## Bibliography

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

- J. S. Alper and M. Bridger, Mathematics, models and Zeno’s paradoxes, Synthese 110 (1997) 143-–166.
- 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.
- J. L. Bell, Infinitesimals, Synthese 75 (1988) 285–-315.
- M. Berz, Non-archimedean analysis and rigorous computation,

International Journal of Applied Mathematics 2 (2000) 889–930. - C. B. Boyer, Cavalieri, limits and discarded infinitesimals’ Scripta Mathematica 8 (1941) 79-–91.
- F. Cajori, Indivisibles and ‘ghosts of departed quantities’ in the history of mathematics, Scientia 37 (1925) 301–-306.
- 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. - P. J. Cohen, Comments on the foundations of set theory, Proc. Sym. Pure Math. 13 no.1 (1971) 9–-15.
- J. Copeland, Accelerating Turing Machines, Minds and Machines 12 (2002) 281-–301.
- 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.
- D. Corfield, Towards a Philosophy of Real Mathematics. Cambridge University Press, 2003. ISBN 0521817226.
- 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.
- J. W. Dauben, Marx, Mao and mathematics: The politics of infinitesimals, Documenta Mathematica, Extra Volume ICM 1998, III, 799–809.
- R. Dedekind, Essays on the theory of numbers. I: Continuity and irrational numbers. II: The nature and meaning of numbers, Dover, 1963.
- J. Dieudonn\'{e}, A Panorama of Pure Mathematics: As seen by N. Bourbaki. Academic Press, New York, 1982.
- 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.
- 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.
- 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.
- P. Ernest, The Philosophy of Mathematics Education. Routledge Falmer, 1991. ISBN 1850006679.
- P. Ernest, Social Constructivism as a Philosophy of Mathematics. State University of New York Press, 1998. ISBN 0791435873.
- R. Falk, Infinity: A Cognitive Challenge, Theory and Psychology 4 no. 1(1994) 35–60.

DOI: 10.1177/0959354394041002. - J. Ferreir\’os, “What fermented in me for years”: Cantor’s discovery of transfinite numbers, Historia Mathematica 22 (1995) 33–-42.
- 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.
- E. Fischbein, D. Tirosh, and P. Hess, The intuition of infinity, Educational Studies in Mathematics 10 no. 1 (1979) 3–40.
- E. Fischbein, Tacit models and infinity, Educational Studies in Mathematics 48 no. 2–3 (2001) 309–329.
- E. Fischbein1 and M. Baltsan, The mathematical concept of set and the ‘collection’ model, Educational Studies in Mathematics 37 no. 1 (1998) 1–22.
- C. Ford, Dmitrii Egorov: Mathematics and religion in Moscow, The Mathematical Intelligencer 13 no. 2 (1991) 24–30.
- H. Freudenthal, Didactical Phenomenology of Mathematical Structures, Reidel, Holland, 1983.
- J. W. Grabiner, The origins of Cauchy’s rigorous calculus, MIT Press, Cambridge, Mass. 1981.
- J. W. Grabiner, Who gave you the epsilon? Cauchy and the origins of rigorous calculus, American Mathematical Monthly 90 (1983) 185–-194.
- A. Grunbaum, Modern science and Zeno’s paradoxes, George Allen and Unwin Ltd., London, 1968.
- V. Harnik, Infinitesimals from Leibniz to Robinson: time to bring them back to school, Mathematical Intelligencer 8 no. 2 (1986) 41–-47, 63.
- B. Hasson, The existence of group preference functions, Public Choice 28 (1976) 89–100.
- L. Henkin, On mathematical induction, The American Mathematical Monthly 67 no. 4 (1960) 323–338.
- G. Hjorth, Borel equivalence relations, in Handbook on Set Theory (M. Foreman and A. Kanamori, eds.). Springer, 2006. ISBN 978-1-4020-4843-2.
- S. Jackson, A.S. Kechris, and A. Louveau, Countable Borel equivalence relations, J. Math. Logic 2 (2002) 1–-80.
- D. Jacquette, A dialogue on Zeno’s paradox of Achilles and the Tortoise, Argumentation 7 (1993) 273–290.
- H. N. Jahnke, Cantor’s cardinal and ordinal infinities: an epistemological and didactic view, Educational Studies in Mathematics 48 (2001) 175–-197.
- A. Kanamoru, Zermelo and set theory, Bull. Symbolic Logic 10 no. 4 (2004) 487–553.
- H. J. Keisler, Elementary calculus: An approach using infinitesimals (Experimental version). Bodgen & Quigley, 1971.
- J. Keisler, Foundations of Infinitesimal Calculus, Prindle, Weber and Schmidt, Boston, 1976.
- I. Kleiner, History of the infinitely small and the infinitely large in calculus, Ed. Stud. Math. 48 (2001)

