This month, we sit down with Patricia Blanchette to discuss the work of Gottlob Frege, one of the most influential philosophers of the 20th century. Click here to listen to our conversation.
We saw in our episode on the philosophy of mathematics how difficult it was to say what numbers are. What is the number three, and how do I come to know things about it? (Like that it’s odd, that it’s prime, that it’s the lowest number that’s both odd and prime, that it’s a factor of 135, and so on for all the things a mathematician might teach you about it.) Frege thought we could make some headway on these questions if we could show that arithmetic was really just complicated logic. And one way of demonstrating that arithmetic is just complicated logic is by showing that you can translate any statement about arithmetic into a statement about logic without changing its meaning.
Logic, you ask? What do numbers have to do with logic? Well, logicians study what are called valid inferences: inferences in which the premises, if true, guarantee the conclusion no matter what. For example, if Jane is riding a bus, it follows that someone is riding a bus–no matter what. If it’s true that Jane is riding the bus, then it has to be true that someone is.
What does this have to do with numbers? Well, think back to what we said about the number zero during our interview with Agustin Rayo. ‘The number of dinosaurs is zero’ really just means ‘there are no dinosaurs’–or, more stiltedly, ‘it is not the case that something is a dinosaur.’ Logicism, the idea that arithmetic is really just disguised logic, is based on the idea that any statement about numbers can be translated into a ‘something’ statement, in more or less the way we did for statements involving the number zero.
Tune in to hear Patricia Blanchette explain how the whole thing works!