Does Euclidean geometry implicitly assume an observer?

Question

Let's consider a point particle at rest in space relative to an observer. Without a coordinate system, the observer can still determine that the particle occupies a specific point in space. However, for an observer who is moving relative to the original observer, the point particle appears to occupy multiple points in space over time. This raises the question of whether it makes sense to talk about points in space independently of an observer's reference frame. Does this mean that Euclidean geometry, which relies on points and distances between them, requires an implicit observer?

In other words, the idea of a point in space is relative to an observer's reference frame. Without a reference frame, it is impossible to talk about the location of a point in space. Am i correct in assume tha this implies that Euclidean geometry is not an absolute, objective truth about the nature of space, but rather a tool for describing the relationships between points as they appear to an observer?

Thanks in advance

Answer 1

Euclidean geometry does not turn on an observer, because it does not deal with point particles. Only with the mathematical abstraction of point.

Because it is an abstraction, it can not actually be observed visually; only representations of it could be. Therefore there can not be two observers.

Answer 2

Yes, a mathematical object such as a Euclidean space is an abstraction that can be useful to deal with some physical situations, but does not have any special claim to "truth" as far as the world goes.

A mathematical structure such as a Euclidean space does not have an explicit notion of an "observer". Implicitly, all of the space is accessible simultaneously (there is no notion of time either).

Answer 3

Your question should answer itself if you follow your own train of thought properly. You have introduced the idea of two observers and talked yourself into believing that their contrasting perspectives cause a degree of confusion. So why doesn't that tell you to leave observers out of it altogether?

Answer 4

It does not, and I don't think you can make that claim. Euclidean geometry is described in a constructed, imaginary universe. The explicit postulates of that universe do not mention observers. The best you could do is argue that the universe cannot exist without someone to observe it, but that is a controversial claim.

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Euclidean geometry is so named because it was described by Euclid of Ancient Greece, about 2-3 centuries before Christ. Euclid took the simple, and self-evident ideas of points, lines and distances, and "naively" followed these to their various logical conclusions. The famous book detailing these conclusions is called the Elements (he must surely have written many things, but they haven't all survived to our day).

Euclid was not writing about point particles, reference frames or coordinate systems. To my knowledge, the Elements does not delve into questions about whether an object exists if there is no one to observe it. Euclid was not writing about physical objects in the real world, but about the idea of a point or a line and what that idea must imply. It is not hard to imagine that a point exists even in absence of an observer, and therefore this premise is part of the basis of Euclid's geometry. His conception of distance is invariant to observation.

Am i correct in assume tha this implies that Euclidean geometry is not an absolute, objective truth about the nature of space

Euclidean geometry is essentially a bunch of "if these assumptions are true, then the following things must be true". Euclid spelled out these assumptions, but some are controversial even in terms of what they actually mean, if anything. Beyond that, nobody knows if these assumptions are, in fact, true for the space we inhabit, or indeed any space at all, so it would be quite irrational to claim that Euclid's conclusion are "absolute, objective truth".

At best you can talk about whether it's true that the assumptions would imply the conclusions. Most people agree that they do. But whether the assumptions are true is another matter.

In our physical universe, it seems that some of Euclid's assumptions are probably not true, and they are not even true in everyday life. However, in some very limited contexts, such as drawings on small pieces of flat paper or constructing wooden furniture, they do not seem to be violated.

Note also that the word "geometry" literally means the "measurement of the ground", in the sense of things like square and triangle plots of land. Most of us humans currently live on Earth, which is round, so Euclid's ideas are decidedly incorrect when applied to, well, the measurement of the ground. Of course the word geometry now refers to measuring other kinds of objects and spaces as well.