A Dialogue on Infinity

between a mathematician and a philosopher

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Bibliography

This is (a germ of) bibliography for our future book A Dialogue on Infinity.

1. J. S. Alper and M. Bridger, Mathematics, models and Zeno’s paradoxes, Synthese 110 (1997) 143-–166.
2. J. Baez and J. Dolan, Categorification, in Higher Category Theory  (Ezra Getzler and Mikhail Kapranov, eds.). Contemp. Math. 230, American Mathematical Society, Providence, Rhode Island, 1998, pp. 1–36.
3. J. L. Bell, Infinitesimals, Synthese 75 (1988) 285–-315.
4. M. Berz, Non-archimedean analysis and rigorous computation,
   International Journal of Applied Mathematics 2 (2000) 889–930.
5. C. B. Boyer, Cavalieri, limits and discarded infinitesimals’ Scripta Mathematica 8 (1941) 79-–91.
6. F. Cajori, Indivisibles and ‘ghosts of departed quantities’ in the history of mathematics, Scientia 37 (1925) 301–-306.
7. P. Cameron, The random graph revisited, in European Congress of Mathematics, Barcelona, July 10-14, 2000, Volume II (C. Casacuberta, R. M. Miró-Roig, J. Verdera and S. Xambó-Descamps
   eds.), Birkhäuser, Basel, 2001, pp. 267–274.
8. P. J. Cohen, Comments on the foundations of set theory, Proc. Sym. Pure Math. 13 no.1 (1971) 9–-15.
9. J. Copeland, Accelerating Turing Machines, Minds and Machines 12 (2002) 281-–301.
10. J. Copeland, Hypercomputation in the Chinese Room, in Unconventional Models of Computation 2002 (C.S. Calude et al. eds.). Lect. Notes Comp. Sci. 2509  (2002) 15–-26.
11. D. Corfield, Towards a Philosophy of Real Mathematics. Cambridge University Press, 2003. ISBN 0521817226.
12. J. W. Dauben, Abraham Robinson and nonstandard analysis: history, philosophy, and foundations of mathematics, in W. Aspray and P. Kitcher (eds.), History and Philosophy of Modern Mathematics, University of Minnesota Press, Minneapolis, 1988, pp. 177–-200.
13. J. W. Dauben, Marx, Mao and mathematics: The politics of infinitesimals, Documenta Mathematica, Extra Volume ICM 1998, III, 799–809.
14. R. Dedekind, Essays on the theory of numbers. I: Continuity and irrational numbers. II: The nature and meaning of numbers, Dover,   1963.
15. J. Dieudonn\'{e}, A Panorama of Pure Mathematics: As seen by N. Bourbaki. Academic Press, New York, 1982.
16. E. Dubinsky, K. Weller, M. McDonald and A. Brown,  Some historical issues and paradoxes regarding the infinity concept: an APOS analysis, Part 1. Educational Studies in Mathematics, Educational Studies in Mathematics 58 (2005) 335–359.
17. E. Dubinsky, K. Weller, M. McDonald and A. Brown, Some historical issues and paradoxes regarding the infinity concept: an APOS analysis, Part 2. Educational Studies in Mathematics, Educational Studies in Mathematics 60 (2005) 253–266.
18. P. Ehrlich, The rise of non-Archimedean mathematics and the roots of a misconception I: The emergence of non-Archimedean systems of magnitudes, Arch. Hist. Exact Sci. 60 (2006) 1–-121.
19. P. Ernest, The Philosophy of Mathematics Education. Routledge Falmer, 1991. ISBN 1850006679.
20. P. Ernest, Social Constructivism as a Philosophy of Mathematics. State University of New York Press, 1998. ISBN 0791435873.
21. R. Falk, Infinity: A Cognitive Challenge, Theory and Psychology 4 no. 1(1994) 35–60.
    DOI: 10.1177/0959354394041002.
22. J. Ferreir\’os, “What fermented in me for years”: Cantor’s discovery of transfinite numbers, Historia Mathematica 22 (1995) 33–-42.
23. U. Fidelman,  The hemispheres of the brain and the learning of standard and non-standard analysis, International Journal of Mathematical Education in Science and Technology 18 no. 3 (1987) 445 — 452.
24. E. Fischbein, D.  Tirosh,  and P. Hess, The intuition of infinity, Educational Studies in Mathematics 10 no. 1 (1979) 3–40.
25. E. Fischbein, Tacit models and infinity, Educational Studies in Mathematics 48 no. 2–3 (2001) 309–329.
