Labels for Non-Individuals?

Original Paper

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Quasi-set theory is a first-order theory without identity, which allows us to cope with non-individuals in a sense. A weaker equivalence relation called “indistinguishability” is an extension of identity in the sense that if x is identical to y then x and y are indistinguishable, although the reciprocal is not always valid. The interesting point is that quasi-set theory provides us with a useful mathematical background for dealing with collections of indistinguishable elementary quantum particles. In the present paper, however, we show that even in quasi-set theory it is possible to label objects that are considered as non-individuals. This is the first paper of a series that will be dedicated to the philosophical and physical implications of our main mathematical result presented here.

Key words:

quasi-sets non-individuality labels quantum mechanics 


  1. 1.
    1. Adams, R., ‘Primitive thisness and primitive identity,’ J. Phil. 76, 5–26 (1979).Google Scholar
  2. 2.
    2. Da Costa, N. C. A., and R. Chuaqui, ‘On Suppes' set theoretical predicates,’ Erkenntnis 29, 95–112 (1988).Google Scholar
  3. 3.
    3. Da Costa, N. C. A., and A. S. Sant'Anna, ‘The mathematical role of time and spacetime in classical physics,’ Found. Phys. Lett. 14, 553–563 (2001).CrossRefMathSciNetGoogle Scholar
  4. 4.
    4. Da Costa, N. C. A., and A. S. Sant'Anna, ‘Time in thermodynamics,’ Found. Phys. 32, 1785–1796 (2002).CrossRefMathSciNetGoogle Scholar
  5. 5.
    5. Dalla Chiara, M. L., and G. Toraldo di Francia, ‘Individuals, kinds and names in physics’, in G. Corsi et al., eds., Bridging the Gap: Philosophy, Mathematics, Physics, pp. 261–283 (Kluwer Academic, Dordrecht, 1993).Google Scholar
  6. 6.
    6. French, S., ‘Identity and individuality in quantum theory,’ The Stanford Encyclopedia of Philosophy, Edward N. Zalta, ed., URL = (2004).Google Scholar
  7. 7.
    7. Huggett, N., ‘Atomic metaphysics,’ J. Phil. 96, 5–24 (1999).CrossRefGoogle Scholar
  8. 8.
    8. Krause, D., ‘On a quasi-set theory,’ Notre Dame J. Formal Logic 33, 402–411 (1992).MATHMathSciNetGoogle Scholar
  9. 9.
    9. Krause, D., A. S. Sant'Anna, and A. G. Volkov, ‘Quasi-set theory for bosons and fermions: quantum distributions,’ Found. Phys. Lett. 12, 51–66 (1999).CrossRefMathSciNetGoogle Scholar
  10. 10.
    10. Mandel, L., ‘Coherence and indistinguishability,’ Optics Lett. 16, 1882–1884 (1991).ADSCrossRefGoogle Scholar
  11. 11.
    11. Manin, Yu. I., ‘Problems of present day mathematics I: Foundations,’ in Browder, F. E., ed., Mathematical Problems Arising from Hilbert Problems, Proceedings of Symposia in Pure Mathematics XXVIII (AMS, Providence, 1976), pp. 36–36.Google Scholar
  12. 12.
    12. Mendelson, E., Introduction to Mathematical Logic (Chapman & Hall, London, 1997).MATHGoogle Scholar
  13. 13.
    13. Ryder, L. H., Quantum Field Theory (Cambridge University Press, Cambridge, 1996).MATHGoogle Scholar
  14. 14.
    14. Sakurai, J. J., Modern Quantum Mechanics (Addison-Wesley, Reading, 1994).Google Scholar
  15. 15.
    15. Sant'Anna, A. S., and D. Krause, ‘Indistinguishable particles and hidden variables,’ Found. Phys. Lett. 10, 409–426 (1997).MathSciNetGoogle Scholar
  16. 16.
    16. Sant'Anna, A. S., and A. M. S. Santos, ‘Quasi-set-theoretical foundations of statistical mechanics: a research program,’ Found. Phys. 30, 101–120 (2000).CrossRefMathSciNetGoogle Scholar
  17. 17.
    17. Suppes, P., Representation and Invariance of Scientific Structures (CSLI, Stanford, 2002).MATHGoogle Scholar
  18. 18.
    18. Weyl, H. (1949), Philosophy of Mathematics and Natural Science (Princeton University Press, Princeton).MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of PhilosophyUniversity of South CarolinaColumbia

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