A PHYSICO-MATHEMATICAL RIGOUR 759
dimensional A language and apply such an 'all' to such an infinite process then we simply produce a self-contradiction. If we apply to such a semantic process a four^dimensional 'all with a date', then we have arrested, for the 'time' being, the process, or taken a static cross section of the infinite process at that date; but then we deal with a finite. Once we are constantly conscious of abstracting in different orders, these subtle differences become quite clear and the solutions of the problems of infinity follow a similar path as the older problems of the 'infinitesimal', which also was self-contradictory, unnecessary for mathematics. When treated as a variable finite it was satisfactory and sufficient, and has proven to be a most creative notion in mathematics. In the problems of the irrational, continuity. , similar subtle identifications or non-discriminations of el, A, three-dimensional terms with , non-el, four-dimensional terms occur, which once eliminated, clear up not only the paradoxes, but some self-contradictory, often unconscious, postulates of some parts of mathematics.
Lack of space does not allow me to go into further details, except to suggest how some subtle discriminations may help to eliminate identification. In my -system the differentiation between orders of abstractions on physiological grounds, the introduction of multiordinality of terms, four-dimensional considerations . , as a structural necessity for all languages, makes the theory of types unnecessary.
For a better understanding of the present work we must at least differentiate:
A) Between numerical experience and mathematics.
B) Between languages with content and languages without content.
C) Between creative building of verbal schemes which, for the sake of generality, have no content, called pure mathematics, and the application of these schemes to actual problems, with content, called applied mathematics.
D) Between the contentless mathematics and the investigation of the foundation of mathematics which represents the investigation of the s.r of mathematicians and belongs to a future non-el psycho-logics with content.
E) Between different forms of complex adjustment which we have in common with the primitive man, and even the higher animals, and reasoning which starts with conscious observations, passing to descriptions and inferences , .
F) Between the dynamic process of relating ('thinking') on the unspeakable semantic levels, and the verbal expression of 'relations'.
G) Between the use of negative terms, disagreement, contradiction; and self-contradiction. In a -system contradictions take the form of self-contradictions.
H) Between el 'logics' expressed in terms of 'true', 'false', and modality, and the investigations of one-, two-, three-, and co-valued s.r, which become a general theory of values, and which may some day include all human interests.
I) Between the inherent circularity of 'human knowledge', which must start with sets of undefined terms, and so start with some knowledge, and circular definitions or explanations which define or explain nothing.