Dr. Carl Wagner
Professor
Department of Mathematics
The University of Tennessee
Knoxville, TN 37996
Office: (865) 974-4305
Fax: (865) 974-6576
wagner@math.utk.edu

Biography

I was born in Newark, New Jersey. I studied mathematics and philosophy as an undergraduate at Princeton, graduating in 1965, and completed my doctorate in mathematics at Duke in 1969, with a dissertation on p-adic function theory directed by Leonard Carlitz. I joined the mathematics faculty at the University of Tennessee in 1969. Visiting appointments have included a year as a Fellow at the Center for Advanced Study in the Behavioral Sciences at Stanford in 1978-79, and semesters as a Visiting Fellow in Philosophy at Princeton in 1990 and the University of Arizona in 1991.

Research Interests

Enumerative combinatorics (combinatorics of modular binomial lattices; analysis of statistics on discrete structures). Applications of probability theory to philosophy of science and decision theory (e.g., the old evidence problem, and the problem of revising a probability when conditioning is inapplicable).

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Publications

Recent papers: (Please note that to view the pdf files below, you must have Adobe Acrobat Reader.)

_____, Simpson's Paradox and the Fisher-Newcomb problem, Grazer Philosophische Studien 40(1991), pp. 185-194. (pdf version)

_____, Misadventures in conditional expectation: the two-envelope problem, Erkenntnis 51 (1999), 233-241.

_____, Old evidence and new explanation II, Philosophy of Science 66 (1999), 283-288.

_____ and S. Mulay, Verifying the independence of partitions of a probability space, Bulletin of the Australian Mathematical Society 61 (2000), 263-266.

_____, The Carlitz lattice path polynomials, Discrete Mathematics 222(2000), 291-298.

_____, Old evidence and new explanation III, Philosophy of Science 68(2001), S165-S175..

_____, Probability kinematics and commutativity, Philosophy of Science 69(2002), 266-278.

_____, Modus Tollens Probabilized, British Journal for the Philosophy of Science, 55(2004), 747-753. (pdf version)

_____, Commuting probability revisions: the uniformity rule, Erkenntnis 59 (2003), 349-364. (pdf version)

_____, Partition Statistics and q-Bell Numbers (q = - 1), Journal of Integer Sequences, 7(2004), Article 04.1.1 (12 pages). (pdf version)

_____ and M. Shattuck, Parity Theorems for Statistics on Domino Arrangements, The Electronic Journal of Combinatorics 12(2005), # N10 (13 pages). (pdf version)

Teaching Materials:

Math 323/507
_____, CHOICE, CHANCE, AND INFERENCE: An Introduction to Combinatorics, Probability and Statistics. (pdf version)

Math 421
_____, BASIC COMBINATORICS (pdf version)

Complete list of publications:

BOOKS:

1. Carl Wagner and Keith Lehrer, RATIONAL CONSENSUS IN SCIENCE AND SOCIETY, Reidel, Dordrecht-Boston, 1981.

ARTICLES:

2. _____, "Interpolation series for continuous functions on -adic completions of GF(q, x)," Acta Arithmetica 17(1971), 384-406.

3. _____, "Linear operators in local fields of prime characteristic," Journal fur die Reine und Angewandte Mathematik 251(1971), 153-160.

4. _____, "Interpolation series in local fields of prime characteristic," Duke Mathematical Journal 39(1972) 203-210.

5. _____, "On the factorization of some polynomial analogues of binomial coefficients," Archiv der Mathematik 24(1973), 50-52.

6. _____ and T. Hearne, "Minimal covers of finite sets," Discrete Mathematics 5(1973), 247-251.

7. _____, "Automorphisms of p-adic number fields," American Mathematical Monthly 81(1974), 51-52.

8. _____, "Covers of finite sets," Congressus Numerantium 8(1973), 515-520.

9. _____ and T. Hearne,"Boolean subtractive algebras," Notre Dame Journal of Formal Logic 15(1974), 317-324.

10. _____, "Differentiability in local fields of prime characteristic," Duke Mathematical Journal 41 (1974), 285-290.

