What does 'length' mean?

Question

Just came to know about Aristotle's Wheel paradox. I read and watched some explanations. While understanding the explanations, a completely new question is bothering me. It kind of shook my mathematical understanding of certain things.

So here's what confusing me. There are an equal number of points between [0,1] and [0,2] but still respective lengths are different. Same amount of points covering different lengths. So what exactly 'length' is?

I would love some simple intuitive explanation. Or do I have to re-read my measure theory notes :p?

Answer 1

I think you have diagnosed the root of your problem when you wrote, in a comment, "I feel I have never really understood [the real numbers]."

I gather you have looked at formal treatments (e.g., Dedekind cuts), but still don't feel comfortable with $\mathbb{R}$.

A few remarks that may help.

First, the apocryphal quote from the famous mathematician von Neuman: "Young man, in mathematics you don't understand things. You just get used to them."

Next, the "official" answer to your question: the cardinality of a subset of $\mathbb{R}$ and its measure (if it is measurable) are two different things. In particular, we can have $m(A)\neq m(B)$ although $\#A=\#B$. That's just how it is.

Next, your image of $A$ and $B$ as rows of dots, where "counting the dots" should tell you something about the measures (lengths) of $A$ and $B$. Before the modern understanding of the real line was developed, some mathematicians did think in these terms, and were puzzled with paradoxes like the one you mention. (See "the method of indivisibles", aka Cavalieri's principle, mid 1600's; to quote Wikipedia, "Today Cavalieri's principle is seen as an early step towards integral calculus,").

But we could ask, if each dot has size zero, why doesn't a line segment made of dots have length (size) zero? One's image is of a finite number of dots, each with nonzero size. I can't truly visualize a point as having length zero.

In addition, given any two real numbers, there are an infinite number of real numbers between them, so there's no such thing as placing them "next to each other".

In short, we have to let go of this crude picture in order to understand, or at least get used to, the modern understanding of $\mathbb{R}$.

You mention studying measure theory. The standard notion used in modern mathematics is the Lebesgue measure. However, on the real line, Lebesgue measure essentially starts by defining the length of an interval $[a,b]$ to be $b-a$.

In mathematics, visualization makes a good friend but a bad master.

Answer 2

In order to define the length of any (smooth) curve in 3-dimensional (Euclidean) space, we must first define the length of straight line segments. This is accomplished by the Pythagorean theorem (which is more of a definition than a theorem in this context). Namely, given a straight line segment $L$ with endpoints $A=(x_0, y_0, z_0)$ and $B=(x_1, y_1, z_1)$, we define the length of $L$ to be the quantity \begin{align*} l=\sqrt{(x_1-x_0)^2 + (y_1-y_0)^2 + (z_1-z_0)^2}. \end{align*} This quantity is often written as $l=d(A, B)$ for simplicity. In the case $L=[0, 1]$, we have $A=(0, 0, 0)$ and $B=(1, 0, 0)$, so $l=1$. Similarly, in the case $L=[0, 2]$, we get $l=2$. So we see that for straight line segments, length has nothing to do with any members of the set aside from the endpoints. This is just a definition, not a theorem.

Now, say $\Gamma$ is a (smooth) curve in 3-dimensional (Euclidean) space which twists and winds and knots around itself. How might we define the length of $\Gamma$? Say $\Gamma$ has endpoints $A$ and $B$. If you zoom in on any point of $\Gamma$, it looks like a straight line. So what we do is we place a massive number of points on $\Gamma$ in between $A$ and $B$, packed densely together. Call these points \begin{align*} A=x_0, x_1, ..., x_n = B. \end{align*} Since $\Gamma$ looks like a straight line when we zoom in, and these points are placed very close together, we can estimate the length of $\Gamma$ by adding up the lengths of the straight line segments connecting subsequent points: \begin{align*} l\approx d(x_0, x_1)+d(x_1, x_2)+...+d(x_{n-1}, x_n). \end{align*} As we take denser and denser sets of points $x_0, ..., x_n$, this approximation becomes increasingly accurate and converges to a particular value. This value is defined to be the length of $\Gamma$.

Answer 3

Distance is what a ruler measures. It’s how tired you get if you walk the path and how much graphite you use when drawing it.

Distance has a number of nice properties - chaining multiple lines adds the lengths. Shifting it around doesn’t change the length.

In particular, if you just have points on a line, say and interval $[1,3]$, then you just take the difference between the end and the beginning getting $3-1=2$ as the distance.

When you are in normal Euclidean space, you can you use the Pythagorean Theorem to figure out the length of diagonal lines. If you are in weirder metric spaces that are stretched out like taffy, or curved, or just using a different coordinate system, you may need a more complicated formula (called a metric) to determine distance.

When you have curved lines, you can approximate it with a bunch of smaller straight lines. Summing those up gives you the length of the whole thing. You can make this rigorous with limits.

When your line has gaps, then you can just subtract out the gaps. When there’s lots and lots of gaps, then it’s ambiguous what exactly you want and there’s a number of different mathematical approaches some of which are simpler or have nicer properties while others are defined in wider variety of odd cases. The Lebegue measure is a standard choice. Any measure you see is normally expressed in multiple dimensions since an area or volume can be split lots of little rectangles of area or cubes of a volume in exactly the same way.

Answer 4

the usual diagram..................