The Automation of Proof: A Historical and Sociological Exploration

IEEE Annals of the History of Computing

Volumn 17, Issue 3, 1995, Pages 7-29

  Mackenzie, Donald ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

ARTIFICIAL INTELLIGENCE; COMPUTER APPLICATIONS; COMPUTER HARDWARE; COMPUTER SCIENCE; COMPUTER SOFTWARE; HISTORY; MATHEMATICAL TECHNIQUES; SYSTEMS ANALYSIS; THEOREM PROVING;

AUTOMATION OF PROOF; MATHEMATICAL LOGIC; SOCIOLOGY; VERIFICATION;

AUTOMATION;

EID: 0029379080     PISSN: 10586180     EISSN: None     Source Type: Journal    
DOI: 10.1109/85.397057     Document Type: Article

References (186)

1. M. Davis, “The Prehistory and Early History of Automated Deduction,” in Automation of Reasoning: Classical Papers on Computational Logic, J. Siekmann and G. Wrightson, eds. Springer. Berlin, 1983, Vol. 1, pp. 1-28.
2. L. Wos and L. Henschen, “Automated Theorem Proving, 1965-1970.” in Automation of Reasoning: Classical Papers on Computational Logic, J. Siekmann and G. Wrightson, eds., Springer. Berlin, 1983, Vol. 2. pp. 1–24.
3. D.W. Loveland. “Automated Theorem Proving: A Quarter Century Review.” Contemporary Mathematics. Vol. 29, 1984, pp. 1-45.
4. D.J. O’Leary, “Principia Mathematica and the Development of Automated Theorem Proving,” in Perspectives on the History of Mathemtuical Logic. T. Drucker, ed., Boston, Birkhäuser, 1991, pp. 48–53.
5. G. Lolli La Macchina e le Dimostruzioni: Matematica Logica e Informatica il Mulino, Bologna, 1987.
6. T.L. Heath, The Thirteen Books of Euclid's Elements, Cambridge Univ. Press Cambridge, 1908.
7. Heath, Euclid's Elements, p. 155.
8. G. Leibniz, “Elements of a Calculus,” in Leibniz: Logical Papers — A Selection G.H.R. Parkinson, ed. Clarendon Press. Oxford, 1966, pp. 17-24 quotation on p. 18.
9. G. Frege, Begriffsschrift eine der arithmetischen nachge-bildete Formelsprache des reinen Denkens, Nebert, Halle, 1879.
10. A.N. Whitehead and B. Russell Principia Mathematica (second edition), Cambridge Univ. Press, Cambridge, 1925, Vol. 1, pp. 94–95 and 132.
11. K. Gödel “Uber Vollständigkeit und Widerspruchsfreiheit Ergebnisse eines mathematischen Kolloquiums, Vol. 3, 1931, pp. 12-13.
12. K. Gödel “Uber formal unentscheidbare Sätze der Principia mathematica und verwandter Systeme I,'’ Monatshefte für Mathematik und Physik, Vol. 38.1931, pp. 173–198.
13. Lolli, La Macchina e le Dimostrazioni, pp. 13–25.
14. H. Mehrtens, Symbolische Imperative: Zu Natur und Beherrschungsprogramm des wissenschaftlichen Moderne. In Die Modernisierung moderner Gesellschaften: Verhandlungen des 25. Deutschen Soziologentages in Frankfurt am Main 1990, Campus. Frankfurt am Main, 1991, pp. 604–616.
15. H. Poincaré Science et Méthode Flammarion Paris, 1908, p. 157 (my translation).
16. H. Mehrtens, Moderne Sprache Mathematik: Eine Geschichte des Streits um die Grundlagen der Disziplin und des Subjekts formaler Systeme. Suhrkamp, Frankfurt am Main. 1990.
17. A.M. Turing, “On Computable Numbers, with an Application to the Entscheidungsproblem,” Proc. London Mathematical Soc., Series 2. Vol. 42,1939. pp. 230–265.
18. N. Bourbaki, Elements of Mathematics: Theory of Sets, Addison Wesley, Reading. Mass., 1968. p 11.
19. R.S. Boyer and J.S. Moore, “A Theorem Prover for a Computational Logic,” keynote address to 10th Conf. on Automated Deduction, 1990, p. 8.
