MATHEMATICS AND THE WORLD 263
In daily language a similar device is extremely useful and has very far-reaching psycho-logical semantic effects. Thus, if we say 'pencili', 'pencila', . . . 'pencil*', we have indicated structurally two main characteristics: (1) the absolute individuality of the" object, by adding the indefinitely individualizing subscript I, 2, ... n; and (2) we have also complied with the nervous higher order abstracting characteristics, which establish similarity in diversity of different 'pencils'. From the point of view of relations, these are usually found empirically; besides, they may be invariant, no matter how changing the world may be.
In general terms, the structure of the external world is such that we deal always on the objective levels with absolute individuals, with absolute differences. The structure of the human nervous system is such that it abstracts, or generalizes, or integrates., in higher orders, and so finds similarities, discovering often invariant (sometimes relatively invariant) relations. To have 'similar structure', a language should comply with both structural exigencies, and this characteristic is found in the mathematical notation ofwhich can be enlarged to the daily language as 'Smith*', 'Fido*'., where * 1, 2, 3,. .. n.
Further objective enquiry shows that the world and ourselves are made up of processes, thus, '' is quite a different person from
' To be convinced, it is enough to look over old photographs
of ourselves, the above remark being structurally entirely general. A language of 'similar structure' should cover these facts. We find such a language in the vocabulary of 'function', 'prepositional function', as already explained, involving also four-dimensional considerations.
As words are not objectsand this expresses a structural factwe see that the 'is' of identity is unconditionally false, and should be entirely abolished as such. Let us be simple about it. This last semantic requirement is genuinely difficult to carry out, because the general el structure of our language is such as to facilitate identification. It is admitted that in some fields some persons identify only a little; but even they usually identify a great deal when they pass to other fields. Even science is not free from identification, and this fact introduces great and artificial semantic difficulties, which simply vanish when we stop identification or the confusion of orders of abstractions. Thus, for instance, the semantic difficulties in the foundations of mathematics, the problems of 'infinity', the 'irrational'., the difficulties of Einstein's theory, the difficulties of the newer quantum theory, the arguments about the 'radius of the universe', 'infinite velocities', the difficulties in the present theory., ., are due, in the main, to semantic blockages or commitment to the structure of the old languagewe may call it 'habit'which says structurally very little,