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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:

  1. 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.

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

  3. 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.

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I don't understand the last paragraph, but "rotations" depend on the underlying euclidean structure i.e. the dot product. –  Olivier Bégassat Oct 18 '11 at 2:28
    
The beginnings of an answer would take many pages. The Pythagorean Theorem, in Elements, does not involve a quantitative notion of distance at all. Indeed, it is not connected with distance, but with a non-quantitative notion of area. –  André Nicolas Oct 18 '11 at 2:31
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(cont'd.) But by all means keep these questions in mind as you learn more. The ability to question assumptions is a valuable skill and a great producer of new ideas. Just as long as you don't let doubt cripple everything you do. –  Henning Makholm Oct 18 '11 at 2:41
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To modify a line from Numerical Recipes: the reals are important because they show up in the problems mathematicians (and others) like to solve. –  Ross Millikan Oct 18 '11 at 2:58
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@JohnJamesSmith: No explicit numbers are ever mentioned. No formal definition of equality is made, but the underlying intuition is equidecomposability. Area is simpler than length. As a partial illustration, if you have taken calculus, the area below $y=f(x)$, above the $x$-axis, between $x=a$ and $x=b$, exists for any continuous $f$, while arclength can be a nightmare. –  André Nicolas Oct 18 '11 at 3:47
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3 Answers

"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.

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If you care about the idea of a set of a single object, you care about the reals (the reals can be constructed just from sets of sets of empty sets.) –  JeremyKun Oct 18 '11 at 16:13
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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.

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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 plain Euclidean geometry.

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