Logica Universalis

, Volume 8, Issue 2, pp 193–214 | Cite as

Toward a Clarity of the Extreme Value Theorem

Article

Abstract

We apply a framework developed by C. S. Peirce to analyze the concept of clarity, so as to examine a pair of rival mathematical approaches to a typical result in analysis. Namely, we compare an intuitionist and an infinitesimal approaches to the extreme value theorem. We argue that a given pre-mathematical phenomenon may have several aspects that are not necessarily captured by a single formalisation, pointing to a complementarity rather than a rivalry of the approaches.

Mathematics Subject Classification (2010)

Primary 26E35 Secondary 00A30 01A85 03F55 

Keywords

Benacerraf Bishop Cauchy constructive analysis continuity extreme value theorem grades of clarity hyperreal infinitesimal Kaestner Kronecker law of excluded middle ontology Peirce principle of unique choice procedure trichotomy uniqueness paradigm 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bair, J., Błaszczyk, P., Ely, R., Henry, V., Kanovei, V., Katz, K., Katz, M., Kutateladze, S., McGaffey, T., Schaps, D., Sherry, D., Shnider, S.: Is mathematical history written by the victors? Notices Am. Math. Soc. 60(7), 886–904 (2013). See http://www.ams.org/notices/201307/rnoti-p886.pdf and http://arxiv.org/abs/1306.5973
  2. 2.
    Bair, J., Błaszczyk, P., Ely, R., Henry, V., Kanovei, V., Katz, K., Katz, M., Kutateladze, S., McGaffey, T., Schaps, D., Sherry, D., Shnider, S.: Interpreting Euler’s infinitesimal mathematics (2014), in preparationGoogle Scholar
  3. 3.
    Bascelli T.: Galileo’s quanti: understanding infinitesimal magnitudes. Arch. Hist. Exact Sci. 68(2), 121–136 (2014)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Bascelli, T., Bottazzi, E., Herzberg, F., Kanovei, V., Katz, K., Katz, M., Nowik, T., Sherry, D., Shnider, S.: Fermat, Leibniz, Euler, and the gang: The true history of the concepts of limit and shadow. Notices Am. Math. Soc. (2014), to appearGoogle Scholar
  5. 5.
    Beeson, M.: Foundations of Constructive Mathematics. Metamathematical Studies. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], 6. Springer, Berlin (1985)Google Scholar
  6. 6.
    Bell J.: A Primer of Infinitesimal Analysis. 2nd edn. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar
  7. 7.
    Bell, J.: Continuity and infinitesimals. Stanford Encyclopedia of philosophy. Revised 20 July 2009Google Scholar
  8. 8.
    Benacerraf P.: What numbers could not be. Philos. Rev. 74, 47–73 (1965)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Berger J., Ishihara H.: Brouwer’s fan theorem and unique existence in constructive analysis. MLQ Math. Log. Q. 51(4), 360–364 (2005)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Berger J., Bridges D., Schuster P.: The fan theorem and unique existence of maxima. J. Symbolic Logic 71((2), 713–720 (2006)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Bishop E.: Foundations of Constructive Analysis. McGraw-Hill Book Co., New York (1967)MATHGoogle Scholar
  12. 12.
    Bishop, E.: The crisis in contemporary mathematics. In: Proceedings of the American Academy Workshop on the Evolution of Modern Mathematics (Boston, Mass.). Historia Math. 2(1975), no. 4, 507–517 (1974)Google Scholar
  13. 13.
    Bishop, E.: Schizophrenia in contemporary mathematics [published posthumously; originally distributed in 1973]. In Errett Bishop: reflections on him and his research (San Diego, Calif., 1983), pp. 1–32, Contemp. Math. 39, Am. Math. Soc., Providence, RI (1985)Google Scholar
  14. 14.