137–174. - A. P. Kirman and D. Sondermann, Arrow’s theorem, many agents, and invisible dictators, Journal of Economic Theory 5 (1972) 267–-277.
- P. Kitcher, Fluxions, limits, and infinite littlenesse: a study of Newton’s presentation of the calculus, Isis 64 (1973) 33–-49.
- M. Kline, Euler and infinite series, Mathematics Magazine 56 (1983) 307–-314.
- E. Knobloch, Galileo and Leibniz: Different approaches to infinity, Archive for History of Exact Sciences 54 (1999) 87–-99.
- J. P. S. Kung, Combinatorics and nonparametric mathematics, Annals of Combinatorics 1 (1997) 105–106.
- 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.
- G. Lakoff and R. N\'{u}nez, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics Into Being. Basic Books, New York, 2000.
- E. Landau, Foundations of Analysis, Chelsea, 1966.
- D. Laugwitz, On the historical development of infinitesimal mathematics, I, II, American Mathematical Monthly 104 (1997) 447–-455, 660–-669
- 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.
- R. Lutz and L. G. Albuquerque, Modern infinitesimals as a tool to match intuitive and formal reasoning in analysis, Synthese 134 (2003) 325–-351.
- A. I. Malcev, Algebraic Systems, Springer-Verlag, 1973.
- Yu.~I.~Manin, Mathematics and Physics, Birkh\”{a}user, 1981.
- A. Mamolo and R. Zazkis, Paradoxes as a window to infinity, Research in Mathematics Education 10 no. 2 (2008) 167–182.
- G. Mar and P. St. Denis, What the Liar taught Achilles, Journal of Philosophical Logic 28 (1999) 29-–46.
- J. E. Marsden and A. Weinstein, Calculus Unlimited. Benjamin-Cummings Publishing Co., Longman, US, 1981.
- B. Mazur, When is one thing equal to some other thing?
http://www.math.harvard.edu/~mazur/preprints/when_is_one.pdf.

- 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.
- J. Monaghan, Young peoples’ ideas of infinity, Educational Studies in Mathematics 48 no. 2–3 (2001) 239–257.
- A. W. Moore, A brief history of infinity, Scientific American 272 no. 4 (1995) 112–-116.
- A. W. Moore, The Infinite, 2nd ed., Routledge and Paul, London, 1999.
- G. Moore, Hilbert on the infinite: The role of set theory in the evolution of Hilbert’s thought’ Historia Mathematica 2 (2002) 40–-64.
- A. Moreno and G. Waldegg, The conceptual evolution of actual mathematical infinity,

Educational Studies in Mathematics 22 no. 3 (1991) 211–231. - E. Nelson, Internal set theory: A new approach to nonstandard analysis, Bull. Amer. Math. Soc. 83 (1977) 1165-–1198.
- E. Nelson, The syntax of non standard analysis, Ann. Pure and Appl. Logic 38 (1988) 123–-134.
- 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. - G. Peano, {The principles of arithmetic, presented by a new method, in From Frege to G\”odel,

(Jean van Heijenoort, editor), Harvard, 1967. - D. Pierce, Induction and recursion. A talk at Logic Colloquium 2008—ASL European Summer Meeting, Bern, Switzerland, 3–8 July 2008.
- K. Popper, On the possibility of an infinite past: a reply to Whitrow, Brit. J. Phil. Sci. 29 (1978), 47–60.
- A. Robert, Analyse non standard, Presses Polytechniques Romandes, Lausanne, 1985.
- A. Robinson, Non-standard Analysis (Princeton Landmarks in Mathematics and Physics). Princeton University Press, Princeton, 1996.
- D. G. Saari and Z. Xia, Off to infinity in finite time, AMS Notices 42 (1995) 538–546.
- W. C. Salmon, Zeno’s Paradoxes, Bobbs Merrill, New York, 1970.
- A. Sen, Internal consistency of choice, Econometrica 61 3 (1993) 495–521.
- D. Tall, The notion of infinite measuring number and its relevance in the intuition of infinity,
*Educational Studies in Mathematics*, 11 (1980) 271–284. - D. Tall, Intuitions of infinity,
*Mathematics in School*10 3 (1981) 30–33. - D. Tall, Infinitesimals constructed algebraically and interpreted geometrically,
*Mathematical Education for Teaching*4 (1981) 34–53. - D. Tall, Natural and formal infinities, Educational Studies in Mathematics 2001 48 (2 & 3) 199–-238.
- D. Tall, A Child Thinking about Infinity.
*Journal of Mathematical Behavior*2001. - D. Tall and D. Tirosh, Infinity – the never-ending struggle, Educational Studies in Mathematics 48 no. 2–3 (2001) 129–136.
- A. Tarski, A decision method for elementary algebra and geometry. Rand, 1948.
- S. Thomas, Cayley graphs of finitely generated groups, Proc. Amer. Math. Soc. 134 (2006) 289–294.
- D. Tirosh, Finite and infinite sets: Definitions and intuitions, International Journal of Mathematical Education in Science and Technology 30 no. 3 (1999) 341–-349.
- D. Tirosh, The role of students’ intuitions of infinity in teaching the Cantorian theory, Mathematics Education Library III (2002) 199–214.
- 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.
- 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.
- 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.
- 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.
- L. van den Dries, Tame Topology and o-Minimal Structures, London Mathematical Society Lecture Notes Series, vol.~248. Cambridge University Press, Cambridge, 1998.
- K. Weller, A. Brown, E. Dubinsky, M. McDonald and C. Stenger, Intimations of infinity, Notices AMS 51 no. 7 (2004) 741–750.
- 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.
- 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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