Not precisely mathematics, but this week's Sunday reflection by physicist Max Tegmark on why we need to be careful when it comes to programming artificial intelligence:
"If you're walking on the sidewalk and there's an ant there, would you actively go and stomp on it just for kicks? (Me: 'No.')
"Now, suppose you're in charge of this big hydroelectric plant that's gonna bring green energy to a large region of the U.S. And just before you turn the water on, you discover there's an anthill right in the middle of the flood zone. What are you gonna do? It's too bad for the ants, right? It's not that you hate ants. It's not that you're an evil ant-killer. It's just that your goals weren't aligned with the goals of the ants, and you were more powerful than the ants. Tough luck for the ants. We want to design AI in the future so that we don't end up being those ants."