Mathematics has a very interesting role in theoretical physics.
Let me use as an example the Rydberg equation for calculating the electron energies for the hydrogen atom.
Rydberg observed the spectral lines of hydrogen and found an equation which predicted the various frequencies. This was found with no underlying theoretical basis for choosing the equation.
Many years later Niels Bohr supplied a theory which resulted in that same equation.
This raises the following question: "Does the lack of a theory render the mathematics INCORRECT?" Another question is: "Does the existence of a theory render the mathematics CORRECT?"
I would claim that the answer to both questions is: "No."
My rationale lies in the fact that it would appear that every equation describing the behavior of the physical world is simply an approximation based on various simpifications and assumptions.
For example, so-called "Newtonian physics" offers calculations which would allow one to predict the observed velocity of one object when viewed from the frame of reference of a second, moving object. Would such mathematical calculations be "correct"? The answer is that they are good enough for some circumstances and not for others.
Einstein developed a different mathematical expression for calculating relative velocities. His equation works just as well as Newton's but also is accurate for relative and observed velocities close to the speed of light.
Does this mean Newton's equations are WRONG and Einstein's equations are RIGHT?
The best answer to that question is that "It depends." For many circumstances, Newton is just as good as Einstein. When the day comes that we understand what "dark matter" really is, we just might find Einstein's equations as lacking then as we sometimes find Newton's lacking today.
The point I am trying to make is that it is just possible that mathematics NEVER describes any existing physical system exactly. But for some purposes, the math is really quite good. As for your original question, the Rydberg equation was probably able to predict the frequency of spectral lines that Rydberg was unable to observe.
That question wasn’t for you. I know all this! It was for the person on the thread denigrating math in theoretical physics.
May as well address this to both.
There is a qualitative difference in the way math is applied in physics today vs. in the past.
In the past, the applied math had real physical ties to reality.
With the relativity era that all changed. With relativity, one changes reality to match the math. It’s a fundamentally different process than the one used to derive equations from observation in the absence of underlying theory.
Today, the underlying theory is the starting point and observations are both altered and outright fabricated in order to make them match the theory.
Enter the math.
Math reduces the physical world to symbols. However, the symbol is only meaningful when used as an accurate proxy for the physical phenomena underlying it.
Today what we see is a systematic misuse of mathematical symbols in defiance of the physical realities. The presence of infinities, the abuse of time, requirements for extra dimensions, and so on and so forth.
To me, what is disrespectful of the impact of mathematics on astrophysics is not the challenging of the current paradigm, but its uncritical acceptance.
Throw in accelerating institutional corruption and peer review practices of increasingly poor quality, and a cutting edge consisting largely of claims derived from unrepeatable experiments, and what we have here is a false description of the universe on its final legs, as observations continue to blow irreparable holes in the model.
In the big picture, the fundamental problems are even higher level that that, though. The big picture problem is that our conception of the universe derives from an era before electricity. Which is kind of a big deal in terms of how the universe actually operates.
The mechanistic gravity-centric universe has failed just as completely as the geo-centric one. It is time for new conceptions that include from the start the new forces that we now know exist abundantly in creation.