Mathematics has always provided a swath of useful tools in science’s engagements with describing the natural world. Many great theoretical discoveries and experimental verifications have come about through manipulation of scientific statements expressed in the language of mathematics. However, the foundations of mathematics are developed in an abstract, ideal world, riddled with various assumptions about the nature of the elements being manipulated. In the example I will look into shortly, the primary mathematical assumption will be continuity, or when applied to a physical situation, the continuity of space. When on uses these mathematical tools to deduce new information from a given set of hypotheses, there is an implicit understanding that the assumptions made by the mathematical methods don’t violate the properties of the physical system under investigation. However, as our understanding of the fundamentals of physics continues to evolve, scientists are confronted with challenges to these mathematical idealizations. With these new challenges, I wish to examine the implications that this issue will have on the way mathematical methods are used in science and the role they will have in the development of new hypothesis and anticipated observations.
Under the theory of loop quantum gravity, space itself is quantized in such a way that there exists a fundamental, smallest possible volume that can physically exist (I must admit, I am omitting a lot of details here, but this is the most important result of the theory for the purposes of the entry). In this way, space can be seen as network of immensely small ‘grains’. Following from this, the notion of an infinitely small point, length, or area is not something that can be physically realizable.
This present an issue to the use of infinitesimal calculus in physical calculations. How exactly can one say, in the process of approximating a volume, that one is taking an infinite sum of infinitely thin slices of some shape, when these infinitely thin slices are something that are not something that can actually exist. In a sense, one could make the move from approximation to exact calculation by utilizing the lowest possible bound for volume.
Of course, mathematical tools need not correspond directly to the physical reality in most cases. They are birthed from a idealized world for a reason, after all. However, when these mathematical concepts are used in the process of scientific inference, the accordance between the math and the real world becomes something much more worthy of note. For example, as the conceptions of things like time and space evolve, the mathematical models one uses to describe them must change accordingly. This isn’t a mere exercise in word games, as this very change was observed with the transition from euclidean to noneuclidean geometry with the genesis of general relativity. In this case, the use of different mathematical schemas can been seen to have a direct correlation with the properties of the physical system it is being used to describe, but there is no major conceptual conflict between the tools used and the physical system itself.
This is not the case when considering LQG and the infinitesimal. Since one of the fundamental assumptions of infinitesimal calculus is the existence of an arbitrarily small volume, there exists a conflict between the conceptual basis of the mathematical and the physical system itself. What this conflict means, in essence, is that there is no way that the mathematical system can be applied to solve physical problems without modification. While in most cases, the results that come from a mathematically idealized method and the physically adapted method are negligible, when we begin to work at scales where these differences make a significant difference, this incongruity will become quite the issue.
Quite frankly, this poses not a huge issue to the progression of mathematical methods in the physical sciences. In fact, there have been a variety of elegant solutions to this issue. In many ways, since we use different tools for quantitative manipulation at different scale sizes, it makes little sense to worry about the apparent mismatch between infinitesimal calculus and LQG. Quantum calculus (also known as calculus without limits) is one solution that allows for the employment of calculus-like methods in discrete systems, but there also exists a variety of other mathematical structures like spintegers that can work just as effectively.