I have a question about Whitehead and Russell "Principia Mathematica".
Can mathematics be reduced to formal logic?
Let's narrow the question a bit: can arithmetic be reduced to logic? If arithmetic can't be so reduced, then certainly mathematics more generally can't be. What would count as giving a reduction of arithmetic to logic? Well, We would need to give explicit definitions (or perhaps some other kind of bridging principles) relating the concepts of arithmetic to logical concepts. Otherwise we won't get arithmetical concepts into the picture at all. We would need to show how the theorems of arithmetic can in fact be derived from logical axioms plus those definitions or bridge principles. But what kind of definitions in terms of what sorts of concepts are we allowed at step (1)? And what kinds of logical principle can we draw on at step (2)? Suppose you think that the notion of a set is a logical notion. And suppose that you define zero to be the empty set, one to be the set of all singleton sets, two to be the set of all pairs, and so on. We can then define the...
- Log in to post comments