Mathematics and the scientific method

Mathematics is essential to many sciences. One important function of mathematics in science is the role it plays in the expression of scientific models. Observing and collecting measurements, as well as hypothesizing and predicting, often require mathematical models and extensive use of mathematics. Mathematical branches most often used in science include calculus and statistics, although virtually every branch of mathematics has applications, even "pure" areas such as number theory and topology. Mathematics is fundamental to the understanding of the natural sciences and the social sciences, all of which rely heavily on statistics. Statistical methods, comprised of accepted mathematical formulas for summarizing data, allow scientists to assess the level of reliability and the range of variation in experimental results.
Whether mathematics itself is properly classified as science has been a matter of some debate. Some thinkers see mathematicians as scientists, regarding physical experiments as inessential or mathematical proofs as equivalent to experiments. Others do not see mathematics as a science, since it does not require experimental test of its theories and hypotheses. In practice, mathematical theorems and formulas are obtained by logical derivations which presume axiomatic systems, rather than a combination of empirical observation and method of reasoning that has come to be known as scientific method. In general, mathematics is classified as formal science, while natural and social sciences are classified as empirical sciences.