MATHEMATICS AND THE WORLD 259
times greater than b, this relation is asymmetrical, exact, unique, and specific.
The above simple remarks are not entirely orthodox. That 5/1 ■ - 5 is very orthodox, indeed; but that numbers, in general, represent indefinitely many exact, specific, and unique, and, in the main, asymmetrical relations is a structural notion which necessitates the revision of the foundations of mathematics and their rebuilding on the basis of new semantics and a future structural calculus. When we say 'indefinitely many', this means, from the reflex point of view, 'indefinitely flexible', or 'fully conditional' in the semantic field, and, therefore, a prototype of human semantic reactions (see Part VI). The scope of the present work precludes the analysis of the notion of the lately disputed 'irrational'; but we must state that this revision requires new psycho-logical and structural considerations of fundamental 'logical' postulates and of the problems of 'infinity'. If, by an arbitrary process, we postulate the existence of a 'number' which alters all the while, then, according to the definition given here, such expressions should be considered as functions, perhaps, but not as a number, because they do not give us unique and specific relations.
These few remarks, although suggestive to the mathematician, do not, in any way, exhaust the question, which can only be properly presented in technical literature in a postulational form.
It seems that mathematicians, no matter how important the work which they have produced, have never gone so far as to appreciate fully that they are willy-nilly producing an ideal human relational language of structure similar to that of the world and to that of the human nervous system. This they cannot help, in spite of some vehement denials, and their work should also be treated from the semantic point of view.
Similarly with measurement. From a functional or actional and semantic point of view, measurement represents nothing else but a search for empirical structure by means of extensional, ordered, symmetrical, and asymmetrical relations. Thus, when we say that a given length measures five feet, we have reached this conclusion by selecting a unit called 'foot', an arbitrary and unspeakable affair, then laying it end to end five times in a definite extensional order and so have established the asymmetrical, and, in each case, unique and specific relation, that the given entity represents, in this case, five times as many as the arbitrarily selected unit.
Objection may be raised that the formal working out of a definition of numbers in terms of relations, instead of classes, would be very