ON THE NOTION OF INFINITY 207
Existing mathematical symbolism and formalism lead to identification of both fundamentally different semantic processes and introduce a great deal of avoidable confusion. A A orientation will allow us to retain mathematical symbolism and formalism, but will not allow the identification of the semantic process of passing from number to number, which passing is not a number, with the result of this process which, in each case, becomes a definite and finite number.
It becomes obvious that the A terminology and present standard notions of 'number' identify the semantic process with its result, an identification which must ultimately be disastrous. The semantic process is thus potentially infinite, but the passing from n to n + 1 characterizes the semantic process, not number; numbers representing only finite results of the indefinitely extended semantic process.
A A analysis without identification discloses, then, that only the semantic process can be indefinitely extended, but that the results of this process, or a number in each case, must be finite. To speak about an 'infinite' or, as it is called, 'transfinite' 'number', is to identify entirely different issues, and involves very definite self-contradictions in m.o terms. The existing mathematical terminology has been developed without the realization of A issues and the multiordinality of terms and leads automatically to such identifications. As long as mathematicians do not consider A issues, the problems of mathematical infinity will remain unsolved and hopeless; and yet, without a scientific theory of infinity, all of mathematics and most of science would be entirely impossible. A A clarification of these problems involves a new semantic definition of numbers and mathematics, given in Chapter XVIII, which eliminates a great many mysteries in connection with mathematics and does not allow these dangerous and befogging identifications.
From a A point of view, we must treat infinity in the first can-torian sense; namely, as a variable finite, the term variable pertaining to the semantic process but not to number, the term finite pertaining to both the semantic arrest of the infinite semantic process, and so characterizing also its result; namely - a number.
In the meantime, the numerical technique is indefinitely flexible in the sense that no matter how great a number we take, we always can, by a semantic process, produce a greater number, and no matter how small the difference between two numbers might be, we always can find a third number which will be greater than the smaller, and smaller than the given greater number. Thus, we see that the numerical technique is such as to correspond in flexibility exactly to the semantic processes, but