260 V. MATHEMATICS A A LANGUAGE
laborious, and would also involve a revision of the foundations of mathematics. This can hardly be denied; but, in the discussions of the foundations, the confusion of orders of abstractions is still very marked, thereby resulting in the manufacture of artificial semantic difficulties. Moreover, the benefits of such a definition, in eliminating the mysteries about mathematics, are so important that they by far outweigh the difficulties.
As the only possible content of knowledge is structural, as given in terms of relations and multiordinal and multi-dimensional order; numbers, which establish an endless array of exact, specific, and, in each case, unique, relations are obviously the most important tools for exploring the structure of the world, since structure can always be analysed in terms of relations. In this way, all mysteries about the importance of mathematics and measurement vanish. The above understanding will give the student of mathematics an entirely different and a very natural feeling for his subject. As his only possible aim is the study of structure of the world, or of whatever else, he must naturally use a relational tool to explore this complex of relations called 'structure'. A most spectacular illustration of this is given in the internal theory of surfaces, the tensor calculus., described in Part VIII.
In all measurements, we select a unit of a necessary kind, for a given case, and then we find a unique and specific relation as expressed by a number, between the given something and the selected unit. By relating different happenings and processes to the same unit-process, we find, again, unique and specific interrelations, in a given case, between these events, and so gather structural (and most important, because uniquely possible) wisdom, called 'knowledge', 'science',.
If we treat numbers as relations, then fractions and all operations become relations of relations, and so relations of higher order, into the analysis of which we cannot enter here, as these are, of necessity, technical.
It should be firmly grasped, however, that some fundamental human relations to this world have not been changed. The primitive may have believed that words were things (identification) and so have established what is called the 'magic of words' (and, in fact, the majority of us still have our s.r regulated by some such unconscious identifications) ; but, in spite of this, the primitive or 'civilized' man's words are not, and never could be, the things spoken about, no matter what semantic disturbances we might have accompanying their use, or what delusions or illusions we may cherish in respect to them.