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Is Euclidean geometry a mathematical theory, or is it a theory of empirical science?

If taking it to be a mathematical theory would it be due to having alternative geometries? If so, is it in some way related to underdetermination? Alternatively, if we consider it to be an empirical theory would it mean that we assume something about the space?

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    [Could have asked the same question about any of the non-Euclidean geometries. Used Euclidean geometry as an example]. – L.M. Student Nov 28 '15 at 3:41
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    Geometry is about entities that do not, and can not, empirically exist. Points have no dimensions, straight lines have only one, planes have only two. Such things cannot have a material existence, they are mental abstractions; they may provide a useful model of how reality "works", bu they cannot express the truth of the real world. – Luís Henrique Sep 18 '16 at 15:18
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It is both, or rather it has aspects that are mathematical, and aspects that are empirical (the same is true of Newtonian mechanics or special relativity, but more obviously so). If we take Euclidean geometry as the science of space, as Greeks understood it, it is an empirical theory that may or may not match observations, and is subject to revision based on them. On the other hand, if we take it as the collection of theorems derivable from Hilbert's axioms (which complete those of Euclid) then it is a purely mathematical theory. This is related to existence of alternative geometries only in the sense that we can not conclude a priori that empirical geometry is Euclidean, because alternatives do exist.

Underdetermination enters in that observations allow us to test empirical geometry only to some precision for example, so small curvature may not be detectable. More subtly, as Poincare pointed out observations only depend on the pair geometry + physics, not each one separately. So we could keep Euclidean geometry of space by altering the physical part of the theory, and still get a theory equivalent to general relativity, but with rather unattractive physics. As a simpler example, Einstein's special relativity and Lorentz's theory of ether are empirically equivalent, but one is set in Minkowski 4D spacetime, whereas the other in the Newtonian 3D space times 1D time, but with physics that compensates for the absence of ether wind with specialized dynamical effects, like the length contraction and time dilation.

See Is Logic Empirical? for a related discussion.

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    Would the hypothetical equivalent theory of general relativity (arriving at by adopting Euclidean geometry of space and altering the physical part of the theory accordingly) be unattractive physics due to it not being mathematically aesthetic (symmetry, simplicity, etc.)? If so - is it a matter of convention to choose to give up common sense (of space which is captured in Euclidean geometry)? – L.M. Student Nov 30 '15 at 7:53
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    @student The conventionalist view was advocated by Poincare, Einstein criticized it on physical grounds books.google.com/… (p.405). I am not aware of anyone actually trying to formulate a Euclidean equivalent of GR, but something like that on flat Lorentzian background was done by Logunov, and is also unpopular with physicists. en.wikipedia.org/wiki/… – Conifold Nov 30 '15 at 20:25
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    @student To answer your question more directly let's take special relativity. There is a Euclidean version of it, Lorentz's theory of ether. Discarding ether does "simplify" SR and makes it symmetries more explicit, but those weren't Einstein's reasons for discarding it. It was that it is "unphysical", no conceivable experiment could detect its presence. – Conifold Dec 2 '15 at 20:24
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    One could say externally that physicists look for conceptual simplicity and unity, but they would resist leaving it at that. Physicists develop ideas about reality, "physical intuition", a relative of common sense but informed by advanced physical theories and experiments. Physical intuition certainly often favors conceptual simplicity and unity, but it may come to clash with common common sense independently of that. – Conifold Dec 2 '15 at 20:26
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    @student Realism hopes that theories are co-determined by context and interpretation, both conscious (conventions, norms) and unconscious (intuition, tradition). Then in the totality of context the ether became untenable. An alternative view is that there is no unique determinate out there for theory to match, because reality is not a theory of itself. A theory then is not a picture of reality, but a template for observer's interaction with reality. As such it is not meant to be objective, it is meant to be effective. And the ether ceased to be an effective physical heuristic. – Conifold Dec 5 '15 at 0:23
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Euclidean geometry is a mathematical theory, not a scientific theory.

It starts from definitions and axioms and makes deductions from the axioms. It is not necessary to go out, to draw lines and angles and to measure real geometric figures.

On the other hand, Euclid like everybody until the 19th century thought that Euclidean geometry also describes our world.

Prompted by the question whether the parallel postulate can be deduced from the other axioms, mathematicians in the 19th century (Gauss, Bolyai, Lobachevski) detected non-Euclidean geometries. This shows that the parallel postulate is independent from the other axioms.

The proof goes by showing that models exists which satisfy all axioms but not the parallel postulate. Hence the question came up whether Euclidean geometry is the only geometry to be used in natural science. As we know today thanks to the General Theory of Relativity, near large mass distribution spacetime is curved and not Euclidean. Here Euclidean geometry does not apply. A nice example is gravitation lensing, see

https://en.wikipedia.org/wiki/Gravitational_lens#/media/File:A_Horseshoe_Einstein_Ring_from_Hubble.JPG

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    @Penguin All three, formalism, intuitionism, and logicism do not consider mathematics part of natural science. – Jo Wehler Nov 28 '15 at 10:55
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    Thanx. Thought you assumed ahead that geometry is syntactic (not semantic) and therefore purely mathematical. – user18096 Nov 28 '15 at 10:59
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Geometry is a part of mathematics. It starts with axioms, draws conclusions from them, and so on. "Alternative geometries" has nothing much to do with it being part of mathematics; people had used a set of axioms that they believed to be sufficient and it turned out not to be, and it turned out that adding any one of three possible additional axioms would produce one distinct mathematical geometry out of three possible ones.

On the other hand, practical geometry can be seen as either engineering or as science. There are many aspects that have nothing to do with mathematical geometry. For example, building precise instruments, how these instruments are affected by temperature, atmosphere pressure and humidity along a line that is measured, effects of the earth's curvature and of relativity.

For example, when you determine the location of points precisely using measurements in geometry, it all starts working fine just in agreement with mathematical geometry, but then you measure more precisely and the results disagree because (1) the earth is not flat but a ball, (2) because it is not a ball but a rotational ellipsoid, (3) because it is not a rotational ellipsoid but a flattened rotational ellipsoid, (4) because it is not a flattened rotational ellipsoid but a deformed flattened rotational ellipsoid. And then you take into account relativity...

So practical geometry is very much a science.

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We now regard geometry as a mathematical theory: one makes assumptions and deduces, strictly logicall, the consequences of those assumptions, with only passing interest in the accuracy of those assumptions.

But when Hilbert was growing up, mathematicians and scientists almost all took the opposite point of view. As late as 1880, nearly everybody considered Geometry as an empirical science, a branch of Physics treated with mathematical tools. (They also thought of Probability in the same way: a branch of Physics.) Some Italians around Peano, I forget who, and Pasch, were pioneers of the new approach, which is now universal. Hilbert learned from them and in the 1880's recast their work, which for the first time produced completely rigorous axioms even for Euclidean geometry. He was also partly inspired by work on non-Euclidean geometry, which at the the time was still completely controversial since it didn't seem physical. Now we know that it is physical, but people like Klein and Hilbert championed the view that Geometry was a purely abstract, deductive, logical part of Mathematics, which is now the dominant view among mathematicians.

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