Wolfgang Schwarz

Why Did Frege Give Up?

When I prepared for my exam, I noticed something curious.

Richard Heck, in "The Julius Caesar Objection", claims that

In a letter to Russell, Frege explicitly considers adopting Hume's Principle as an axiom, remarking only that the 'difficulties here' are not the same as those plaguing Axiom V [p.274 in Language, Thought and Logic].

The claim is repeated by Crispin Wright and Bob Hale in the introduction to The Reason's Proper Study (p.11f., fn.21). The letter Heck, Wright and Hale refer to is xxxvi/7 from July 1902.

Now the main difficulty plaguing Axiom V is of course that it is inconsistent (in standard second-order logic). Frege's remark suggests that he knew that Hume's Principle, unlike Axiom V, is consistent. And that he even knew this less then two months after receiving Russell's first letter. This would be very surprising, for how could he know? The fact that Russell's Paradox can't be easily derived certainly couldn't convince Frege. Nor could the fact that there are informal arguments that show the consistency of Hume's Principle, for similar arguments seem to show the consistency of Axiom V. The only good reason to believe that HP is consistent is the recognition that any derivation of a contradiction from HP can be turned into a derivation of a contradiction from theories like standard analysis whose consistency is not in doubt. But this kind of reasoning was alien to Frege (even more than to all his pre-1931 contemporaries, as the correspondence with Hilbert shows).

Of course, Frege rejected HP as a primitive proposition in Grundlagen for reasons other than doubts about its consistency. Namely, for the Julius Caesar reasons: HP does not succeed in defining "the number of", because it does not settle the truth conditions of all well-formed expressions in which "the number of" occurs. It has often been noted that the same is true for Axiom V, however, which looks very much like HP. The solution to this puzzle is that Frege never thought of Axiom V as defining "extension". Rather, he took it for granted that the concept "extension" is already known (as he explicitly says in the footnote on p.80 of Grundlagen), and hence that it is already known that Julius Caesar is not an extension. Axiom V was merely meant to state a general fact about extensions, not to define them. Frege probably didn't want to say the same about HP, because taking it for granted that the concept "the number of" is already known means giving up much of the logicist project.

So the "other difficulties" Frege speaks of in his letter can't simply be the Julius Caesar problems (as Heck, Wright and Hale suggest), because these affect Axiom V as well. They can only be the problem that taking "the number of" for granted undermines a certain philosophical project. But if Frege really knew that arithmetic can be reduced to the simple and consistent HP, this would still have served the more Fregean project of showing how much in mathematics can be done without Kantian intuition. -- After all, Frege was not motivated by any empiricist dogmas; he even accepted that geometry is based on intuition. So why didn't he just say that what he had proved was that if intuition is required in arithmetic, then it is only required for HP? Why did he completely give up his project?

The reason, I think, is that he did not believe that HP is consistent.

Then why does he speak of "other difficulties" in the letter? He doesn't. At least he does not in the letter as it is published in the Wissenschaftlicher Briefwechsel. What he says here about taking something like HP as primitive is

Die Schwierigkeiten sind hierbei aber dieselben wie bei der Umsetzung der Allgemeinheit einer Gleichheit in eine Werthverlaufsgleichheit [p.224].

which means that "the difficulties here are the same as [those plaguing Axiom V]".

So either Heck, Wright and Hale (oddly) read a negation where there is no negation, or the English translation they refer to (Frege's correspondence, edited by G.Gabriel, p.141) contains a serious (and odd) error, or I am wrong and the German publication contains such an error. The latter is rather unlikely as the English translation is probably based on the Wissenschaftlicher Briefwechsel edition. Could somebody look up the English translation in the Gabriel edition and tell me whether it contains the negation?