Math is the basis to all science. It wasn't just created as something you should learn in school. The sole reason it was created was to explain the universe and its mechanics. Newton created the fundamentals of calculus to define his laws. He knew that a force was holding massive celestial bodies together but until he put math to it it was just an unexplainable phenomenon. With his calculations we know how strong Gravity is and what effect it has on its surroundings. Math is not only a science but it is the foundation to all science. It is the language of the working of our universe, an observation of how it works.
Mathematics is scientific beyond all doubt. To be scientific, something must be "related to or pertaining to science." Due to the fact that mathematics is used in nearly scientific paper (anything with an equation or statistic), there is very little doubt about the scientific nature of mathematics. Mathematics doesn't need to use the scientific method to be scientific, only to be science. There is a difference!
It certainly is. Maths and Science are interdependent. If Mathematics wasn't there Physics and large parts of Chemistry would be entirely non-existent. Mathematics is one of my most favorite subjects and it also behaves as the backbone of Science. All the mathematicians posses sound knowledge on physics and the physicist posses sound knowledge on Mathematics. What I am trying to say is that they are interlinked.
Just that. Mathematics. It exists independent of matter and energy. It is a thing unto itself. Science is the study of the natural world, and though it uses mathematics, mathematics is not science. It is a tool that science uses. So, no. It is not science. Hope that clears things up.
P.S. Since mathematics is independent of matter and energy and universal in nature, one might ask themselves where the laws of mathematics came from. A material universe could not have created them. I would debate anyone on this topic. Laws require a lawgiver. I could make similar arguments for the laws of logic, which are also immaterial, and information and high level languages. Information requires will and intelligence. DNA is a high level language. Where did this language come from? Once again, it could not have been created by a material universe. The law of entropy forbids it. And there are no exceptions to this law. Despite what evolutionists would have you believe.
Mathematics is the basis of most scientific subjects it's what supports them and not part of them (at least the most essential and abstract forms of it). This is where I distinguish it from Applied Mathematics which may be of scientific core.
By saying that Mathematics is not a scientific subject I do not mean it doesn't resort to scientific method, just that at its centre lies a pure Logical/Philosophical meaning.
If I had to call it a "Science" I'd say a "Science aside the others".
Science is a specific system of investigation, which relies on empirical observations. Mathematics is the formal study of logical consequence. It does not have anything to do with empirical reality, and can often contradict empirical reality. But that's fine. So long as we have a set of axioms, we can see where those axioms lead, regardless of what empirical evidence tells us.
On the other hand, in order to really accept science, we need to formalize it, and all formalizations, are mathematical. Inductive methodologies rely on Bayesian inference. Deductive methodologies rely on assumption of logical consistency and extended probability theory (including possible and impossible rather than the typical 0 - 1).
So is mathematics a field of science? No. But science may be a field of mathematics.