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So, it's my understanding that Russell and Whitehead's project of logicism in the Principia Mathematica didn't work out. I understand that two reasons for this are (1) that some of their axioms don't seem to be derivable from pure logic and (2) Gödel's incompleteness theorems. However, particularly since symbolic logic and the philosophy of mathematics are not my area, it's hard for me to see how 1 & 2 work and defeat the project.

I agree with Richard's and Alex's general remarks about "logicism" and what counts as "logical". It would indeed be far too quick to reject every form of logicism just because it makes the existence of an infinite number of objects a matter of "logic". Still, it is perhaps worth reiterating (as Richard indeed does) that Principia gets its infinity of objects by theft rather than honest toil: it just asserts an infinity of objects as a bald axiom rather than trying to conjure them out of some more basic logical(?) principles in a more Fregean way. So I'd still want to say that, whatever the fate of other logicisms, Russell and Whitehead 's version -- given it is based on theft! -- can't really be judged an honest implementation of the original logicist programme as e.g. described in the Principles , even prescinding from incompleteness worries. But for all that, three cheers for Principia in its centenary year!

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In the Principles of Mathematics, Russell boldly asserts "All mathematics deals exclusively with concepts definable in terms of a very small number of logical concepts, and ... all its propositions are deducible from a very small number of fundamental logical principles." Principia , a decade later, is an attempt to make good on that programmatic "logicist" claim. Now, one of the axioms of Principia is an Axiom of Infinity which in effect says that there is an infinite number of things. And you might very well wonder whether that is a truth of logic . (If someone thinks the number of things in the universe is finite, are they making a logical mistake?) Another axiom is the Axiom of Reducibility, which I won't try to explain here, but which is even less obviously a logical law -- and indeed Russell himself argued that we should accept it only because it has nice mathematical consequences in the context of the rest of Principia's system. Still, there is some room for...

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