Logic

Did Bertrand Russell or any of the logicists ever reply or address Goedel's incompleteness theorem directly?

Comments

I do not think Russell every addressed it directly, and Frege died before Gödel did his work. It is possible that some of the positivistic logicists, like Hempel, did, but not so far as I know.

That said, incompleteness has been raised as a problem for contemporary forms of logicism, generally known as "neo-logicism". (For an introduction, see this paper of mine or this paper, by Fraser MacBride.) I think the response to this objection is pretty straightforward, however. Take neo-logicism to be the view that the truths of arithmetic (we'll focus on arithmetic) are all logical consequences of some core set of principles that, though not truths of logic in any sense now available to us, have some claim to be regarded as "conceptual" truths, or "analytic" truths, or something in this general vicinity. The incompleteness theorem tells us that there can be no algorithmically computable ("recursive") set of principles from which all truths of mathematics follow, if but only if we assume that what follows from a recursive set of principles must be algorithmically listable ("recursively enumerable"): That is, if it would be possible to write a computer program that would spit out all the infinitely many logical consequences of that set of principles, one after the other. This is true in the case of so-called "first-order" logic, but most forms of neo-logicism assume that the background logic will be some sort of second-order logic, and then the principle just mentioned is false: Second-order logic is itself incomplete, and there is no reason that all truths of arithmetic cannot be second-order logical consequences even of some finite set of axioms. Indeed, they are. (This is a consequence of the so-called categoricity of second-order arithmetic. See this SEP entry for more information.)

A different sort of reply, which I myself would prefer, modifies the definition of logicism. Suppose we had good reason to believe that the following statements about cardinal numbers (numbers that answer "How many?" questions) were all somehow equivalent to truths of logic:

Every number has one and only one number that comes right after it.
Every number other than zero comes right after one and only one number.
Zero does not come right after any number.
The sum of n and zero is n.
The sum of n and the number that comes right after m is the number that comes right after the sum of n and m. (I.e., n+(m+1)=(n+m)+1.)
The product of n and zero is zero.
The product of n and the number that comes right after m is the sum of the product of n and m and n. (I.e., n*(m+1)=(n*m)+n).)

These are the axioms of a theory known as "Robinson arithmetic", and, though it is very weak in some ways (it does not imply that addition is commutative, for example), it is a substantial bit of mathematics, in a well-defined sense. The claim that all of these are truths of logic would, it seems to me, be very interesting and very important, if true, if only because this theory clearly implies that there are infinitely many numbers. And its importance is completely independent of whether any other arithmetical truths count as logical. So this sort of view certainly isn't vulnerable to objections based on the incompleteness theorem.

Read another response by Richard Heck

Read another response about Logic