Assignment 7. Additive Measures

For submission

1. Additive measures allow us to find the measure of some totality from the measure of its parts. That is, when we join two systems together, the measure assigned to the joined system is the arithmetic sum of the measures assigned to each part. Which of the following are additive measures:

length, weight, area, volume, density, temperature?

Explain why those that aren't additive measures fail to be so.

2. In some fictional matter theory, a body consists of infinitely many points of no mass.

a. If the body consists of a countable infinity of points, what is the mass of the body?
b. If the body consists of any uncountable (e.g. continuum sized) infinity of points, what can we say about its mass?

3. How does your answer to 2b. assist in resolving the classical paradoxes of measure?

For discussion

Not for submission

A. Democritus used Zeno's paradox of measure as an argument for the existence of atoms. How cogent is this argument? Does it force us to conclude that space must consist of non-zero magnitude spatial atoms and that time consists of non-zero magnitude temporal atoms?

B. Is a line such as the circumference of a circle nothing but a set of points? If so, why are the two circumferences not the same in all their properties, including length? But if they differ, what extra property is there that distinguishes them?

C. Consider an account of measure that allows only finite additivity, but not countable additivity. What do we gain? What do we lose?

D. Is there no way to sum an uncountable infinity of measures? Perhaps we should just try harder?

E. Consider the three paradoxes resolved in this chapter:
          Zeno's paradoxes of measure;
          the stadium and;
          the geometric version of Aristotle's wheel.
They use notions derived from modern measure theory. Might it be possible to explain to thinkers in antiquity how the paradoxes are resolved in terms compatible with their methods?

To make this last question a little more vivid, imagine that you are transported by a time machine back to ancient Greece (and can now speak ancient Greek). What would you say to the ancient Greek philosophers to explain how the paradoxes of measure are escaped? They might be impatient, so you might need to find something short and quick.