256 V. MATHEMATICS A A LANGUAGE
system and may even mark the beginning of a new period in standards of evaluation, in which mathematics will take the place which it ought to have. Certainly, there must be something the matter with our epis-temologies and 'psychologies' if they cannot cope with these problems.
A simple explanation is given by a new A analysis and a semantic definition of numbers. What follows is written, in the main, for non-mathematicians, as the word 'semantic' indicates, but it is hoped that professional mathematicians (or some, at least) may be interested in the meanings of the term 'number', and that they will not entirely disregard it. As semantic, the definition seems satisfactory; but, perhaps, it is not entirely satisfactory for technical purposes, and the definition would have to be slightly re-worded to satisfy the technical needs of the mathematicians. In the meantime, the gains are so important that we should not begrudge any amount of labour in order to produce finally a mathematical and, this time, A semantic definition of numbers.
As has already been mentioned, the importance of notation is paramount. Thus the Roman notation for numberI, II, III, IV, V, VI., was not satisfactory and could not have led to modern developments in mathematics, because it did not possess enough positional and structural characteristics. Modern mathematics began when it was made possible by the invention or discovery of positional notation. We use the symbol '1' in 1, 10, 100, 1000., in which, because of its place, it had different values. In the expression '1', the symbol means 'one unit'; in 10, the symbol '1' means ten units; in 100, the symbol '1' means one hundred units,.
To have a positional notation, we need a symbol '0', called zero, to indicate an empty column and, at least, one symbol '1'. The number of special symbols for 'number' depends on what base we accept. Thus,
For more details on notation, the interested reader is referred to the fascinating and elementary book of Professor Danzig, Number the Language of Science. Here we only emphasize what is necessary for our purpose. Every system has its advantages and drawbacks. Thus, in