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Mathematics is the abstract study of topics such as quantity (numbers),[1] structure,[2] space,[1] and change.[3][4][5] There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.[6][7]

Mathematicians seek out patterns[8][9] and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry.


  1. 1.0 1.1 "mathematics, n.". Oxford English Dictionary. Oxford University Press. 2012. Retrieved June 16, 2012. "The science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis." 
  2. Kneebone, G.T. (1963). Mathematical Logic and the Foundations of Mathematics: An Introductory Survey. Dover. pp. 4. ISBN 0-486-41712-3. "Mathematics ... is simply the study of abstract structures, or formal patterns of connectedness." 
  3. LaTorre, Donald R., John W. Kenelly, Iris B. Reed, Laurel R. Carpenter, and Cynthia R Harris (2011). Calculus Concepts: An Informal Approach to the Mathematics of Change. Cengage Learning. pp. 2. ISBN 1-4390-4957-2. "Calculus is the study of change—how things change, and how quickly they change." 
  4. Ramana (2007). Applied Mathematics. Tata McGraw–Hill Education. p. 2.10. ISBN 0-07-066753-5. "The mathematical study of change, motion, growth or decay is calculus." 
  5. Ziegler, Günter M. (2011). "What Is Mathematics?". An Invitation to Mathematics: From Competitions to Research. Springer. pp. 7. ISBN 3-642-19532-6. 
  6. Mura, Robert (Dec 1993). "Images of Mathematics Held by University Teachers of Mathematical Sciences". Educational Studies in Mathematics 25 (4): 375–385. 
  7. Tobies, Renate and Helmut Neunzert (2012). Iris Runge: A Life at the Crossroads of Mathematics, Science, and Industry. Springer. pp. 9. ISBN 3-0348-0229-3. "It is first necessary to ask what is meant by mathematics in general. Illustrious scholars have debated this matter until they were blue in the face, and yet no consensus has been reached about whether mathematics is a natural science, a branch of the humanities, or an art form." 
  8. Steen, L.A. (April 29, 1988). The Science of Patterns Science, 240: 611–616. And summarized at Association for Supervision and Curriculum Development,
  9. Keith Devlin, Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe (Scientific American Paperback Library) 1996, ISBN 978-0-7167-5047-5