section_path: Philosophy/Proposed definitions
section_name: Proposed definitions
paragraph_text: Proposed definitions
  
  There is no general consensus about a definition of mathematics or its epistemological status — that is, its place among other human activities. A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. There is not even consensus on whether mathematics is an art or a science. Some just say, "mathematics is what mathematicians do". This makes sense, as there is a strong consensus among them about what is mathematics and what is not. Most proposed definitions try to define mathematics by its object of study.
  
  Aristotle defined mathematics as "the science of quantity" and this definition prevailed until the 18th century. However, Aristotle also noted a focus on quantity alone may not distinguish mathematics from sciences like physics; in his view, abstraction and studying quantity as a property "separable in thought" from real instances set mathematics apart. In the 19th century, when mathematicians began to address topics — such as infinite sets — which have no clear-cut relation to physical reality, a variety of new definitions were given. With the large number of new areas of mathematics that appeared since the beginning of the 20th century and continue to appear, defining mathematics by this object of study becomes an impossible task. For example, in lieu of a definition, Saunders Mac Lane in Mathematics, form and function summarizes the basics of several areas of mathematics, emphasizing their inter-connectedness, and observes:
  
  the development of Mathematics provides a tightly connected network of formal rules, concepts, and systems. Nodes of this network are closely bound to procedures useful in human activities and to questions arising in science. The transition from activities to the formal Mathematical systems is guided by a variety of general insights and ideas.
  
  Another approach for defining mathematics is to use its methods. So, an area of study can be qualified as mathematics as soon as one can prove theorems — assertions whose validity relies on a proof, that is, a purely-logical deduction. Others take the perspective that mathematics is an investigation of axiomatic set theory, as this study is now a foundational discipline for much of modern mathematics.