The language of physics is mathematics. In fact, mathematics is the language of science as a whole, as I reiterated recently in an engaging exchange with one of our readers. (Another part of the exchange is outlined in the article "Can Science Prove Anything?")
As mentioned in that earlier article, this reader was concerned about how I instructed people to "analyze the data" in my Introduction to Scientific Method article, when I said:
use proper mathematical analysis to see if the results of the experiment support or refute the hypothesis.
His concern is that mathematics can be misleading, used to obfuscate results. In fact, he was concerned that scientists apply "arbitrary constants" to make the equations work out the way they want to. After all, he suggested, wasn't this exactly what Albert Einstein did when he proposed his cosmological constant?
Well, let's see about that ...
The role of mathematics in physics
Let's begin with the note that I said "use proper mathematical analysis." Certainly, there are many types of mathematical analysis which are not proper for a given set of data, and I am not suggesting using those. Assigning arbitrary constants would fall in line, by my way of thinking, as an improper mathematical analysis.
The reason mathematics is the language of choice for science is twofold.
First, mathematics allows for a great deal of precision. This precision isn't just in the use of a specific number to a given degree of precision, but also in the way the formal structure of the language allows those numbers to interact in very specific ways. It should always be the goal of a scientist to make their predictions and analyze their data with as much precision as possible.
Second, the mathematical structure is one of the best ways to discern patterns within the data, which non-mathematical analyses would often overlook. This isn't to say that there aren't occasionally some benefits to a non-mathematical approach, but in aggregate they are far outweighed by the benefits gained through mathematical analysis.
The concern that physicists assign arbitrary constants also deserves some greater investigation. At first, I dismissed this, because physicists actually take a great deal of care to conduct experiments to measure the value of their constants to as much precision as possible, especially for the fundamental physical constants.
However, there is a point to be made here. In the realm of theoretical physics, there are some people who make all kind of theoretical predictions, and they can do so because the values and properties of certain portions of physics equations are not yet really well-defined, because we can't yet conduct experiments that would allow us to nail down those properties. Physicists can really make a large number of fairly outlandish claims and even come up with the mathematics to support it. You can create models of inflationary universes, oscillating universes, multiple universes, fractal universes, and so on ... all by making different assumptions about the mathematics involved in describing our universe.
In fact, the case of Einstein's cosmological constant is classic demonstration of a scientist going awry in a highly mathematical theory ... or is it?
Einstein's Big Blunder: Cosmological Constant
Shortly after developing his theory of general relativity, Einstein was informed of a problem. It turned out that his theory resulted in a universe that was inherently unstable and would either expand or collapse, based upon the curvature of spacetime.
The problem is that Einstein believed (along with every other physicist and astronomer of the day) that the universe was essentially static and did not, overall, change over time. It certainly didn't expand or collapse.
In order to fix this, Einstein introduced an extra term into his equation which represented, in the words of Brian Green (in his new book The Hidden Reality), "the amount of energy stitched into the very fabric of space itself." If this value was exactly right, it would result in the static universe that he and everyone else believed existed, based on the known evidence of the time.
Years later, when the expansion of space was discovered, Einstein called this the biggest blunder of his life.
However, is this really a case where the mathematics can be seen as the failure? I don't think so.
Think about it: Einstein's mathematical theory of general relativity is the very theory that predicted the universe should be expanding. Until that point, no one had given any thought to this possibility.
Note: Some had, since the time of Sir Isaac Newton, wondered why the universe didn't collapse under gravity's attraction, but that's another situation.
The failure was not a failure of mathematics, because the mathematical prediction yielded a new insight about the changing nature of the universe which, ultimately, turned out to be true. The failure was that Einstein did not trust his mathematics enough to even investigate this possibility, but instead sought a way to negate it. Why? Because he had no evidence to support such a prediction.
In this case, the mathematics was correct while the evidence was incomplete and misleading!
So, instead of proving the case that a proper mathematical analysis isn't part of the scientific method, the case of Einstein's cosmological constant is a perfect example of mathematics being essential to scientific investigation ... when applied correctly.
In fact, Einstein was even less off base than he knew, because it appears that the universe does have a certain amount of energy stitched into its very fabric. The cosmological constant is still in play even today ... in the form of dark energy, one of the newest and most intriguing mysteries of physics!