Mathematics is the art of making vague intuitive ideas precise and then studying the result. Many examples can be given of the wealth of interesting matter that has arisen when a vague intuitive idea has been made precise. Half the solution to a problem is to state it precisely. Among these vague intuitive ideas is that of natural number and that of preciseness itself, there is also the vague intuitive idea of correctness. In this book we are mainly concerned with making these three vague intuitive ideas precise and with inventing a method whereby our thoughts can be either communicated to others or stored for our own memory.

It may be that our concept of natural number arises from our perception of our own heart beats; this gives us a linear (and as far as we can perceive) unending progression, without conscious beginning or ending. The concept of an unending progression of distinct things with a definite starting entity and never returning to any entity previously encountered is the essence of the concept of natural number; this concept is given to us by our perception of our own heart beats. May-be our perception of time arises from our sensing of the circulation of the blood in the brain. This gives a linear background to our thoughts and sense data. It is amusing to imagine a creature with a two-dimensional flow of fluid through its body, such a creature might have a two-dimensional conception of time and be quite unable to conceive of natural numbers as we do.