Foundations of Science

, Volume 18, Issue 2, pp 259–296 | Cite as

Tools, Objects, and Chimeras: Connes on the Role of Hyperreals in Mathematics

  • Vladimir Kanovei
  • Mikhail G. KatzEmail author
  • Thomas Mormann


We examine some of Connes’ criticisms of Robinson’s infinitesimals starting in 1995. Connes sought to exploit the Solovay model \({\mathcal{S}}\) as ammunition against non-standard analysis, but the model tends to boomerang, undercutting Connes’ own earlier work in functional analysis. Connes described the hyperreals as both a “virtual theory” and a “chimera”, yet acknowledged that his argument relies on the transfer principle. We analyze Connes’ “dart-throwing” thought experiment, but reach an opposite conclusion. In \({\mathcal{S}}\), all definable sets of reals are Lebesgue measurable, suggesting that Connes views a theory as being “virtual” if it is not definable in a suitable model of ZFC. If so, Connes’ claim that a theory of the hyperreals is “virtual” is refuted by the existence of a definable model of the hyperreal field due to Kanovei and Shelah. Free ultrafilters aren’t definable, yet Connes exploited such ultrafilters both in his own earlier work on the classification of factors in the 1970s and 80s, and in Noncommutative Geometry, raising the question whether the latter may not be vulnerable to Connes’ criticism of virtuality. We analyze the philosophical underpinnings of Connes’ argument based on Gödel’s incompleteness theorem, and detect an apparent circularity in Connes’ logic. We document the reliance on non-constructive foundational material, and specifically on the Dixmier trace \({-\hskip-9pt\int}\) (featured on the front cover of Connes’ magnum opus) and the Hahn–Banach theorem, in Connes’ own framework. We also note an inaccuracy in Machover’s critique of infinitesimal-based pedagogy.


Axiom of choice Dixmier trace Hahn–Banach theorem Hyperreal Inaccessible cardinal Gödel’s incompleteness theorem Infinitesimal Klein–Fraenkel criterion Leibniz Noncommutative geometry P-point Platonism Skolem’s non-standard integers Solovay models Ultrafilter 

Mathematics Subject Classification (2000)

Primary 26E35 Secondary 03A05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Albeverio, S., Høegh-Krohn, R., Fenstad, J., & Lindstrøm, T. (1986) Nonstandard methods in stochastic analysis and mathematical physics. (Pure and Applied Mathematics, Vol. 122). Orlando, FL: Academic Press, Inc.Google Scholar
  2. Atiyah M. (2006) The interface between mathematics and physics: A panel discussion sponsored by the DIT & the RIA. Irish Mathematical Society Bulletin 58: 33–54Google Scholar
  3. Barner, K. (2011). Fermats “adaequare”—und kein Ende? Mathematische Semesterberichte, 58(1), 13–45. See
  4. Bell J. L., Machover M. (1977) A course in mathematical logic. North-Holland Publishing Co., Amsterdam–New York–OxfordGoogle Scholar
  5. Bell J., Slomson A. (1969) Models and ultraproducts: An introduction. North-Holland Publishing Co., Amsterdam-LondonGoogle Scholar
  6. Benci, V., Horsten, L., & Wenmackers, S. (2011) Non-archimedean probability. Milan Journal of Mathematics, to appear. See
  7. Bishop E. (1977) Review: H. Jerome Keisler, elementary calculus. Bulletin of the American Mathematical Society 83: 205–208CrossRefGoogle Scholar
  8. Blass A., Laflamme C. (1989) Consistency results about filters and the number of inequivalent growth types. Journal of Symbolic Logic 54(1): 50–56CrossRefGoogle Scholar
