, Volume 16, Issue 4, pp 424–459 | Cite as

The Meaning of Category Theory for 21st Century Philosophy

  • Alberto Peruzzi


Among the main concerns of 20th century philosophy was that of the foundations of mathematics. But usually not recognized is the relevance of the choice of a foundational approach to the other main problems of 20th century philosophy, i.e., the logical structure of language, the nature of scientific theories, and the architecture of the mind. The tools used to deal with the difficulties inherent in such problems have largely relied on set theory and its “received view”. There are specific issues, in philosophy of language, epistemology and philosophy of mind, where this dependence turns out to be misleading. The same issues suggest the gain in understanding coming from category theory, which is, therefore, more than just the source of a “non-standard” approach to the foundations of mathematics. But, even so conceived, it is the very notion of what a foundation has to be that is called into question. The philosophical meaning of mathematics is no longer confined to which first principles are assumed and which “ontological” interpretation is given to them in terms of some possibly updated version of logicism, formalism or intuitionism. What is central to any foundational project proper is the role of universal constructions that serve to unify the different branches of mathematics, as already made clear in 1969 by Lawvere. Such universal constructions are best expressed by means of adjoint functors and representability up to isomorphism. In this lies the relevance of a category-theoretic perspective, which leads to wide-ranging consequences. One such is the presence of functorial constraints on the syntax–semantics relationships; another is an intrinsic view of (constructive) logic, as arises in topoi and, subsequently, in more general fibrations. But as soon as theories and their models are described accordingly, a new look at the main problems of 20th century’s philosophy becomes possible. The lack of any satisfactory solution to these problems in a purely logical and set-theoretic setting is the result of too circumscribed an approach, such as a static and punctiform view of objects and their elements, and a misconception of geometry and its historical changes before, during, and after the foundational “crisis”, as if algebraic geometry and synthetic differential geometry – not to mention algebraic topology – were secondary sources for what concerns foundational issues. The objectivity of basic geometrical intuitions also acts against the recent version of structuralism proposed as ‘the’ philosophy of category theory. On the other hand, the need for a consistent and adequate conceptual framework in facing the difficulties met by pre-categorical theories of language and scientific knowledge not only provides the basic concepts of category theory with specific applications but also suggests further directions for their development (e.g., in approaching the foundations of physics or the mathematical models in the cognitive sciences). This ‘virtuous’ circle is by now largely admitted in theoretical computer science; the time is ripe to realise that the same holds for classical topics of philosophy.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Artin M.,, Grothendieck A. and Verdier J. (1972). Théorie des Topos et Cohomologie Etale des Schémas I, Séminaire de géométrie algébrique du Bois-Marie, 4 (SGA4). Springer, Berlin Google Scholar
  2. Bell J. L. (1988). Topoi and Local Set Theories. Oxford University Press, Oxford Google Scholar
  3. Bell J. L. (1998). A Primer of Infinitesimal Analysis. Cambridge University Press, Cambridge Google Scholar
  4. Bell J. L (2004). Whole and Part in Mathematics. Axiomathes 14: 285–294 CrossRefGoogle Scholar
  5. Bell, J. L.: ‘Cover Schemes, Frame-valued Sets and their Potential Uses in Spacetime Physics in Spacetime Physics Research Trends’, in: Albert Reimer (ed.), Horizons in World Physics, Volume 248, Nova Publications, New York (in press)Google Scholar
  6. Bourbaki N. (1962). L’architecture des mathématiques. In: Le Lionnais, F. (eds) Les Grands Courants de la Pensée Mathématique, pp 35–47. Albert Blanchard, Paris Google Scholar
  7. Cartier P. (1997–1998). Notes sur l’histoire et la philosophie des mathématiques I-III. Prépublications de l’IHES, ParisGoogle Scholar
  8. Geroch R. (1984). Mathematical Physics. University of Chicago Press, ChicagoGoogle Scholar
  9. Ghilardi S. and Meloni G. (1988). Modal and tense predicate logics: models in presheaves and categorical conceptualisation. In: Borceaux, F. (eds) Categorical Algebra and its Applications, pp 130–142. Springer, Berlin CrossRefGoogle Scholar
  10. Grothendieck A. (1970). Catégories fibrées et descente. In: Grothendieck, A. (eds) Revêtements Etales et Group Fundamental (SGA 1), pp 145–194. Springer, Berlin Google Scholar