26. E. Fischbein1 and M. Baltsan, The mathematical concept of set and the ‘collection’ model, Educational Studies in Mathematics 37 no. 1 (1998) 1–22.
27. C. Ford, Dmitrii Egorov: Mathematics and religion in Moscow, The Mathematical Intelligencer 13 no. 2 (1991) 24–30.
28. H. Freudenthal, Didactical Phenomenology of Mathematical Structures, Reidel, Holland, 1983.
29. J. W. Grabiner, The origins of Cauchy’s rigorous calculus, MIT Press,  Cambridge, Mass. 1981.
30. J. W. Grabiner, Who gave you the epsilon? Cauchy and the origins of rigorous calculus, American Mathematical Monthly 90 (1983) 185–-194.
31. A. Grunbaum,  Modern science and Zeno’s paradoxes, George Allen and Unwin Ltd., London, 1968.
32. V. Harnik, Infinitesimals from Leibniz to Robinson: time to bring them back to school, Mathematical Intelligencer 8 no. 2 (1986) 41–-47, 63.
33. B. Hasson, The existence of group preference functions, Public Choice 28 (1976) 89–100.
34. L. Henkin, On mathematical induction, The American Mathematical Monthly 67 no. 4  (1960) 323–338.
35. G. Hjorth, Borel equivalence relations, in Handbook on Set Theory (M. Foreman and A. Kanamori, eds.). Springer, 2006.  ISBN 978-1-4020-4843-2.
36. S. Jackson, A.S. Kechris, and A. Louveau, Countable Borel equivalence relations, J. Math. Logic 2 (2002) 1–-80.
37. D. Jacquette, A dialogue on Zeno’s paradox of Achilles and the Tortoise, Argumentation 7 (1993) 273–290.
38. H. N. Jahnke, Cantor’s cardinal and ordinal infinities: an epistemological and didactic view, Educational Studies in Mathematics 48 (2001) 175–-197.
39. A. Kanamoru, Zermelo and set theory, Bull. Symbolic Logic 10 no. 4 (2004) 487–553.
40. H. J. Keisler,  Elementary calculus: An approach using infinitesimals (Experimental version). Bodgen & Quigley, 1971.
41. J. Keisler, Foundations of Infinitesimal Calculus, Prindle, Weber and Schmidt, Boston, 1976.
42. I. Kleiner, History of the infinitely small and the infinitely large in calculus, Ed. Stud. Math. 48 (2001)
    137–174.
43. A. P. Kirman and D. Sondermann, Arrow’s theorem, many agents, and invisible dictators, Journal of Economic Theory 5 (1972) 267–-277.
44. P. Kitcher, Fluxions, limits, and infinite littlenesse: a study of Newton’s presentation of the calculus, Isis 64 (1973) 33–-49.
45. M. Kline, Euler and infinite series, Mathematics Magazine 56 (1983) 307–-314.
46. E. Knobloch, Galileo and Leibniz: Different approaches to infinity, Archive for History of Exact Sciences 54 (1999) 87–-99.
47. J. P. S. Kung, Combinatorics and nonparametric mathematics, Annals of Combinatorics  1 (1997) 105–106.
48. I. Lakatos, Cauchy and the continuum: the significance of non-standard analysis for the history and philosophy of mathematics, Mathematical Intelligencer 1 (1978) 151–-161.
49. G. Lakoff and R. N\'{u}nez, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics Into Being. Basic Books, New York, 2000.
50. E. Landau, Foundations of Analysis, Chelsea, 1966.
51. D. Laugwitz, On the historical development of infinitesimal mathematics, I, II, American Mathematical Monthly 104 (1997) 447–-455, 660–-669
52. T. Leinster, A survey of definitions of $n$-category, math.CT/0107188, 2001; also Theory and Applications of Categories 10  no. 1 (2002) 1–70.
53. R. Lutz and L. G. Albuquerque, Modern infinitesimals as a tool to match intuitive and formal reasoning in analysis, Synthese 134 (2003) 325–-351.
54. A. I. Malcev, Algebraic Systems, Springer-Verlag, 1973.
55. Yu.~I.~Manin, Mathematics and Physics, Birkh\”{a}user, 1981.
56. A. Mamolo and R. Zazkis, Paradoxes as a window to infinity, Research in Mathematics Education  10 no. 2 (2008) 167–182.
57. G. Mar and P. St. Denis, What the Liar taught Achilles, Journal of Philosophical Logic 28 (1999) 29-–46.
58. J. E. Marsden  and A. Weinstein, Calculus Unlimited. Benjamin-Cummings Publishing Co., Longman, US, 1981.
59. B. Mazur, When is one thing equal to some other thing?