20. P.H. Nidditch. Introductory Formal Logic of Mathematics, Univ. Tutorial Press, London, 1957, pp. v, 1, and 6–7.
21. W.S. Jevons, “On the Mechanical Performance of Logical Inference,” Philosophical Trans. of the Royal Soc., Vol. 160, 1870, pp. 497-518.
22. P McCorduck, Machines Who Think: A Personal Inquiry into the History and Prospects of Artificial Intelligence, Freeman, San Francisco, 1979, pp. 104, 145.
23. H. Simon. Models of My Life, Basic Books, New York, chap. 13.
24. H. Simon, interviewed by A. Dale. Pittsburgh, Apr. 21, 1994.
25. McCorduck, Machines Who Think, p. 116.
26. A. Newell J.C. Shaw, and H. Simon, “Empirical Explorations of the Logic Theory Machine: A Case Study in Heuristic,” Proc. Western Joint Computer Conf., Vol. 15, 1957, pp. 218-239, quotations on p. 219.
27. Newell, Shaw and Simon, “Empirical Explorations of the Logic Theory Machine,” p. 221.
28. O’Leary, “Principia Mathematica and the Development of Automated Theorem Proving,” p. 52.
29. B. Russell, 1etter to Herbert Simon. Nov. 2, 1956, quoted in O'Leary “Principia Mathematica and the Development of Automated Theorem Proving,” p. 52.
30. McCordack. Machines Who Think. p. 142.
31. Newell, Shaw, and Simon. “Empirical Explorations of the Logic Theory Machine,” p. 220.
32. Simon, interviewed by Dale.
33. D. Crevier Al: The Tumultuous History of the Search for Artificial Intelligence. Basic Books. New York, 1993, pp. 52–54.
34. H.L. Gelernter and N. Rochester, “Intelligent Behavior in Problem-Solving Machines,” IBM J. Research and Development. Vol. 2. 1958. pp. 336–345.
35. H. Gelernter J.R. Hansen, and D.W. Loveland, “Empirical Explorations of the Geometry-Theorem Proving Machine.” in Computers and Thought, E.A. Feigenbaum and J. Feldman. eds., McGraw-Hill New York, 1963, pp. 153–163, quotation on p. 153.
36. Simon. interviewed by Dale.
37. H. Wang. “Toward Mechanical Mathematics,” IBM J. Research and Development. Vol. 4. 1960, pp. 2–21, quotation on p. 3.
38. Wang, “Toward Mechanical Mathematics,” p. 18.
39. K. Gödel “Die Vollständigkeit der Axiome des logischen Funktionenkalküls, ’’ Monatshefte für Mathematik und Physik. Vol. 37, 1930, pp. 349–360.
40. B. Dreben, “On the Completeness of Quantification Theory.” Proc. National Academy of Sciences, Vol. 38. 1952, pp.1047-1052.
41. W.V. Quine , “A Proof Procedure for Quantification Theory,” J. Symbolic Logic. Vo1. 20. No. 2, June 1955, pp. 141–149.
42. D. Prawitz “Preface to ‘A Mechanical Proof Procedure “.” in Automation of Reasoning: Classical Papers on Computational Logic, J. Siekmann and G. Wrightson, eds., Springer, Berlin, 1983. Vol. 1. pp. 200–201.
43. M. Presburger “Uber die Vollständigkeit eines gewissen Systems der Arithmetik ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt.” in Sprawozdanie z I Kongresu matemaryków krajów slowians-kich Warszawa 1929. Sklad Glówny Warsaw, 1930, pp. 92–101.
44. M. Davis, interviewed by A.J. Dale, New York, Apr. 29, 1994.
45. M. Davis, ‘'A Computer Program for Presburger's Algorithm.” paper presented to Cornell Summer Institute for Symbolic Logic, 1957, in Automation of Reasoning: Classical Papers on Computational Logic, J. Siekmann and G. Wrightson, eds., Springer, Berlin, 1983, Vol. 1, pp. 41–48.
