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Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of "figures and numbers".

In the formalist view, it is the investigation of axiomatically defined abstract structures using symbolic logic and mathematical notation; other views are described in the article on the philosophy of mathematics.

Mathematics might be seen as a simple extension of spoken and written languages, with an extremely precisely defined vocabulary and grammar, for the purpose of describing and exploring physical and conceptual relationships.

Although mathematics itself is not usually considered a natural science, the specific structures that are investigated by mathematicians often have their origin in the natural sciences, most commonly in physics.

However, mathematicians also define and investigate structures for reasons purely internal to mathematics, because the structures may provide, for instance, a unifying generalization for several subfields, or a helpful tool for common calculations.

Finally, many mathematicians study the areas they do for purely aesthetic reasons, viewing mathematics as an art form rather than as a practical or applied science. Einstein referred to the subject as the Queen of the Sciences in his book Ideas and Opinions. Mathematics is considered absolute, without any reference.