Soap_Dude wrote: > Hi, > > I've developed great interests in many areas related to pure and > analitical sciences, especially physics, philosophy and math. This > forum is a perfect place for me to learn some new materials. Since this > is my first post in the forum, I'd like to introduce myself. I'm 17 > year old studen from Canada. > > So to the topic of set theory. I didn't get a chance ot read the whole > of the arguments, but here is a fundamental problema friend of mine > (also in high school) has raised. > When numbers were invented by our ancestors, whether it is based on > digits or otherwise, they were intuitively meant to be discrete. A man > only had an integer amount of fingers. Well, that's a given. But when > the Greeks finallized the concept of geometry which deals exclusively > with space (which is assumed, axiomatically or not, to be continuous). > In orde to give quantative measures of space, they employed the > abstract concept of numbers. The mistake that the Greeks made was, in > my opinion, that they employed a throughly discrete system of > measurement to describe something that is by nature continuous. To the > humble limits of my knowledge, no one bothered ask whether discrete > systems are fit to describe continuity. A point has no dimension and > thus, no length. A ray, however, has a one sided limit, but under our > current mathematical systems there is no way to locate it if it is > located at some real location. Hence, the concept of limits developed. > > I hightly believe in the computational view of mathematics. It is not > that continuity does not exist, but our number system, which is based > on the repeation of natural numbers (think PI or), is unable to locate > it precisely. The process of constructing real numbers , or realizing > them, undoutedly arises from manipulating natural numbers. The golden > ratio for example ((1+5^(1/2))/2). The adaptations of the natural > numbers into the real numbers inevitably raises problems about > continuity, and perhaps about infinity as well. > > So in conclusion, there is a possibility that our number system might > not be suitable to describe continuity without raising problems. Thank > you for reading this. I hope I hadn't wasted you time.
Yes, I think you are right. Numbers alone aren't very satisfactory when dealing with the continuum. I believe that both arithmetic and diagrammatic concepts in Euclidean geometry are fundamental. E.g. a line segment is a fundamental geometric entity and is not to be visualized as being a set of points. See Sec. 4 my paper <http://arxiv.org/abs/math.LO/0506475> which defines a limited version of real analysis using this idea.