  9. Błaszczyk, P., Katz, M., & Sherry, D. (2012). Ten misconceptions from the history of analysis and their debunking. Foundations of Science (online first). See doi: 10.1007/s10699-012-9285-8 and
  10. Borovik, A., Jin, R., & Katz, M. (2012). An integer construction of infinitesimals: Toward a theory of Eudoxus hyperreals. Notre Dame Journal of Formal Logic, 53(4), 557–570. Scholar
  11. Borovik A., & Katz M. (2012). Who gave you the Cauchy–Weierstrass tale? The dual history of rigorous calculus. Foundations of Science, 17(3), 245–276. See doi: 10.1007/s10699-011-9235-x and
  12. Bråting K. (2007) A new look at E. G. Björling and the Cauchy sum theorem. Archive for History of Exact Sciences 61(5): 519–535CrossRefGoogle Scholar
  13. Breger H. (1994) The mysteries of adaequare: A vindication of Fermat. Archive for History of Exact Sciences 46(3): 193–219CrossRefGoogle Scholar
  14. Breuillard, E., Green, B., & Tao, T. (2011) The structure of approximate groups, Publications Mathématiques. Institut de Hautes études Scientifiques, to appear. See
  15. Brukner, Č., & Zeilinger A. (2005). Quantum physics as a science of information, in Quo vadis quantum mechanics?. In Frontiers Collection (pp. 47–61). Berlin: Springer.Google Scholar
  16. Cantor, G. (1932). Foundations of a general theory of manifolds. (Grundlagen einer allgemeinen Mannigfaltigkeitslehre.) Leipzig. Teubner, 1883, 47 S. Reproduced in Georg Cantor, Gesammelte Abhandlungen, (pp. 165–209) Berlin: Springer.Google Scholar
  17. Carey A., Phillips J., Sukochev F. (2003) Spectral flow and Dixmier traces. Advances in Mathematics 173(1): 68–113CrossRefGoogle Scholar
  18. Cassirer E. (1957) The philosophy of symbolic forms. Yale University Press, New Haven and LondonGoogle Scholar
  19. Chang C. C., Keisler H. J. (1992) Model Theory (3rd ed.). North Holland, AmsterdamGoogle Scholar
  20. Choquet G. (1968) Deux classes remarquables d’ultrafiltres sur \({\mathbb{N}}\). Bulletin des Sciences Mathématiques (2) 92: 143–153Google Scholar
  21. Christensen, J. (1974). Topology and Borel structure. Descriptive topology and set theory with applications to functional analysis and measure theory. (North-Holland Mathematics Studies, Vol. 10). (Notas de Matemática, No. 51). North-Holland Publishing Co., Amsterdam-London; American Elsevier Publishing Co., Inc., New York.Google Scholar
  22. Cifoletti, G. (1990). La méthode de Fermat: son statut et sa diffusion. Algèbre et comparaison de figures dans l’histoire de la méthode de Fermat. In: Cahiers d’Histoire et de Philosophie des Sciences. Nouvelle Série 33. Paris:Société Française d’Histoire des Sciences et des Techniques.Google Scholar
  23. Connes, A. (1969/70) Ultrapuissances et applications dans le cadre de l’analyse non standard. 1970 Séminaire Choquet: 1969/70, Initiation à à l’Analyse Fasc. 1, Exp. 8, 25 pp. Paris: Secrétariat mathématique.Google Scholar
  24. Connes A. (1970) Détermination de modèles minimaux en analyse non standard et application. Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences. Séries A et B 271: A969–A971Google Scholar
  25. Connes A. (1976) Classification of injective factors Cases. II 1, II , III λ, λ ≠ 1. Annals of Mathematics (2) 104(1): 73–115CrossRefGoogle Scholar
  26. Connes, A. (1990). Essay on physics and noncommutative geometry. The interface of mathematics and particle physics (Oxford, 1988). In: The Institute of Mathematics and its Applications Conference Series. New Series 24 (pp. 9–48). New York: Oxford University Press.Google Scholar
