250 V. MATHEMATICS A A LANGUAGE
The existing definitions of mathematics are not entirely satisfactory. They are either too broad, or too narrow, or do not emphasize enough the main characteristics of mathematics. A semantic definition of mathematics should be broad enough to cover all existing branches of mathematics ; should be narrow enough to exclude linguistic disciplines which are not considered mathematical by the best judgement of specialists, and should also be flexible enough to remain valid, no matter what the future developments of mathematics may be.
I have said that mathematics is the only language, at present, which in structure, is similar to the structure of the world and the nervous system. For purposes of exposition, we shall have to divide our analysis accordingly, remembering, in the meantime, that this division is, in a way, artificial and optional, as the issues overlap. In some instances, it is really difficult to decide under which division a given aspect should be analysed. The problems are very large, and for full discussion would require volumes; so we have to limit ourselves to a suggestive sketch of the most important aspects necessary for the present investigations
From the point of view of general semantics, mathematics, having symbols and propositions, must be considered as a language. Fr,om the psychophysiological point of view, it must be treated as an activity of the human nervous system and as a form of the behaviour of the organisms called humans.
All languages are composed of two kinds of words: (1) Of names for the somethings on the un-speakable level, be they external objects., or internal feelings, which admittedly are not words, and (2) of relational terms, which express the actual, or desired, or any other relations between the un-speakable entities of the objective level.
When a 'quality' is treated physiologically as a reaction of an organism to a stimulus, it also becomes a relation. It should be noticed that often some words can be, and actually are, used in both senses; but, in a given context, we can always, by further analysis, separate the words used into these two categories. Numbers are not exceptions; we can use the labels 'one', 'two'., as numbers (of which the character will be explained presently) but also as names for anything we want, as, for instance, Second or Third Avenue, or John Smith I or John Smith II. When we use numbers as names, or labels for anything, we call them numerals; and this is not a mathematical use of 'one', 'two'., as these names do not follow mathematical rules. Thus, Second Avenue and Third Avenue cannot be added together, and do not give us Fifth Avenue in any sense whatever.
Names alone do not produce propositions and so, by themselves, say nothing. Before we can have a proposition and, therefore, meanings,