Our common use of the word “mind” invests the term with considerable obscurity and ambiguity. This is partly because, whilst many philosophers, psychologists and scientists deny any reality to the notion, many others have regarded it as the only reality Others again have regarded both mind and matter as having reality but reality of quite different types, e.g. subjective reality and objective reality. Without going into these controversies I must begin by pointing out that for a psychologist, whatever his philosophical beliefs, the mental processes of the people he is studying are as real and as objective as the interior of a star is to an astrophysicist—and as inaccessible to direct observation. Any other assumption would make nonsense not only of psychology but of the whole of education. And it has a special bearing on the problem of determining the nature of mathematics.