  27. Connes A. (1994) Noncommutative geometry. Academic Press, Inc, San Diego, CAGoogle Scholar
  28. Connes A. (1995) Noncommutative geometry and reality. Journal of Mathematical Physics 36(11): 6194–6231CrossRefGoogle Scholar
  29. Connes A. (1997) Brisure de symétrie spontanée et géométrie du point de vue spectral. [Spontaneous symmetry breaking and geometry from the spectral point of view]. Journal of Geometry and Physics 23(3-4): 206–234CrossRefGoogle Scholar
  30. Connes, A. (2000a). Noncommutative geometry–year 2000. GAFA 2000 (Tel Aviv, 1999). Geometric and Functional Analysis 2000, Special Volume, Part II, pp. 481–559.Google Scholar
  31. Connes, A. (2000b). Noncommutative geometry year 2000. Preprint (2000), see
  32. Connes A. (2000c) A short survey of noncommutative geometry. Journal of Mathematical Physics 41(6): 3832–3866CrossRefGoogle Scholar
  33. Connes A. (2000d) Interview: la réalité mathématique archaï que. La Recherche, 2000. See
  34. Connes, A. (2004). Cyclic cohomology, noncommutative geometry and quantum group symmetries. In item (Connes et al. 2004), (pp. 1–71).Google Scholar
  35. Connes, A. (2007). An interview with Alain Connes. Part I: conducted by Catherine Goldstein and Georges Skandalis (Paris). European Mathematical Society. Newsletter 63, 25–30. See
  36. Connes, A. (2007). Non-standard stuff. Blog. See
  37. Connes, A. (2009). Private communication. January 12, 2009Google Scholar
  38. Connes, A. (2012a). Private communication. June 17, 2012Google Scholar
  39. Connes A. (2012b) Private communication. July 2, 2012Google Scholar
  40. Connes, A., Cuntz, J., Guentner, E., Higson, N., Kaminker, J., & Roberts, J. (2004). Noncommutative geometry. Lectures given at the C.I.M.E. Summer School held in Martina Franca, September 3–9, 2000. In S. Doplicher & R. Longo (Eds.), Lecture Notes in Mathematics, 1831. Springer-Verlag, Berlin: Centro Internazionale Matematico Estivo (C.I.M.E.), Florence.Google Scholar
  41. Connes, A., Lichnerowicz, A., & Schützenberger M. (2001) Triangle of thoughts. (Translated from the 2000 French original by Jennifer Gage). Providence, RI: American Mathematical Society.Google Scholar
  42. Corfield D. (2003) Towards a philosophy of real mathematics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  43. Cutland N., Kessler C., Kopp E., Ross D. (1988) On Cauchy’s notion of infinitesimal. The British Journal for the Philosophy of Science 39(3): 375–378CrossRefGoogle Scholar
  44. Davies E. B. (2011) Towards a philosophy of real mathematics (book review of item Corfield 2003). Notices of the American Mathematical Society 58(10): 1454–1457Google Scholar
  45. Davis, M. (1977). Applied nonstandard analysis. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. Reprinted: Dover, NY, 2005, see
  46. Davis M. (2006) The incompleteness theorem. Notices of the American Mathematical Society 53(4): 414–418Google Scholar
  47. Davis, M. (2012a), Pragmatic Platonism. Preprint.Google Scholar
  48. Davis, M. (2012b). Private communication. July 1, 2012.Google Scholar
  49. Dennett D. (1991) Real patterns. Journal of Philosophy 88(1): 27–51CrossRefGoogle Scholar
  50. Dieks, D. (2002). MathSciNet review of item (Connes et al. 2001). See
