Skip to main content
SpringerLink
Log in

Toward a Clarity of the Extreme Value Theorem

  • Published:
Logica Universalis Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  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. 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 preparation

  3. Bascelli T.: Galileo’s quanti: understanding infinitesimal magnitudes. Arch. Hist. Exact Sci. 68(2), 121–136 (2014)

    Article  MathSciNet  Google Scholar 

  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 appear

  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)

  6. Bell J.: A Primer of Infinitesimal Analysis. 2nd edn. Cambridge University Press, Cambridge (2008)

    Book  Google Scholar 

  7. Bell, J.: Continuity and infinitesimals. Stanford Encyclopedia of philosophy. Revised 20 July 2009

  8. Benacerraf P.: What numbers could not be. Philos. Rev. 74, 47–73 (1965)

    Article  MathSciNet  Google Scholar 

  9. Berger J., Ishihara H.: Brouwer’s fan theorem and unique existence in constructive analysis. MLQ Math. Log. Q. 51(4), 360–364 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Berger J., Bridges D., Schuster P.: The fan theorem and unique existence of maxima. J. Symbolic Logic 71((2), 713–720 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  11. Bishop E.: Foundations of Constructive Analysis. McGraw-Hill Book Co., New York (1967)

    MATH  Google Scholar 

  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)

  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)

  14. Bishop, E., Bridges, D.: Constructive Analysis. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 279. Springer, Berlin (1985)

  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)

  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. 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)

  18. Bridges, D.: Constructive Functional Analysis. Research Notes in Mathematics 28. Pitman (Advanced Publishing Program), Boston, Mass.-London (1979)

  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)

  20. Bridges D.: Continuity and Lipschitz constants for projections. J. Log. Algebr. Program. 79(1), 2–9 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  21. Cauchy, A. L.: Cours d’Analyse de L’Ecole Royale Polytechnique. Première Partie. Analyse algébrique (Paris: Imprimérie Royale, 1821)

  22. Kock, A.: Synthetic Differential Geometry. 2nd edn. London Mathematical Society Lecture Note Series, vol. 333. Cambridge University Press, Cambridge (2006)

  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. Diener H., Loeb I.: Sequences of real functions on [0,1] in constructive reverse mathematics. Ann. Pure Appl. Logic 157(1), 50–61 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  25. Edwards H.: Kronecker’s algorithmic mathematics. Math. Intell. 31(2), 11–14 (2009)

    Article  MATH  Google Scholar 

  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)

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  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)

  30. Gauthier, Y.: Internal logic. Foundations of mathematics from Kronecker to Hilbert. In: Synthese Library, vol. 310. Kluwer Academic Publishers Group, Dordrecht (2002)

  31. Gauthier Y.: Classical function theory and applied proof theory. Int. J. Pure Appl. Math. 56(2), 223–233 (2009)

    MATH  MathSciNet  Google Scholar 

  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. Gispert-Chambaz, H.: Camille Jordan et les fondements de l’Analyse. Publications Mathematiques d’Orsay 82-05

  34. Grzegorczyk A.: Computable functionals. Fundamenta Mathematicae 42, 168–202 (1955)

    MATH  MathSciNet  Google Scholar 

  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. 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)

  37. Hatcher W.: Calculus is algebra. Amer. Math. Monthly 89(6), 362–370 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  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. Hellman G.: Mathematical constructivism in spacetime. British J. Philos. Sci. 49(3), 425–450 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  40. Hewitt E.: Rings of real-valued continuous functions. I. Trans. Amer. Math. Soc. 64, 45–99 (1948)

    Article  MATH  MathSciNet  Google Scholar 

  41. Ishihara H.: An omniscience principle, the König lemma and the Hahn-Banach theorem. Z. Math. Logik Grundlag. Math. 36(3), 237–240 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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)

    Article  MathSciNet  Google Scholar 

  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. 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. Katz M., Tall D.: A Cauchy-Dirac delta function. Found. Sci. 18(1), 107–123 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  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. Keisler, H.J.: Elementary Calculus: an Infinitesimal Approach. 2nd edn. Prindle, Weber & Schimidt, Boston (1986)

