Remember what we said mathematics was and what its purpose was. Mathematics is the science of method that allows us to identify attributes and relationships, in particular, those that cannot be directly perceived. Much of physics deals with these attributes and relationships. Much of physics pertains to attributes and relationships which we cannot directly perceive. But which we can measure using mathematics.
Only once we perceive these attributes and relationships can we make any study of them. Once we have measured these can we identify the nature of these attributes and relationships. Only once we have measured them can we understand any causal relationships between entities and their actions.
We cannot grasp these important causal relationships until we have measured the attributes of entities and the various relationships between them. It is only by studying such, that we can grasp such casual relationships. It is only by using mathematics that we can understand the attributes of physical entities and the relationships between them.
Without mathematics, physics would consist only of directly observable attributes of entities and the directly observable relationships between them. We would know that fire is hot, but we would be unable to know at what point water vaporizes.
Similarly, we would know that if we drop things that they fall. We would not know that objects fall at the same rate and we would be unable to form the concept of gravity pulling things down.
Without calculus, Newton would not have been able to formulate his laws of motion and we would know very little or nothing about mechanics.
If we were not able to apply mathematics, we would have no ability to measure the properties on the scale of atoms or molecules. We would, therefore, know almost nothing about the atomic world or chemistry. In fact, we would be unable to measure the attributes of most things physics deals with. Nor would be be able to measure the relationships between many of these things.
This is why physics is such an inherently mathematical subject. Because so many facts of physics must first be uncovered by using mathematics to identify attributes of things and the relationships between them.
But, is this where physics ends? Once we identify the laws of motion, should we just take them as a given and not attempt to explain them? Once we have stated that this is how gravity works, should we just leave it there?
No, of course not. Mathematics describes attributes and relationships. But it does not tell us the nature of the things we are describing and why they seem to work the way they do.
In physics, we want to understand the nature of things and the mechanisms of what we observe. We want to understand how electricity works, why the atom works the way it does, how gravity works and so on. By understanding the nature of the entities involved and how they interact.
Once we understand the entities involved and how they interact, we can explain what we observe. We can explain mechanisms and why the physical world works as it does.
But, how far can we go? Say we reduce the atom to, say, the actions of ropes. We show that everything in physics can be shown to be the result of intertwined ropes. Do we need to explain what the ropes are made of and why they work the way they are? What if something else makes up the ropes? Well, what is that made out of? Are we trapped constantly explaining every new thing we find?
(We are not saying that this is necceasirly true. The Rope Hypothesis may well not be true. This is simply for illustrative purposes.)
No. There is such a thing as the most basic physical entities, the existence of which must be accepted. Not everything is made out of more simple things. At some point, there must be “the most fundamental thing” of which everything else is made or which simply exists as an independent form of matter.
There is a story where a Hindu was asked what holds up the World Turtle, upon which the world rests. This person apparently retorted that it is turtles all the way down!
There is no such thing as physics all the way down. Eventually, we would reach a point where we have to stop trying to explain. We understand the most basic forms of existents there are and we cannot go any further down.
That is it, these are the most simplest things they are. They have a nature, they work this way, that is it. There is no explaining how or why, that is just how it is. We must accept it as an axiom.
Now, will we ever get to that point? I am not so sure. I am not sure how we would even know we had. It seems it would be difficult to show this. But, there is such a point, even if we do not reach it.
So, in short, physics is the study of matter. Not concepts. And not of magic. Not of mathematical equations. Matter is in a sense, all that really matters in physics…