Question on Einstein's opening passage of relativity

[s__1000]

Homework Statement

Einstein says ' we cannot ask whether it is true that only one line goes through two points' - What does he mean? and secondly, can this be proved?

Relevant Equations

This is from 'Physical Meaning of Geometrical Propositions'

Einstein says ' we cannot whether it is true that only one line goes through two points' ' We can only say Euclidean Geometry deals with things called straight lines to each of which is ascribed the property of being uniquely determined by two points situated on it, - What does he mean? and secondly, can this be proved?

 

[valenumr]

It is a comment on an axiom of euclidean geometry.

 

[s__1000]

It is a comment on an axiom of euclidean geometry.

so it cannot be proved - this axiom of Euclidean geometry

 

[valenumr]

so it cannot be proved - this axiom of Euclidean geometry

I think that's what you are supposed to think about, at least int the context of Einstein's text: https://mathcs.clarku.edu/~djoyce/elements/bookI/post1.html

 

[PeroK]

so it cannot be proved - this axiom of Euclidean geometry

With mathematics you have to start with some axioms. Euclid was probably the first to understand this and developed geometry as an axiomatic system. The axioms are here:

https://en.wikipedia.org/wiki/Euclidean_geometry#Axioms

So, it's not a question of proving it. It's an axiom on which you build your mathematical system.

 

[s__1000]

With mathematics you have to start with some axioms. Euclid was probably the first to understand this and developed geometry as an axiomatic system. The axioms are here:

https://en.wikipedia.org/wiki/Euclidean_geometry#Axioms

So, it's not a question of proving it. It's an axiom on which you build your mathematical system.

yes I understand but I am asking generally can it be proved -

 

[Orodruin]

yes I understand but I am asking generally can it be proved -

Axioms, by definition, cannot be proved. That is why they are axioms.

 

[valenumr]

yes I understand but I am asking generally can it be proved -

This is what you must think about to answer the homework problem.

 

[s__1000]

Ah but I don't understand his latter paragraph: where I think he asks the truth of the geometrical proposition that two points on a practically rigid body always correspond to the same distance independent of any changes in the body's position. Here we can ask the truth of a geometrical proposition - it's correspondence to nature.' And then can't we say this is not correct because of the length contraction of a body due to relativistic speeds." If, in pursuance of our habit of thought, we now supplement the propositions of Euclidean geometry by the single proposition that two points on a practically rigid body always correspond to the same distance (line-interval), independently of any changes in position to which we may subject the body, the propositions of Euclidean geometry then resolve themselves into propositions on the possible relative position of practically rigid bodies.1) Geometry which has been supplemented in this way is then to be treated as a branch of physics. We can now legitimately ask as to the "truth" of geometrical propositions interpreted in this way, since we are justified in asking whether these propositions are satisfied for those real things we have associated with the geometrical ideas. In less exact terms we can express this by saying that by the "truth" of a geometrical proposition in this sense we understand its validity for a construction with rule and compasses."

 

[valenumr]

Perhaps the key takeaway is that "truth" is only relative to the framework (i.e. euclidean geometry), but not universal.

 

[PeroK]

yes I understand but I am asking generally can it be proved -

If you ask whether an axiom can be proved, then you do not understand!

 

[PeroK]

" If, in pursuance of our habit of thought, we now supplement the propositions of Euclidean geometry by the single proposition that two points on a practically rigid body always correspond to the same distance (line-interval), independently of any changes in position to which we may subject the body, the propositions of Euclidean geometry then resolve themselves into propositions on the possible relative position of practically rigid bodies.1) Geometry which has been supplemented in this way is then to be treated as a branch of physics. We can now legitimately ask as to the "truth" of geometrical propositions interpreted in this way, since we are justified in asking whether these propositions are satisfied for those real things we have associated with the geometrical ideas. In less exact terms we can express this by saying that by the "truth" of a geometrical proposition in this sense we understand its validity for a construction with rule and compasses."

In more modern language this means.

Euclidean geometry is pure mathematics and need not represent the geometry of physical space. Whether the universe does have a Euclidean geometry or not can be tested.

 

[Orodruin]

In more modern language this means.

I think this is a key point. While reading Einstein’s writing may be of historical interest, it is certainly not the best way to learn relativity (or geometry). A modern textbook would serve this purpose much better.

