characteristics. In the formulation of the last sentence, we cannot make the 'training in discovery' an educational discipline. The opposite is true in a -system, based on non-identity, as we can train simply and effectively in non-Identity, which ultimately leads to differentiation, and so discovery.
Because of the elementary, and purely semantic character of the following pages, I have often restrained myself from giving technical, supposedly 'rigorous', and often A rationalizations, which we occasionally call definitions. In a semantic and A treatment, at this pioneering stage, stressing old definitions Would be seriously confusing; and I wished to avoid such witty wittgensteinian 'definitions' as 'A point in space is a place for an argument'. In a number of instances, and for my purpose, I often avoided unsatisfactory formal definitions, preferring to depend upon the ordinary meanings of words.
For the reader who wishes to acquaint himself with an elementary theory of limits and corresponding sets of definitions, I would suggest the book of the late Professor J. G. Leathern, Elements of the Mathematical Theory of Limits (London and Chicago, 1925). This theory is based on Pascal's Calcolo Infini-tesimale, Borel's Theorie des fonctions, and Godefroy's Theorie des series. Lea them's book has been printed under the supervision of Professor H. F. Baker, F.R.S., of the University of Cambridge, and Professor E. T. Whittaker, F.R.S., of Edinburgh. I give these names for professional mathematicians, to indicate the semantic trend which underlies this particular treatment of limits and which does not greatly conflict with a outlook. This outlook may be summarized in part, in the words of Borel somewhat as follows: 'To the evolution of physics should correspond an evolution of mathematics, which, without abandoning the classical and well-tried theories, should develop however, with the results of experiments in view'. This statement implies vaguely the 'similarity of structure'., and so requires as a modus operandi the rejection of identity.
There seems to be little doubt that a complete and radical revision of the semantic aspects of human mathematical behaviour is pending. Such a revision appears to be laborious and difficult, and should be undertaken from the point of view of the theory of the unique and specific relations, called numbers. I doubt if a single man could accomplish this revision. Such an undertaking will probably be the result of group activities, and may, in the beginning, be unified by the formulation of one fundamental principle of non-identity, the disregard of which, from science down to 'mental' ills, can be found at the bottom of practically all avoidable human difficulties.
The problems are very complicated and extremely difficult, and need to be treated from many angles. At present, we have many scientific societies, grouped by their specialties; but we do not have a scientific society composed of many different specialists whose work could be unified by some common and general principle. There can be no doubt that the principle of 'identity', or 'absolute sameness in all aspects', is invariably false to facts. The main problem is to trace this semantic disturbance of improper evaluation in all fields of science and life, and this requires a new co-ordinating scientific body of many specialists, with branches in all universities. Each group would meet, say monthly, to