no. 2
Aggregate theory versus set theory
Philosophia Scientiae, Tome 9 (2005) no. 2, pp. 131-144.

Les arguments de Maddy avancés en 1990 contre la théorie des agrégats se trouvent affaiblis par le retournement qu'elle opère en 1997. La présente communication examine cette théorie à la lumière de ce retournement ainsi que des récentes recherches sur les “Nouveaux axiomes pour les mathématiques”. Si la théorie des ensembles est la théorie de la partie-tout des singletons, identifier les singletons à leurs membres singuliers ramène la théorie des ensembles à la théorie des agrégats. Toutefois si les singletons ne sont pas identiques à leurs membres singuliers, ce ne sont donc pas des objets extensionnels et ils ne peuvent former une base pour la Science. Dans tous les cas, l'hypothèse d'un continuum n'a aucun intérêt sur le plan physique. Je montre d'abord que, parce qu'il y a des prédicats non-sortaux, la tentative de Frege de faire dériver l'arithmétique de la logique bute dès ses premiers pas. Car il y a des propriétés sans nombre et la contingence de cette condition signifie que la définition du zéro donnée par Frege ne peut s'obtenir à partir de la logique. Ce résultat révèle le besoin de considérer davantage les propriétés sans nombre et donc de générer une théorie des continuums en se basant sur les agrégats méréologiques plutôt que sur des ensembles contenant des nombres de choses.

Maddy's 1990 arguments against Aggregate Theory were undermined by the shift in her position in 1997. The present paper considers Aggregate Theory in the light this, and the recent search for ‘New Axioms for Mathematics'. If Set Theory is the part-whole theory of singletons, then identifying singletons with their single members collapses Set Theory into Aggregate Theory. But if singletons are not identical to their single members, then they are not extensional objects and so are not a basis for Science. Either way, the Continuum Hypothesis has no physical interest. I first show that, because there are non-sortal predicates, Frege's attempt to derive Arithmetic from Logic stumbles at its very first step. For there are properties without a number, and the contingency of that condition means Frege's definition of zero is not obtainable from Logic. This result then points to the need to consider more fully properties without a number, and so to generate a theory of continua based on mereological aggregates rather than sets containing numbers of things.

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Slater, Hartley. Aggregate theory versus set theory. Philosophia Scientiae, Tome 9 (2005) no. 2, pp. 131-144. http://www.numdam.org/item/PHSC_2005__9_2_131_0/

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