46. Wang. “Toward Mechanical Mathematics.
47. H. Wang, “Proving Theorems by Pattern Recognition,” Comm. ACM, Vol. 4, No. 3, 1960, pp. 229–243.
48. Wang, “Toward Mechanical Mathematics,” p. 4.
49. E.W. Beth, “On Machines which Prove Theorems,” Simon Stevin Wisen Natur-kundig Tijdschrift, Vol. 32, 1958, pp. 49–60, quotation on p. 59.
50. A. Robinson, “Proving a Theorem (as Done by Man, Logician, or Machine),” in Automation of Reasoning: Classical Papers on Computational Logic, J. Siekmann and G. Wrightson, eds.,Springer, Berlin, 1983, Vol. 1, pp. 74–76, quotation on p. 74.
51. P.C. Gilmore, “A Proof Method for Quantification Theory: Its Justification and Realization,” IBM J. Research and Development, Vol. 4, 1960. pp, 28–35.
52. Wang, “Toward Mechanical Mathematics.
53. D. Prawitz, H. Prawitz and N. Voghera, “A Mechanical Proof Procedure and its Realization in an Electronic Computer,” J. ACM, Vol. 7,1960, pp. 102–128.
54. Gilmore, “Proof Method for Quantification Theory.
55. J.A. Robinson, interviewed by A.J. Dale, Edinburgh, Feb. 15 and 16. 1994.
56. W.F Miller, letter to author, June 18, 1993.
57. M. Davis and H. Putnam, '’ A Computing Procedure for Quantification Theory,” J. ACM, Vol. 7, No.3, 1960. pp. 201–215.
58. D. Prawitz, “An Improved Proof Procedure,” Theoria, Vol. 26,1960, pp. 102–139.
59. J.A. Robinson, “A Machine-Oriented Logic Based on the Resolution Principle,'’ J. ACM, Vol. 12, No.1, 1965, pp. 23–41, quotations on pp. 23 and 24.
60. Robinson. interviewed by Dale.
61. J.A. Robinson, “Logic and Logic Programming,” Comm. ACM, Vol. 35. No.3, 1992. pp. 41–65.
62. J. Fleck, “Development and Establishment in Artificial Intelligence, “ in Scientific Establishments and Hierarchies: Sociology of the Sciences Yearbook, Vol. VI, N. Elias, H. Martins, and R. Whitley, eds., Reidel, Dordrecht, 1982, pp. 169–217.
63. D.W. Loveland, Automated Theorem Proving: A Logical Basis, North-Holland, Amsterdam, 1978.
64. Robinson, “Logic and Logic Programming,” p. 45.
65. Crevier, AI, p. 64.
66. S. Papert, interviewed by A.J. Dale, Cambridge. Mass., June 7,1994.
67. M. Minsky, S. Papert, and staff, Proposal to ARPA for Research on Artificial Intelligence at M. I. T, 1971-1972 MIT Artificial Intelligence Memo No. 245,1971, pp. 13–14.
68. P.J. Hayes, “A Critique of Pure Treason,” Computational Intelligence, Vol. 3,1987, pp. 179–185, quotations on p. 183.
69. R.A. Kowalski, “The Early Years of Logic Programming,” Comm. ACM, Vol. 31, No.1, 1988, pp. 38–42.
70. Sir J. Lighthill, “Artificial Intelligence: A General Survey,” in Artificial Intelligence: A Paper Symposium. Science Research Council, London, 1973. p. 1 0.
71. S.A. Cook. “The Complexity of Theorem-Proving Procedures,” Proc. Third Ann. ACM Symp. Theory of Computing. ACM. New York, 1971. pp. 151–158.
72. C. Cherniak “Computational Complexity and the Universal Acceptance of Logic,” J. Philosophy, Vol. 81, No. 12, Dec. 1984. pp. 739–758.
73. W.W. Bledsoe “I Had a Dream: AAAI Presidential Address, 19 Aug. 1985.” AI Magazine. Vol. 7,1986, pp. 57- 61. quotation on p. 57.
74. A.O. Boyer and R.S. Boyer, “A Biographical Sketch of W.W. Bledsoe,” in Automated Reasoning: Essays in Honor of Woody Bledsoe. R.S. Boyer, ed., Kluwer, Dordrecht, 1991, pp. 1-29.
75. W.W. Bledsoe, “Non-Resolution Theorem Proving,” Artificial Intelligence, Vol. 9, 1977, pp. 1–35, quotation on pp. 2–3.
