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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.2011-10-18
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    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.2011-10-18
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    These are good question. Unfortunately they may not have really good answers. To a first approximation, an answer in each case could be, "because these abstractions turn out to be more useful than others we could have made". But that just pushes the question back a level: what is it about these specific abstractions that makes them particularly useful? A grokkable answer to _that_ requires that one learns both about the applications of the standard constructions, _and_ about the alternatives that have been proposed for special situations.2011-10-18
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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.2011-10-18
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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.2011-10-18
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    @Henning Though that's disappointing, thanks for the encouragement!2011-10-18
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    @Olivier That's just why I'm concerned about the justification of the Euclidean norm by appealing to the fact that circles defined by $x^2 + y^2 = r^2$ have rotational symmetry, since dot products seem to be the motivation for the Euclidean norm in the first place.2011-10-18
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    @André Isn't Euclid's notion of area quantitative, given that the squares are somehow added? I thought any rigorous notion of area would require loads of definitions that would seem far more difficult to make non-arbitrary than definitions of length.2011-10-18
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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.2011-10-18

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