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I am aware of the standard method of summoning the real numbers into existence -- by considering limits of convergent sequences of quotients.

But I never actually think of real numbers in this way. I think of a real number as a vector in 1D Euclidean space, the good old number line. A signed distance. My intuition is that $\mathbb R$ is something at least as fundamental as $\mathbb Q$: It is the fabric upon which $\mathbb Q$ is drawn. That $\mathbb Q$ is embedded in $\mathbb R$, rather than extending to it.

Or maybe side-by-side we could have geometry (defining $\mathbb R$), and $\mathbb Q$ (constructed by algebra). And we could demonstrate an isomorphism between our $\mathbb Q$ and a matching subset of our $\mathbb R$.

The classical extension of $\mathbb Q$ to $\mathbb R$ feels awkward and perverse to me. It doesn't seem right that $\mathbb Q$ should have some prior (or more fundamental) existence.

In the same way that any computational system can be simulated on a Conway Game of Life with appropriate starting grid, we could postulate this as an axiomatic setup. But it looks contrived, some kind of entertaining mental contortion. Possibly interesting, but certainly not fundamental to understanding of the science.

I like to build my mathematical universe from common sense constructs. A piece of paper (Euclidean plane) equipped with a straightedge and compass. And a counting system.

I'm currently looking at Geometric Algebra, which is magnificent! But it requires a scalar field $\mathbb R$. Which means underneath this simplicity is the intricacy of the classical construction of $\mathbb R$ from $\mathbb Q$. I wonder if this is avoidable...

It isn't clear to me that one cannot use geometry to define $\mathbb R$. Why should a Euclidean plane be a less valid foundation than $\mathbb Q$?

Maybe signed area could be used for multiplication. So if I have a vector -3, I need a perpendicular vector -$1 \over 3$ to ensure the signed area is 1. etc.

Is there some effort towards a geometric definition for $\mathbb R$, such that Algebra and Geometry may exist as two wings of the same bird?

P i
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    recommend reading Hartshorne Geometry: Euclid and Beyond – Will Jagy Jul 07 '16 at 22:04
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    Conway's surreals build $\Bbb{R}$ with no particular stop-off at $\Bbb{Q}$ (see On Numbers And Games, first half). – Mark Fischler Jul 07 '16 at 23:15
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    My point is only that you don't have to oversell the complications caused by the construction — tons of people "do geometry" every day and don't think about the construction at all. The axioms are natural enough and easy to use; one barely even thinks about them. Thumbing through Artin's "Geometric Algebra" I don't see any point where he really uses the construction apart from an aside proving that all archimedean ordered fields embed in $\mathbb R$, and I suspect it may not even be needed there. Maybe someone will set me straight. – Hoot Jul 08 '16 at 00:09
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    Also, it seems that knowledge of $\mathbb R$ includes knowledge of $\mathbb Q$. It you believe that one should start from a minimal set of objects and assumptions then it seems hard to call $\mathbb R$ the more fundamental of the two. – Hoot Jul 08 '16 at 00:11
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    Is there something wrong with the synthetic construction of the reals by defining it to be an ordered field with the lub property. This doesnt really make reference to the rationals. I guess its not an explicit construction... – Andres Mejia Jul 08 '16 at 00:19
  • @Hoot, but minimal in what sense? I believe classical thinking prefers not to mix axioms from separate domains (algebra, geometry). But what if we wish to minimise complexity? Geometric Algebra mixes domains for it's axiom-base (I think?). So why not go to geometry for a definition for $\mathbb R$? – P i Jul 08 '16 at 00:20
  • @AndresMejia, well, yes. Namely that since its not a construction, it doesn't prove the existence of $\mathbb{R}$. – goblin GONE Jul 08 '16 at 01:04
  • @goblin i'm not sure what you mean. One can " construct" the real numbers from the rationals, which are constructed from the integers, which are given by the natural numbers, which are also given axiomatically. Why does the construction "prove the existence" of the reals? – Andres Mejia Jul 08 '16 at 01:25
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    You could probably reword Tarski's axioms in the language of geometry, if you were so inclined. I don't know that I see any benefit in essentially just hiding the algebra, though... –  Jul 08 '16 at 02:10
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    (+1) for a good question. Please notice my followup at : Non-Euclidean Geometrical Algebra for Real times Real? . – Han de Bruijn Oct 03 '16 at 12:35
  • I've been working on a theory of magnitudes - it is the type of primitive geometric object I think you are interested in. See

    https://math.stackexchange.com/questions/2963140/translating-tarskis-axiomatization-logic-of-mathbb-r-to-the-theory-of-magnit

     – CopyPasteIt Nov 27 '18 at 03:57

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