76. C. Hewitt. Description and Theoretical Analysis (using Schemas) of Planner: A Language for Proving Theorems and Manipulating Models in a Robot. PhD thesis, MIT, Cambridge, 1971.
77. Bledsoe. “Non-Resolution Theorem Proving,” p. 12.
78. Loveland, “Automated Theorem Proving,’ p. 24.
79. W.W. Bledsoe, “Splitting and Reduction Heuristics in Automatic Theorem Proving,” Artificial Intelligence, Vol. 2, 1971. pp. 55–77. quotations on p. 55.
80. A.P. Morse, A Theory of Sets. Academic Press, New York, 1965.
81. Bledsoe. “Splitting and Reduction Heuristics,” p. 66.
82. W.W. Bledsoe. R.S. Boyer, and W.H. Henneman, “Computer Proofs of Limit Theorems,” Artificial Intelligence, Vol. 3, 1972. pp. 27–60, quotations on pp. 28 and 35, and details of the procedure on pp. 35–38.
83. L.T. Wos et al., “The Concept of Demodulation in Theorem Proving.'’ J. ACM, Vol. 14, No.4. Oct. 1967, pp. 698–709.
84. G. Robinson and L. Wos “Paramodulation and Theorem-Proving in First-Order Theories with Equality,” Machine Intelligence, Vo1. 4. 1969, pp. 135–150.
85. S. Winker and L. Wos. “Automated Generation of Models and Counterexamples and its Application to Open Questions in Ternary Boolean Algebra,” Proc. Eighth Int'l Symp. Multiple-valued Logic, Rosemont, Ill., IEEE and ACM, New York, 1978, pp. 251–256.
86. S. Winker. L. Wos and E.L. Lusk, “Semigroups Antiautomorphism and Involutions: A Computer Solution to an Open Problem. I” Mathematics of Computation. Vol. 37,1981. pp. 533–545.
87. S. Winker, “Generation and Verification of Finite Models and Counterexamples Using an Automated Theorem Prover Answering Two Open Questions,” J. ACM, Vol. 29, No.2. 1982. pp, 273–284.
88. L. Wos. “Solving Open Questions with an Automated Theorem-Proving Program.” in Proc. Sixth Conf. Automated.
89. Deduction, New York, June 7–9, 1982. D.W. Loveland. ed.
90. L. Wos et al. “Questions Concerning Possible Shortest Single Axioms for the Equivalential Calculus: An Application of Automated Theorem Proving to Infinite Domains:’ Notre Dame J. Formal Logic, Vol. 24, No. 2, 1983, pp. 205–223.
91. T.S. Kuhn, The Structure of Scientific Revolutions. 2nd ed., Univ. of Chicago Press, Chicago, 1970.
92. L. Wos, “Transition to the Future.” J. Automated Reasoning, Vol. 9, No.1. 1992, p. iii.
93. J.R. Guard et al., “Semi-Automated Mathematics,” J. ACM, Vol. 16, 1969, pp. 49–62, quotation on p. 49.
94. R. Bumcrot, “On Lattice Complements.” Proc. Glasgow Mathematical Assn. Vol. 7, 1965, pp. 22–23.
95. Guard et al., “Semi-Automated Mathematics,” p. 58.
96. N.G. de Bruijn, “Automath: A Language for Mathematics,” in Automation of Reasoning: Classical Papers on Computational Logic, J. Siekmann and G. Wrightson eds., Springer. Berlin ,1983 Vol. 2, pp. 159–200, quotation on p. 159.
97. E.G.H. Landau, Grundlagen der Analysis, Akademische Verlag. Leipzig, 1930.
98. L.S.van Benthem Jutting. Checking Landau's “Grundlagen” in the Automath System. PhD thesis, Technische Hogeschool, Eindhoven, 1977.
99. P.J. Davis and R. Hersh, “Rhetoric and Mathematics,” in The Rhetoric of the Human Sciences. J.S. Nelson et al., eds., London, Univ. of Wisconsin Press, 1987, pp. 53–68, quotation on pp. 63–64.