  51. Dixmier J. (1966) Existence de traces non normales. Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences. Séries A et B 262: A1107–A1108Google Scholar
  52. Durham, I. (2011) In search of continuity: thoughts of an epistemic empiricist. See
  53. Earman J. (1975) Infinities, infinitesimals, and indivisibles: the Leibnizian labyrinth. Studia Leibnitiana 7(2): 236–251Google Scholar
  54. Ehrlich P. (2006) The rise of non-Archimedean mathematics and the roots of a misconception. I. The emergence of non-Archimedean systems of magnitudes. Archive for History of Exact Sciences 60(1): 1–121CrossRefGoogle Scholar
  55. Ehrlich P. (2012) The absolute arithmetic continuum and the unification of all numbers great and small. Bulletin of Symbolic Logic 18(1): 1–45CrossRefGoogle Scholar
  56. Engliš M., Zhang G. (2010) Hankel operators and the Dixmier trace on strictly pseudoconvex domains. Documenta Mathematica 15: 601–622Google Scholar
  57. Erdös P., Gillman L., Henriksen M. (1955) An isomorphism theorem for real-closed fields. The Annals of Mathematics (2) 61: 542–554CrossRefGoogle Scholar
  58. Fahey C., Lenard C., Mills T., Milne L. (2009) Calculus: A Marxist approach. The Australian Mathematical Society Gazette 36(4): 258–265Google Scholar
  59. Farah I., Shelah S. (2010) A dichotomy for the number of ultrapowers. Journal of Mathematical Logic 10(1-2): 45–81CrossRefGoogle Scholar
  60. Foreman M., Wehrung F. (1991) The Hahn–Banach theorem implies the existence of a non-Lebesgue measurable set. Fundamenta Mathematicae 138(1): 13–19Google Scholar
  61. Fraenkel, A. (1946). Einleitung in die Mengenlehre. Dover Publications, New York, NY, [originally published by Springer, Berlin, 1928].Google Scholar
  62. Fraenkel A. (1967) Lebenskreise. Aus den Erinnerungen eines jüdischen Mathematikers. Deutsche Verlags-Anstalt, StuttgartGoogle Scholar
  63. Gayral, V., Iochum, B., & Sukochev, F. (2012). (Org.): Traces Singulières et leurs Applications du 02/01/2012 au 06/01/2012. CIRM, Marseille. See
  64. Gierz G., Hofmann K., Keimel K., Lawson J., Mislove M., Scott D. (2003) Continuous lattices and domains. Encyclopedia of Mathematics and its applications. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  65. Gödel, K. (1940). The consistency of the axiom of choice and of the continuum hypothesis with the axioms of set theory. In: Annals of Mathematics Studies (Vol. 3, pp. 66) Princeton: Princeton University Press.Google Scholar
  66. Goldblatt, R. (1998). Lectures on the hyperreals. An introduction to nonstandard analysis. Graduate Texts in Mathematics (Vol. 188) New York: Springer-VerlagGoogle Scholar
  67. Goldbring I. (2010) Hilbert’s fifth problem for local groups. Annals of Mathematics (2) 172(2): 1269–1314CrossRefGoogle Scholar
  68. Goldenbaum, U., Jesseph, D. (Eds.) (2008) Infinitesimal differences: Controversies between Leibniz and his contemporaries. Walter de Gruyter, Berlin-New YorkGoogle Scholar
  69. Goodstein R. (1944) On the restricted ordinal theorem. Journal of Symbolic Logic 9: 33–41CrossRefGoogle Scholar
  70. Grabiner J. (1981) The origins of Cauchy’s rigorous calculus. MIT Press, Cambridge, Mass-LondonGoogle Scholar
  71. Hacking, I. (2013). The mathematical animal: philosophical thoughts about proofs, applications, and other mathematical activities. Cambridge University Press, (forthcoming).Google Scholar
  72. Halmos P. (1985) I want to be a mathematician. An automathography. Springer-Verlag, New YorkCrossRefGoogle Scholar
