Historian and Philosopher of Logic Stephen Read on the history of paradoxes, semantic paradoxes, and its direct connection to the foundations of mathematics
How have paradoxes been used throughout history? How have paradoxes influenced the study of mathematics? In what ways do paradoxes challenge knowledge and truth? These and other questions are answered by Historian and Philosopher Stephen Read.
If we hang him, he’ll have spoken the truth and so we should let him cross the bridge. But, if we let him cross the bridge he will have lied, and so we should have hung him. So, Sancho Panza, how shall we judge this case? And it takes a while for Sancho Panza to appreciate the paradox, but eventually he gives his judgement which is to hang the half of him that lied and let the half of him that spoke the truth cross the bridge.
You could have a set of numbers, you could have a set of sets, you could have a set of sets that are members of themselves, you could perhaps have a set of sets that are not members of themselves. But, then he thought, hang on. If you had a set of sets that weren’t members of themselves, would that set be a member of itself or not?
I’ve described a number of semantic paradoxes mostly to do with truth. I’ve then shown that they are very similar to some set theoretical paradoxes at the heart of the foundations of mathematics, and actually they ramify out. There are also epistemic paradoxes which apply to concepts like knowledge as well as concepts like truth.