, Volume 18, Issue 1, pp 1–24 | Cite as

Infinity in Ontology and Mind

  • Nino B. CocchiarellaEmail author
invited paper


Two fundamental categories of any ontology are the category of objects and the category of universals. We discuss the question whether either of these categories can be infinite or not. In the category of objects, the subcategory of physical objects is examined within the context of different cosmological theories regarding the different kinds of fundamental objects in the universe. Abstract objects are discussed in terms of sets and the intensional objects of conceptual realism. The category of universals is discussed in terms of the three major theories of universals: nominalism, realism, and conceptualism. The finitude of mind pertains only to conceptualism. We consider the question of whether or not this finitude precludes impredicative concept formation. An explication of potential infinity, especially as applied to concepts and expressions, is given. We also briefly discuss a logic of plural objects, or groups of single objects (individuals), which is based on Bertrand Russell’s (1903, The principles of mathematics, 2nd edn. (1938). Norton & Co, NY) notion of a class as many. The universal class as many does not exist in this logic if there are two or more single objects; but the issue is undecided if there is just one individual. We note that adding plural objects (groups) to an ontology with a countable infinity of individuals (single objects) does not generate an uncountable infinity of classes as many.


Formal ontology Ontology Universals Conceptual realism Conceptualism Nominalism Logical realism Natural realism Plural objects Infinity Potential infinity Quantum mechanics Multiverse String theory 


  1. Campbell K (1981) The metaphysics of abstract particulars. Midwest Stud Philos 6:477–488CrossRefGoogle Scholar
  2. Cocchiarella NB (1987) Logical studies in early philosophy. Ohio State University Press, ColumbusGoogle Scholar
  3. Cocchiarella NB (2007) Formal ontology and conceptual realism, synthese library, vol 339. Springer, DordrechtGoogle Scholar
  4. Dedekind R (1888) Was sind und was sollen dies Zahlen? BraunschweigGoogle Scholar
  5. De Witt B, Graham N (eds) (1973) The many-worlds interpretation of quantum mechanics. Princeton University Press, PrincetonGoogle Scholar
  6. Gatlin LL (1972) Information theory and the living system. Columbia University press, NYGoogle Scholar
  7. Greene B (1999) The elegant universe. Random House Inc., NYGoogle Scholar
  8. Greene B (2004) The fabric of the cosmos, space, time, and the texture of reality. Alfred A Knopf, NYGoogle Scholar
  9. Gribbin J (2001) Hyperspace, our final frontier. DK Pub Inc., NYGoogle Scholar
  10. Henkin L (1962) Nominalistic analysis of mathematical language. In: Nagel E, Suppes P, Tarski A (eds) Logic, methodology and philosophy of science, Proceedings of the 1960 international congress. Stanford University Press, Stanford, pp 187–193Google Scholar
  11. Luminet J-P, Starkman GD, Weeks JR (1999) Is space finite? Sci Am 280:90–97CrossRefGoogle Scholar
  12. McCall S (1994) A model of the universe: space-time, probability, and decision. Clarendon Press, OxfordGoogle Scholar
  13. Penrose R (2004) The road to reality, a complete guide to the laws of the universe. Alfred A Knopf, NYGoogle Scholar
  14. Quine WV (1953) From a logical point of view. Harvard University Press, CambridgeGoogle Scholar
  15. Russell B (1903) The principles of mathematics, 2nd edn. (1938). Norton & Co, NYGoogle Scholar
  16. Shannon CE, Weaver W (1949) The Mathematical theory of communication. University of Illinois Press, UrbanaGoogle Scholar
  17. Steinhardt P, Turok N (2007) Endless universe. Doubleday, NYGoogle Scholar
  18. Tegmark M (2003) Parallel universes. In: Barrow JD, Davies PCW, Harper CL (eds) Science and ultimate reality: from quantum to cosmos, honoring John Wheeler’s 90th birthday. Cambridge University Press, CambridgeGoogle Scholar
  19. Tarski A (1924) Sur les ensembles finis. Fundamenta Mathematicae 6:43–95Google Scholar
  20. Williams DC (1953) The elements of being. Rev Metaphys 6:1–18Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of PhilosophyIndiana UniversityBloomingtonUSA

Personalised recommendations