The purpose of this monograph is to demonstrate the connection between integrability theory and the twistor theory of R. Penrose. The unifying role is played by the self-dual Yang-Mills equations. Another aim is to reduce the mystery surrounding the topics under consideration. Why are there so many mysteries in the mathematics of mathematical physics? The answer is simple. Modern mathematics is the study of abstract structures where existence is synonymous with freedom from contradictions. On the other hand, physics is the study of objects and phenomena which actually exist physically, and for a physicist mathematics is merely a language for the description of these objects and phenomena. As our understanding of these is necessarily incomplete and imperfect, this is bound to be reflected in the mathematical description. It is now traditional in mathematical physics to use ideas of modern physics not only in the strict restricted meanings they had in their contradiction free definitions, but in a vaguely enlarged sense which appeals to the intuition. The strict rules of modern mathematics become immediately blurred and here lies the root of most of the mysteries which abound in modern mathematical physics.
This book is a clear demonstration of this thesis. It is nowhere explained what the reader is already supposed to know and definitions are scant in the text and often appear after a particular concept has already been used a number of times. The few definitions that are there lack the crispness which is to be found in definitions appearing in modern mathematical texts. Propositions often contain conjectures, pious hopes, guesses and plausibility arguments disguised as deductions. This is not necessarily a criticism of the book: even some serious pure mathematicians find the rigidity of modern mathematics too restrictive and welcome the freedom that comes with vagueness. What cannot be discovered with rigid definitions and deductions can often be formulated and used with remarkable success with vague ideas: Dirac’s delta function and Penrose’s original tensors are outstanding examples of this phenomenon.
The present book is a useful addition to the literature which could be of great assistance to those who are engaged in studying these topics with the aim of finding a contradiction free description devoid of mysticism. As far as this book is concerned, even the term integrability in its title has only a vague meaning that is open to interpretation and thus adds to the general mystery.