Chapter XX (part 5)

Consequently, the number of points in the cube, a 3 is equal to ∞ ∞∞ , this expression is equal to the expression ∞ ∞ and ∞, i.e., this means that an infinity continues to grow, remaining at the same time unchanged . ______ Thus in transfinite numbers, we see that two magnitudes equal separately to a third, can be not equal to each other. Generally speaking, we see that the fundamental axioms of our mathematics do not work there, are not there valid. We have therefore a full right to establish the law, that the fundamental axioms of mathematics enumerated above are not applicable to transfinite numbers, but are applicable and valid only for finite numbers. We may also say that the fundamental axioms of our mathematics p. 251 are valid for constant magnitudes only. Or in other words they demand unity of time and unity of place . That is, each magnitude is equal to itself at a given moment. But if we take a magnitude which varies, and take it in different moments, then it will not be equal to itself. Of course, we may say that changing, it becomes another magnitude, that it is a given magnitude only so long as it does not change. But this is precisely the thing that I am talking about. The axioms of our usual mathematics are applicable to finite and constant magnitudes only. Thus quite in opposition to the usual view, we must admit that the mathematics of finite and constant magnitudes is unreal, i.e., that it deals with the unreal relations of unreal magnitudes; while the mathematics of infinite and fluent magnitudes is real, i.e., that it deals with the real relations of real magnitudes. Truly the greatest magnitudes of the first mathematics has no dimension whatever, it is equal to zero , or a point, in comparison with any magnitude of the second mathematics, ALL MAGNITUDES OF WHICH, DESPITE THEIR DIVERSITY, ARE EQUAL AMONG THEMSELVES. Thus both here, as in logic, the axioms of the new mathematics appear as absurdities: A magnitude can be not equal to itself . A part can be equal to the whole, or it can be greater than the whole . One of two equal magnitudes can be infinitely greater than another . All DIFFERENT magnitudes are equal among themselves . A complete analogy is observed between the axioms of mathematics and those of logic. The logical unit—a concept—possesses all the properties of a finite and constant magnitude. The fundamental axioms of mathematics and logic are essentially one and the same. They are correct under the same conditions, and under the same conditions they cease to be correct. Without any exaggeration we may say that the fundamental axioms of mathematics and of logic are correct only just as long as mathematics and logic deal with magnitudes which are artificial , conditional , and which do not exist in nature. p. 252 The truth is that in nature there are no finite , constant magnitudes, just as also there are no concepts . The finite, constant magnitude, and the concept are conditional abstractions, not reality, but merely the sections of reality, so to speak. How shall we reconcile the idea of the absence of constant magnitudes with the idea of an immobile universe? At first sight one appears to contradict the other. But in reality this contradiction does not exist. Not this universe is immobile, but the greater universe, the world of many dimensions, of which we know that perpetually moving section called the three-dimensional infinite sphere. Moreover, the very concepts of motion and immobility need revision, because, as we usually understand them with the aid of our reason, they do not correspond to reality. Already we have analyzed in detail how the idea of motion follows from our time-sense, i.e., from the imperfection of our space-sense .