100. P. Naur and B. Randell, eds., Software Engineering: Report of a Conference Sponsored by the NATO Science Committee, Garnisch, Germany, 7–11 Oct. 1968, NATO, Brussels, 1969.
101. E. Peláez A Gift from Pandora's Box: The Software Crisis, PhD thesis. Univ, Edinburgh, 1988.
102. H.H. Goldstine and J. von Neumann, “Planning and Coding of Problems for an Electronic Computing Instrument,” in John von Neumann: Collected Works, Vol. V, A.H. Taub, ed., Pergamon, Oxford. 1963, pp. 80-151.
103. A.M. Turing, “Checking a Large Routine,” in Report of a Conference on High Speed Automatic Calculating Machines, Cambridge Univ. Mathematical Laboratory, Cambridge, 1949. pp. 67–69, reprinted in F.L. Morris and C.B. Jones. “An Early Program Proof by Alan Turing,” Annals of the History of Computing, Vol. 6, No.2, Apr. 1984, pp. 139–143.
104. C.B. Jones, The Search for Tractable Ways to Reason about Programs. Univ, Manchester, Dept. of Computer Science, UMCS-92-4-4. 1992. p. 8.
105. J. McCarthy, “Towards a Mathematical Science of Computation,” in Information Processing 1962: Proc. IFIP Congress 62, C.M. Popplewell , e d., North Holland, Amsterdam, 1963, pp. 21–28.
106. P. Naur, “Proof of Algorithms by General Snapshots,” BIT, Vol. 6,1966, pp. 310–316.
107. R.W. Floyd, “Assigning Meanings to Programs,” Mathematical Aspects of Computer Science: Proc. Symposia in Applied Mathematics, Vol. 19,1967, pp. 19–32.
108. C.A.R. Hoare.”An Axiomatic Basis for Computer Programming,” Comm. ACM, Vol. 12. No. 10. 1969. pp. 576-580 and 583.
109. J. McCarthy and J. Painter. “Correctness of a Compiler for Arithmetic Expressions.” in Mathematical Aspects of Computer Science: Proc. Symposia in Applied Mathematics, Vo1. 19. 1967, pp. 33–41.
110. R.L. London, “Computer Programs Can Be Proved Correct.” in Theoretical Approaches to Non-Numerical Problem Solving: Proc. Fourth Systems Symp, at Case Western Reserve Univ. Springer. Berlin, 1970. pp. 281 -302.
111. J.C. King, A Program Verifier, PhD thesis, Carnegie Mellon Univ. l969, p. 166.
112. King. Program Verifier.
113. D.I. Good, Toward a Man-Machine Svstem for Proving Program Correctness. PhD thesis, Univ of Wisconsin, 1970.
114. D.L Good, personal communication.
115. D.E. Bell and L.J. LaPadula, Secure Computer Systems: Mathematical Foundations. Mitre Corp. Bedford, Mass. MTR-2547,1973.
116. D.I. Good, “Mechanical Proofs about Computer Programs,” Philosophical Trans. of the Royal Soc. of London, Series A. Vol. 312, 1984, pp. 389–409, quotation on p. 389.
117. W.W. Bledsoe and P. Bruell.”A Man-Machine Theorem-Proving System.” in Advance Papers of Third Int'l Joint Conf. Artificial lntelligence, W.W. Bledsoe, ed. Vol. 5. Part 1. 1974, pp. 51–72.
118. D.I. Good, “Proof of a Distributed System in Gypsy,” in Formal Specification: Proc. Joint IBM/Univ. of Newcastle upon Tyne Seminar, 7-10 Sept., 1983. M.J. Elphick. ed., Univ. of Newcastle upon Tyne, Computing Laboratory. 1983, pp. 44–89. quotation on p. 61.
119. D.I. Good, interviewed by E. Pelaez, Austin. Tex., May 16, 1991.
120. Good. “Proof of a Distributed System in Gypsy.
121. M. Schaefer. “Symbol Security Condition Considered Harmful.” Proc. 1989 IEEE Symp, Security and Safety, IEEE CS Press, Los Alamitos, Calif. 1989, pp. 20-46, especially p. 26.
122. Department of Defense, Trusted Computer System Evaluation Criteria, National Computer Security Center, Fort Meade, Md., DOD 5200.28-STD, 1985.
