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First of all, it's clear that any theorem of synthetic geometry can be proven in analytic geometry. But I'm not so sure that the opposite holds. Perhaps if our synthetic geometry had a notion of measure with the entire set of real numbers, we could use that to "mimic" the coordinate system.

However, if it didn't have any sort of measurement (i.e. Hilbert's axiomatization), I'm thinking that maybe you could take a line segment to be the "unit length", and somehow construct the rational lengths and some irrational lengths. The problem is, I don't know whether being able to give coordinate names to points actually gives us any power to carry out analytical proofs. Also, we won't even have the entire real line using this method, since there are numbers that aren't constructible. So essentially what I'm asking is: Is synthetic geometry logically complete? Moreover, is it possible to mimic analytic proofs in coordinates by constructing a coordinate system out of a synthetic geometry without measurement?

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    $\begingroup$ Luke that depends on if you are using what is known as Constructive Geometry. Incidentally while natural and rational numbers can be constructed, not all irrational numbers can be constructed (like $\sqrt [3]{2}$). $\endgroup$ – Sentinel135 Jun 30 '17 at 22:55
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To give a clear answer we need to define what you mean by synthetic geometry and analytic geometry.

I assume that we use the axioms of Hilbert to define synthetic geometry, and that analytic geometry corresponds to the real Cartesian plane $\mathbb{R}²$. If you assume only the first four groups of axioms (I-IV), you don't even have all the ruler and compass constructible points. There are propositions which are provable in $\mathbb{R}²$ but not using axioms I-IV. For example Euclid's proposition I 22 is not provable using only group I-IV. If you add an axiom about intersection of lines and circles then you get all ruler and compass geometry (see Chapter 10 of Franz Rothe's book).

But, if you assume the group V of Hilbert's axioms then the axiom system is categorical, the only model is the real Cartesian plane.

It is possible "to mimic analytic proofs in coordinates by constructing a coordinate system out of a synthetic geometry", it was first done by Descartes and with a more rigorous presentation by Hilbert in Foundations of Geometry (the proofs can be found in Chapter 4.21 of Hartshorne's book: Geometry: Euclid and beyond or Chapter 18 of Franz Rothe's book). The axiom of continuity (Dedekind and Archimedes) are not necessary for the arithmetization of geometry. Without continuity axioms, you obtain a Cartesian plane over a Pythagorean field.

It is not true that Hilbert's axioms does not have any sort of measurement, it is not explicit, but from Archimedes' axiom from Group V, it is possible to build a measurement function (see Chapter 8 of Franz Rothe's book).

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If we are using the constructive numbers. We can prove up to denseness of the constructables(the constructables aren't dense). Other than that you have to use a different form of constructive logic (one that uses more than the straightedge and compass) or another, axiom or set of them, to prove more postulates in analytical geometry. For instance, if we were to use projective geometry, then we should be able to prove just about everything in analytical geometry.

However, the use of coordinates to prove things is not the approach that synthetic geometry uses to prove well anything except by disproving a conjecture. If you want to disprove something, then using a model with a coordinate system works wonders.

I digress, what I am trying to say is no Synthetic Geometry isn't logically incomplete. This is due to it being logically consistent so far. Lastly, a form of measure is required to mimic any analytical proof. Also Synthetic Geometry has axioms that form a measure relating to both area and angles, and sometimes distance too.

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