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We all begin gathering mathematical ideas in early childhood, when we discover that our two hands match, and later when we learn that other children also have grandmothers, so that this is an abstract relationship that a child might bear to an older person, and then that ‘uncle’ and ‘cousin’ are of this type also…

As the reader goes through it, this book [Conceptual Mathematics] may add some treasures to the collection, but that is not its goal. Rather we hope to show how to put the vast storehouse in order, and to find the appropriate tool when it is needed, so that the new ideas and methods collected and developed as one goes through life can find their appropriate places as well. There are in these pages general concepts that cut across the artificial boundaries dividing arithmetic, logic, algebra, geometry, calculus, etc. There will be little discussion about how to do specialized calculations, but much about the analysis that goes into deciding what calculations need to be done, and in what order. Anyone who has struggled with a genuine problem without having been taught an explicit method knows that this is the hardest part.

This book could not have been written fifty years ago; the precise language of concepts it uses was just being developed. True, the ideas we’ll study have been employed for thousands of years, but they first appeared only as dimly perceived analogies between subjects. Since 1945, when the notion of ‘category’ was first precisely formulated, these analogies have been sharpened and have become explicit ways in which one subject is transformed into another. It has been the good fortune of the authors to live in these interesting times, to see how the fundamental insight of categories has led to clearer understanding, thereby better organizing, and sometimes directing the growth of mathematical knowledge and its applications.

Preliminary versions of this book have been used by high school students and university classes, graduate seminars, and individual professionals in several countries. The response has reinforced our conviction that people of widely varying backgrounds can master these important ideas.

F. W. Lawvere & S. H. Schanuel (1997). Conceptual Mathematics: A First Introduction to Categories. Cambridge University Press, p. xiii.

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