Warning: This piece is not meant to be readable. I’m just trying to explore some ideas about the foundations of math and belief.

How is it that we can imagine things that are logically impossible? I’ve always been a little perturbed by this. It was strange to me that we can imagine or even believe things that are logically impossible.

It makes sense to me that we can imagine things that are physically impossible. Our imagination is not constrained by the laws of physics. We can imagine physical objects which don’t exist or cannot exist. I can imagine someone traveling faster than the speed of light. Or I can imagine teleporting into the future or past instantaneously.

Contrast this with the relationship between our imagination and the laws of logic. At first glance it seems like our imagination must be constrained by logic. I can’t imagine 17 being an even number. I can’t imagine “I am alive” being both false and true at the same time. A proposition that is logically impossible is something that necessarily cannot be true, independent of what is or isn’t allowed by the laws of physics.

Yet on some level, I can imagine something that is logically impossible. I can believe that 17 is divisible by 2, or that “the sky is blue” is both false and true at the same time. I can state those beliefs, and I can act as if they were true. In other words, my beliefs or thinking can be irrational.

Now, you could argue that when I say “I am imagining 17 being even” I’m not actually imagining that, because again, it’s logically impossible. I just think that I’m imagining it; but in reality, I’m imagining something else.

The fact that it’s possible to think about or believe things that are logically impossible—in other words, the fact that it’s possible to be irrational—seems both interesting and significant. Again, it could just be that all irrationality—all our claims to be able to imagine a logical contradiction being true—are merely errors in our ability to describe what is going on in our mind. But that in itself is significant. We have the ability to be mistaken about our understanding of the computations going on in our brain.


I asked this question on Twitter and got some excellent replies:

I’m more convinced now that we can’t actually imagine logically impossible things.