I am reading a rather dull philosophical tomb on vagueness. It is not really a tomb but it feels like one. I am just at the beginning, reading about the sorites paradox and borderline cases, but I thought I would blog a few thoughts about it.
The sorites paradox - this addresses the question of what constitutes a heap of sand or a bald man or a tall woman. If I have one piece of sand, and I keep adding sand to it, at which point do I have a heap of sand? There is a further 'philosophical' problem with these terms, namely how do you determine the validity of a statement that uses one of these vague terms. If I a am tall is someone .10 inch shorter than me also tall?
There are all sorts of philosophical solutions to these problems - most of which seem like utter mental masturbation and contrary to common sense. But what interests me is the intersection between this issue and other issues - in particular mathematics, and in particular thoughts about infinity (set theory) and thoughts about series (infinite series perhaps).
First let me address infinity and set theory. Josef Cantor came up with the idea that there are infinite number of infinite sets.
What does this mean? For example lets look at the set of natural numbers. It goes 1,2,3.... indefinitely. We would say the size of the set of natural numbers is infinity. Well is there anything larger than the infinity of the natural numbers. Cantor said (this was one of his proofs), that there were a greater number of rational numbers between 0 and 1 than in the natural number set. These are different types of sets. Natural numbers are countable, I can generate another number by applying some application to a lower number (such as adding 1, or multiplying by 2). For the numbers between 0 and 1 all I need to do is add a number to the denominator. 1/10, 1/100, 1/1000. I am not generating an infinite set by a mathematical operation but by constructing a new number. (I could go on but I want to get back to the paradox)
Also remember there are more irrational numbers than rational numbers (more holes in the number line than points) but I digress.
Anyway all this talk of different sizes of infinity. his seems like a bizarre question to ask at first, because infinity is not really a size but a description of a limiting circumstance. And does it really make sense to say the infinity of natural numbers is different than the infinity of the rational numbers, or irrational numbers. True the way we generate the number set is different but infinity is infinity.
This reminds me of the sorites paradox. When does a set become infinite, or infinite of a particular order? In this case each is defined by a different mathematical expression. A countable infinite set can be expressed by a function, and a non countable set can be defined by a transformation. But does this mean that the infiniteness is different. If I become bald by shaving my head or by genetics does that change the nature of baldness. Infinity is a limiting condition - it describes what differentiates the set from other sets that with definite boundaries.
Ok and my second thought about sorites - namely series, or rather limits. Perhaps baldness is that which is approached but never reached, like the limit of an infinite series. Rather than express the distance from baldness, or the way we reach baldness, we just default and call the condition baldness. Baldness is never actually reached, so it really does not make sense to ask if one hair added or subtracted makes a person bald. Rather, every hair lost brings someone closer to the condition of baldness which is never actually reached. I had more coherent thoughts on the airplane to LA when I was thinking about all these things - but you will have to deal with the limits of my memory.