  73. Hamkins, J. (2012a). Is the dream solution of the continuum hypothesis attainable? See
  74. Hamkins, J. (2012b). The set-theoretic multiverse. The Review of Symbolic Logic, 5: 416–449. See doi: 10.1017/S1755020311000359.Google Scholar
  75. Hersh, R. (1997). What is mathematics, really? New York: Oxford University Press.Google Scholar
  76. Herzberg F. (2007) Internal laws of probability, generalized likelihoods and Lewis’ infinitesimal chances–a response to Adam Elga. The British Journal for the Philosophy of Science 58(1): 25–43CrossRefGoogle Scholar
  77. Hirschfeld J. (1990) The nonstandard treatment of Hilbert’s fifth problem. Transactions of the American Mathematical Society 321(1): 379–400Google Scholar
  78. Hörmander L. (1976) Linear partial differential operators. Springer Verlag, Berlin-New YorkGoogle Scholar
  79. Hrbáček K. (1978) Axiomatic foundations for nonstandard analysis. Fundamenta Mathematicae 98(1): 1–19Google Scholar
  80. Hrushovski E. (1996) The Mordell-Lang conjecture for function fields. Journal of the American Mathematical Society 9(3): 667–690CrossRefGoogle Scholar
  81. Hrushovski E. (2012) Stable group theory and approximate subgroups. Journal of the American Mathematical Society 25: 189–243CrossRefGoogle Scholar
  82. Isaacson D. (2011) The reality of mathematics and the case of set theory. In: Novák Z., Simonyi A. (Eds.) Truth, reference and realism. Central European University Press, Budapest, pp 1–76Google Scholar
  83. Ishiguro H. (1990) Leibniz’s philosophy of logic and language (2nd ed.). Cambridge University Press, CambridgeGoogle Scholar
  84. Jesseph, D. (2012). Leibniz on the Elimination of infinitesimals: Strategies for finding truth in fiction. In N. B. Goethe, P. Beeley & D. Rabouin (Eds.), To appear in Leibniz on the interrelations between mathematics and philosophy, (Archimedes Series, 27 pages). Springer VerlagGoogle Scholar
  85. Kalton N., Sedaev A., Sukochev F. (2011) Fully symmetric functionals on a Marcinkiewicz space are Dixmier traces. Advances in Mathematics 226(4): 3540–3549CrossRefGoogle Scholar
  86. Kanovei V. (1980) The set of all analytically definable sets of natural numbers can be defined analytically. Mathematics of the USSR, Izvestija 15: 469–500CrossRefGoogle Scholar
  87. Kanovei V. (1991) Undecidable hypotheses in Edward Nelson’s Internal Set Theory. Russian Mathematical Surveys 46(6): 1–54CrossRefGoogle Scholar
  88. Kanovei, V., & Reeken, M. (2004). Nonstandard analysis, axiomatically. Springer Monographs in Mathematics. Berlin: Springer, xvi+408 pp.Google Scholar
  89. Kanovei V., Shelah S. (2004) A definable nonstandard model of the reals. Journal of Symbolic Logic 69(1): 159–164CrossRefGoogle Scholar
  90. Kanovei V., Uspensky V. (2006) Uniqueness of nonstandard extensions. Moscow University Mathematics Bulletin 61(5): 1–8Google Scholar
  91. Kantor I. (1972) Certain generalizations of Jordan algebras. Trudy Seminara po Vektornomu i Tenzornomu Analizu s ikh Prilozheniyami k Geometrii, Mekhanike i Fizike 16: 407–499Google Scholar
  92. Katz, K., & Katz, M. (2011a). Cauchy’s continuum. Perspectives on Science, 19(4), 426-452. See and
  93. Katz, K., & Katz, M. (2011b). Meaning in classical mathematics: is it at odds with Intuitionism? Intellectica, 56(2), 223–302. See
  94. Katz, K., & Katz, M. (2012a). Stevin numbers and reality. Foundations of Science, 17(2), 109–123. See and doi: 10.1007/s10699-011-9228-9.