 

[italicus]

Going back to the Einstein sentence. Consider the Earth, and its poles. It is possible to connect them by means of infinite meridians , not only one. Meridians are “ Straight lines “ for a bidimensional being living on the surface.
But consider also two any points P and Q on the sphere. A maximum circle passes through both. There are two arcs of geodesic from P to Q, one left side of P , the other right side.
Conclusion: when the space is curved, there isn’t one “straight line “ only, connecting P and Q. Here geometry isn’t Euclidean in the large.

 

[Steve4Physics]

so it cannot be proved - this axiom of Euclidean geometry

Some thoughts...

Any proposition, P (e.g. that the shortest distance between two points is along a unique straight line), can be assumed, without proof, to always be true. If P is then used in some thought-system, P is an axiom of the system.

Of course, if P is incorrect, results derived using it may be incorrect.

And there is nothing to stop someone from trying to prove P in terms of some more basic/primitive set of axioms. For example you might, starting with the axioms of differential geometry, prove P is true only in specific circumstances.

(But beware of recursively worrying about more and more basic/primitive sets of axioms, which will give you a headache!)

 

[PeroK]

Some thoughts...

Any proposition, P (e.g. that the shortest distance between two points is along a unique straight line), can be assumed, without proof, to always be true. If P is then used in some thought-system, P is an axiom of the system.

Of course, if P is incorrect, results derived using it may be incorrect.

And there is nothing to stop someone from trying to prove P in terms of some more basic/primitive set of axioms. For example you might, starting with the axioms of differential geometry, prove P is true only in specific circumstances.

(But beware of recursively worrying about more and more basic/primitive sets of axioms, which will give you a headache!)

This seems muddled to me. It's not clear mathematical thinking in any case. There's a difference between a proposition and an axiom; and, there's a difference between a proposition and a qualifier. For example:

A group axiom is that all elements have a (multiplicative) inverse.

Not all matrices are invertible, hence the set of all matrices is not a group.

"All matrices are invertible" is a proposition that can be disproved. But, it cannot be a valid axiom.

What you can do is consider the set of all invertible matrices - which is a multiplicative group.

 

[PeroK]

PS the problem with the axioms of Euclidean geometry is that people have a pre-conceived notion of what a "straight line" is. And, they confuse that pre-conceived notion with the abstract concept. This is a modern idea that Euclid himself, although he invented axiomatic mathematics, didn't realize.

In fact, it took until the 19th Century for mathematicians to realize this distinction.

 

[jbriggs444]

Going back to the Einstein sentence. Consider the Earth, and its poles. It is possible to connect them by means of infinite meridians , not only one. Meridians are “ Straight lines “ for a bidimensional being living on the surface.

If you are constructing a non-Euclidean geometry on a spherical surface, you will likely want to adjust the notion of "point" so that it is modeled as a pair of antipodal locations on the globe.

Now if you try to select as two "points", the North pole and the South pole and note that they do not uniquely define a line, you have not found a violation of the axiom because you have not selected two distinct points. Instead, you have selected the north/south pole pair twice.

Under this interpretation, the truth of the axiom that for any two distinct "points" there is a unique "line" containing those "points" is upheld.

[Note that having an interpretation of the mathematical theory is important. One has to be able to map the mathematical entities within the theory to measurable physical entities before experiment can be relevant to the "truth" of any mathematical proposition.]

One can proceed to see that this geometry works just fine. However it will violate the parallel postulate ("given a line L and a point P not on that line, there is exactly one line parallel to L and containing P"). In this geometry there is no such thing as a pair of parallel lines. Instead, one could assert that "given a line L and a point P not on that line, every line through P will intersect with L".

 

[Steve4Physics]

This seems muddled to me. It's not clear mathematical thinking in any case. There's a difference between a proposition and an axiom; and, there's a difference between a proposition and a qualifier. For example:

A group axiom is that all elements have a (multiplicative) inverse.

Not all matrices are invertible, hence the set of all matrices is not a group.

"All matrices are invertible" is a proposition that can be disproved. But, it cannot be a valid axiom.

What you can do is consider the set of all invertible matrices - which is a multiplicative group.

I think an axiom (not only in maths) is simply a statement which is declared to be true (and in the mind of the declarer, probably seems self-evident, a decent guess or just interesting to pursue). The axiom can then be used with other axioms as a basis for developing further ideas.

At least informally, I think that ‘postulate' and ‘axiom’ are often used interchangeably, though preference for one over the other probably depends on the subject/context.