Topics Mathematics

Question I recently heard about mathematical paradoxes and I have a perhaps strange question: It seems to me that the goal is to figure out what the fundamental problem is, i.e. what gives rise to the paradox, so we can perhaps rewrite the axioms so that the problems disappear. But why not just say: "Well, paradoxes arise when you talk about sets that contain every set, so let's avoid talk about sets that contain empty sets." (Kind of like saying that bad things happen when you divide something with zero, so don't do it!)

Accepted:
October 4, 2005

Comments

It's not so easy to know if you've avoided talk that could lead to a contradiction. Is it OK to talk about a set that contains all sets but one? All sets but two? No--it turns out those sets lead to contradictions too. What if you don't explicitly refer to a set that contains all sets, but such a set is used implicitly in some piece of reasoning? Where do you draw the line? How do you know if you've crossed the line? Rewriting the axioms is a way of drawing the line.

One way to see why avoiding contradictions is so important is to think about proof by contradiction. To prove a statement P, mathematicians sometimes assume that P is false and then try to deduce a contradiction. This method of proof is based on the idea that if you can deduce a contradiction, then the assumption that P was false must be incorrect, so P must be true. But if contradictions can arise even if you haven't made a false assumption, then you'll be able to use proof by contradiction to prove false statements. (In fact, this is the basis for the fact of logic that from contradictory premises you can prove anything.)

Also: Even if, as a practical matter, it weren't so important to avoid contradictions, isn't it more intellectually satisfying to try to track down the source of the contradiction, rather than just avoiding certain kinds of sets without really understanding why you have to avoid them?

By the way, division by 0 is not just something that mathematicians avoid. In the case of division, mathematicians also have clear lines that say what is allowed and what isn't allowed, and they have reasons for drawing those lines. For example, suppose we define "c divided by d" to be the unique solution for x in the equation dx = c. Then it is a theorem, provable using basic properties of multiplication of real numbers, that for any numbers c and d, if d is not equal to 0 then the equation dx = c has a unique solution, and if d = 0 then the equation dx = c has either no solution or infinitely many solutions. So you can prove that division by any number other than 0 is defined, and division by 0 is undefined.

Let me add one other thing. I thought the first thing you said was aboslutely right: "the goal is to figure out what the fundamental problem is, i.e. what gives rise to the paradox". The reason is that it is supposed that our being led to paradox in the case of, say, sets or truth or vagueness shows us that there is something about sets or truth or and vagueness that we don't really understand. If we understood things properly, we would understand how the paradox could be avoided, and not simply because we put our heads in the sand. So paradoxes are manifestations of our lack of understanding, and it is the lack of understanding that we really want to remedy.