  95. Katz, K., & Katz, M. (2012b). A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography. Foundations of Science, 17(1), 51–89. See doi: 10.1007/s10699-011-9223-1 and
  96. Katz, M. (1995). A proof via the Seiberg-Witten moduli space of Donaldson’s theorem on smooth 4 -manifolds with definite intersection forms. R.C.P. 25, Vol. 47 (Strasbourg, 1993–1995), 269–274, Prépubl. Inst. Rech. Math. Av., 1995/24, Univ. Louis Pasteur, Strasbourg, See
  97. Katz, M., Leichtnam, E. (2013). Commuting and non-commuting infinitesimals. American Mathematical Monthly (to appear).Google Scholar
  98. Katz, M., Schaps, D., & Shnider, S. (2013). Almost equal: The method of adequality from diophantus to fermat and beyond. Perspectives on Science 21(3), (to appear).
  99. Katz, M., & Sherry, D. (2012a) Leibniz’s infinitesimals: Their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond. Erkenntnis (online first), see doi: 10.1007/s10670-012-9370-y and
  100. Katz, M., & Sherry, D. (2012b). Leibniz’s laws of continuity and homogeneity. Notices of the American Mathematical Society, 59(11), (to appear)Google Scholar
  101. Kawai, T. (1983) Nonstandard analysis by axiomatic methods. In: Southeast Asia Conference on Logic, Singapore 1981, Studies in Logic and Foundations of Mathematics (Vol. 111, pp. 55–76). North Holland.Google Scholar
  102. Keisler H.J. (1986) Elementary calculus: An infinitesimal approach. (2nd ed.). Boston: Prindle, Weber & Schimidt See
  103. Keisler H. J. (1994) The hyperreal line. In: Ehrlich P. (Ed.) Real numbers generalizations of reals, and theories of continua. Kluwer Academic Publishers, Dordrecht, pp 207–237CrossRefGoogle Scholar
  104. Klein, F. (1908) Elementary Mathematics from an Advanced Standpoint. Vol. I. Arithmetic, Algebra, Analysis. Translation by E. R. Hedrick and C. A. Noble [Macmillan, New York, 1932] from the third German edition [Springer, Berlin, 1924]. Originally published as Elementarmathematik vom höheren Standpunkte aus (Leipzig, 1908).Google Scholar
  105. Kunen, K. (1980). Set theory. An introduction to independence proofs. Studies in Logic and the Foundations of Mathematics (Vol. 102). Amsterdam-New York: North-Holland Publishing Co.Google Scholar
  106. Lakoff G., Núñez R. (2000) Where mathematics comes from. How the embodied mind brings mathematics into being. Basic Books, New YorkGoogle Scholar
  107. Larson P. (2009) The filter dichotomy and medial limits. Journal of Mathematical Logic 9(2): 159–165CrossRefGoogle Scholar
  108. Levey, S. (2008). Archimedes, Infinitesimals and the Law of Continuity: On Leibniz’s Fictionalism. In: Goldenbaum et al. [68], pp. 107–134.Google Scholar
  109. Lord, S., & Sukochev, F. (2010) Measure theory in noncommutative spaces. SIGMA Symmetry Integrability Geom. Methods Appl. 6(Paper 072):36Google Scholar
  110. Lord S., Sukochev F. (2011) Noncommutative residues and a characterisation of the noncommutative integral. Proceedings of the American Mathematical Society 139(1): 243–257CrossRefGoogle Scholar
  111. Lord S., Potapov D., Sukochev F. (2010) Measures from Dixmier traces and zeta functions. Journal of Functional Analysis 259(8): 1915–1949CrossRefGoogle Scholar
  112. Łoś, J. (1955). Quelques remarques, thérèmes et problèmes sur les classes définissables d’algèbres. In Mathematical interpretation of formal systems (pp. 98–113). Amsterdam: North-Holland Publishing Co.Google Scholar
