Operationally defined, no infinite sets can exist, and the idea of infinite sets of different sizes is also impossible. However, what is possible, is that in the generation of positional names, some sets of names must be produced faster and some slower, so that at any given moment the size of the sets will differ. A number is the name of a position. A position can be applied to any dimension, and any number of dimensions. We can create mathematics using triangulation (the greek model) beginning with sizes, and using ratios of sizes, or we can create mathematics of positions and apply them to lengths in any dimension. This property defines the reciprocity of measurement. So we can start with the mathematics of triangulation, then apply it to scale (just as we apply money to prices as a standard unit), and then we can scale the unit reciprocally.

If we taught mathematics operationally then all this fictionalism that makes math nonsense to so many people would disappear.

So for example, we can measure art by triangulation but we cannot measure it by numbers. We can measure distances by either triangulation or numbers. There are phenomenon we cannot measure by numbers other than probability (causal density is too high).

If we taught mathematics operationally then all this fictionalism that makes math nonsense to so many people would disappear.

So for example, we can measure art by triangulation but we cannot measure it by numbers. We can measure distances by either triangulation or numbers. There are phenomenon we cannot measure by numbers other than probability (causal density is too high).