An essay on the foundations of geometry · Bertrand Russell
THE AXIOMS OF METRICAL GEOMETRY.
Now a number, to begin with, is a whole consisting of smaller units, all of these units being qualitatively alike. In order, therefore, that a continuous quantity may be expressible as a number, it must, on the one hand, be itself a whole, and must, on the other hand, be divisible into qualitatively similar parts. In the aspect of a whole, the quantity is _intensive_; in the aspect of an aggregate of parts, it is _extensive_. A purely intensive quantity, therefore, is not numerable--a purely extensive quantity, if any such could be imagined, would not be a single quantity at all, since it would have to consist of wholly unsynthesized particulars. A measurable quantity, therefore, is a whole divisible into similar parts. But a continuous quantity, if divisible at all, must be _infinitely_ divisible. For otherwise the points at which it could be divided would form natural barriers, and so destroy its continuity. But further, it is not sufficient that there should be a possibility of division into mutually external parts; while the parts, to be perceptible as parts, must be mutually external, they must also, to be knowable as _equal_ parts, be capable of overcoming their mutual externality.
For this, as we have seen, we require superposition, which involves Free Mobility and homogeneity--the absence of Free Mobility in time, where all other requisites of measurement are fulfilled, renders direct measurement of time impossible. Hence infinite divisibility, free mobility, and homogeneity are necessary for the possibility of measurement in _any_ continuous manifold, and these, as we have seen, are equivalent to our three axioms. These axioms are necessary, therefore, not only for spatial measurement, but for all measurement. The only manifold given in experience, in which these conditions are satisfied, is space. All other exact measurement--as could be proved, I believe, for every separate case--is effected, as we saw in the case of time, by reduction to a spatial correlative. This explains the paramount importance, to exact science, of the mechanical view of nature, which reduces all phenomena to motions in time and space. For number is, of all conceptions, the easiest to operate with, and science seeks everywhere for an opportunity to apply it, but finds this opportunity only by means of spatial equivalents to phenomena[177].
Ideas discussed in this chapter
Text from Project Gutenberg, public domain in the United States.