Computation in Science (Second Edition). Konrad Hinsen

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      The use of computation for understanding has been emphasized in the context of physics by Sussman and Wisdom [13]. They developed an original approach to teaching classical mechanics by means of computation [14], which is available online, as is complementary material from a corresponding MIT course. The authors used the same approach in a course on differential geometry [15], available online as well.

      The Bootstrap project uses games programming for teaching mathematical concepts to students in the 12–16 year age range. An example of computing as a teaching aid in university-level mathematics has been described by Ionescu and Jansson [16].

      Computer programming is also starting to be integrated into science curricula because of its utility for understanding. See the textbook by Langtangen [17] for an example.

      A very accessible introduction to the ideas of digital physics is given by Stephen Wolfram in his essay ‘What is ultimately possible in physics?’ [18].

References

[1] Crossley J N and Henry A S 1990 Thus spake al-Khwārizmı̄: A translation of the text of Cambridge University Library Ms. Ii.vi.5 Hist. Math. 17 103–31

[2] Risch R H 1969 The problem of integration in finite terms Trans Am. Math. Soc 139 167–88

[3] Bronstein M 1990 Integration of elementary functions J. Symb. Comput. 9 117–73

[4] Knuth D E 1974 Computer science and its relation to mathematics Am. Math. Mon. 81 323–43

[5] Victor B Kill Math http://worrydream.com/KillMath/.

[6] Victor B Media for Thinking the Unthinkable http://worrydream.com/#!/MediaForThinkingTheUnthinkable.

[7] Downey A B 2012 Think Complexity: Complexity Science and Computational Modeling 1st edn (Sebastopol, CA: O’Reilly)

[8] Xing J, Mather W and Hong C 2014 Computational cell biology: past, present and future Interface Focus 4 20140027

[9] Prusinkiewicz P, Hanan J S, Fracchia F D, Lindenmayer A, Fowler D R, de Boer M J M and Mercer L 1996 The Algorithmic Beauty of Plants The Virtual Laboratory (New York: Springer)

[10] Zuse K 1970 Calculating Space (Boston, MA: MIT Press) Technical Translation AZT-70-164-GEMIT https://www.worldscientific.com/doi/abs/10.1142/9789814374309_0036

[11] Zenil H (ed) 2013 A Computable Universe: Understanding and Exploring Nature as Computation (Singapore: World Scientific)

[12] Ifrah G 2000 The Universal History of Numbers: From Prehistory to the Invention of the Computer vol 1 (New York: Wiley)

[13] Sussman G J and Wisdom J 2002 The role of programming in the formulation of ideas Technical Report AIM-2002-018 MIT Artificial Intelligence Laboratory http://hdl.handle.net/1721.1/6707

[14] Sussman G and Wisdom J 2001 Structure and Interpretation of Classical Mechanics (Cambridge, MA: MIT Press)

[15] Sussman G J, Wisdom J and Farr W 2013 Functional Differential Geometry (Cambridge, MA: The MIT Press)

[16] Ionescu C and Jansson P 2016 Domain-specific languages of mathematics: presenting mathematical analysis using functional programming Electron. Proc. Theor. Comput. Sci. 230 pp 1–15

[17] Langtangen H P 2016 A Primer on Scientific Programming with Python Texts in Computational Science and Engineering vol 6 (Berlin: Springer)

[18] Wolfram S 2012 What Is ultimately possible in physics? A Computable Universe ed H Zenil (Singapore: World Scientific) pp 417–33 https://www.stephenwolfram.com/publications/what-ultimately-possible-physics/

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