242 IV. STRUCTURAL FACTORS IN A LANGUAGES
geometry. For instance, we have to deal with different frames or references, or different systems of co-ordinates. We find that they represent different languages and that they may introduce purely verbal statements which have nothing to do with the subject of our analysis (extrinsic characteristics). We have also found that some characteristics may appear in one form of representation and not appear so readily in another. For instance, we know that every line, except the X axis through a point 0 which is the intersection of a parabola with its X axis, cuts the curve a second time. This fact, important for us to know, appears clearly in the polar co-ordinate form of representation, but does not appear in the rectangular form of equations, although, when once a characteristic of a curve is discovered, it can be usually translated into the other co-ordinate languages. In such cases as this, a language of new structure has a kind of creative character, in that it makes some structural discoveries easier.
But the co-ordinate methods were not quite satisfactory. They introduced, too easily, too many extraneous, extrinsic characteristics, belonging to the language and not to the subject. Mathematicians decided to get away from these metaphysical 'outside' references by referring the entity to itself to become more experimental. They invented the internal theory of surfaces, a vector language where they refer the entity to itself, its curvature, its length and direction. Finally, in the extension of the vector language which is called the tensor calculus, they achieved a still larger kind of independence. Having invented three languages in which we can speak about one issue, we are now able to meet the problem of comparison of these languages. At once, most important structural and methodological problems arise.
The newer quantum mechanics are also an epoch-making linguistic structural innovation, not only in physics, but also in methodology. We have, at present, three, or, perhaps, more, quantum mechanics which speak about one subject, but in entirely different languages. I say 'three or more', because, from a methodological point of view, it is very hard, at present, to be precise, as the problems are too new and, as yet, too little analysed. Similar remarks apply to systems. Before two systems can be compared, a second system must be produced. Then we can compare them.
In our brief verbal analysis of 'space', and 'time', and 'matter', we have seen that these represent terms, or linguistic means, not objects. We have seen, also, that these antique forms of representation have very unsatisfactory structural implications. They introduce a verbal elementalism structurally absent in nature, and by a process of objecti-