While the relationship between mathematics and science is a contented one

While the relationship between mathematics and science is a contented one, some arguing that mathematics itself is a branch of science it is imperative to take into consideration the fundamental nature of each one before coming to any conclusions of what distinguishes them both from each other.
The philosophical drive behind science is empiricism meaning; for deductions to be made or a hypothesis to be proven some sort of verification is needed. These are generally obtained by conducting researches, noting down observations and other means of experimenting. The problems which arise with this sort of reasoning that in theory, scientific hypotheses are never really proven but are nothing more than guesses, or preferably an educated and testable predication derived from experimentation and observation which can later be disproved in light of new tests or when a hypothesis fails to arrive at the predicted result.
Contrary to science, mathematics applies a more abstract reasoning resulting in the removal of all but the necessary elements required allowing these elements to be generalized and understandable. Based on abstraction a mathematical model can be drawn up to explain systems by testing the relationship of different components in it, make behavioral predictions, etc. Mathematical models can be incorporated and aid in supporting a hypothesis.