    Bishop, E., Bridges, D.: Constructive Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 279. Springer, Berlin (1985)Google Scholar
  15. 15.
    Boniface, J., Schappacher, N.: “Sur le concept de nombre en mathématique”: cours inédit de Leopold Kronecker à à Berlin (1891). [“On the concept of number in mathematics”: Leopold Kronecker’s 1891 Berlin lectures] Rev. Histoire Math. 7(2), 206–275 (2001)Google Scholar
  16. 16.
    Borovik, A., Katz, M.: Who gave you the Cauchy–Weierstrass tale? The dual history of rigorous calculus. Found. Sci. 17(3), 245–276 (2012). See http://dx.doi.org/10.1007/s10699-011-9235-x and http://arxiv.org/abs/1108.2885
  17. 17.
    Bottazzini, U.: The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass. Translated from the Italian by Warren Van Egmond. Springer, New York (1986)Google Scholar
  18. 18.
    Bridges, D.: Constructive Functional Analysis. Research Notes in Mathematics 28. Pitman (Advanced Publishing Program), Boston, Mass.-London (1979)Google Scholar
  19. 19.
    Bridges, D.: A Constructive look at the Real Number Line. In: Real numbers, generalizations of the reals, and theories of continua, pp. 29–92, see item [26] (1994)Google Scholar
  20. 20.
    Bridges D.: Continuity and Lipschitz constants for projections. J. Log. Algebr. Program. 79(1), 2–9 (2010)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Cauchy, A. L.: Cours d’Analyse de L’Ecole Royale Polytechnique. Première Partie. Analyse algébrique (Paris: Imprimérie Royale, 1821)Google Scholar
  22. 22.
    Kock, A.: Synthetic Differential Geometry. 2nd edn. London Mathematical Society Lecture Note Series, vol. 333. Cambridge University Press, Cambridge (2006)Google Scholar
  23. 23.
    Davis, M.: Applied Nonstandard Analysis. Pure and Applied Mathematics. Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1977. Reprinted: Dover, New York (2005). see http://store.doverpublications.com/0486442292.html
  24. 24.
    Diener H., Loeb I.: Sequences of real functions on [0,1] in constructive reverse mathematics. Ann. Pure Appl. Logic 157(1), 50–61 (2009)CrossRefMATHMathSciNetGoogle Scholar
  25. 25.
    Edwards H.: Kronecker’s algorithmic mathematics. Math. Intell. 31(2), 11–14 (2009)CrossRefMATHGoogle Scholar
  26. 26.
    Ehrlich, P. (ed.): Real numbers, generalizations of the reals, and theories of continua. In: Ehrlich, P. (ed.) Synthese Library, vol. 242. Kluwer Academic Publishers Group, Dordrecht (1994)Google Scholar
  27. 27.
    Ehrlich P.: 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(1), 1–121 (2006)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Feferman, S.: Relationships between constructive, predicative and classical systems of analysis. In: Proof theory (Roskilde, 1997), Synthese Lib., vol. 292, pp. 221–236. Kluwer Acad. Publ., Dordrecht (2000)Google Scholar
  29. 29.
    Freudenthal, H.: Cauchy, Augustin-Louis. In: Gillispie, C.C. (ed.) Dictionary of Scientific Biography, vol. 3, pp. 131–148. Charles Scribner’s sons, New York (1971)Google Scholar
  30. 30.
    Gauthier, Y.: Internal logic. Foundations of mathematics from Kronecker to Hilbert. In: Synthese Library, vol. 310. Kluwer Academic Publishers Group, Dordrecht (2002)Google Scholar
  31. 31.
    Gauthier Y.: Classical function theory and applied proof theory. Int. J. Pure Appl. Math. 56(2), 223–233 (2009)MATHMathSciNetGoogle Scholar
  32. 32.
    Gauthier, Y.: Kronecker in contemporary mathematics. General arithmetic as a foundational programme. Reports on mathematical logic 48, 37–65 (2013). See http://dx.doi.org/10.4467/20842589RM.13.002.1254
  33. 33.
    Gispert-Chambaz, H.: Camille Jordan et les fondements de l’Analyse. Publications Mathematiques d’Orsay 82-05Google Scholar
