After Math: The Work of Proof in the Age of Human-Machine Collaboration

ISIS

Volumn 102, Issue 3, 2011, Pages 494-505

  Dick, Stephanie ^a  

^a HARVARD UNIVERSITY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ARTICLE; ARTIFICIAL INTELLIGENCE; COMPUTER INTERFACE; HISTORY; HUMAN; MASS COMMUNICATION; MATHEMATICS; RESEARCH; UNITED STATES;

ARTIFICIAL INTELLIGENCE; CHICAGO; DIFFUSION OF INNOVATION; HISTORY, 20TH CENTURY; HUMANS; MATHEMATICS; RESEARCH; USER-COMPUTER INTERFACE;

EID: 80053612544     PISSN: 00211753     EISSN: 15456994     Source Type: Journal    
DOI: 10.1086/661623     Document Type: Article

References (70)

1. Alan Turing, "Computing Machinery and Intelligence," Mind, 1950, 59:433-460, on p. 442.
2. ATP was in the spirit of Charles Babbage and George Boole, who shared the conviction that human thought, and especially mathematical thought, could be automated. A more specific call to arms for ATP research came from John McCarthy, who coined the term "Artificial Intelligence" in 1955. In 1958 he proposed that researchers attempt to develop automated theorem-proving programs.Note.
3. Pamela McCorduck, Machines Who Think: A Personal Inquiry into the History and Prospects of Artificial Intelligence (Natick, Mass.: Peters, 2004), p. 51.
4. ATP research was not limited to traditional academic centers. Some important early engines were developed at the RAND Corporation, the Argonne National Laboratory, and IBM research labs. Early ATP research sites, engines, and researchers are well documented in Donald Loveland, "Automated Theorem Proving: A Quarter Century Review," in Automated Theorem Proving: After Twenty-five Years, ed. W. W. Bledsoe and Loveland (Contemporary Mathematics, 29) (Providence, R.I.: American Mathematical Society, 1984), pp. 1-46. Note.
5. Some members of the team, among them Steve Winker and Ross Overbeek, were initially employed at Northern Illinois University. According to Overbeek, a quarrel with the university administration in the early 1980s regarding the trajectory of their then rapidly growing Department of Computer Science provoked his (and many others') departure. Until that time, the Northern group was quite separate from, and even in competition with, the Argonne group. Many Northern researchers eventually took up positions at Argonne, however, essentially merging the two teams. This information comes from a telephone interview I conducted with Ross Overbeek, 10 Nov. 2010. Note.
6. The gesture made in the title of this essay to Walter Benjamin's seminal paper "The Work of Art in the Age of Mechanical Reproduction" is not motivated merely by the fact that AURA is so named. Benjamin's concept "aura" refers to a historically grounded notion of authenticity for art objects that cannot be (re)produced by mechanical means. The creators of AURA similarly believed that something of human mathematical practice could not be translated into mechanical processes. Note.
7. Walter Benjamin, "The Work of Art in the Age of Mechanical Reproduction" (1936), in Illuminations, ed. Hannah Arendt, trans. Harry Zohn (London: Harcourt Brace Jovanovich, 1968), pp. 217-242.
8. In response to the prospect of automation, ATP researchers articulated various beliefs and assumptions about what human mathematical work and thought consists in. As such, ATP is a "powerful disclosing agent" for images and understandings of mathematics. This concept comes from Lucy Suchman, who proposes that emotive robotics is a "powerful disclosing agent" for assumptions about the nature of human emotions. Note.
9. Lucy Suchman, Human-Machine Reconfigurations: Plans and Situated Actions (Learning in Doing: Social, Cognitive, and Computational Perspectives) (Cambridge: Cambridge Univ. Press, 2006), p. 226.
10. Larry Wos and Steve Winker, "Open Questions Solved with the Assistance of AURA," in Automated Theorem Proving, ed. Bledsoe and Loveland (cit. n. 2), pp. 73-84, on p. 74.
