Chapter VII (part 1)

p. 73 CHAPTER VII The impossibility of the mathematical definition of dimensions. Why does not mathematics sense dimensions? The entire conditionality of the representation of dimensions by powers. The possibility of representing all powers on a line. Kant and Lobachevsky. The difference between non-Euclidian geometry and metageometry. Where shall we find the explanation of the three-dimensionality of the world, if Kant's ideas are true? Are not the conditions of the three-dimensionality of the world confined to our receptive apparatus, to our psyche? NOW that we have studied those "relations which our space itself bears within it" we shall return to the questions: But what in reality do the dimensions of space represents—and why are there three of them ? The fact that it is impossible to define three-dimensionality mathematically must appear most strange. We are little conscious of this, and it seems to us a paradox, because we speak of the dimensions of space, but it remains a fact that mathematics does not sense the dimensions of space. The question arises, how can such a fine instrument of analysis as mathematics not feel dimensions, if they represent some real properties of space? Speaking of mathematics, it is necessary to recognize first of all, as a fundamental premise, that correspondent to each mathematical expression is always the relation of some realities . If there is no such a thing, if it be not true—then there is no mathematics. This is its principal substance, its principal contents. To express the correlations of magnitudes is the problem of mathematics. But these correlations must be between something. Instead of algebraical a, b and c it must be possible to substitute some reality. This is the ABC of all mathematics; a, b and c are credit bills; they can be good ones only if behind them there is a real something , and they can be counterfeited if behind them there is no reality whatever. p. 74 "Dimensions" play here a very strange rôle. If we designate them by the algebraic symbols a, b and c, they have the character of counterfeit credit bills. For this a, b and c it is impossible to substitute any real magnitudes which are capable of expressing the correlations of dimensions. Usually dimensions are represented by powers: the first, the second, the third; that is, if a line is called a, then a square, the sides of which are equal to this line, is called a 2 , and a cube, the face of which is equal to this square, is called a 3 . This among other things gave Hinton the foundation on which he constructed his theory of tesseracts , four-dimensional solids—a 4 . But this is pure fantasy. First of all, because the representation of "dimensions" by powers is entirely conditional. It is possible to represent all powers on a line. For example, take the segment of a line equal to five millimetres; then a segment equal to twenty-five millimetres will be the square of it, i.e., a 2 and a segment of one hundred and twenty-five millimetres will be the cube—a 3 . How shall we understand that mathematics does not feel dimensions—that it is impossible to express mathematically the difference between dimensions? It is possible to understand and explain it by one thing only—namely, that this difference does not exist . We really know that all three dimensions are in substance identical, that it is possible to regard each of the three dimensions either as following the sequence, the first, the second, the third , or the other way about. This alone proves that dimensions are not mathematical magnitudes. All the real properties of a thing can be expressed mathematically as quantities, i.e., numbers, showing the relation of these properties to other properties.