123. M.H. Cheheyl et al. “Verifying Security,” Computing Surveys, Vol. 13, No. 3, 1981.pp. 279–339.
124. C.L. Gold, S.E. Goodman, and B.G. Walker, “Software: Recommendations for an Export Control Policy,” Comm. ACM. Vol. 23. No.4. 1980. pp. 199–207, quotations on pp. 199 and 203.
125. G. Pottinger, Proof Requirements in the Orange Book: Origins Implementation, and Implications, typescript. Feb. 11, 1994.
126. Boyer and Moore, “Theorem Prover for a Computational Logic,” p. 1.
127. R.S. Boyer, interviewed by A.J. Dale, Austin, Tex., Apr. 8, 1994.
128. J.S. Moore, interviewed by A.J. Dale, Austin, Tex., Apr. 7, 1994.
129. R.S. Boyer and J.S. Moore, A Computational Logic, Academic Press. New York, 1979, p. xiii.
130. R. Burstall, “Proving Properties of Programs by Structural Induction,” Computer J., Vol. 12, 1969, pp. 41–48.
131. R.S. Boyer and J.S. Moore, “Proving Theorems about LISP Functions.” J. ACM, Vol. 22, No.1, 1975, pp. 129–144.
132. Boyer and Moore. A Computational Logic, p. xiv.
133. Boyer, interviewed by Dale.
134. Boyer and Moore, “Theorem Prover for a Computational Logic,” p. 4.
135. Boyer, interviewed by Peláez.
136. R. Pollack, interviewed by A.J. Dale, Edinburgh, Nov. 15, 1993.
137. Boyer and Moore. “Theorem Prover for a Computational Logic.” p. 3.
138. R.S. Boyer and J.S. Moore, A Computational Logic Handbook, Academic Press, San Diego, 1988, p. xiii.
139. N. Shankar, Proof-checking Metamathematics, PhD thesis, Univ. of Texas at Austin, 1986.
140. R. Milner, interviewed by A.J. Dale, Edinburgh, Nov. 17 and 30,1993.
141. R. Milner, Logic for Computable Functions: Description of a Machine Implementation, Stanford Univ. Stanford Artificial Intelligence Project. AIM-169, 1972.
142. M. Gordon, R. Milner, and C. Wadsworth, Edinburgh LCF, Univ. of Edinburgh, Dept. of Computer Science, CSR-11-77, 1977.
143. Stuart Anderson, personal communication.
144. R. Milner, LCF: A Way of Doing Proofs with a Machine, Univ. of Edinburgh, Dept. of Computer Science, CSR-41-79, 1979. p. 3.
145. Gordon, Milner, and Wadsworth, Edinburgh LCF, p. 5.
146. L.C. Paulson. Logic and Computation: Interactive Proof with Cambridge LCF, Cambridge Univ. Press, Cambridge, 1987, p.209.
147. W.P. van Stigt, Brouwer's Intuitionism, North-Holland, Amsterdam, 1990.
148. R.L. Constable et al., Implementing Mathematics with the Nuprl Proof Development System, Prentice Hall, Englewood Cliffs, N.J., 1986. especially p. 13.
149. F.K. Hanna and N. Daeche, “Specification and Verification Using Higher-Order Logic: A Case Study.” in Formal Aspects of VLSI Design. G.J. Milne and P.A. Subrahmanyam, eds., North-Holland, Amsterdam. pp. 179–213. quotation on p. 180.
150. P.B. Andrews et al., “Automating Higher-Order Logic,” Contemporary Mathematics, Vol. 29,1984, pp. 169–192.
151. M. Gordon, “HOL: A Machine Oriented Formulation of Higher-Order Logic,” Univ. of Cambridge, Computer Laboratory, Tech. Report 103, 1985.
152. A. Cohn, “A Proof of Correctness of the Viper Microprocessor: The First Level.” Univ. of Cambridge, Computer Laboratory, Tech. Report No. 104, 1987.
153. M. Gordon, interviewed by D. MacKenzie, Cambridge, Jan. 10,1991.
154. Loveland,”Automated Theorem Proving.
155. W.W. Bledsoe. “A New Method for Proving Certain Presburger Formulas,” Advance Papers of Fourth Int'l Joint Conf. Artificial Intelligence, Tbilisi, 3-8 Sept. 1975, pp. 15-21.