  11. Hyland J. M. (1982). The Effective Topos. In: Troelstra, A. S. (eds) The Brouwer Centenary Symposium, pp 165–216. North-Holland, AmsterdamGoogle Scholar
  12. Isham, C. and J. Butterfield, A topos perspective on the Kochen-Specker theorem: I-II, resp. arXiv:quant-ph/98035055 v4 13 Oct 1998 and arXiv:quant-ph/9808067 v2 8 Nov 1998Google Scholar
  13. Jacobs B. (1999). Categorical Logic and Type Theory. North Holland, Amsterdam Google Scholar
  14. Johnstone P. (1982). Stone Spaces. Cambridge University Press, Cambridge Google Scholar
  15. Lambek J (1968). Deductive Systems and Categories I. Syntactic Calculus and Residuated Categories. Mathematical Systems Theory 2: 287–318 CrossRefGoogle Scholar
  16. Lambek J. and Scott P. J. (1986). Introduction to Higher Order Categorical Logic. Cambridge University Press, CambridgeGoogle Scholar
  17. Lawvere F. W (1963). Functorial Semantics of Algebraic Theories. Proceedings of the National Academy of Science USA 50(5): 869–872CrossRefGoogle Scholar
  18. Lawvere, F. W.: 1964, ‘An Elementary Theory of the Category of Sets’, Proceedings of the National Academy of Science, USA 52, 1506–1511Google Scholar
  19. Lawvere, F. W.: 1969, Diagonal Arguments and Cartesian Closed Categories, Lecture Notes in Mathematics no. 92, Berlin: Springer, pp. 134–145Google Scholar
  20. Lawvere F. W (1969). Adjointness in Foundations. Dialectica 23: 281–296Google Scholar
  21. Lawvere F. W. (1975). Continuously variable sets: algebraic geometry = geometric logic. In: Rose, H. E. and Shepherdson, J. C. (eds) Logic Colloquium ’73., pp 135–156. North Holland, AmsterdamGoogle Scholar
  22. Lawvere, F. W.: 1986, Introduction to Categories in Continuum Physics, Lecture Notes in Mathematics no. 1174, Berlin: Springer, pp. 1–18Google Scholar
  23. Lawvere F. W (1989). Qualitative Distinctions Between Some Topoi of Generalized Graphs. Contemporary Mathematics 92: 261–299Google Scholar
  24. Lawvere F. W. ‘Tools for the Advancement of Objective Logic: Closed Categories and Topoi’, in [33], 43–56Google Scholar
  25. Lawvere F. W (1999). Categorie e Spazio: Un Profilo. Lettera Matematica Pristem 31: 35–50Google Scholar
  26. Lawvere F. W. and Rosebrugh R. (2003). Sets for Mathematics. Cambridge University Press, Cambridge Google Scholar
  27. Lawvere F. W. and Schanuel S. (1997). Conceptual Mathematics. Cambridge University Press, New YorkGoogle Scholar
  28. Longo, G.: ‘Space and Time in the Foundations of Mathematics, or Some Challenges in the Interactions with Other Sciences’. Invited lecture, First AMS/SMF meeting, Lyon, July 2001, downloadable at
  29. Mac Lane S. 1979, Review of Topos Theory by P. T. Johnstone, Bull. Am. Math. Soc. (N.S.) 1, 1005–1014Google Scholar
  30. Mac Lane S. (1986). Mathematics: Form and Function. Springer, BerlinGoogle Scholar
  31. Mac Lane S. and Moerdijk I. (1992). Sheaves in Geometry and Logic. Springer, New York Google Scholar
  32. McLarty C (1990). Uses and Abuses of Topos Theory. British Journal for the Philosophy of Science 41: 351–375CrossRefGoogle Scholar
  33. McLarty C (1988). Defining Sets as Sets of Points of Spaces. Journal of Philosophical Logic 17: 75–90CrossRefGoogle Scholar
  34. McLarty. C.: 2003. ‘Exploring Mathematical Structuralism’, Philosophia Mathematica, 11, to appearGoogle Scholar
  35. (1994). Logical Foundations of Cognition. Oxford University Press, OxfordGoogle Scholar
  36. Peruzzi A (1989). The Theory of Descriptions Revisited. Notre Dame Journal of Formal Logic 30: 91–104CrossRefGoogle Scholar
  37. Peruzzi A (1989). Towards a Real Phenomenology of Logic. Husserl Studies 6: 1–24CrossRefGoogle Scholar
  38. Peruzzi A. (1991). Meaning and Truth: The ILEG Project. In: Wolf, E., Nencioni, G. and Tscherdanzeva, H. (eds) Semantics and Translation, pp 53–59. Moscow Academy of Sciences, Moscow Google Scholar
  39. Peruzzi A. (1991). Categories and logic. In: Usberti, G. (eds) Problemi Fondazionali nella Teoria del Significato, pp 137–211. Olschki, FlorenceGoogle Scholar
  40. Peruzzi, A.: 1998. On the Logical Meaning of Precategories, Talk at the Open Seminar 1992–1993, Department of Philosophy, University of Florence, 1998
  41. Peruzzi, A.: 1994, ‘From Kant to Entwined Naturalism’, Annali del Dipartimento di␣Filosofia IX [1993], Olschki, Florence: University of Florence, pp. 225–334Google Scholar
  42. Peruzzi A. (2000). The geometric roots of semantics. In: Albertazzi, L. (eds) Meaning and Cognition, pp 169–201. John Benjamins, AmsterdamGoogle Scholar
  43. Peruzzi A (2002). ILGE-interference patterns in semantics and epistemology. Axiomathes 13: 39–64CrossRefGoogle Scholar
  44. (1995). Mind as Motion: Explorations in the Dynamics of Cognition. MIT Press, Cambridge MAGoogle Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  • Alberto Peruzzi

There are no affiliations available

Personalised recommendations