Note 147 — Introductory material

There is, however, in such a view, as I remarked above, a logical error as to the nature of magnitude. This error has been already pointed out in dealing with Erdmann[155], and need only be briefly repeated here. A judgment of magnitude is essentially a judgment of comparison: in unmeasured quantity, comparison as to the mere more or less, but in measured magnitude, comparison as to the precise how many times. To speak of differences of magnitude, therefore, in a case where comparison cannot reveal them, is logically absurd. Now in the case contemplated above, two magnitudes, which appear equal in one position, appear equal also when compared in another position. There is no sense, therefore, in supposing the two magnitudes unequal when separated, nor in supposing, consequently, that they have changed their magnitudes in motion. This senselessness of our hypothesis is the logical ground of the mathematical indeterminateness as to the law of variation. Since, then, there is no means of comparing two spatial figures, as regards magnitude, except superposition, the only logically possible axiom, if spatial magnitude is to be self-consistent, is the axiom of Free Mobility in the form first given above.