11. I propose that "mathematically uninteresting" developments ought to be studied in the history of mathematics for many reasons parallel to those offered by pioneers of the sociology of scientific knowledge who favored adopting a symmetry between the "winners" and the "losers" of scientific controversies. Note.
12. Steven Shapin and Simon Schaffer, Leviathan and the Air-Pump: Hobbes, Boyle, and the Experimental Life (Princeton, N.J.: Princeton Univ. Press, 1985).
13. Limiting the history of mathematics to those problems now considered interesting reinforces the vision of mathematics as continuous and cumulative. What mathematicians consider interesting and significant is part of what needs to be explained, as is the appearance of continuity. Note.
14. Peter Galison, Image and Logic: A Material Culture of Microphysics (Chicago: Univ. Chicago Press, 1987);
15. Donna Haraway, "A Cyborg Manifesto: Science, Technology, and Socialist-Feminism in the Late Twentieth Century," in Simians, Cyborgs, and Women: The Reinvention of Nature (New York: Routledge, 1991), pp. 149-181;
16. Bruno Latour, "Where Are the Missing Masses? The Sociology of a Few Mundane Objects," in Shaping Technology, Building Society: Studies in Sociotechnical Change, ed. Wiebe Bijker and John Law (Cambridge, Mass.: MIT Press, 1992), pp. 225-258;
17. Latour, Science in Action: How to Follow Scientists and Engineers through Society (Cambridge, Mass.: Harvard Univ. Press, 1988);
18. Andrew Pickering, The Mangle of Practice: Time, Agency, and Science (Chicago: Univ. Chicago Press, 1995);
19. Hans-Jörg Rhein-Berger, Toward a History of Epistemic Things: Synthesizing Proteins in the Test Tube (Stanford, Calif.: Stanford Univ. Press, 1997);
20. Suchman, Human-Machine Reconfigurations (cit. n. 3).
21. Amir Alexander, Geometrical Landscapes: The Voyages of Discovery and the Transformation of Mathematical Practice (Stanford, Calif.: Stanford Univ. Press, 2002);
22. Donald MacKenzie, Mechanizing Proof: Computing, Risk, and Trust (Cambridge, Mass.: MIT Press, 2001);
23. Herbert Mehrtens, "Mathematical Models," in Models: The Third Dimension of Science, ed. Soraya de Chadarevian and Nick Hopwood (Stanford, Calif.: Stanford Univ. Press, 2004), pp. 276-306;
24. Reviel Netz, The Shaping of Deduction in Greek Mathematics: A Study in Cognitive History (Cambridge: Cambridge Univ. Press, 1999);
25. Brian Rotman, Ad Infinitum... The Ghost in Turing's Machine: Taking God Out of Mathematics and Putting the Body Back In: An Essay in Corporeal Semiotics (Stanford, Calif.: Stanford Univ. Press, 1993);
26. Andrew Warwick, "The Laboratory of Theory; or, What's Exact about the Exact Sciences?" in The Values of Precision, ed. M. Norton Wise (Princeton, N.J.: Princeton Univ. Press, 1995), pp. 311-351;
27. Warwick, Masters of Theory: Cambridge and the Rise of Mathematical Physics (Chicago: Univ. Chicago Press, 2003).
28. MacKenzie, Mechanizing Proof, p. 78. AURA did eventually provide useful applications in optimizing circuit design and program correctness proving, but this was not the original goal of the project.
29. Jack Holl, Argonne National Laboratory, 1946-96 (Chicago: Univ. Chicago Press, 1997), p. 125.
30. Argonne continued to host a highly successful and robust team engaged in enrolling computers in the production of mathematical knowledge until about 2006. Around that time, the DOE cut funding to the group and several key researchers either retired or accepted positions elsewhere: Ross Overbeek to Stephanie Dick, email, 6 Jan 2011. Note.
31. The axioms and propositions were expressed in terms of Boolean formulas-called "clauses"-with disjunction as the primary logical operator.
32. Larry Wos, Daniel Carson, and George Robinson, "The Unit Preference Strategy in Theorem Proving," in The Collected Works of Larry Wos, Vol. 1 (Hackensack, N.J.: World Scientific, 2001), pp. 17-28, on p. 20 (originally published in American Federation of Information Processing Society, Proceedings, 1964, 26:615- 621).