11. _____, "Linear pseudo-polynomials over GF[q, x]," Archiv der Mathematik 25 (1974), 385-390.

12. _____, "Polynomials over GF(q, x) with integral-valued differences," Archiv der Mathematik 27 (1976), 495-501.

13. _____, "Newton's inequality and a test for imaginary roots," Two-Year College Mathematics Journal 8 (1977), 145-147.

14. _____, "Consensus through respect: a model of rational group decisionmaking," Philosophical Studies 34 (1978), 335-349.

15. _____, "Minimal multiplicative covers of an integer," Discrete Mathematics 24 (1978), 87-94.

16. _____, "The formal foundations of Lehrer's theory of consensus," in PROFILE: KEITH LEHRER, Radu Bogdan, editor, Reidel Publishing Company, 1980.

17. _____, "Jonathan Cohen's THE PROBABLE AND THE PROVABLE: A Review," Duke Law Journal (1979), 1071-1081.

18. _____ and J. Aczel, "A characterization of weighted arithmetic means," SIAM Journal on Algebraic and Discrete Methods 1 (1980) pp. 259-60.

19. _____, "Allocation, Lehrer models, and the consensus of probabilities," Theory and Decision 14 (1982), 207-220.

20. _____, J. Aczel, Pl. Kannappan, and C.T. Ng, "Functional equations and inequalities in rational group decisionmaking" in GENERAL INEQUALITIES 3, E. Beckenbach and W. Walter, eds., 1983.

21. _____ and J. Aczel, "Rational group decision making generalized: The case of several unknown functions, C.R. Math. Rep. Acad. Sci. Canada 3 (1981), 139-142.

22. _____, "Enumeration of generalized weak orders," Archiv der Mathematik 39 (1982), 147-152.

23. _____ and Keith Lehrer, "Probability amalgamation and the independence issue: A Reply to Laddaga," Synthese 55 (1983), 339-346.

24. _____ and Ralph Gonzalez, "Moments and semi-invariants of the interclass Mahalanobis distance," Proceedings of the IEEE Computer Society Conference on Pattern Recognition and Image Processing (1982), 12-17.

25. _____, "Anscombe's paradox and the rule of three-fourths," Theory and Decision 15 (1983), 303-308.

26. _____, J. Aczel, and C.T. Ng, "Aggregation theorems for allocation problems," SIAM Journal on Algebraic and Discrete Methods 5 (1984), 1-8.

27. _____ and Ralph Gonzalez, "Moments of the interclass Mahalanobis distance," IEEE Transactions on Systems, Man, and Cybernetics, SMC 13 (1983), 1135-1139.

28. _____ and Ralph Gonzalez, "Semi-invariants of the interclass Mahalanobis distance," IEEE Transactions on Systems, Man, and Cybernetics, SMC 14 (1984), 534-538.

29. _____, "Aggregating subjective probabilities: some limitative theorems," Notre Dame Journal of Formal Logic 25 (1984), 233-240.

30. _____, "Avoiding Anscombe's paradox," Theory and Decision 16 (1984), 233-238.

31. _____, "On the formal properties of weighted averaging as a method of aggregation," Synthese 62 (1985), 97-108.

32. _____, and Keith Lehrer, "Intransitive indifference: the semi-order problem," Synthese 65 (1985), 249-256.

33. _____, "Evaluating social welfare functionals: a reply to Nurmi," Conceptus 19, No. 47 (1985), 76-82.

34. _____ and Carl Sundberg, "A functional equation arising in multi-agent statistical decision theory," Aequationes Mathematicae 32 (1987), 32-37.

35. _____ and Christian Genest, "Further evidence against independence preservation in expert judgement synthesis," Aequationes Mathematicae 32 (1987), 74-86.

36. _____ and John Long, "On the distribution of the number of optimal elements for various generalized weak orders," Archiv der Mathematik 53 (1989), 87-94.

37. _____, "Consensus for belief functions and related uncertainty measures," Theory and Decision 26(1989), 295-304.

38. _____ and John Long, "Random variables related to a class of ordered structures," Advances in Applied Mathematics 11 (1990), 49-62.

39. _____ and B. Tonn, "Diagnostic conditionalization," Mathematical Social Sciences 19 (1990), 159-165.

