During the 1960s and 1970s in the UK, there was considerable debate regarding the nature and relevance of child-centred learning, particularly with regard to what it meant in terms of mathematics education. Forty years later, we now have a more clearly defined distinction between what is referred to as "process objectives" (investigational skills) and "content objectives" (the body of knowledge) and the controversy has now abated almost entirely. But how I wish that this book had been widely available thirty or forty years ago, for it gives eloquent expression (albeit implicitly) to the philosophy of the investigational approach to the learning of mathematics — as exemplified by the way in which the author taught himself mathematics as a young child and after.

Having read early parts of the book (the word "chapter" has been avoided), readers will see why Philip Davis has tended towards applied mathematics since childhood years and thereafter. One of his first excursions into the realm of applications was his role as bookkeeper for his cousin who kept racehorses. Then there was his fascination with number games and mathematical puzzles etc. Books on elementary mathematics were a source of recreation to the young Davis and one wonders how his teachers reacted to the very young student who was so able to think things through for himself. I am sure that he would have been a delight to teach, even though he was largely self-taught! There is a later section with the heading "A Can of Peaches," used by the author to illustrate the amount of latent mathematics revealed by the study of many everyday objects, which leads him on to the wider discussion on the relevance of mathematics to the world at large and the relation of mathematics to society. But analysis of the can of peaches illustrates perfectly the means by which the subject can be taught in a creative and exploratory way.

The title of this book may suggest an autobiographical emphasis and, to a great extent this turns out to be the case. But the reader soon discovers that it is not a biography in the usual sense, for there is no expatiation upon family genealogy nor attempts to describe those semi-fictional images masquerading as "earliest memories." However, the structure is largely, but not strictly, chronological, commencing with Davis's earliest recollections of his mathematical learning. He describes his high school years, his undergraduate training, his war-time experience and his subsequent post-graduate studies at Harvard, where faculty members included such noted mathematicians such as Garrett Birkhoff, Lynn Loomis, Saunders MacLane (not to mention visiting speakers such as Henri Cartan).

Philip Davis is now very nearly eighty years of age. His life has spanned an era of enormous social, political, scientific and technological change, perhaps without parallel in any other period of history. He has long been a mathematician of international repute; he has published prolifically on a range of themes and mathematical specialisms and at least two of his books are regarded as "classics." This, then, is the context in which the author seeks to reflect upon his own personal and mathematical development, whilst simultaneously attempting to make sense of the tumultuous times through which he has lived — no small task for any autobiographer but, then again, Davis had significant help from a ghost-writer by the name of Thomas Jefferson (deceased), which shall later be explained.

Having read later parts of this book, readers will perceive that it could easily have been given an alternative title such as "From Math to Polymath," for it treats many profound topics of a philosophical/mathematical/artistic nature. For example, it leads us towards dialectical consideration of themes such as:

Mathematical symbols vs. Icons.

The Deterministic vs. the Probabilistic.

Platonic Philosophies of Mathematics vs. Humanistic Philosophies.

etc.

The author places mathematics in a historical context and acquaints us with many mathematicians of the distant and recent past. He considers, amongst other issues, the place of mathematics in society and speculates upon its future development. He reflects upon other cultural disciplines (art, literature) and how they stand in relation to his own area of study. He attempts to portray the development of mathematics since the time of Thomas Jefferson, with whose ghost he carries out a one-sided correspondence, addressing letters, which aspire to update Jefferson concerning the myriad of mathematical innovations since the 18th century.

The book is organised into thirteen parts, the first five of which describe the author's mathematical education whilst simultaneously portraying the personal circumstances and the political and social milieu in which it took place. One gets a very vivid impression of the various social and political tensions in the USA during the 1930s, the second world war, the cold war and through to the present day. My reaction upon reading the early parts of the book was one of considerable delight. Descriptions of family, friends and teachers, whom he encountered up to his time of graduation, are warmly and perceptively written. Also, the accounts of his mathematical learning, and his references to a variety of mathematical topics, are such that non-mathematical readers, who would be able to enjoy much of this book, may possibly be inspired to commence study of the subject as a source of enjoyment and intellectual stimulation. Many later parts of the book could be construed as "essays" on the sort of themes described above and they can be read in isolation or as part of a continuous reading of the whole.

It seems a contradiction in terms to say that this book is readable and enjoyable whilst being intellectually challenging but the art of great popularisers of mathematics is to entertain and motivate yet never trivialise. Accordingly, I wholeheartedly recommend the purchase of a copy of this warmly humane and thought-provoking book since its total message cannot possibly be encapsulated within a review such as this. It is a book that deserves to be revisited time and again.

Peter Ruane ([email protected]) has been involved with the teaching of mathematics at many levels since 1966. He has taught children from five to eighteen years old and he was employed in the field of mathematics education of teachers for the past twenty years. He is a regular reviewer for the UK journal, the *Mathematical Gazette*, from which he won the Seventh Annual Writing Award for his article 'The Curious Rectangles of Rollet and Rees'. His other publications are in the field of mathematics education.

Prior to his career in mathematics education he was variously employed, for a period of twelve years, as a docker (stevedore), slaughterman, bartender, farm labourer, hotel worker, shop assistant, etc.