Logicism and its Philosophical Legacy by William Demopoulos (review)

As is well known, Gottlob Frege’s logicist program was an effort to show that arithmetic can be reduced to logic, or in Frege’s words, that arithmetical truths are analytic a priori. The word ‘logicism’ was also used for a view held by several antipsychologists, such as Hermann Lotze, Edmund Husserl, and Frege, who considered numbers and other mathematical objects to be independent of the human mind. William Demopoulos focuses on one aspect of Frege’s logicism. He emphasizes that, unlike the Kantian mathematical tradition, in which arithmetic was seen to depend on the provinces of geometry and kinematics, Frege wished to show that arithmetic is independent of those fields. He notes that Frege did not regard intuition as a potentially faulty guide to truth; instead, Frege’s epistemological interests were broadly architectonic (170–76). This is an important point of view, to which Frege’s last writings in 1924 and 1925 could give more support. In those writings, the interest in architectonic is strongly present, even if the logicist program has disappeared. Frege there distinguishes between three sources of knowledge, sense-perception, the logical source of knowledge, and the geometrical source of knowledge, and suggests that arithmetic has its basis in geometry (Nachgelassene Schriften, Hamburg: Felix Meiner Verlag, 1969, 298).

The main idea of Demopoulos’s study is that Frege’s work—and logicism in general—“contains insights that bear on fundamental problems of interpretation that arise in the context of empirical theories” (1). Demopoulos seeks to argue that there are different conceptions of analysis that have their source in the logicist tradition and that are relevant to the philosophy of mathematics and the philosophy of science. His special point of interest is what he calls the structuralist thesis: “The theoretical component of what our theories express is wholly captured by statements which depend only on the logical category of their constituent concepts” (161). He considers Bertrand Russell, Frank Ramsey, and Rudolf Carnap to be supporters of the thesis. In his view, for Russell, the thesis has the consequence that our knowledge of events which transcend our acquaintance is knowledge only of their structure; for Ramsey the role of theory is to provide organizing parameters; and for Carnap it leads to neutrality in the debate between realism and anti-realism (161, 162).

The book opens with a chapter on Frege’s analysis of arithmetical knowledge and continues with one on Carnap’s thesis that certain applied mathematical theories are not factual. The Ramsey-sentence reconstruction of theories and Carnap’s alleged position in the realism-instrumentalism controversy are discussed in several chapters. Carnap’s distinction between internal and external questions is analyzed in detail. The distinction transforms traditional metaphysical questions into problems of choosing a linguistic framework. Unlike Carnap, Demopoulos does not regard the controversy between realism and instrumentalism as requiring neutrality (68). Nor does he hold the view that truth reduces to empirical adequacy. He argues that the key metaphysical disagreement is a disagreement over the concept of truth. Demopoulos shows that the Carnap-Ramsey program does not work. Russell’s structural realism is also criticized convincingly, especially taking into account M. H. A. Newman’s “Mr. Russell’s Causal Theory of Perception” (Mind 1928).

The last chapters on Frege and Russell focus on contextual definitions and contextual analysis. The collection ends with a chapter on propositional functions. No concluding chapter is added to the book.

There is new material, but most of the chapters were published earlier as articles, which may be the main reason for overlap. Demopoulos gives new light on what is often presented as a history of the empiricist philosophy of science by considering it a part of the legacy of logicism. One who is not familiar with technical tools is likely to lose some of the arguments for the theses presented, but can certainly follow the main lines of the study. Metaphysics comes up here and there in the book, and one might have wished to see...