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Proofs and Predictions in Human Problem Solving

  • K. Vela Velupillai
Article
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Accepted:

Abstract

This paper suggests that Herbert Simon’s concept of proof and predictions, in the solution of problems by human’s, considered as Information Processing Agents subject to boundedly rational behaviour and satisficing objectives, is to be interpreted in terms of constructive mathematics.

Keywords

Herbert Simon Proofs Predictions Constructive mathematics Jordan curve theorem Chess GO 

JEL Classification

B31 B41 C63 C65 
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References

  1. Appel, K., & Haken, W. (1978). The four-color problem. In L. A. Steen (Ed.), Mathematics today: Twelve informal essays (pp. 153–180). New York: Springer.CrossRefGoogle Scholar
  2. Berg, G., Julian, W., Mines, R., & Richman, F. (1975). The constructive Jordan curve theorem. Rocky Mountain Journal of Mathematics, 5(2), 225–236.CrossRefGoogle Scholar
  3. Bishop, E. (1985). Schizophrenia in contemporary mathematics. In M. Rosenblatt (Ed.), Errett Bishop: Reflections on him and his research (Vol. 39)., Contemporary mathematics Rhode Island: American Mathematical Society, Providence.Google Scholar
  4. Bosch, R. A., & Smith, J. A. (1998). Separating hyperplanes and the authorship of the disputed federalist papers. The American Mathematical Monthly, 105(7), 601–608.CrossRefGoogle Scholar
  5. Bourbaki, N. (2004). Theory of sets. Berlin: Springer (This is the edition I have access to).CrossRefGoogle Scholar
  6. De Groot, A. D. (1946). Het Denken van den Schaker. Amsterdam: North-Holland.Google Scholar
  7. De Groot, A. D. (1978). Thought and choice in chess (2nd English ed.). The Hague: Mouton Publishers.Google Scholar
  8. Dowek, G. (2015). Computation, proof, machine—Mathematics enters a new age. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  9. Euwe, M. (1929). Mengentheoretische Betrachtungen über das Schachspiel. Koninklijke Nederlandske Akademie van Wetenschappen, 32, 633–642.Google Scholar
  10. Feferman, S. (1998). What does logic have to tell us about mathematical proofs? In S. Feferman (Ed.), In the light of logic, chapter 9 (pp. 177–186). Oxford: Oxford University Press.Google Scholar
  11. Fritsch, R., & Fritsch, G. (1998). The four-color theorem—History, topological foundations and idea of proof. New York: Springer.Google Scholar
  12. Gonzáles-Velasco, E. A. (1980). On limit cycles of two-dimensional analytic flows. Funkcialaj Ekvacioj, 23, 351–355.Google Scholar
  13. Hacking, I. (1973). Leibniz and descartes: Proof and eternal truths. In Dawes Hicks lecture on philosophy, Proceedings of the British Academy (Vol. LIX, 16 pages). London: Oxford University Press.Google Scholar
  14. Hales, T. C. (2007). The Jordan curve theorem, formally and informally. The American Mathematical Monthly, 114(10), 882–894.CrossRefGoogle Scholar
  15. Hales, T. C. (2013). Mathematics in the age of the Turing machine. In R. G. Downey (Ed.), Turing’s legacy. ASL lecture notes in logic (pp. 253–298). Cambridge: Cambridge University Press.Google Scholar
  16. Hardy, G. H. (1929). Mathematical proof. Mind, 38(149), 1–25.CrossRefGoogle Scholar
  17. Hassabis, D. (2017). Artificial intelligence: Chess match of the century. Review of Deep thinking: Where machine intelligence ends and human creativity begins by Gary Kasparov. Nature, 544(7651), 413–414.CrossRefGoogle Scholar
  18. Hsu, F. H., Anantharaman, T., Campbell, M., & Nowatzyk, A. (1990). A grandmaster chess machine. Scientific American, 263(4), 44–50.CrossRefGoogle Scholar
  19. Ilyashenko, Y., & Yakovenko, S. (1995). Concerning the Hilbert sixteenth problem, chapter 1. In Y. Ilyashenko & S. Yakovenko (Eds.), Concerning the Hilbert 16th problem (Vol. 165, pp. 1–19)., American Mathematical Society Translations, Series 2 Providence: American Mathematical Society.Google Scholar
  20. Jones, J. P. (1981). Classification of quantifier prefixes over diophantine equations. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 27, 403–410.CrossRefGoogle Scholar
  21. Kasparov, G. (2010). The chess master and the computer. The New York Review of Books, 57(2), 16–19.Google Scholar
  22. King, M. (2016). The end of alchemy: Money, banking and the future of the global economy. London: Little, Brown Book Group.Google Scholar
