274 V. MATHEMATICS A A LANGUAGE
ing. That is why we are groping - the only method possible under such conditions being the animalistic trial-and-error methods, swaying masses by inflammatory speeches because reason has nothing to offer, being tied up by the old verbal structure to the older consequences based on animalistic and fundamentally false-for-man assumptions.
4) In the old A, el, two-valued system, agreement is theoretically impossible; so one of the main, and perhaps revolutionary, semantic departures from the old system is the fact that a non-el oo-valued A-system, based on fundamental negative premises, leads to a theory of universal agreement, which is based on a structural revision of our languages, producing new and undisturbed s.r, which eliminate the copying of animals in our nervous reactions.
The subject matter of this chapter divides, naturally, into three interconnected semantic parts. In the first, we shall recall a few general notions, known in the main but seldom taken into consideration, reformulated in a language of different structure. In the second, I shall indicate how the most important mathematical disciplines, which traditionally and, in the opinion of the majority, could hardly be called mathematical, represent a scientific and exact formulation of the general 'thinking' process. In this connection, a few words will be said about the theory of aggregates, and a little more about the theory of groups. This latter theory, with its implications and applications, leads to a reformulation of mathematics on quite obvious psycho-logical grounds, bringing mathematics into the closest relationship to the general processes of mentation. Finally, in the third part, I shall indicate the astonishing and quite unexpected physiological fact of the similarity of the structure of mathematics with the structure and function of our nervous system.
The intelligent layman should be reminded that, although he needs to know about mathematics, the minimum given here, supplemented, perhaps, by a few most elementary and fascinating books on mathematical philosophy, given in the bibliography to this volume, yet he does not need, and probably never will need, more technical mathematics than is given in the high schools and supplemented by the fundamental notions of the differential calculus. For directly we treat all languages, mathematics included, from a more general (and, at present, perhaps, the most general) aspect; namely, structure; the reader will obtain all the essential psycho-logical benefits of modern science by absorbing the ^-system and habits, which will result in completely novel standards of evaluation and distinctly modern and adult s.r.
The last is of extreme and unrealized importance. In fact, its importance cannot be fully appreciated until we actually acquire such reactions,