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I have a question about Whitehead and Russell "Principia Mathematica".

Can mathematics be reduced to formal logic?

July 25, 2008

Response from Peter Smith on July 28, 2008

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,

  1. 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.
  2. 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 successor, addition and multiplication functions in reasonably neat ways, and so get translations of arithmetical claims into claims about sets. Now suppose you also allow some basic principles about sets to count as logical axioms. Then, lo and behold, you'll be able to derive arithmetical truths from the definitions by using 'logic' (a distant echo here of how Whitehead and Russell in fact proceeded).

Job done? Well, most of us nowadays would say "no", because we wouldn't take the notion of a set to be a purely logical one, or set theory to be logic. But Frege thought otherwise, taking the notion of the extension of a predicate to be a logical notion. And although Russell traded in extensions for "propositional functions", he did think that a rich theory of such entities counts as logic. This isn't the place to argue for the modern narrower view of the scope of logic as against Frege or Russell. But mentioning this divergence highlights one key issue involved in answering the initial question: how much "logic" are we allowed?

With hindsight, we can see that well before Principia, Frege got to what now seems a really key result. Suppose you take the bridge principle (unhistorically called "Hume's Principle")

The number of Fs = the number of Gs if and only if there is a one-one correspondence between the Fs and the Gs.

Then this time define zero to the the number of things that aren't self-identical, one to be the number of things identical to zero, two to be the number of things identical to either zero or one, and so on. We can again add definitions of successor, addition and multiplication functions in reasonably neat ways. And then -- and this is "Frege's Theorem" -- the bridge principle plus those definitions plus second-order logic gives us arithmetic.

Now even if we balk at counting set theory as logic, maybe second-order logic is properly so-called (though actually, quite a few will dissent from this, following Quine in counting second-order logic as actually just some set theory in sheep's clothing -- but let that pass.) However, we still have to ask: is Hume's Principle an acceptable sort of principle to use in a reductive endeavour? It seems that its left hand side introduces entities (numbers) which don't appear on the right-hand side (which just talks of their being a suitable one-one function with domain the Fs and range the Gs). So Hume's Principle, many will say, can't be a genuine definition as it imports new ontology. Hence Frege's Theorem doesn't really reduce arithmetic to something else, i.e. logic plus definitions. At best, though still interestingly, it reduces arithmetic to just one core arithmetical principle (plus logic and definitions). But again, this isn't the place to argue about the status of Hume's Principle. But mentioning the question now highlights another issue involved in answering the initial question: what kind of bridge-principles linking arithmetic to logic are we allowed?

If we are a bit stern about what counts as logic, and stern about the kinds of bridge principles that are allowed in a successful reduction, then no, arithmetic surely can't be reduced to logic. But some (especially so-called neo-logicists) do think we can be less stern and rescue something true and important from the old attempts to reduce arithmetic to logic.


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