  34. 34.
    Grzegorczyk A.: Computable functionals. Fundamenta Mathematicae 42, 168–202 (1955)MATHMathSciNetGoogle Scholar
  35. 35.
    Guillaume, M.: “Review of Katz, M.; Sherry, D. Leibniz’s infinitesimals: their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond. Erkenntnis 78(3), 571–625 (2013)” Math. Rev. (2014). See http://www.ams.org/mathscinet-getitem?mr=3053644
  36. 36.
    Hardy, G., Wright, E.: An introduction to the theory of numbers. 6th edn. Revised by D. R. Heath-Brown and J. H. Silverman. Oxford University Press, Oxford (2008)Google Scholar
  37. 37.
    Hatcher W.: Calculus is algebra. Amer. Math. Monthly 89(6), 362–370 (1982)CrossRefMATHMathSciNetGoogle Scholar
  38. 38.
    Havenel J.: Peirce’s clarifications of continuity. Trans. Charles S. Peirce Soc. Q. J. Am. Philos. 44(1), 86–133 (2008)Google Scholar
  39. 39.
    Hellman G.: Mathematical constructivism in spacetime. British J. Philos. Sci. 49(3), 425–450 (1998)CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Hewitt E.: Rings of real-valued continuous functions. I. Trans. Amer. Math. Soc. 64, 45–99 (1948)CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    Ishihara H.: An omniscience principle, the König lemma and the Hahn-Banach theorem. Z. Math. Logik Grundlag. Math. 36(3), 237–240 (1990)CrossRefMATHMathSciNetGoogle Scholar
  42. 42.
    Kästner, A.G.: Anfangsgründe der Analysis endlicher Größen. Witwe Vandenhoeck, Göttingen (1760). See http://gdz.sub.uni-goettingen.de/no_cache/dms/load/img/?IDDOC=116776
  43. 43.
    Katz, K., Katz, M.: A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography. Found. Sci. 17(1), 51–89 (2012). See http://dx.doi.org/10.1007/s10699-011-9223-1 and http://arxiv.org/abs/1104.0375
  44. 44.
    Katz M., Schaps D., Shnider S.: Almost equal: the method of adequality from diophantus to fermat and beyond. Perspect. Sci. 21(3), 283–324 (2013)CrossRefMathSciNetGoogle Scholar
  45. 45.
    Katz, M., Sherry, D.: Leibniz’s laws of continuity and homogeneity. Notices Am. Math. Soc. 59(11), 1550–1558 (2012). See http://www.ams.org/notices/201211/ and http://arxiv.org/abs/1211.7188
  46. 46.
    Katz, M., Sherry, D.: Leibniz’s infinitesimals: their fictionality, their modern implementations, and their foes from Berkeley to Russell and beyond. Erkenntnis 78(3), 571–625 (2013). See http://dx.doi.org/10.1007/s10670-012-9370-y and http://arxiv.org/abs/1205.0174
  47. 47.
    Katz M., Tall D.: A Cauchy-Dirac delta function. Found. Sci. 18(1), 107–123 (2013)CrossRefMATHMathSciNetGoogle Scholar
  48. 48.
    Katz, V.: “Review of Bair et al., Is mathematical history written by the victors? Notices Amer. Math. Soc. 60 (2013), no. 7, 886–904.” Math. Rev. (2014). See http://www.ams.org/mathscinet-getitem?mr=3086638
  49. 49.
    Keisler, H.J.: Elementary Calculus: an Infinitesimal Approach. 2nd edn. Prindle, Weber & Schimidt, Boston (1986)Google Scholar
  50. 50.
    Keisler, H.J.: The hyperreal line. In: Real Numbers, Generalizations of the Reals, and Theories of Continua, pp. 207–237 (see item Ehrlich 1994 [26])Google Scholar
  51. 51.
    Klein, F.: 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]Google Scholar
  52. 52.
    Knobloch, E.: Leibniz’s rigorous foundation of infinitesimal geometry by means of Riemannian sums. Found. Formal Sci., 1 (Berlin, 1999). Synthese 133(1–2), 59–73 (2002)Google Scholar
  53. 53.
    Knobloch, E.: Galileo and German thinkers: Leibniz. In: Galileo and the Galilean school in universities in the seventeenth century (Italian), vol. 14, pp. 127–139, Studi Cent. Interuniv. Stor. Univ. Ital. CLUEB, Bologna 2011Google Scholar
