Posted by: Alexandre Borovik | August 23, 2008

Preface to “Shadows of the Truth”

To give you an idea of my project under development, a book Shadows of the Truth: Metamathematics of Elementary Mathematics, I reproduce here recently updated Preface.


Toutes les grandes personnes ont d’abord été des enfants

(Mais peu d’entre elles s’en souviennent)

Antoine de Saint Exupéry, Le Petit Prince.

This book  is my first attempt to look at  mathematics from a new and somewhat unusual point of view. I have started to systematically record and analyse mathematically various psychological difficulties experienced by children in their early learning of mathematics. I hope that my approach will eventually allow me to gain a better understanding of how we — not only children, but adults, too — do  mathematics. This explains the title of the book: metamathematics is mathematics applied to study of mathematics. I chase shadows: I am trying to identify and clearly describe hidden structures of elementary mathematics which may intrigue, puzzle,  and — like shadows in the night — sometimes scare an inquisitive child.

The real life material in my research is limited to stories  that my fellow mathematicians have chosen to tell me; they represent tiny but personally significant episodes from their childhood. I directed my inquiries to mathematicians, for an obvious reason: only mathematicians  posses an adequate language which allows them to describe in some depths their experiences of learning mathematics.  So far my approach is justified mostly by warm welcome it found in my mathematician friends, and I am most grateful to them for their support. For some reason (and the reason which deserves a study on its own) my colleagues know what I am talking about.

I hope that my notes could be useful to specialists in mathematical education and in psychology of education. But I wish to make it absolutely clear: I am not from making any recommendations on mathematics teaching. Moreover, I emphasise that the primary aim of my project is to understand the nature of hardcore mainstream “research” mathematics.

The emphasis on child’s experiences makes my programme akin to linguistic and cognitive science. However, when a linguist studies formation of speech in a child, he studies language, not the structure of linguistic as a scientific discipline. When I propose to study formation of mathematical concepts in a child, I wish to get insights into interplay of mathematical structures in mathematics. Mathematics has an astonishing power of reflection, and a self-referential study of mathematics by mathematical means plays an increasingly important role within mathematical culture. I simply suggest to make a step further (or aside, or back in life) and take a look back in time, in one’s child years.

A philosophically inclined reader will immediately see a parallel with Plato’s Allegory of the Cave: children in my book see shadows of the Truth and sometimes find themselves in a psychological trap because their teachers and other adults around them see neither Truth, nor its shadows. But I am not doing philosophy; I am a mathematician and I stick to a concise mathematical reconstruction of what the child had actually seen.

My book is also an attempt  to trigger the chain of memories in my readers: every, even most minute, recollection of difficulties and paradoxes of their early mathematical experiences are most welcome. Please write to me at

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