156. D. Loveland, interviewed by D. MacKenzie, Edinburgh, Aug. 25,1993.
157. Loveland. “Automated Theorem Proving.
158. A.J. Nevins,”A Human Oriented Logic for Automatic Theorem-Proving.” J. ACM, Vol. 21, No.4, 1974, pp. 606–621.
159. Robinson, “Machine-Oriented Logic Based on the Resolution Principle.
160. L. Wos, “Automated Reasoning Answers Open Questions,” Notices of the American Mathematical Soc., Vol. 40, No.1, 1993, pp 15-26, quotation on p. 16.
161. D.B. Lenat, AM: An Artificial Intelligence Approach to Discovery in Mathematics as Heuristic Search. PhD thesis. Stanford Univ. 1976.
162. H. Wang, “Computer Theorem Proving and Artificial Intelligence.” Contemporary Mathematics. Vo1. 29, 1984. pp. 49–70, quotation on pp, 49–50.
163. Simon, interviewed by Dale.
164. Wang, “Toward Mechanical Mathematics,” p. 3.
165. J.A. Robinson, “Formal and Informal Proofs,” in Automated Reasoning: Essays in Honor of Woody Bledsoe, R.S. Boyer, ed. Kluwer, Dordrecht, 1991, pp. 267–282.
166. Wang, “Computer Theorem Proving and Artificial Intelligence.” p. 60.
167. H.A. Simon and A. Newell, “Heuristic Problem Solving: The Next Advance in Operations Research,” Operations Research. 6, 1958, pp. 1–10, quotation on p. 7.
168. E.A. Feigenbaum and J. Feldman. eds. Computers and Thought, McGraw-Hill, New York, 1963.
169. Anon. “The Talk of the Town,” The New Yorker, June 11 1966, pp, 27–28.
170. H.L. Dreyfus, What Computers Can't Do: The Limits of Artificial Intelligence. Harper & Row, New York, 1979, p. 293.
171. Wang, “Toward Mechanical Mathematics,” p. 3.
172. Dreyfus, What Computers Can't Do. p. 300.
173. Wang, “Computer Theorem Proving and Artificial Intelligence.” p. 65.
174. Lighthill, “Artificial Intelligence.” pp. 10 and 20.
175. Robinson, “Formal and Informal Proofs,” p. 269.
176. D. Hilbert, “The Foundations of Mathematics.” in From Frege to Gödel: A Source Book in Mathematical Logic, 1879– 1931, J. van Heijenoort. ed. Harvard Univ. Press. Cambridge. Mass., 1967, pp. 464–479, quotation on p. 465.
177. D. Gorenstein, “The Enormous Theorem, “ Scientific American, Vol. 253. No.6, Dec. 1985. pp. 92–103. quotation on p. 92.
178. R. DeMillo, interviewed by A.J. Dale, Chicago, Sept. 1994.
179. R. Lipton, interviewed by A.J. Dale, Princeton, N.J. May 4, 1994.
180. R. DeMillo, R. Lipton, and A. Perlis, “Social Processes and Proofs of Theorems and Programs,” Comm. ACM, Vol. 22, 1979, pp. 271–280. quotation on p. 275.
181. K. Appel and W. Haken, “Every Planar Map is Four Colorable,” Illinois J. Mathematics, Vol. 21, 1977. pp. 429–567.
182. F.F. Bonsall, “A Down-to-Earth View of Mathematics,” American Mathematical Monthly, Vol. 89, 1982, pp. 8-14, quotations on pp. 13-14.
183. D.I.A. Cohen, “The Superfluous Paradigm.” in The Mathematical Revolution Inspired by Computing. J.H. Johnson and M.J. Loomes, eds. Clarendon, Oxford, 1991, pp. 323–329, quotation on p. 328.
184. D.I.A. Cohen. interviewed by A.J. Dale. New York. May 2. 1994.
185. M. Schaefer. interviewed by Carrel Pottinger. Rockville, Md. Mar. 23. 1903.
186. D. MacKenzie. “The Fangs of the VIPER.” Nantrr. Vol. 352. No. 6335. Aug. 8. 1991, pp. 467-468.