33. Loveland, "Automated Theorem Proving" (cit. n. 2). The debates about how much of mathematical knowledge and practice could be articulated and formalized in order to permit automation resonate with Harry Collins's discussion in Artificial Experts: Social Knowledge and Intelligent Machines (Cambridge, Mass.: MIT Press, 1992).
34. Collins proposes that only those social practices that have themselves been sufficiently stabilized as to resemble mechanical processes can be exported to machines as intelligent purveyors of the associated knowledge.
35. Most, if not all, of the Logic camp ATP engines were based on a powerful inference rule called "resolution," developed by Alan Robinson, who often visited Argonne as a researcher. Resolution was not based on any known human practice and was in fact difficult and counterintuitive for humans to understand. Note.
36. J. A. Robinson, "A Machine-Oriented Logic Based on the Resolution Principle," Journal of the Association for Computing Machinery, 1965, 12:25-41;
37. MacKenzie, Mechanizing Proof (cit. n. 6), pp. 77-83.
38. He studied first at the University of Chicago and received his Ph.D. from the University of Illinois at Urbana-Champaign. Wos's mathematics training was, however, not traditional in one important respect: he has been blind since birth. In order to pursue his mathematics education, he and his professors had to develop a Braille system with which he could encounter visual and textual mathematical objects. Wos's blindness was also a factor in his turn from traditional mathematics to ATP research at Argonne. In spite of offering to share his salary with someone who could do the "board work" in his classes, and feeling fully supported by the Department of Mathematics at the University of Illinois at Urbana-Champaign, Wos was denied a teaching position by a dean at the university because of perceived obstacles presented by his blindness. Note.
39. Tim A. Obermiller, "Top of His Game," University of Chicago Magazine, 1997, 89:4 (online edition); further infor mation comes from a telephone interview I conducted with Larry Wos, 4 Nov. 2010.
40. Wos recounted one story in which the primary realization for his development of quad arithmetic came to him while he was sleeping: "It just came to me. So if you want to give credit, if you want to get pedantic as hell-I've got a big imagination that works when I'm asleep!" Wos interview, 4 Nov. 2010. Wos's position resembles that of Jacques Hadamard in The Psychology of Invention in the Mathematical Field (Princeton, N.J.: Princeton Univ. Press, 1945). Note.
41. Wos and Winker, "Open Questions Solved with the Assistance of AURA" (cit. n. 4), p. 74 (quotation); and Overbeek interview, 10 Nov. 2010. AURA is therefore somewhat ironically related to Benjamin's "aura"-the former denotes an engine that captures everything but the unmechanizable essence of human practice. Note.
42. Different positions appear to be, at least in part, motivated by the different backgrounds from which ATP practitioners came. Further research is required before more specific claims about these positions can be made. Note.
43. The weighting mechanism is described in Brian Smith, Reference Manual for the Environmental Theorem Prover: An Incarnation of AURA (Springfield, Va.: National Technical Information Service, U.S. Department of Commerce, 2010) (originally printed as an Argonne National Laboratory Technical Report, 1988), p. 18. Note.
44. Ross Overbeek, "Wos and Automated Deduction at ANL: The Ethos," in Automated Reasoning and Its Applications: Essays in Honor of Larry Wos, ed. Robert Veroff and Gail Pieper (Cambridge, Mass.: MIT Press, 1997), pp. 1-12, on p. 3.
45. Assembly Language is a low-level (close to the hardware) symbolic representation of instructions and data that users wish to communicate to the computer. See Ned Chapin, 360/370 Programming in Assembly Language, 2nd ed. (New York: McGraw-Hill, 1973) (the first edition appeared in 1968). Note.
46. For complete specifications see Smith, Reference Manual for the Environmental Theorem Prover (cit. n. 17).