  113. Luxemburg, W. (1964). Nonstandard analysis. Lectures on A. Robinson’s Theory of infinitesimals and infinitely large numbers, Second corrected ed. Pasadena: Mathematics Department, California Institute of Technology.Google Scholar
  114. Luxemburg, W. (1963). Addendum to “On the measurability of a function which occurs in a paper by A. C. Zaanen”. Nederl. Akad. Wetensch. Proceedings of Series A 66 Koninklijke Nederlandse Akademie van Wetenschappen. Indagationes Mathematicae, 25, 587–590.Google Scholar
  115. Luxemburg, W. (1973). What is nonstandard analysis? Papers in the foundations of mathematics. American Mathematical Monthly 80(6), part II, 38–67.Google Scholar
  116. Machover M. (1993) The place of nonstandard analysis in mathematics and in mathematics teaching. The British Journal for the Philosophy of Science 44(2): 205–212CrossRefGoogle Scholar
  117. Mac Lane S. (1986) Mathematics, form and function. Springer-Verlag, New YorkCrossRefGoogle Scholar
  118. Margenau, H. (1935). Methodology of Physics, 2 parts. Philosophy of Physics, 2, 48–72, 164–187.Google Scholar
  119. Margenau H. (1950) The nature of physical reality. A philosophy of modern physics. McGraw-Hill Book Co., Inc, New York, NYGoogle Scholar
  120. Marquis J.-P. (1997) Abstract mathematical tools and machines for mathematics. Philosophia Mathematica. Series III 5(3): 250–272CrossRefGoogle Scholar
  121. Marquis, J.-P. (2006). A path to the epistemology of mathematics: homotopy theory. In The architecture of modern mathematics (pp. 239–260). Oxford: Oxford University PressGoogle Scholar
  122. Meyer, P. (1973). Limites médiales, d’après mokobodzki, séminaire de probabilités, VII (Univ. Strasbourg, année universitaire 1971–1972) Lecture Notes in Mathematics (Vol. 321, pp. 198–204) Berlin: Springer.Google Scholar
  123. Mokobodzki, G. (1967/68). Ultrafiltres rapides sur N. Construction d’une densité relative de deux potentiels comparables. 1969 Séminaire de Théorie du Potentiel, dirigé par M. Brelot, G. Choquet et J. Deny: 1967/68, Exp. 12, 22 pp. Secrétariat mathématique, Paris.Google Scholar
  124. Morley M., Vaught R. (1962) Homogeneous universal models. Mathematica Scandinavica 11: 37–57Google Scholar
  125. Nelson E. (1977) Internal set theory: A new approach to nonstandard analysis. Bulletin of the American Mathematical Society 83(6): 1165–1198CrossRefGoogle Scholar
  126. Otte, M. (1994). Das Formale, das Soziale, und das Subjektive. Eine Einführung in die Philosophie und Didaktik der Mathematik. Frankfurt/Main: Suhrkamp Verlag.Google Scholar
  127. Novikov P. S. (1963) On the consistency of some propositions of the descriptive theory of sets. American Mathematical Society Translations (2) 29: 51–89Google Scholar
  128. Pawlikowski J. (1991) The Hahn–Banach theorem implies the Banach–Tarski paradox. Fundamenta Mathematicae 138(1): 21–22Google Scholar
  129. Proietti C. (2008) Natural numbers and infinitesimals: A discussion between Benno Kerry and Georg Cantor. History and Philosophy of Logic 29(4): 343–359CrossRefGoogle Scholar
  130. Raussen M., Skau C. (2010) Interview with Mikhail Gromov. Notices of the American Mathematical Society 57(3): 391–403Google Scholar
  131. Resnik M. (1994) Mathematics as a Science of Patterns. Oxford University Press, OxfordGoogle Scholar
  132. Robinson A. (1966) Non-standard analysis. North-Holland Publishing Co, AmsterdamGoogle Scholar
  133. Rudin, W. (1956). Homogeneity problems in the theory of Čech compactifications. Duke Mathematical Journal, 23, 409–419 and 633.Google Scholar
