The Outer Limits of Reason: What Science, Mathematics, and Logic Cannot Tell Us

by Noson S. Yanofsky
The MIT Press, Cambridge, MA, 2013
424, illus. 118 b/w. Trade, $29.95
ISBN: 978-0-262-01935-4.

Reviewed by Thomas Colin
Marie Curie Fellow
‘CogNovo’, Cognition Institute

thomas.colin@plymouth.ac.uk

The Outer Limits of Reason is for truth-seekers, whether they are scientists, philosophers, or just curious individuals. Its stated aim is to trace the current and the ultimate frontiers of reason – the lines separating what we already know, what we do not know, and what we cannot possibly know. For example, we already know that the earth orbits the sun; but we do not know how to express quantum mechanics and general relativity in a single coherent theory (many physicists are trying as you read this). More surprisingly, there is a third category: we cannot possibly predict the natural evolution of the weather ten weeks from now. This third category, the fundamentally unknowable, is the most interesting: how can we be sure that something cannot be proved, shown, calculated? Can there be such a thing as unknowable truths? Some such questions will be answered within the pages of The Outer Limits of Reason; others have not been answered yet, and are left for the reader to ponder.

It serves to write a few words about the author, Noson S. Yanofsky. He is a professor in computer science and a man filled with curiosity whose interests range far beyond computers. Knowledgeable in physics, philosophy, and mathematics, he is first and foremost a skilled and enthusiastic popular science writer. He always strives to keep his readers entertained and interested. Indeed, the book contains very few equations, and those present are not there to be explained but to help explaining. This makes for a book that is accessible to any inquiring mind with high-school-level knowledge of the sciences.

The contents reflect the scope of the author’s knowledge and interests. The focus is on certain academic branches of knowledge: in particular physics, information theory, philosophy, and mathematics. Other seemingly relevant areas, in particular neuroscience and psychology, are omitted – this is because Yanofsky is interested in the limits of `Science, Mathematics, and Logic` - not those of individual scientists, mathematicians, and logicians.

Within the disciplines covered, the author occasionally expresses and defends his personal opinions on matters still in dispute. In his defence, whenever this occurs, Yanofsky is careful to mention the existence of a dispute; nonetheless, this leads to a biased account of the state of the art, particularly when discussing philosophical issues. Furthermore, where the author is not a specialist, but merely a knowledgeable and intelligent amateur, the most inquisitive readers will find a few propositions to be poorly substantiated, and may even discover a couple mistakes in the demonstrations.

A still more fundamental problem is the lack of a strong principle behind the various limits of reason. In biology, evolution plays this role, giving coherence to the study of the myriad of living species. But there is no over-arching, systematic principle valid across the “myriad unsolvable problems and paradoxes” described by Yanofsky, and so it remains a disorderly collection. To be sure, some paradoxes follow patterns across the disciplines (especially the self-referential ones); but others seem unique. Thus the book is not an elucidation of its subject matter; rather, it is an exploration of it. The study of all the specific examples fails to reach a resolution, an elegant synthesis.

Despite these limitations, every reader should learn something from the Outer Limits of Reason – whether it is a non-mathematical introduction to general relativity, a reminder on Zeno’s paradoxes, or accessible insights about Gödel’s incompleteness theorem. The scope of the material covered is so wide, and the writing so clear and intuitive, that all readers will learn something new and stimulating. Where their interest is excited, it is up to them to find more advanced material – but the basic introductions covered in the book, even by themselves, constitute wonderful food for thought.