A prediction, or any assertion, that cannot be defended might still be true, but an explanation that cannot be defended is not an explanation. - David Deutsch

That progress is both possible and desirable is perhaps the quintessential idea of the Enlightenment. It motivates all traditions of criticism, as well as the principle of seeking good explanations. But it can be interpreted in two almost opposite ways, both of which, confusingly, are known as ‘perfectibility’. One is that humans, or human societies, are capable of attaining a state of supposed perfection – such as the Buddhist or Hindu ‘nirvana’, or various political utopias. The other is that every attainable state can be indefinitely improved. Fallibilism rules out that first position in favour of the second.

Mathematical knowledge may, just like our scientific knowledge, be deep and broad, it may be subtle and wonderfully explanatory, it may be uncontroversially accepted; but it cannot be certain.

Think of all our knowledge-generating processes, our whole culture and civilization, and all the thought processes in the minds of every individual, and indeed the entire evolving biosphere as well, as being a gigantic computation. The whole thing is executing a self-motivated, self-generating computer program. More specifically it is, as I have mentioned, a virtual-reality program in the process of rendering, with ever-increasing accuracy, the whole of existence.