  54. 54.
    Knobloch, E.: “Review of: Katz, M.; Schaps, D.; Shnider, S. Almost equal: the method of adequality from Diophantus to Fermat and beyond. Perspectives on Science 21(3), 283–324 (2013).” Math. Rev. (2014). See http://www.ams.org/mathscinet-getitem?mr=3114421
  55. 55.
    Kohlenbach U.: Effective moduli from ineffective uniqueness proofs. An unwinding of de La Valleé Poussin’s proof for Chebycheff approximation. Ann. Pure Appl. Logic 64(1), 27–94 (1993)CrossRefMATHMathSciNetGoogle Scholar
  56. 56.
    Kohlenbach, U.: Applied proof theory: proof interpretations and their use in mathematics. In: Springer Monographs in Mathematics. Springer, Berlin (2008)Google Scholar
  57. 57.
    Kreinovich, V.: Categories of space-time models (Russian). PhD dissertation, Soviet Academy of Sciences, Siberian Branch, Institute of Mathematics (1979)Google Scholar
  58. 58.
    Kreinovich, V.: Review of D.S. Bridges, Constructive functional analysis, Pitman, London 1979 (see item [18] above). Zbl 401:03027; Math Reviews 82k:03094Google Scholar
  59. 59.
    Kronecker, L.: Über den Begriff der Zahl in der Mathematik. 1891 lecture. First published in Boniface & Schappacher 2001 [14]Google Scholar
  60. 60.
    Laugwitz D.: Definite values of infinite sums: aspects of the foundations of infinitesimal analysis around 1820. Arch. Hist. Exact Sci. 39(3), 195–245 (1989)CrossRefMATHMathSciNetGoogle Scholar
  61. 61.
    Lawvere F.: Toward the description in a smooth topos of the dynamically possible motions and deformations of a continuous body. Third colloquium on categories (Amiens, 1980), Part I. Cahiers Topologie Géom. Différentielle 21(4), 377–392 (1980)MATHMathSciNetGoogle Scholar
  62. 62.
    Lichtenberg, G.: Aphorisms. Translated by R. J. Hollingdale. Penguin Books (1990). [Book A is dated 1765–1770]Google Scholar
  63. 63.
    Lightstone A.: Infinitesimals. Amer. Math. Monthly 79, 242–251 (1972)CrossRefMATHMathSciNetGoogle Scholar
  64. 64.
    Lindstrøm, T.: An Invitation to Nonstandard Analysis. Nonstandard Analysis and its Applications (Hull, 1986), pp. 1–105, London Math. Soc. Stud. Texts 10, Cambridge Univ. Press, Cambridge (1988)Google Scholar
  65. 65.
    Łos, J.: Quelques remarques, théorèmes et problèmes sur les classes définissables d’algèbres, in Mathematical interpretation of formal systems, pp.98–113, North-Holland Publishing Co., Amsterdam (1955)Google Scholar
  66. 66.
    Mormann, T., Katz, M.: Infinitesimals as an issue of neo-Kantian philosophy of science. HOPOS J. Int. Soc. History Philos. Sci. 3(2), 236–280 (2013). See http://www.jstor.org/stable/10.1086/671348 and http://arxiv.org/abs/1304.1027
  67. 67.
    Novikov, S.: The second half of the 20th century and its results: the crisis of the society of physicists and mathematicians in Russia and in the West. (Russian) Istor.-Mat. Issled. (2) No. 7(42), 326–356, 369 (2002)Google Scholar
  68. 68.
    Novikov, S.: The second half of the 20th century and its conclusion: crisis in the physics and mathematics community in Russia and in the West. Amer. Math. Soc. Transl. Ser. 2, 212, Geometry, topology, and mathematical physics, 1–24, Amer. Math. Soc., Providence, RI, 2004. (Translated from Istor.-Mat. Issled. (2) No. 7(42) (2002), 326–356, 369; by A. Sossinsky)Google Scholar
  69. 69.
    Peirce, C. S.: Three grades of clearness. In: The Logic of Relatives. in The Monist vol. 7, pp. 161–217 (1897)Google Scholar
  70. 70.
    Reeder, P.: Infinitesimals for metaphysics: consequences for the ontologies of space and time. Degree Doctor of Philosophy, Ohio State University, Philosophy (2012)Google Scholar
