Don’t you just love a fancy title?

Well, I do. That’s why I wrote it.

I came up with this idea a couple of years ago, dismissed it as paranoia, and forgot about it. But it kept popping up in my head again.

Let me explain.

Gödel’s incompleteness theorem states that:

“No consistent system of axioms whose theorems can be listed by an effective procedure is capable of proving all truths about the relations of the natural numbers. For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system.”

Comprendez-vous? Chouette!

Yeah. An axiom is a statement that you assume from the beginning to be true. They are supposed to be self-evident, and then you use various logical methods to extract other statements from your list of axioms. The trouble is, it’s very hard to get further than “I think therefore I am.” You see, in order to say that you ought to also have a rule that says “everything which thinks is.” It might seem obvious, but you then start to get into defining thought and existence, and it all ends up in a horrible mess.

That is the trouble with rationality: it can make you go mad.

Yes, anyway. Gödel says that if you have a list of axioms, you can’t prove every truth using only that system that axioms. For instance, you can’t prove your axioms, which you ought to do, really. At least, I think so.

If you start with only one axiom, and then derive something else from it, and then derive something else from that ad nauseum, and then you have something with which you can prove your first axiom, you might think you have proved all of your axioms, but now your proof of one of the axioms relies on that axiom itself. And:

But it’s cheating.

Now for the paranoia. Consider the dictionary. What does it do? It gives definitions of words. How does it do that? Using words, of course. So how is possible to *know* for sure that you have fitted all the words together into their pattern in the same way as everybody else? How do you know that the word “the” means the same to you as it does to anybody else? Let alone complicated words like “counter-current multiplier mechanism” (which is something that happens in kidneys, by the way).

With no external reference point, like mime or pointing at solid objects, would it be possible to get words swapped around in such a way that a dictionary was completely self-consistent? I don’t know, I haven’t even read all the way through my dictionary yet, so I haven’t even started trying to switch things about. That would be tedious, complicated, and almost certainly futile. Still, I find it a scary thought.

Luckily, external reference points do exist. We learn our first words by associating them with changing environmental factors. Foods are usually mentioned just before they appear. People’s names are usually used when the person talking is looking at a specific somebody else. So the chances of getting everything skewed is even lower than exceptionally low.

The colours thing could be true, though. People could have shifted perception so that their brain sees green at the wavelengths that other people see as blue. And because we can only communicate about colours by their names or wavelengths or something else completely arbitrary (basically not a mind-meld) there would be no way to tell. I don’t find that quite so scary though.

Imagine if everybody else were talking a different language that just happened to have the same words but with different meanings. How scary would that be? And what would have to happen for you to notice? It’s crazy. Maybe I’m crazy. Probably. I have things that need doing. Like proving that the sum of the reciprocals of any three positive real numbers is greater than or equal to twice the sum of the reciprocals of each of the three pairs of the same numbers. I could write it like this if you like: (1/a)+(1/b)+(1/c) ≥ 2(1/(a+b)+1/(a+c)+1/(b+c))

So long.