Let us consider what the domain of science covers. There is a hierarchy in science arranged like a tree: physics, chemistry, geology, biochemistry, biology. Each one is constrained by the rules of its parent, yet at the same time exhibits behaviors that go beyond those of the parent.
But physics is itself the child of mathematics, and hence mathematics is the sine qua non of science. An event that cannot be counted, weighed, timed or otherwise measured is not amenable to the procedures of science; it is a one time event, a fluke, a chance confluence of nature. Science only considers phenomena that can be treated mathematically in some way.
Some observers claim that mathematics is a language created by humans to describe the world around them. This seems unlikely. Mathematicians take simple starting points and construct concepts that have no connection with reality – until one day, it is discovered that they do. Eugene Wigner wrote of this in his paper “The Unreasonable Effectiveness of Mathematics in the Natural Sciences“. Mathematics was not created to explain the world around us; instead, that world is found to match mathematics. This is a very peculiar state of affairs.
But it is not just this uncanny matching that is strange – it is the fact that there is any matching at all. Why should gravity obey an inverse square law? Why is road damage proportional to the fourth power of the axle weight? The physical universe and mathematics are inextricably intertwined, and I suggest that the two are in fact one – that the physical universe is the manifestation of mathematics. At the macroscopic level, the equations are impossibly complex, but at the level of an elementary particle such as an electron or photon, there is no difference between the particle and its mathematical description – the particle is its wave function.
Several conclusions follow from this. One is that the universe is, in some sense, eternal. The laws of mathematics are not going to be repealed in the future, and there was no time when they did not apply.
Secondly, consider Gödel’s Incompleteness Theorem, which states that in a mathematical system of sufficient complexity, there are propositions which cannot be proved true or false. The system can be extended to cover these propositions, but further undecidable propositions will always arise.
That is to say that science is necessarily an incomplete explanation of the world.