The emancipating effect of confronting the undecidable

Pretty well everyone knows of some of the great breakthroughs that occurred in the twentieth century. We’re all aware, even if we don’t understand, the work of Albert Einstein, Erwin Schrödinger, Sigmund Freud, James Watson (or should that be Rosalind Franklin?)

Curiously, though, hardly anyone seems to have heard of the work of Kurt Gödel, and yet he made a discovery that seems to me even more disturbing and transformational than those others.

Kurt Gödel
A devastating but liberating truth

Gödel established that there are certain so-called “undecidable” questions in mathematics. Such questions are unanswerable, by which I don’t mean unanswered: it’s not a matter of a lack of knowledge – we don’t know the answer – but a matter of full, exhaustive knowledge – we know they can’t be answered. Indeed, if we’re good at enough at maths, we can prove they can’t be answered.

These are often simple questions, which is not to say their subject matter is simple: the only one I know about concerns transcendental mathematics, and the word “transcendental” is a bit of a give-away, isn’t it? Something transcendental isn’t likely to be easy. The particular question I’m thinking of concerns the relative size of different infinities, and I imagine that you would agree with me in thinking that the idea that two things can be infinite but one is bigger than the other, is pretty mind-blowing in itself.

The question is simple only in the sense that the answer is yes or no. Which is where Gödel comes in. Because there’s no third answer, and he showed that both answers are wrong. Rigorously proved it: answer “yes” and you get to a logical contradiction, answer “no” and you get to another logical contradiction. So the question is simply unanswerable or, to use the preferred term, undecidable.

Think about that for a moment. There is no other field which, like mathematics, always has a right of a wrong answer. And yet even in mathematics, there are areas where there is neither right nor wrong.

These days we’re frequently told that our societies are paralysed by a terrible moral relativism. People beat pulpits, metaphorical or real, and bewail the passing of absolute truths. They hanker for the days when we knew, always and undeniably, right from wrong and could teach it to our children.

And Gödel showed that the quest isn’t just unachievable, it’s meaningless. In even the most abstract of all fields, mathematics, there is ultimately no absolute truth. If that’s true of mathematics, how much more so must it be the case for the messier disciplines in which all the uncertainties of reality cloud the picture further.

So that means we’re forced back onto our own resources. There is no simple guide that can tell us what’s right, at all times, under all conditions. Which is what those of us who rather suspected that relativism made more sense had always suspected. In the end, you have to go with what works, in any particular set of circumstances.

“Thous shalt not kill”? Well, as a general rule, thou shouldn’t indeed. But maybe in certain circumstances euthanasia, or abortion, or military action, or even execution can be justified – may, indeed, be the only option.

Which means that instead of a helpful if inflexible, general if monolithic guide to behaviour, we’re forced back on to the need to exercise judgement in the light of real circumstances. Forced to take our own decisions. Then live by the consequences.

That this may be not so much a choice as essential to the very nature of our lives is the breakthrough Gödel made. It seems to me more people should know about it.

Because I, at least, I find the notion liberating.