Gödel’s theorem showing the impossibility of proving the consistency of arithmetic raises serious questions about the role of logic in mathematics and indeed in science in general. One consequence of this is that the axiomatic approach to discovery is of necessity somewhat limited. The goal of encoding knowledge within a system that would permit the logical deduction of all truths is no longer viable, if indeed it ever was. What then can serve as a “logic of discovery”?

In this book, the author attempts to answer this question. To this end, he surveys the history of logic in science before suggesting ways in which the problems raised by Gödel can be addressed. The author makes two essential points. First, reasoning can be either analytic, in which case an argument proceeds from known hypotheses, or synthetic, in which case problems are solved by forming hypotheses that would be sufficient to resolve the problem and then trying to show the hypotheses. The second point is that doing science requires a generative logic that can deliver hypotheses. Logic as traditionally understood is not generative, since in a certain sense the conclusions are contained in the axioms.

The historical survey, which occupies the first two-thirds of the book, begins with the Greeks and covers the development of logic up through the work of Frege and Gentzen. The reader who is interested in understanding the basic philosophical issues of a logic of discovery will find this a rewarding read. The chapter on Aristotle’s work on syllogisms is particularly clearly written and rewards close attention, as does the material on Bacon, Descartes, Leibniz, and Kant.

The final third of the book is devoted to the reconsideration of the role of logic. The author considers the role that emotion and evolution might play in reasoning. Can it be the case that logical reasoning has evolved, or is logic fixed? What about emotion and intuition? The author is somewhat careful about intuition, as it does not seem to be amenable to logical interpretation.

The final part of the book introduces a series of what the author calls “rules of discovery.” Unlike logic, these are intended to be “ampliative,” in the sense that they offer conclusions that are not embedded in the premises. Broadly speaking, the rules are either forms of induction or forms of analogy. (In a lengthy discussion, the author argues that abduction is not ampliative because the conclusion is embedded in the hypotheses.) The section illustrates each rule with examples that show how particular scientists or mathematicians may have exploited the rules to arrive at their conclusions. He admits that these are somewhat speculative, given the convention that authors give deductive arguments for their conclusions rather than revealing the route that led them to the conclusion. (The author quotes Poincaré on how he discovered certain results, and rightly points out that he did not provide a recipe for discovery!) This final section is the one that a computer scientist interested in machine learning may find of greatest interest, precisely because it suggests structures that can be used to amplify the knowledge in a system by forming new hypotheses.

The book, which is primarily a work of philosophy, is written clearly and proceeds in a logical fashion. The author’s conclusions do not entirely resolve the issues created by Gödel’s work. However, as the author himself says, hypotheses are only tentative and are subject to being replaced.