40. _____, "Corroboration and conditional positive relevance," Philosophical Studies 61 (1991), 295-300.

41. _____ and B. Tonn, "Risk assessment and lower probabilities," Journal of Environmental Systems 20:2 (1991), 169-177.

42. _____, "Simpson's paradox and the Fisher-Newcomb problem," Grazer Philosophische Studien 40(1991), 185-194.

43. _____, "Generalizing Jeffrey conditionalization," in Uncertainty in Artificial Intelligence, D. Dubois, et al, eds., Morgan Kaufmann Publishers, San Mateo, CA, 1992.

44. _____, "Generalized probability kinematics," Erkenntnis 36(1992), 245-257.

45. _____ and C. Sundberg, "Characterizations of monotone and two monotone capacities," Journal of Theoretical Probability 5(1992), 159-167.

46. _____ and C. Sundberg, "Generalized finite differences and Bayesian conditioning of Choquet capacities," Advances in Applied Mathematics 13 (1992), 262-272.

47. _____ and B. Tonn, "Constructing lower probabilities," in Uncertainty in Artificial Intelligence, D. Heckerman, et al, eds. Morgan Kauffman Publishers, San Mateo, CA, 1993, 515-518.

48. _____ and R. Davis, "Covering algebras and q-binomial generating functions," Discrete Mathematics, 128 (1994), 95-111.

49. _____"Imprecise probability and expert forecasting," Proceedings of the Sixth International Conference on Tools with Artificial Intelligence (Nov. 6-9, 1994), IEEE Computer Society Press, 668-671.

50. _____, "Surjections, differences, and binomial lattices," Studies in Applied Mathematics, 93 (1994), 15-27.

51. _____"Generalized Stirling and Lah numbers," Discrete Mathematics, 160 (1996), 199-218.

52. _____"Old evidence and new explanation," Philosophy of Science 64 (1997), 677-690.

53. _____"Combinatorial proofs of power sum formulas," Archiv der Mathematik 68 (1997), 464-467

54. _____ and S. Mulay, "Finitary bases and formal generating functions," Discrete Mathematics 190 (1998), 177-189.

55. _____, B. Tonn, M. Schweitzer, G. Godfrey, and D. MacGregor, Methodology for assessing future environmental legislative, regulatory, and judicial events, "Futures Research Quarterly 14(1998), 47-75.

56. _____, Misadventures in conditional expectation: the two-envelope problem, Erkenntnis 51 (1999), 233-241.

57. _____, Old evidence and new explanation II, Philosophy of Science 66 (1999), 283-288.

58. _____ and S. Mulay, Verifying the independence of partitions of a probability space, Bulletin of the Australian Mathematical Society 61 (2000), 263-266.

59. _____, The Carlitz lattice path polynomials, Discrete Mathematics 222(2000), 291-298.

60. _____, Old evidence and new explanation III, Philosophy of Science 68(2001), S165-S175.

61. _____, Probability kinematics and commutativity, Philosophy of Science 69(2002), 266-278.

62. _____, Two dogmas of probabilism, in The Epistemology of Keith Lehrer, Erik J. Olsson (ed.), Kluwer, Dordrecht (2003), 143-152.

63. _____, Commuting probability revisions: the uniformity rule, Erkenntnis 59 (2003), 349-364.

64. _____, Modus tollens probabilized, British Journal for the Philosophy of Science, 55(2004), 747-753.

65. _____, Partition Statistics and q-Bell numbers (q=-1), Journal of Integer Sequences, 7(2004), Article 04.1.1 (12 pages).

66. _____ and M. Shattuck, Parity Theorems for Statistics on Domino Arrangements, The Electronic Journal of Combinatorics 12(2005), # N10 (13 pages).

 

Teaching and Office Schedule

My teaching and office schedule are available through this link.

Useful Links

-- American Mathematical Society (AMS)

-- Institute for Mathematics and Its Applications (IMA)

-- Society for Industrial and Applied Mathematics (SIAM)

--Probabilistic Thinking, by Richard Jeffrey

-- Imprecise Probabilities

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Last update: 5/07
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