What is the difference (or relationship) between geometric length and arithmetic numbers?

Question

In Abbott's Understanding Analysis there was a phrase like, "Ancient Greeks did not understand the difference (or relationship) between geometric length and arithmetic numbers." What is this difference (or relationship)? Could this misunderstanding be the main reason behind the Zeno's paradoxes?

Answer 1

Preliminaries

There have been two simple constructions in geometry known since ancient times. (1) Halving a line segment. (2) Copying a line segment from here to there (onto a straight line from a given point). Based on these two operations one can compare line segments; one can say, for instance that $AB$ is shorter or longer than $CD$. Claiming that two segments are of equal length is a much harder problem!

The first Chapter

Given a segment of "unit length" (this is just a name for now) one can use the two constructions above to "measure" the length of a given segment with arbitrary precision but not necessarily with complete precision. Already the ancient Greeks discovered that the hypotenuse of a right triangle cannot be measured with complete precision in terms of the legs of the same triangle. For them the distribution of the points on a straight line must have seemed to be too sparse for they could not find the exact end points of the hypotenuse of certain triangles.

For the Greeks only the rational numbers were "rational". So, the length of the hypotenuse mentioned above must have been a mystery for them -- if not a mystery then at least something "irrational". But the ancient Greeks were not able to overcome this intellectual challenge.

The Second Chapter

Already Archimedes observed that there was another problem with the concept length. He nailed down the axiom (named after him) that there were no segments of infinitesimal length. As he put it: "Given a line segment $AB$ shorter than another line segment $CD$ there is a natural $n$ such that copying the segment $AB$ $n$ times onto a straight line consecutively one can create a segment longer than $CD$. Archimedes must have noticed that his axiom could not be deduced from the Euclidean axioms.

So Archimedes had to axiomatically exclude the potential point sets consisting of points situated so close to each other such that the segment cannot be used as a measuring rod.

The Third Chapter

We have seen so far that the Euclidean points on an Euclidean straight line are not dense enough and at the same time their distribution can be (locally) too dense. At least, this is how I imagine the mind set of the pre-Archimedian and post-Euclidean Greek geometer.

The Fourth Chapter

It took more than two thousand years in geometry to arrive to the Dedekind axiom that regulated the density of the points on a straight line so it don't be neither too dense nor too sparse.

The Fifth Chapter

The development of the real numbers is very similar to the development of the ordering and regulating the density of the points of a straight line. The concept of an ordered field does not grab the concept of the real number without the Dedekind axiom, the density regulator.

Today it is easy to see the isometry between the Euclidean line and the set of Reals. Be aware that the common central concepts of the geometry of the straight line and the topology of the reals are: ordering and regulating density. (For instance: Non standard analysis comes from changing the density regulations.)

Epilogue The concept of real numbers and the concept of length in geometry are two notions with a parallel development process. We can see clearly the differences and the similarities.

Answer 2

Starting with a line segment of any length, and designate it as having length one, the Greeks believed it's always possible to precisely measure geometric lengths using this unit length. For example, to measure a length of 1.2, you can (after some trial and error) divide the unit length into five subunits of length 0.2, so that 1.2 is comprised of exactly 6 such subunits. This implicitly assumes that lengths can be expressed as the ratios of two numbers, i.e. lengths (corresponding to real numbers) are rational, thus it is impossible to measure irrational lengths like $\sqrt 2$ because no unit can fit evenly into these lengths. As Abbott wrote, the Greeks "were forced to accept that number was a strictly weaker notion than length". In modern terms, $\mathbb{Q} \subset\mathbb{R}$.

Citing Wikipedia: "The Pythagorean method ... claimed that there must be some sufficiently small, indivisible unit that could fit evenly into one of these lengths as well as the other. However, Hippasus, in the 5th century BC, was able to deduce that there was in fact no common unit of measure, and that the assertion of such an existence was in fact a contradiction. He did this by demonstrating that if the hypotenuse of an isosceles right triangle was indeed commensurable with a leg, then one of those lengths measured in that unit of measure must be both odd and even, which is impossible."