Through the flaw

This book is a collection of articles which make various attempts to salvage things of interest from the ruins of Gottlob Frege's pioneering enterprise of proving that the whole of number theory can be reduced to the concepts and laws of pure logic. Russell's discovery in 1903 of a contradiction in Frege's intuitively attractive and simple set theory demonstrated a serious technical flaw in Frege's execution of the formal proof of the so-called "logicist thesis", and Frege himself never succeeded in remedying this defect.

His reaction was to abandon the attempt to execute his programme. Others have generally written off his philosophy of mathematics as being something of negligible interest: a bold but incoherent idea. By contrast, the system of logic that he invented for proving the logicist thesis is the foundation of modern formal logic, and his elucidations of its leading ideas have been treated with the greatest respect and interest. This collection of articles is meant to correct the disparity of esteem accorded to these two aspects of Frege's thinking.

The 18 articles fall into three groups, arranged in ascending order of technicality and difficulty. The first five discuss the intellectual background to Frege's logicism. The second four analyse the mathematical content of his initial attempt to execute the programme, as outlined in the Foundations of Arithmetic (1884). The final nine focus mostly on his analysis of real numbers in The Basic Laws of Arithmetic (1893 and 1903). Many of the articles contain a number of formal proofs and a great deal of modern logical symbolism and, though generally scholarly, thorough, and well presented, are unlikely to have a wide readership.

New light is thrown on familiar but controversial material. For example, one of the most celebrated features of the Foundations is the so-called "Context Principle", that it is enough for a number-word to have a meaning that every well-formed proposition (or equation) in which it occurs has a definite truth-value. The general principle is commonly now taken as a justification of contextual definitions, and it is treated as a crucial element in Frege's philosophy of language. But a comparison with 19th-century research in geometry suggests that Frege understood its application to arithmetic differently, as a purely mathematical reconstruction of natural numbers which parallels an established pattern for defining points at infinity in projective geometry in terms of unproblematic geometrical concepts. This suggests that Frege considered the context principle as formulating a pattern of mathematical analysis, not as a general principle of a semantic theory.

The second strategy common to many of the articles is taking advantage of the idea that a system of proofs that is inconsistent may contain many proofs that belong to consistent subsystems. This enables one in principle to extract valuable results from the system of The Basic Laws, despite its being globally inconsistent. Various articles vindicate the importance of different parts of Frege's detailed investigations of number theory.

The most striking of these rescue attempts is the demonstration that Frege's appeal to set theory is involved only in his attempt to deduce "Hume's principle", that the number of F-things is identical with the number of G-things if and only if the F-things can be put into one-to-one correspondence with the G-things (where "F" and "G" are variable for properties). His demonstration of the elementary truths of arithmetic from Hume's principle within second-order logic belongs to a consistent system. In this way, almost all of Frege's complicated formal proof of his logicist thesis is isolated from the destructive impact of Russell's paradox.

This collection presents material that has important philosophical implications, but for the most part it leaves the reader to work these out. The articles also apply to the analysis of Frege's work modern ideas in formal logic and number theory; they do not investigate in detail what Frege understood by "the principles of logic" or "logical analysis", a pity in work that purports to clarify Frege's attempt to reduce arithmetic to logic.

The enterprise to which these articles contribute is insufficiently radical. Arguably there is far more to be learned from Frege's philosophy of mathematics than is gained by these attempts at salvaging technical results from the ruins of the Foundations and The Basic Laws.

Gordon Baker is a fellow of St John's College, Oxford.