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In The Non-Euclidean Revolution by Richard Trudeau, he discusses the theorems in Euclidean Geometry. In particular, I was struck by the proof of Theorem 23 (copying an angle). He avoids Euclid's proof because it used Theorem 8/SSS (which Euclid had proved with superposition). What he does instead:

Let $AB$ be the given line, $C$ a point on the line, and $\angle DEF$ the angle. (Hypothesis)

Case 1: If $\angle DEF$ is right,...

Case 2: If $\angle DEF$ is acute,...

Case 3: If $\angle DEF$ is obtuse,...

In each case he gives a construction that copies the angle. Since the three cases exhaust all possibilities then one can copy an angle.

From a construction point-of-view, I feel like this proof is incomplete since it doesn't provide a way to check if an angle is right, acute, or obtuse. I can't think of a proper way to tell the type of an angle that doesn't require observation. What if such a check was impossible? Doesn't this lead to a paradox? The paradox being that a theorem proves you can copy an angle but in reality you can't.

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You are heading towards the hidden depths of constructivist mathematics and a rejection of the axiom of choice and possibly of the use of the law of excluded middle. Take care. –  Henry Dec 11 '12 at 8:23
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up vote 1 down vote accepted

Euclidean geometry as phrased by Euclid does not contain all necessary axioms for its reasoning. There are some intuitively obvious steps (to Euclid), that like you say, was later found to be in need to justification beyond "it is true by observation". This led to various axiomatic formulation of Euclidean geometry in the early 20th century.

For example, in Birkhoff's system, your question would rest upon the postulate of angle measure, which asserts (among other things) that it is possible to compare angles. This would allow in particular comparison of an angle with the right angle to determine whether it is right, acute, or obtuse.

On the other hand, in Hilbert's system we can define acuteness using the notion of "betweenness", a primitive tertiary relation on points. But of course you need to verify that in the construction given in your book, the dependence on acute/obtuse/right can be phrased in a way that is compatible with the proposed definition of acuteness.


In any case, however, this would not lead to a paradox.

It either indicates that the theorem and/or its proof is false, in which case you do not in fact have a theorem, or (if you accept that the theorem is true) that the abstract framework of Euclidean geometry (as given by the axioms) is insufficient to describe "reality" (however you choose to interpret that word).

Or, in other words, this would only lead to a paradox if you add to the axioms of geometry the axiom that "this geometry models reality".

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