How can the geometry (and the reals) be motivated from the bottom up? I'm really not sure that I know what I'm talking about, or if I should just go and learn more math before questioning such things, but I'd like to have answers to the following questions that don't depend on intuitive notions of space, and to be reassured that math is floating on as few cognitive biases as possible:


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*Why should we care about real numbers? The simplest definition on Wikipedia still seems to rely on a bunch of seemingly arbitrary things like fields and how you can't divide by zero.

*Is there any sense in which Euclidean geometry is one of the systems that we should care about?

*What is the very minimum of arbitrary decisions and definitions needed to characterize the standard notions of angles, distances, and the Pythagorean theorem? It seems to me to have the Pythagorean theorem you would need at least a quantitative notion of distance, which would just have to be defined from nothing. I've read some stuff about $\sqrt{a^2 + b^2}$ being special because circles that way are more symmetrical, but that seems rather fishy, since how would you rotate circles without angles, and cosines, and the dot product, and it seems like it's just back to the beginning.
Thanks.
 A: "Why should we care about the real numbers?" I'll give a rephrasing of Yuri's answer. 
To begin with, why should we care about anything? One has to start somewhere; if you don't care about anything, I won't be able to convince you to care about the reals. So, I'm going to assume you care about the number 1. I'll also assume you care about addition and subtraction. Well, now you are committed to caring about the integers, positive, negative, and zero. Now I'll assume you care about division. Well, at this point you care about the rationals. Next, I'll assume that you want bounded, increasing sequences to have limits. Is that arbitrary? All I'm saying is that if you have a sequence of numbers like $3,3.1,3.14,3.141,3.1415,3.14159\dots$, that sequence has 3.2 as an upper bound, but it has 3.15 as a better (smaller) upper bound, and 3.142 as an even better upper bound, and it would be nice for there to be a number you could call its best (that is, least) upper bound. Well, if you care about bounded, increasing sequences having least upper bounds, then you care about the reals. 
In summary, if you care about 1, addition, subtraction, division, and least upper bounds, then you care about the reals. 
A: We should care about real numbers because they show up in tons of natural places! For instance, the number $\pi$ as the ratio of the circumference of a circle to its diameter. We want to understand all real numbers because it helps us understand things like $\pi$, and hence improves our understanding of approximations of mathematical things to the real world, so that we can fly planes and ride bikes and go on facebook.
A: There's surprising a lot of things that Euclid left out, mainly continuity and order. Hilbert provided a full set of axioms for Euclidean geometry. See the books by Moise and Hartshorne mentioned in Book recommendation on plane Euclidean geometry.