  23. Knuth, D. E. (1996). Foreward to: A = B, by Petkovšek, Marko, Herbert S. Wilf & Doron Zeilberger, A K Peters, Wellesley, MA.Google Scholar
  24. Knuth, D. E., & Moore, R. W. (1975). An analysis of alpha-beta pruning. Artificial Intelligence, 6(4), 293–326.CrossRefGoogle Scholar
  25. Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery, edited by John Worrall and Elie Zahar, Cambridge University Press, Cambridge.Google Scholar
  26. Littlewood, J. E. (1957). A mathematician’s miscellany (reprint edition, with ‘minor corrections’). London: Methuen & Co., Ltd.Google Scholar
  27. MacKenzie, D. (2001). Mechanizing proof. Cambridge, MA: The MIT Press.Google Scholar
  28. Newell, A., Shaw, J. C., & Simon, H. A. (1958a). Chess-playing programs and the problem of complexity. IBM Journal of Research and Development, 2, 320–335.CrossRefGoogle Scholar
  29. Newell, A., Shaw, J. C., & Simon, H. A. (1958b). Elements of a theory of human problem solving. Psychological Review, 65(3), 41–56.CrossRefGoogle Scholar
  30. Newell, A., & Simon, H. A. (1958). Heuristic problem solving: The next advance in operations research. Operations Research, 6(1), 1–10.CrossRefGoogle Scholar
  31. Newell, A., & Simon, H. (1972). Human problem solving. Englewood Cliffs, NJ: Prentice-Hall, Inc.Google Scholar
  32. Newman, M. H. A. (1951). Elements of the topology of plane sets of points (2nd ed.). Cambridge: Cambridge University Press.Google Scholar
  33. Polya, G. (1981). Mathematical discovery—On understanding, learning, and teaching problem solving (Combined ed.). New York: Wiley.Google Scholar
  34. Post, E. L. (1936). Finite combinatory processes-formulation 1. Journal of Symbolic Logic, 1(3), 103–105.CrossRefGoogle Scholar
  35. Ramsey, F. P. (1978 (1928)). Universals of law and of fact, chapter 6. A, pp. 128–132. In: D. H. Mellor (Ed.), Foundations—Essays in philosophy, logic mathematics and economics by F. P. Ramsey. London: Routledge & Kegan Paul.Google Scholar
  36. Sieg, W. (2001). Remembrances of Herbert Simon. www.cs.cmu.edu/simon/all.html. Accessed 27 December, 2015.
  37. Silver, D., et al. (2017). Mastering the game of go without human knowledge. Nature, 550(7676), 354–359.CrossRefGoogle Scholar
  38. Simon, H. A. (1954). Bandwagon and underdog effects and the possibility of election predictions. The Public Opinion Quarterly, 18(3), 245–253.CrossRefGoogle Scholar
  39. Simon, H. A. (1977). Models of discovery—And other topics in the methods of science. Dordrecht: D. Reidel Publishing Company.Google Scholar
  40. Simon, H. A. (1989). The scientist as problem solver, chapter 14. In D. Klahr & K. Kotovsky (Eds.), Complex information processing: The impact of Herbert Simon (21st Carnegie-Mellon symposium on cognition) (pp. 375–398). Hillsdale, NJ: Lawrence Earlbaum Associates Inc.Google Scholar
  41. Simon, H. A. (1996). Models of my life. Cambridge, MA: MIT Press.Google Scholar
  42. Stone, R., & Brown, A. (Eds.). (1962). A computable model of economic growth, vol. 1, A programme for growth. London: Chapman and Hall.Google Scholar
  43. Strichartz, R. S. (2000). The way of analysis (Revised ed.). Sudbury, MA: Jones and Bartlett Publishers.Google Scholar
  44. Sundholm, G. (1993). Questions of proof. Manuscrito – Revista Internacional de Filosofia, Campinas, XVI(2), 47–70.Google Scholar
  45. Thom, R. (1971). ”Modern” mathematics: An educational and philosophic error? American Scientist, 59(6), 695–699.Google Scholar
  46. Turing, A. M. (1947). Lecture to the London Mathematical Society on 20 February 1947 (pp. 87–105). In Turing (1992).Google Scholar
  47. Turing, A. M. (1948). Intelligent machinery (pp. 107–127). In Turing (1992).Google Scholar
  48. Turing, A. M. (1950). Computing machinery and intelligence. Mind, 59(236), 433–460.CrossRefGoogle Scholar
  49. Turing A. M. (1953). Digital computers applied to games (pp. 161–185). In Turing (1992).Google Scholar
  50. Turing, A. M. (1992). Collected works of A. M. Turing—Mechanical intelligence, D. C. Ince (Ed.). North-Holland, Amsterdam.Google Scholar
  51. Tymoczco, T. (1979). The four-color problem and its philosophical significance. The Journal of Philosophy, 76(2), 57–83.CrossRefGoogle Scholar
  52. Whitehead, A. N., & Russell, B. (1927). Principia Mathematica (2nd ed., Vol. 1). Cambridge: Cambridge University Press.Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • K. Vela Velupillai
    • 1
  1. 1.SolnaSweden

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