The Apes—Points To Ponder
The Map is not the Territory
There are several things to be concerned about with mathematics. The first is to note that in terms of the models it can create, mathematics can be divided roughly into two parts: discrete mathematics and continuous mathematics.
Discrete mathematics is the study of mathematical structures made up of separate components, individual items, like quanta. So the objects studied in discrete mathematics, such as integers and statements in logic, have distinct, separated values.
In contrast, continuous mathematics deals with objects that vary smoothly, without gaps. An example of this continuity is the simple equation y = 2x. The two linked variables, y and x, in the equation are continuous.
We can ask the question: Is reality continuous or discrete?
The evidence from quantum mechanics is that reality is discrete. So, for example, a spectrometer viewing the light emissions form a particular substance shows us spectral lines rather than a continuity of wavelengths. This, incidentally, accords with the objective science view of reality. However the way our minds model the world is necessarily that way. We see, for example, an iron bar. It is a thing and hence discrete. However we see it and conceive of it as continuous from end to end. We can imagine that it is composed of atoms—discrete things—but we do not know for sure. Even if we possessed a microscope that was powerful enough, and could clearly see the atoms, we could not know for sure whether the space between them was really empty rather than containing some kind of material or energy.
While it is the case that discrete and continuous mathematics can be used together in some contexts, a mathematical model of a real situation always depicts the world as one or the other, discrete or continuous. If the models work well, it will be valued. For example, the mathematical models that were used to calculate artillery tables were valued by the military because there were very accurate within practical parameters, but they were not perfect.
We adopt the same attitude to the problem of infinity—a concept that emerges in both continuous and discrete mathematics. We require the concept, for example, to establish calculus. We cannot demonstrate infinity in the real world, we can only presume it. Nevertheless, if we’re careful in using it, we can employ it productively in mathematics, and employ the mathematical models we create productively. Mathematics can be right within its own context of modeling and mapping. However the map is not the territory.