  134. Russell B. (1903) The principles of mathematics. Cambridge University Press, CambridgeGoogle Scholar
  135. Scott, D. (1961). On constructing models for arithmetic. 1961 Infinitistic Methods (Proceedings of symposium Foundations of Mathematics, Warsaw, 1959) (pp. 235–255). Pergamon, Oxford; Państwowe Wydawnictwo Naukowe, Warsaw.Google Scholar
  136. Shapiro S. (1997) Philosophy of mathematics. Structure and ontology. Oxford University Press, New YorkGoogle Scholar
  137. Shelah, S. (1982). Proper forcing. Lecture Notes in Mathematics (Vol. 940). Berlin-New York: Springer-Verlag.Google Scholar
  138. Shelah S. (1984) Can you take Solovay’s inaccessible away?. Israel Journal of Mathematics 48(1): 1–47CrossRefGoogle Scholar
  139. Sierpiński, W. (1934). Hypothèse du Continu, Monografje Matematyczne, Tome 4, Warszawa-Lwow, Subwencji Funduszu Kultur. Narodowej, v+192 pp. [2nd edition: Chesea, 1956.Google Scholar
  140. Sinaceur H. (1973) Cauchy et Bolzano. Revue d’Histoire des Sciences et de leurs Applications 26(2): 97–112CrossRefGoogle Scholar
  141. Skolem T. (1933). Norsk Matematisk Forenings Skrifter II. Series 1/12: 73–82Google Scholar
  142. Skolem T. (1934). Fundamenta Mathematicae 23: 150–161Google Scholar
  143. Skolem, T. (1955). Peano’s axioms and models of arithmetic. In Mathematical interpretation of formal systems (pp. 1–14). Amsterdam: North-Holland Publishing Co.Google Scholar
  144. Solovay R. (1970) A model of set-theory in which every set of reals is Lebesgue measurable. Annals of Mathematics (2) 92: 1–56CrossRefGoogle Scholar
  145. Stern J. (1985) Le problème de la mesure. Seminar Bourbaki. Astérisque 1983: 325–346Google Scholar
  146. Stillwell J. (1977) Concise survey of mathematical logic. Australian Mathematical Society. Journal. Series A. Pure Mathematics and Statistics 24(2): 139–161CrossRefGoogle Scholar
  147. Stroyan, K., & Luxemburg, W. (1976). Introduction to the theory of infinitesimals. Pure and Applied Mathematics, No. 72. New York-London: Academic Press [Harcourt Brace Jovanovich, Publishers]Google Scholar
  148. Sukochev F., Zanin D. (2011) ζ-function and heat kernel formulae. Journal of Functional Analysis 260(8): 2451–2482CrossRefGoogle Scholar
  149. Sukochev, F. & Zanin, D. (2011b). Traces on symmetrically normed operator ideals. See
  150. Tao T. (2008) Structure and randomness. Pages from year one of a mathematical blog. American Mathematical Society, Providence, RIGoogle Scholar
  151. van den Berg, I., Neves, V. (Eds.) (2007) The strength of nonstandard analysis. Springer, Wien, NewYork, ViennaGoogle Scholar
  152. Wenmackers, S., & Horsten, L. (2012). Fair infinite lotteries. Synthese See doi: 10.1007/s11229-010-9836-x.
  153. Wheeler J. (1994) At home in the universe. Masters of modern physics. American Institute of Physics, Woodbury, NYGoogle Scholar
  154. Wilson M. (1992) Frege: The royal road from geometry. Nous 26: 149–180CrossRefGoogle Scholar
  155. Zelmanov E. (2008) On Isaiah Kantor (1936–2006). Journal of Generalized Lie Theory and Applications 2(3): 111CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  • Vladimir Kanovei
    • 1
    • 2
  • Mikhail G. Katz
    • 3
    Email author
  • Thomas Mormann
    • 4
  1. 1.IPPIMoscowRussia
  2. 2.MIITMoscowRussia
  3. 3.Department of MathematicsBar Ilan UniversityRamat, GanIsrael
  4. 4.Department of Logic and Philosophy of ScienceUniversity of the Basque Country UPV/EHUDonostia San SebastianSpain

Personalised recommendations