    http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf.

60. W. I. McLaughlin and S. L. Miller,  An epistemological use of nonstandard analysis to answer Zeno’s objections against motion, Synthese 92 (1992) 371–384.
61. J. Monaghan,  Young peoples’ ideas of infinity, Educational Studies in Mathematics 48 no. 2–3 (2001)     239–257.
62. A. W. Moore, A brief history of infinity, Scientific American 272 no. 4 (1995) 112–-116.
63. A. W. Moore,   The Infinite, 2nd ed., Routledge and Paul, London, 1999.
64. G. Moore, Hilbert on the infinite: The role of set theory in the evolution of Hilbert’s thought’ Historia Mathematica 2 (2002) 40–-64.
65. A. Moreno  and G. Waldegg,  The conceptual evolution of actual mathematical infinity,
    Educational Studies in Mathematics 22 no. 3 (1991) 211–231.
66. E. Nelson, Internal set theory: A new approach to nonstandard analysis, Bull. Amer. Math.  Soc. 83  (1977) 1165-–1198.
67. E. Nelson, The syntax of non standard analysis, Ann. Pure and Appl. Logic 38 (1988) 123–-134.
68. R. N\'{u}nez, Do real numbers really move? Language, thought and gesture: the embodied cognitive foundations of mathematics. In: 18 Unconventional Essays on the Nature of Mathematics (R.
    Hersh, ed.). Springer, 2005, pp. 160–181.
69. G. Peano, {The principles of arithmetic, presented by a new method, in From Frege to G\”odel,
    (Jean van Heijenoort, editor), Harvard, 1967.
70. D. Pierce, Induction and recursion. A talk at Logic Colloquium 2008—ASL European Summer Meeting, Bern, Switzerland,  3–8 July 2008.
71. K. Popper, On the possibility of an infinite past: a reply to Whitrow, Brit. J. Phil. Sci. 29 (1978), 47–60.
72. A. Robert, Analyse non standard, Presses Polytechniques Romandes, Lausanne, 1985.
73. A. Robinson, Non-standard Analysis (Princeton Landmarks in Mathematics and Physics). Princeton University Press, Princeton, 1996.
74. D. G. Saari and Z. Xia, Off to infinity in finite time, AMS Notices 42 (1995) 538–546.
75. W. C. Salmon, Zeno’s Paradoxes, Bobbs Merrill, New York, 1970.
76. A. Sen, Internal consistency of choice, Econometrica 61 3 (1993) 495–521.
77. D. Tall, The notion of infinite measuring number and its relevance in the intuition of infinity, Educational Studies in Mathematics, 11 (1980) 271–284.
78. D. Tall, Intuitions of infinity, Mathematics in School 10 3 (1981) 30–33.
79. D. Tall, Infinitesimals constructed algebraically and interpreted geometrically, Mathematical Education for Teaching 4 (1981) 34–53.
80. D. Tall, Natural and formal infinities, Educational Studies in Mathematics 2001 48 (2 & 3) 199–-238.
81. D. Tall,  A Child Thinking about Infinity. Journal of Mathematical Behavior 2001.
82. D. Tall and D. Tirosh, Infinity – the never-ending struggle, Educational Studies in Mathematics 48 no. 2–3 (2001) 129–136.
83. A. Tarski, A decision method for elementary algebra and geometry. Rand, 1948.
84. S. Thomas,  Cayley graphs of finitely generated groups,  Proc.  Amer. Math.  Soc. 134 (2006) 289–294.
85. D. Tirosh, Finite and infinite sets: Definitions and intuitions, International Journal of Mathematical Education in Science and Technology 30 no. 3 (1999) 341–-349.
86. D. Tirosh, The role of students’ intuitions of infinity in teaching the Cantorian theory, Mathematics Education Library III (2002) 199–214.
87. D. Tirosh and P. Tsamir, The role of representations in students’ intuitive thinking about infinity, International Journal of Mathematical Education in Science and Technology 27 no. 1 (1996) 33–40.
88. P. Tsamir, The transition from comparison of finite to the comparison of infinite sets: Teaching prospective teachers, Educational Studies in Mathematics 38 (1999) 209–-234.
89. P. Tsamir and T. Dreyfus, Comparing infinite sets — a process of abstraction: The case of Ben, The Journal of Mathematical Behavior 21 no. 1 (2002) 1–23.
90. P. Tsamir and D. Tirosh, Consistency and representations: the case of actual infinity, Journal for Research in Mathematics Education 30 no. 2 (1999) 213–219.
91. L. van den Dries,  Tame Topology and o-Minimal Structures, London Mathematical Society Lecture Notes Series, vol.~248. Cambridge University Press, Cambridge, 1998.
92. K. Weller, A. Brown, E. Dubinsky, M. McDonald and C. Stenger, Intimations of infinity, Notices AMS 51 no. 7 (2004) 741–750.
93. I. Wistedt and M. Martinsson, Orchestrating a mathematical theme: Eleven-year olds discuss the problem of infinity, Learning and Instruction 6 no. 2 (1996) 173–185.
94. B. Zilber,  Pseudo-exponentiation on algebraically closed fields of characteristic zero,  Annals of Pure and Applied Logic 132 no. 1 (2005) 67–95.

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