To begin with, math was invented literally thousands of years before the idea of science was even conceived. There is solid evidence of the use of mathematic way back into ancient times. In contrast science, at best, is only a few hundred years old.
In general, the longer some subject is around the broader and more evolved we would expect that subject to be. Considering that math has been around for much longer than science we would expect math to be much more intricate than science and this is exactly what we find.
The diversity of subjects that fall into a math category is simply enormous. If you can think of a subject that involves using numbers or equations to solve a problem, there most likely has been something written about it.
Science on the other hand is comparatively quite limited. To argue otherwise would be to claim that in a few hundred years, science has been able to amass more information than thousands of years of math did. This is a simple cardinality argument. If you want to play asshole, given that mathematics has named all the integer values and that set is infinite, there are simply names for more things in mathematics that science can ever have.
But playing whose daddy has the bigger dick games is not really telling of much more than the opinions of the various pundits. But a differentiation between math and science can be made on a much more fundamental level. And this appears to be where many scientism fans fail because they don’t have a basic understanding of some of the fundamentals of mathematics.
At their cores mathematics and science are fundamentally different. Science is based on the idea that for something to be valid it must be a falsifiable idea. The idea must be subject to being challenged and there must be a way for that challenge to be tested. Mathematics on the other hand is based on ideas that are not falsifiable (also known as proven). For math the real significance of an idea is when it is shown that it is correct and can never again be challenged.
In fact if you really push a real degreed scientist to show some scientific idea is proven, he will correct you with a statement saying that nothing in science is proven correct. The best science can do is prove something is wrong.
Any idea in science is subject to being changed when new evidence is presented. However, in math, once a theory has been proven, it is not possible for new evidence to be presented. In fact this gets at exactly what a mathematical proof actually is and why the idea of a scientific proof is an oxymoron.
In both science and math, we have what is known as a hypothesis. This can be best described as a half baked idea. It’s generally vague, all the consequences of the idea have not been worked out, and nobody would ever claim that it is in any way proven.
In both fields, once all the consequences of the hypothesis are worked out and we can show that everything is consistent, we reach the level of a theory (a fully baked idea). Notice that at this level, if we assume the theory is correct, we can be assured that everything works and any possible result the theory produces makes sense.
In science the best we can do is begin to look for evidence that a theory is not correct. While evidence that doesn’t contradict some theory is important, by far, evidence that refutes a theory is much more important because it definitively shows that the theory is wrong. But science never achieves a level of certainty where it is legitimate to claim the theory if correct. The best science can do is show that some theory has not been proven wrong.
However, in math, reaching the level of a theory starts the process of trying to prove the theory. While this process may have similarities to the experimental process in science the outcomes are considerably different.
In mathematics, the process of proving a theory is a process where all possible opposing ideas are ruled out leaving the theory as the only possible solution. Because all the alternative ideas have been ruled out, the level of certainty of the theory being correct is enormously larger than what science achieves with its experimentation. In fact when math does succeed in proving a theory, we no longer call it a theory and start calling it a law or a fact to differentiate it from a theory which is by definition unproven.
Because of the rigors of being forced to find all alternative ideas and then rule each and every one out, mathematics guarantees that a proven theory can never be disproven. Put in scientific terms, that means the idea is no longer falsifiable and therefore it is now unscientific.
If math was a part of science, every time it accomplished its goals, another piece of science would suddenly become an invalid idea. While math is a useful tool to science, it is extremely misguided to think that math is a subset of science. It’s not misguided to think that math is a subset of the tools that are at science’s disposal.
But to think math is a subset of science is a kin to thinking a hammer is a subset of a carpenter. At best to make that argument you would have to be rather naive. At worst, it would be a blatant attempt to raise the level of importance of the hammer to the carpenter.
Of course, a carpenter knows what a hammer is good for, doesn’t need a pep-rally on the virtues of the hammer, and surely doesn’t argue that he’s better than everyone because he uses a hammer. Any fool can use a hammer.
If only seems to be the anti-theistic wacko’s, trying to muster some additional credibility that make the argument that math is a part of science. They are obviously only trying to lend the certainty levels found in math to science so they appear to have a better alternative to the theists they seem to hate.
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