47. The weighting mechanism was originally developed by Overbeek for his doctoral dissertation in 1971;
48. Ross Overbeek, "A New Class of Automated Theorem-Proving Algorithms," J. Assoc. Computing Machinery, 1974, 21:191-200.
49. In addition to extensive work on the general implementation, the weighting mechanism was Overbeek's main contribution to the AURA project.
50. Smith, Reference Manual for the Environmental Theorem Prover (cit. n. 17), p. 12 (quotation);
51. Wos interview, 4 Nov. 2010 (weighting examples).
52. In this context, two clauses were called "contradictory" if one contained at least one literal (atomic logical formula) that was the negation of a literal in the other. That is, if one clause contained L and the other contained not-L, these clauses were considered contradictory. Note.
53. Wos et al., "Unit Preference Strategy in Theorem Proving" (cit. n. 10), p. 20.
54. Smith, Reference Manual for the Environmental Theorem Prover (cit. n. 17), pp. 10-11.
55. The humansAURA system qualifies as what Andy Clark and David Chalmers call a "coupled system" of extended cognition (as does the traditional humanspaperpencil system): "If, as we confront a task, a part of the world functions as a process which, were it done in the head, we would have no hesitation in recognizing as part of the cognitive process, then that part of the world is (so we claim) part of the cognitive process.... In these cases, the human organism is linked with an external entity in a two-way interaction, creating a coupled system that can be seen as a cognitive system in its own right."Note.
56. Andy Clark and David Chalmers, "The Extended Mind," in The Extended Mind, ed. Richard Mernary (Cambridge, Mass.: MIT Press, 2010), pp. 27-42, on p. 29.
57. Steve Winker, "Generation and Verification of Finite Models and Counterexamples Using an Automated Theorem Prover Answering Two Open Questions," J. Assoc. Computing Machinery, 1982, 29:273-284.
58. Larry Wos, "Solving Open Questions with an Automated Theorem-Proving Program," Lecture Notes in Computer Science, 1982, 138:1-131, on p. 3.
59. J. Allen and D. Luckham, "An Interactive Theorem-Proving Program," Machine Intelligence, 1970, 5:321-336.
60. Winker K, "Generation and Verification of Finite Models and Counterexamples Using an Automated Theorem Prover Answering Two Open Questions" (cit. n. 26), p. 274.
61. Ibid., p. 278.
62. Steve Winker and Larry Wos, "Automated Generation of Models and Counterexamples and Its Application to Open Questions in Ternary Boolean Algebra," in Proceedings of the Eighth International Symposium on Multiple-Valued Logic (Rosemont, Ill.: IEEE and ACM Publishers, 1978), pp. 251-256.
63. Overbeek, "Wos and Automated Deduction at ANL" (cit. n. 18), pp. 2-3 (emphasis added); and Ross Overbeek to Stephanie Dick, email, 11 Nov. 2010.
64. The problem, suggested by the mathematician Irving Kaplansky, was "Does there exist a finite semigroup admitting a nontrivial antiautomorphism but admitting no nontrivial involutions?" Wos and Winker, "Open Questions Solved with the Assistance of AURA" (cit. n. 4), p. 75. In their first attempt to solve the problem, the group misunderstood the meaning of "involution" (demonstrating their lack of knowledge about finite semi groups). Despite their lack of knowledge, however, the team successfully solved the problem with AURA in their second attempt.Note.
65. Larry Wos and Steve Winker, "Semigroups, Antiautomorphisms, and Involutions: A Computer Solution to an Open Problem," Mathematics of Computation, 1981, 37:533-545;
66. Wos and Winker, "Open Questions Solved with the Assistance of AURA," pp. 75-77.
67. Wos and Winker, "Open Questions Solved with the Assistance of AURA," p. 78.
68. Hans-Jörg Rheinberger's concept of an "experimental system" captures how even very disciplined scientific instruments can surprise their users.
69. Rheinberger, Toward a History of Epistemic Things (cit. n. 5).
70. Wos interview, 4 Nov. 2010.