Abstract
Topos theory plays, in Alain Badiou’s philosophical model, the role of inner logic of mathematics, given its power to explore possible mathematical universes; whereas set theory, because of its axiomatics, plays the role of ontology. However, in category theory, which is a vaster theory, topos theory embodies a particular axiomatic choice, the fundamental consequence of which consists in imposing an internal intuitionist logic, that is a non-contradictory logic which gets rid of the principle of excluded middle. Category theory shows that the dual axiomatic choice exists, namely the one imposing, this time, a logic of the excluded middle which accepts true contradictions without deducing from them everything, and this is called a paraconsistent logic. Therefore, after recalling the basics of category and topos theory necessary to demonstrate the categorical duality of paracompleteness (i.e. intuitionism) and paraconsistency, we will be able to introduce into Badiou’s thought category theory seen as a logic of the possible ontologies, a logic which demonstrates the strong symmetry of the axioms of excluded middle and of non-contradiction.
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Notes
- 1.
This paper is the written version of a talk given, in front of Alain Badiou, at the postgraduate congress “Autour de la pensée d’Alain Badiou”, organised by Alessio Moretti on behalf of the “Association des Thésards en Philosophie de Nice (ATP)” at the University of Nice, France, 19th March 2004 (Alain Badiou was an Invited Speaker at this event).
- 2.
“Plato and/or Aristotle–Leibniz. Set theory and topos theory as seen by the philosopher’s eye”.
- 3.
By classical mathematicians I mean mathematicians who are not partisans or followers of intuitionism. They constitute the overwhelming majority of the mathematical community.
- 4.
One says that a logic is trivialised when it deduces everything, i.e. when inside it everything becomes a theorem (so that the notion of “theoremhood” no longer means anything).
- 5.
They will become even more central in his book Logiques des mondes (2006).
- 6.
Cf Chap. XVI of the Court traité d’ontologie transitoire ([2]).
- 7.
from old Greek “topos”, place, whose old-Greek plural is “topoi”.
- 8.
This notion and the analysis of its internal logics are explored by William James and Chris Mortensen in their book Categories, Sheaves and Paraconsistent Logic.
- 9.
These logic systems were discovered in 1963 by da Costa in Brazil. It is important to remark that all these systems possess a paraconsistent implication, whereas our calculus does not possess implication but possesses a pseudo-difference, because it is the dual of the intuitionist propositional calculus. This special paraconsistent system was discovered more recently by the Brazilian school (da Silva de Queiroz 2001) ([3]).
- 10.
This contradicts the principle of classical logic known as Pseudo-Scot or ex falso quodlibet, which says that from a contradiction such as \(B\wedge\neg B\) we can deduce anything.
- 11.
These logics of sheaves parameterized by topology are one the main themes explored in Alain Badiou’s book Logiques des mondes.
References
Badiou, A.: Platon et/ou Aristote. Théorie des ensembles et théorie des topoi sous l’œil du philosophe. In: Panza M., Salanskis, J.M. (eds.) L’Objectivité mathématique. Masson, Paris (1995)
Badiou, A.: Court traité d’ontologie transitoire. Seuil, Paris (1998)
da Silva de Queiroz, G.: Sobre a dualidade entre intuicionismo e paraconsistência. PhD thesis, Brasil (1998)
Lambek J., Scott, P.J.: Introduction to higher order categorical logic. Cambridge Studies in Advanced Mathematics 7. Cambridge University Press, Cambridge (1986)
Lavendhomme, R.: Lieux du Sujet. Psychanalyse et Mathématiques. Seuil, Paris (2001)
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Angot-Pellissier, R. (2015). The Relation Between Logic, Set Theory and Topos Theory as it Is Used by Alain Badiou. In: Koslow, A., Buchsbaum, A. (eds) The Road to Universal Logic. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-15368-1_7
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