Is Distance an Abstract Concept? A Debate on the Definition of Length

[Werg22]

Me and my teacher have been debating about the concept of distance. While he says that the length of a curve is something that is something that is not abstract, I say that this is false. Distance only applies to lines, the definition of a length of a curve is mearly the limit of the sums of several lines as the we choose the interval to be smaller and smaller. To me, the concept doesn't mean anything unless we give it a purely abstract meaning. Who has the right view?

 

[CRGreathouse]

I think that the length of a curve is no less real than the length of a line. Indeed, a line is just one sort of curve. I suppose, then, that I side with your teacher.

 

[Werg22]

But length is definied with a line. Sure, a line is a curve if you wish to see it as so, but it must be seen as the "fundamental" curve. The length of a curve is defined after that we have defined the length of a line.

 

[Rach3]

Distance on curves is well-defined.
http://mathworld.wolfram.com/ArcLength.html

I'm not sure what your argument is for wanting "not abstract" definitions, seems frivolous.

 

[Rach3]

The length of a curve is defined after that we have defined the length of a line.

What's the problem then?

 

[Werg22]

Distance on curves is well-defined.
http://mathworld.wolfram.com/ArcLength.html

I'm not sure what your argument is for wanting "not abstract" definitions, seems frivolous.

Actually my teacher wants it to be non-abstract, I perfectly agree with the link you gave me.

 

[0rthodontist]

The length of a line segment is equally abstract.

 

[Hurkyl]

Abstract is in the eye of the beholder.

 

[Robokapp]

I'd define distance as anything representing the traveled trajectory of a moving ... something. I mean...there's something called "Angular distance" and it obviously has little to do with a straight line...because it's a circular motion.

 

[MeJennifer]

Distance only applies to lines, the definition of a length of a curve is mearly the limit of the sums of several lines as the we choose the interval to be smaller and smaller. To me, the concept doesn't mean anything unless we give it a purely abstract meaning. Who has the right view?

I do not see how you could argue that distance could only apply to lines.

Assuming a linear coordinate system, one of the differences one could highlight is that the length of a straight line is never a transcendental number while the length of a curve might be.

 

[Data]

well I don't know what a "linear coordinate system" is, but if you mean something like standard cartesian coordinates then there are obviously line segments with transcendental length (take any transcendental number s, then the line segment from 0 to s has length s, obviously).

 

[MeJennifer]

well I don't know what a "linear coordinate system" is, but if you mean something like standard cartesian coordinates then there are trivially line segments with transcendental length (take any transcendental number s, then the line segment from 0 to s has length s, obviously).

Obviously. But a straight line segment with only rational numbers will not become transcendental.

 

[Data]

I don't know what you mean by "a straight line segment with only rational numbers will not become transcendental." It is true that every line segment where, in the cartesian (x_1, ... , x_n) representation, each x_i is rational at the endpoints (or in fact algebraic) will have algebraic length. But that's something like saying "a straight line that has algebraic length has algebraic length." In a well-defined sense, "almost all" line segments have transcendental length (just like "almost all" reals are transcendental).

Edit: Now that I think about this, I am not sure if it is true or not. Take a single point and a vector. Then there are certainly only countably many line segments of algebraic length through that point in the direction of that vector, and uncountably many with transcendental length. But if you instead consider all possible line segments out of that point, the set of all of them with algebraic length will be an uncountable union of disjoint countably infinite sets, which is uncountable. I would not be surprised if the set of all line segments through the point (an uncountable union of disjoint uncountable sets) still had larger cardinality, but I'm not ready to say so positively!

 

[Hurkyl]

I do not see how you could argue that distance could only apply to lines.

Assuming a linear coordinate system, one of the differences one could highlight is that the length of a straight line is never a transcendental number while the length of a curve might be.

Obviously. But a straight line segment with only rational numbers will not become transcendental.

This is actually an interesting point, because it amounts to a proof that the length of a curve is not something you can define in "pure" geometry.

There's a really nice theorem by Tarski that you only need algebraic numbers to do elementary Euclidean geometry. (And similarly, it's all you need for elementary algebra)

So, the fact that a circle defined algebraically (e.g. x² + y² = 1) has a transcendental length is proof that the notion of the "length of a curve" can only be defined through analytical concepts.

I would not be surprised if the set of all line segments through the point (an uncountable union of disjoint uncountable sets) still had larger cardinality, but I'm not ready to say so positively!

Any line segment is completely characterized by its endpoints. So, the set of all line segments is clearly no bigger than the set of all ordered pairs of points. But w² = w for all infinite cardinals, so there are no more line segments than there are points.

 

[matt grime]

Obviously. But a straight line segment with only rational numbers will not become transcendental.

But you did not say anything about restricting to such cases. You said all lengths are algebraic. I think you may have a different definition of a linear coordinate system to what is normally inferred from that phrase.