Prospects for Triviality

Chapter
Part of the Trends in Logic book series (TREN, volume 45)

Abstract

In this paper I argue, contra Mortensen, that there is a case, namely that of a degenerate topos, an extremely simple mathematical universe in which everything is true, in which no mathematical “catastrophe” is implied by mathematical triviality. I will show that either one of the premises of Dunn’s trivialization result for real number theory –on which Mortensen mounts his case– cannot obtain (from a point of view “external” to the universe) and thus the argument is unsound, or that it obtains in calculations “internal” to such trivial universe and the theory associated, yet the calculations are possible and meaningful albeit extremely simple.

Keywords

Triviality Atomic triviality Real number theory Degenerate categories Internal logic 

References

  1. 1.
    Beall, J. C. (1999). From full blooded platonism to really full blooded platonism. Philosophia Mathematica, 7(3), 322–325.CrossRefGoogle Scholar
  2. 2.
    Carnielli, W. A., & Coniglio, M. E. (2013). Paraconsistent set theory by predicating on consistency. Journal of Logic and Computation.Google Scholar
  3. 3.
    Estrada-González, L. Through full-blooded platonism, and what paraconsistentists could find there. Logique et Analyse, 59(235), 283–300.Google Scholar
  4. 4.
    Kabay, P. (2010). On the plenitude of truth. A defense of trivialism. Germany: Lambert Academic Publishing.Google Scholar
  5. 5.
    Kroon, F. (2004). Realism and dialetheism. In G. Priest, J. C. Beall, & B. Armour-Garb (Eds.), The law of non-contradiction. New philosophical essays (pp. 245–263). Toronto: Oxford Clarendon Press.CrossRefGoogle Scholar
  6. 6.
    Mac Lane, S., & Moerdijk, I. (1992). Sheaves in geometry and logic: A first introduction to topos theory. Heidelberg: Springer.CrossRefGoogle Scholar
  7. 7.
    Martínez-Ordaz, M. del R. (2010). Gödel and paraconsistency: The road to non-triviality. Bachelor thesis, Universidad Veracruzana, Mexico (In Spanish).Google Scholar
  8. 8.
    McLarty, C. (1995). Elementary categories. Elementary toposes. Toronto: Oxford Clarendon Press.Google Scholar
  9. 9.
    Mortensen, C. (1989). Anything is possible. Erkenntnis, 30(3), 319–337.CrossRefGoogle Scholar
  10. 10.
    Mortensen, C. (1995). Inconsistent mathematics. Kluwer mathematics and its applications. Berlin: Kluwer Academic Publishers.Google Scholar
  11. 11.
    Mortensen, C. (2000). Prospects for inconsistency. In D. Batens, C. Mortensen, G. Priest, & J.-P. Van Bendegem (Eds.), Frontiers of paraconsistent logic (pp. 203–208). Research Studies Press.Google Scholar
  12. 12.
    Mortensen, C. (2005). It isn’t so, but could it be? Logique et Analyse, 48(189–192), 351–360.Google Scholar
  13. 13.
    Mortensen, C. (2007). Inconsistent mathematics: Some philosophical implications. In A. Irvine (Ed.), Philosophy of mathematics (pp. 631–649). Amsterdam: North Holland.Google Scholar
  14. 14.
    Priest, G. (1998). To be and not to be –that is the answer. On Aristotle on the law of non-contradiction. Philosophiegeschichte und Logische Analyse, 1(1), 91–130.Google Scholar
  15. 15.
    Priest, G. (1998). The trivial object and the non-triviality of a semantically closed theory with descriptions. Journal of Applied Non-Classical Logics, 8(1–2), 171–183.CrossRefGoogle Scholar
  16. 16.
    Priest, G. (2000). Could everything be true? Australasian Journal of Philosophy, 78(2), 189–195.CrossRefGoogle Scholar
  17. 17.
    Priest, G. (2006). Doubt truth to be a liar. Oxford: Oxford University Press.Google Scholar
  18. 18.
    Robles, G. (2008). Weak consistency and strong paraconsistency. In J. María Díaz Nafría & F. Salto Alemany (Eds.), Actas del I Encuentro Internacional de Expertos en Teorías de la Información. Un Enfoque Interdisciplinar (pp. 117–131). León, España: Universidad de León.Google Scholar
  19. 19.
    Weber, Z. (2012). Review of ‘inconsistent geometry’, by Chris Mortensen. Australasian Journal Philosophy, 90(3), 611–614.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2016

Authors and Affiliations

  1. 1.Instituto de Investigaciones FilosóficasUniversidad Nacional Autónoma de MéxicoCoyoacanMexico

Personalised recommendations