Mathematics is something extraordinarily precise. Moreover, it is more accurate in different areas than in some areas we know less about it. People often find mathematics very abstract. But mathematics can describe reality as we understand it.
When we think about solid stuff like an umbrella scientifically, we picture molecules in our heads. We know molecules are made of atoms. Atoms are made out of nuclei and electrons going around. But what about the nucleus? What is an electron? At that stage, the best we can do is to describe some mathematical structure.
Mathematics describes things in the ordinary physical world. We use equations to describe physics. We cannot understand equations without mathematics. And those equations are always extremely accurate. Dirac’s equation describes the electron and corks very precisely. Richard Feynman calculated the distance between New York and Los Angeles with an accuracy of less than human hair thickness. Einstein produced a theory to measure how accurate is this particular system of two stars going around neutron stars, and it’s has a precision of 10¹⁴. These are unbelievably precise.
Those equations and numbers are telling us these are exact theories. Whether we’re talking about the structure of vast entities like neutron stars over great distances or the form of an electron, they are correct. We will always have this extraordinary precision between mathematics on the macroscopic level with neutron stars and at the microscopic level, within the electrons and mathematics.
The equations that mathematics use are mostly small. Yes, they might be difficult to understand, but they are not complicated to get the ideas. For instance, to understand Einstein’s theory, you have to understand curved space, which is not easy to get your mind. But once you get over that, it’s about the simplest thing you could write down.
This precision of mathematics dates back to the ancient Greeks, where they develop mathematical ideas as a field of study stimulated to some degree by physical reality. Their mathematics was extraordinary, but it wasn’t precise as Einstein’s theory. Greek mathematicians didn’t realize that the geometry of the world could have been anything else. They study mathematics as a purely intellectual activity. Geometry was a significant input for them. Then they developed facts of numbers like the properties of numbers, notions of prime numbers, the fact there are infinitely many prime numbers, etc.
Ever since then, mathematics has been had its own life in a sense. It seems to have an independent reality than the ordinary kind of reality, like umbrellas that we usually think of as real.
The mathematical reality is something different. It’s sometimes referred to as a platonic world. People have a lot of trouble thinking of that as real. Mostly, philosophers worry about that. But what does that platonic reality mean? It’s a different kind of reality from the reality of the physical world.
There should be different ways of looking at reality. There’s the reality of our mental experience. This reality interrelates with physical reality. Then the mathematical reality of this platonic world gives reality to these notions.
For instance, the mathematical facts like there is no largest prime number is something independent of ourselves. This fact has always been true. It didn’t become real as soon as somebody found that fact.
Sometimes people think that the reason we have good mathematical laws of physics is the best way we can come to understand the world. But it’s something more than that. I don’t think there’s any way of understanding what we see around us. I don’t know how else to put it structures that would seem impossible to relate to the physical world.
Whether human beings invented mathematics to impose our way of thinking on the physical world or discover it because it’s already out there, mathematics is already there.