  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])

  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]

  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)

  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 2011

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  56. Kohlenbach, U.: Applied proof theory: proof interpretations and their use in mathematics. In: Springer Monographs in Mathematics. Springer, Berlin (2008)

  57. Kreinovich, V.: Categories of space-time models (Russian). PhD dissertation, Soviet Academy of Sciences, Siberian Branch, Institute of Mathematics (1979)

  58. Kreinovich, V.: Review of D.S. Bridges, Constructive functional analysis, Pitman, London 1979 (see item [18] above). Zbl 401:03027; Math Reviews 82k:03094

  59. Kronecker, L.: Über den Begriff der Zahl in der Mathematik. 1891 lecture. First published in Boniface & Schappacher 2001 [14]

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  62. Lichtenberg, G.: Aphorisms. Translated by R. J. Hollingdale. Penguin Books (1990). [Book A is dated 1765–1770]

  63. Lightstone A.: Infinitesimals. Amer. Math. Monthly 79, 242–251 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  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)

  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. 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)

  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)

  69. Peirce, C. S.: Three grades of clearness. In: The Logic of Relatives. in The Monist vol. 7, pp. 161–217 (1897)

  70. Reeder, P.: Infinitesimals for metaphysics: consequences for the ontologies of space and time. Degree Doctor of Philosophy, Ohio State University, Philosophy (2012)

  71. Robinson A.: Non-standard Analysis.. North-Holland Publishing Co., Amsterdam (1966)

    Google Scholar 

  72. Robinson A.: Reviews: foundations of constructive analysis. Am. Math. Monthly 75(8), 920–921 (1968)

    Article  Google Scholar 

  73. Ross, D.: The constructive content of nonstandard measure existence proofs: is there any? pp. 229–239 in reference Schuster et al. [78]

  74. Ross D.: A nonstandard proof of a lemma from constructive measure theory. MLQ Math. Log. Q. 52(5), 494–497 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  76. Sad L., Teixeira M., Baldino R.: Cauchy and the problem of point-wise convergence. Arch. Internat. Hist. Sci. 51(147), 277–308 (2001)

    MATH  MathSciNet  Google Scholar 

  77. Schmieden C., Laugwitz D.: Eine Erweiterung der Infinitesimalrechnung. (German) Math. Z. 69, 1–39 (1958)

    MATH  MathSciNet  Google Scholar 

  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)

  79. Schuster, P.: Unique solutions. MLQ Math. Log. Q. 52(6), 534–539 (2006) (with 53 (2007), no. 2, 214)

  80. Schuster P.: Problems, solutions, and completions. J. Log. Algebr. Program. 79(1), 84–91 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  82. Sherry, D.: The wake of Berkeley’s Analyst: rigor mathematicae?. Stud. Hist. Philos. Sci. 18(4), 455–480 (1987)

  83. Sinaceur H.: Cauchy et Bolzano. Rev. Histoire Sci. Appl. 26(2), 97–112 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  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)

  85. Tarski, A.: Une contribution à à la théorie de la mesure. Fund. Math. 15, 42–50 (1930)

  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)

  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)

  88. Wattenberg F.: Nonstandard analysis and constructivism?. Studia Logica 47(3), 303–309 (1988)

    Article  MathSciNet  Google Scholar 

  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)

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Bar Ilan University, 52900, Ramat Gan, Israel

    Karin U. Katz & Mikhail G. Katz

  2. Department of Mathematics, Lviv National University, Lviv, Ukraine

    Taras Kudryk

Authors
  1. Karin U. Katz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mikhail G. Katz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Taras Kudryk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mikhail G. Katz.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Katz, K.U., Katz, M.G. & Kudryk, T. Toward a Clarity of the Extreme Value Theorem. Log. Univers. 8, 193–214 (2014). https://doi.org/10.1007/s11787-014-0102-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11787-014-0102-8

Mathematics Subject Classification (2010)

Keywords

Search

Navigation