As this book’s title suggests, the author sought to connect the development of the concepts of deduction, formal mathematical proofs, and theories of computation to the development of modern day computers. Von Plato makes the claim right in the first sentence: “if around 1930 Kurt Gödel had not thought very deeply about the foundations of mathematics, there would be no information society in the form in which we have it today.”

In the first chapter, von Plato traces the historical development of the study of logic and the foundations of mathematics, beginning from Aristotle’s syllogisms. He then gives an overall view of how these ideas connected to mathematical developments in the 19th to the early 20th century, giving the readers a glimpse of the contents of the entire book.

The book’s narrative begins with the emergence of the study of the foundations of mathematics — the study of the justifications of procedures and ideas in arithmetic. Here, von Plato expounds on the ideas of Grassmann and Peano — the use of axioms, recursion, and mathematical formalization — which the author suggests is a requisite for subsequent “machine executability.” He then writes about the beginning of algebraic logic, Boole’s “calculus of deductive reasoning,” which has its origins in Aristotle’s syllogisms. Next would be people who would build upon Boole’s mathematics, such as Schröder and Skolem. The author dedicates a whole chapter to Frege and his ideas on formal reasoning: that proofs can be decomposed into simple logical steps and that mathematics can be reduced to logic. Finally, he writes about the contributions of various mathematicians such as Russell, Wittgenstein, Hilbert, Bernays, Gentzen, and, of course, Gödel.

The effort that von Plato put into the book is evident. It is a historical exposition that does not avoid the specialized notations and ideas inherent to this area of mathematics. It presents the evolution of ideas within the context of what was happening in Europe; for example, readers get a glimpse of Bernays’s life, fired from Göttingen because he was a Jew.

The book is not for leisurely reading, however. The suitable audience for this book includes people who have sufficient background in logic or theory of computation. As the author explains, the chapters are based on his lecture notes. What may be lacking, which in no way lessens the substance and significance of this book, is a more careful explanation of the connection claimed by the author, between the study of logic and foundations of mathematics on the one hand and modern day computing on the other. The only chapter that covers this is the prologue, which seems short given that the title suggests a more nuanced explanation of how the areas of mathematics connect to the digital age. Nevertheless, this book is an important contribution to the study of the history of mathematics, and any student, educator, or practitioner of mathematics or computer science, would benefit from reading this work.

Mark Causapin is an Assistant Professor of Mathematics at Aquinas College in Nashville, Tennessee.