Gottlob Frege · Mathematics
As the second volume of the <em>Grundgesetze</em> went to press in 1902, a letter from Bertrand Russell revealed that Frege’s Basic Law V was inconsistent - a contradiction lay at the very foundation of his life’s work. This is the story of that collapse, Frege’s stricken response, and how his logic remade philosophy even as his own dream fell.
By 1902 Frege had spent more than twenty years on a single mission: to prove that arithmetic is logic. The Begriffsschrift had built the logic; the Grundlagen had made the philosophical case; and the two volumes of the Grundgesetze der Arithmetik were to carry out the derivation in full formal rigour, deducing the laws of number from logic alone. But the bridge from pure logic to actual numbers needed a keystone - a principle that would deliver, for any concept, a logical object to serve as its number. Frege’s keystone was Basic Law V. It governs the ‘extension’ of a concept, the object comprising exactly the things that fall under it, and it says, in effect, that two concepts have the same extension precisely when the very same objects fall under them. It looks utterly innocent, almost a triviality. On it Frege built everything. And it was, though he could not see it, a live charge buried in the foundation.
On the sixteenth of June, 1902, Bertrand Russell - then thirty, and one of the very few people alive who had read the Grundgesetze with full understanding - wrote Frege a courteous letter. He admired the work immensely; he agreed with almost all of it; but he had found one difficulty. Consider, Russell wrote, the class of those classes that are not members of themselves. On Frege’s principles this class both must and cannot be a member of itself. In a few lines, Russell had shown that the system was inconsistent - and an inconsistent system is worthless as a foundation, because from a contradiction everything follows; a logic that proves both a statement and its denial proves every statement, true and false alike. The timing could hardly have been crueller. The first volume of the Grundgesetze was long published; the second was already at the printer’s, the culmination of Frege’s life’s work, moving irreversibly toward publication. And into that moment arrived the news that its foundation did not hold.
In the appendix Frege did not merely mourn; he tried to save the system. He proposed a modification of Basic Law V, weakening it so as to block Russell’s derivation, and expressed a guarded hope that the essential theorems of arithmetic might still be recovered. But the repair failed. Frege’s ‘way out,’ as later logicians (Leśniewski, and then Quine in a famous 1955 paper) showed, was itself defective - the weakened law had disastrous consequences of its own, collapsing the domain of objects. There was no easy patch. In time Frege seems to have grasped this, and the effect on him was profound. He never completed the projected third volume; his great productivity faltered; and in his final years he abandoned logicism altogether, coming to believe - reversing the central claim of his life - that arithmetic must after all be founded on geometry, on our intuition of space, much as Kant had said. The man who had set out to prove number was logic died, in 1925, having concluded it was not. It is one of the great tragedies in the history of thought.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned that Russell’s paradox - the class of all classes not members of themselves - showed Frege’s Basic Law V to be inconsistent; that Russell’s letter arrived as the Grundgesetze vol. II went to press; that Frege’s stricken appendix acknowledged the collapse; and that his logic nonetheless remade philosophy th…
Leads to Bertrand Russell.
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