  71. 71.
    Robinson A.: Non-standard Analysis.. North-Holland Publishing Co., Amsterdam (1966)Google Scholar
  72. 72.
    Robinson A.: Reviews: foundations of constructive analysis. Am. Math. Monthly 75(8), 920–921 (1968)CrossRefGoogle Scholar
  73. 73.
    Ross, D.: The constructive content of nonstandard measure existence proofs: is there any? pp. 229–239 in reference Schuster et al. [78]Google Scholar
  74. 74.
    Ross D.: A nonstandard proof of a lemma from constructive measure theory. MLQ Math. Log. Q. 52(5), 494–497 (2006)CrossRefMATHMathSciNetGoogle Scholar
  75. 75.
    Rust, H.: Operational Semantics for Timed Systems: A Non-standard Approach to Uniform Modeling of Timed and Hybrid Systems. Lecture Notes in Computer Science vol. 3456. Springer, Berlin (2005)Google Scholar
  76. 76.
    Sad L., Teixeira M., Baldino R.: Cauchy and the problem of point-wise convergence. Arch. Internat. Hist. Sci. 51(147), 277–308 (2001)MATHMathSciNetGoogle Scholar
  77. 77.
    Schmieden C., Laugwitz D.: Eine Erweiterung der Infinitesimalrechnung. (German) Math. Z. 69, 1–39 (1958)MATHMathSciNetGoogle Scholar
  78. 78.
    Schuster, P., Berger, U., Osswald, H.: (eds.) Reuniting the antipodes—constructive and nonstandard views of the continuum. Proceedings of the symposium held in Venice, May 16–22, 1999. Synthese Library, vol. 306. Kluwer Academic Publishers, Dordrecht (2001)Google Scholar
  79. 79.
    Schuster, P.: Unique solutions. MLQ Math. Log. Q. 52(6), 534–539 (2006) (with 53 (2007), no. 2, 214)Google Scholar
  80. 80.
    Schuster P.: Problems, solutions, and completions. J. Log. Algebr. Program. 79(1), 84–91 (2010)CrossRefMATHMathSciNetGoogle Scholar
  81. 81.
    Schwichtenberg H.: A direct proof of the equivalence between Brouwer’s fan theorem and König’s lemma with a uniqueness hypothesis. J. UCS 11(12), 2086–2095 (2005)MATHMathSciNetGoogle Scholar
  82. 82.
    Sherry, D.: The wake of Berkeley’s Analyst: rigor mathematicae?. Stud. Hist. Philos. Sci. 18(4), 455–480 (1987)Google Scholar
  83. 83.
    Sinaceur H.: Cauchy et Bolzano. Rev. Histoire Sci. Appl. 26(2), 97–112 (1973)CrossRefMATHMathSciNetGoogle Scholar
  84. 84.
    Stroyan, K.: Uniform continuity and rates of growth of meromorphic functions. In: Contributions to non-standard analysis (Sympos., Oberwolfach, 1970), pp. 47–64. Studies in Logic and Foundations of Math., vol. 69. North-Holland, Amsterdam (1972)Google Scholar
  85. 85.
    Tarski, A.: Une contribution à à la théorie de la mesure. Fund. Math. 15, 42–50 (1930)Google Scholar
  86. 86.
    Taylor, R. G.: Review of real numbers, generalizations of the reals, and theories of continua, edited by Philip Ehrlich [see item [26] above]. Modern Logic8, 195–212, Number 1/2 (January 1998–April 2000)Google Scholar
  87. 87.
    Troelstra, A., van Dalen, D.: Constructivism in mathematics. vol. 1. an introduction. In: Studies in Logic and the Foundations of Mathematics, 121. North-Holland Publishing Co., Amsterdam (1988)Google Scholar
  88. 88.
    Wattenberg F.: Nonstandard analysis and constructivism?. Studia Logica 47(3), 303–309 (1988)CrossRefMathSciNetGoogle Scholar
  89. 89.
    Yau, S.-T., Nadis, S.: The Shape of Inner Space. String Theory and the Geometry of the Universe’s Hidden Dimensions. Basic Books, New York (2010)Google Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Karin U. Katz
    • 1
  • Mikhail G. Katz
    • 1
  • Taras Kudryk
    • 2
  1. 1.Department of MathematicsBar Ilan UniversityRamat GanIsrael
  2. 2.Department of MathematicsLviv National UniversityLvivUkraine

Personalised recommendations