The conceptual cut
TL;DR: Any physical theory requires idealizations and assumptions that are hidden in the math
Since one of the goals of our project, Assumptions of Physics, is to provide a general mathematical theory for experimental science, one question we need to answer is exactly what can and cannot be formally captured, and why. There are two key problems, which I call respectively the “web of meaning” and the “conceptual cut.” In this short essay I will concentrate on the latter.
Any description of the world, whether scientific or not, will make use of words and ideas that capture objects, qualities or events. While these are typically useful in their contexts, they are ultimately quite vague, and most of the time are not, at close scrutiny, even self-consistent. The idea is that, since no concept can fully describe reality, we are forced to make a “cut” that separates what is and is not interesting within a specific context.
The easiest form of this problem is the sorites paradox: if you remove a grain from a heap of sand, is it still a heap of sand? If you say yes, then the logical conclusion is that a single grain of sand is also a heap. You could be precise, and say that a heap is more than exactly one thousand grains of sand, but that would be an arbitrary choice. As the concept of heap is hopelessly vague, you may do away with it and just count the grains. However, what exactly is a grain of sand? If I vary the chemical composition of sand by one molecule, would it still be sand? You have the same problem at a deeper level.
The introduction of continuum quantities exacerbates the problem. For example, whether an object is an orange or not may seem like a trivial question to answer. However, oranges develop gradually after the pollination of a flower. At what point shall we say that the object attached to the branch is an actual orange? Since the change is gradual, any moment would be arbitrary. One could say that “orangeness” should have three values: true, false or undetermined. One could say it should be a continuum, going from zero at pollination to one hundred at maturity. All these solutions will have the same problems: when exactly is the property undetermined? When exactly is one hundred? In fact, any model that can be described mathematically can be broken down into propositions that are either true or false, and those binary choices will present the same problem.
If one plays this game, no concept survives. Therefore we should be mindful that all concepts are useful, are meaningful, only within specific contexts. Heaps of sand will suffice as long as we disregard precise quantities. Oranges will be well defined as long as we do not talk about their development. Those cases will be beyond the conceptual cut. A geologist and a botanist, however, may require more refined vocabulary. All knowledge and all theories are contextual.
So, what does this have to do with mathematics? In a sense, nothing. A mathematical model can only be specified after we decide which features we can and want to capture. Therefore, the math itself is not even aware that the cut exists. Whether there is really a thing called grains of sand that can be modelled as a natural number, the math does not know nor care. The issue is, as I like to say, pre-formal. It will forever live in the informal specification of the theory.
The converse is that a formal theory can only be expressed once suitable idealizations are done, once we have made the conceptual cut. Consequently, if one only looks at the formal part of a theory, he will never understand the most important part: its applicability. Therefore, no amount of work on Hilbert spaces will help you understand what quantum mechanics is really describing. No amount of mathematical generalization will help you develop a theory that will work where both quantum mechanics and general relativity fail. Mathematics cannot make the conceptual cut.