubtlessly done. To confirm these circumstances are quite possible this article is composed. We maintain that each point in geometry is not necessarily the least unit, but an object defined as a set of somethings as well, and verify here that the affine geometry in the school-mathematics is never a swindle. Besides,we introduce certain symmetries into a system, or a preaffine space, consisting of the abovementioned somethings.