604 VIII. ON THE STRUCTURE OF MATHEMATICS
for a time-binding class of life. They were an expression of the neurological structure and function and of the tendency toward induction.
In the beginning, names and generalizations were made from the simplest brute facts of life, and our primitive ancestors did not realize, that these crude generalizations might not have a structural validity, which they seldom doubted that they possessed, even as we today, seldom doubt. Those primitive scientists, (and we today differ very little from them), having produced terms, objectified them, and began to speculate about them. Let us examine some examples of such primitive mathematical speculations. Addition, of course, by which we generate numbers,one and one make two, two and one make three. , was all-important. They could not miss the simple fact that three, which is equal to two and one, by definition, is more than two or one. A primitive generalization; namely, that the sum is always more than the summands taken separately, was still further generalized to a postulate that a part is smaller than the whole. This generalization has hampered mathematics almost up to our own day, and for many thousands of years it prevented the discovery of the notion of mathematical infinity, which we have already discussed in Chapter XIV.
It must be noticed that such generalizations involve s.r, which are objective and un-speakable. If verbally formulated they should have a structure similar to that of the facts, otherwise they are fanciful and vicious, because not properly formulated. When formulated they become public structural facts (s.r are personal, individual, non-transmittable, and un-speakable) and so they may be criticized, improved, revised, rejected ,. All human history shows that the correct structural formulation of a problem is usually as good as the solution of it, because sooner or later a solution always follows a formulation.
After many thousands of yearsin fact, practically only the other dayit was found that these primitive generalizations were in general not valid. Negative numbers were invented, and two plus minus-one was no more three but one, The sum was no longer greater than its summands.
The usual tragedy takes place here also. A few people know the facts, but the old primitive structural s.r remain in some of these few, as well as in the great majority of us who did not even know the facts. That such structural s.r do not vanish quickly, or generally, is proven again and again throughout history. We see it very clearly in the problems of 'infinity', orgeometries, or physics. But the most pathetic sight is to see scientists who have rationalized the technique without a deeper re-education of their s.r. This is most clearly seen in the case of many writers on the foundations of mathematics, the Einstein theory or on the newer quantum mechanics. They feel in the old structural way, they rationalize in the new, hence their works are full of self-contradictions. Readers and students alike feel how 'difficult' and messy the whole subject is. As a matter of fact, the new theories are neither messy nor difficult. They are really much simpler and easier than the old theories, provided our structural s.r are purged of the primitive structural tendencies to which every one of us is heir. When this semantic re-education of our structural feelings is accomplished