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

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.

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.

@article{PHSC_2005__9_2_131_0,
     author = {Slater, Hartley},
     title = {Aggregate theory versus set theory},
     journal = {Philosophia Scientiae},
     pages = {131--144},
     publisher = {\'Editions Kim\'e},
     volume = {9},
     number = {2},
     year = {2005},
     language = {fr},
     url = {http://archive.numdam.org/item/PHSC_2005__9_2_131_0/}
}
TY  - JOUR
AU  - Slater, Hartley
TI  - Aggregate theory versus set theory
JO  - Philosophia Scientiae
PY  - 2005
SP  - 131
EP  - 144
VL  - 9
IS  - 2
PB  - Éditions Kimé
UR  - http://archive.numdam.org/item/PHSC_2005__9_2_131_0/
LA  - fr
ID  - PHSC_2005__9_2_131_0
ER  - 
%0 Journal Article
%A Slater, Hartley
%T Aggregate theory versus set theory
%J Philosophia Scientiae
%D 2005
%P 131-144
%V 9
%N 2
%I Éditions Kimé
%U http://archive.numdam.org/item/PHSC_2005__9_2_131_0/
%G fr
%F PHSC_2005__9_2_131_0
Slater, Hartley. Aggregate theory versus set theory. Philosophia Scientiae, Volume 9 (2005) no. 2, pp. 131-144. http://archive.numdam.org/item/PHSC_2005__9_2_131_0/

[1] Black, M. 1970.- The Elusiveness of Sets, Review of Metaphysics, XXIV (1) 614-636.

[2] Boolos, G. 1984.- To be is to be the value of a variable (or to be some values of some variables), Journal of Philosophy, LXXXI, 430-449. | MR

[3] Bostock, D. 1974.- Logic and Arithmetic, volume I., Oxford : O.U.P. | Zbl

[4] Bostock, D. 1979.- Logic and Arithmetic, volume II., Oxford : O.U.P. | MR | Zbl

[5] Bunt, H.C. 1985.- Mass Terms and Model-Theoretical Semantics, Cambridge : C.U.P.

[6] Dummett, M. 1993.- Frege : Philosophy of Mathematics, London : Duckworth. | MR

[7] Fraenkel, A.A. 1965.- Set Theory, in Edwards, P. (ed.), Encyclopedia of Philosophy, New York : Macmillan.

[8] Frege, G. 1968.- The Foundations of Arithmetic, 2 nd edition, Oxford : Blackwell.

[9] Hailperin, T. 1992.- Herbrand Semantics, the Potential Infinite, and Ontology-Free Logic, History and Philosophy of Logic, 13, 69-90. | MR | Zbl

[10] Hale, B. & Wright, C. 2001.- The Reason's Proper Study, Oxford : Clarendon. | MR

[11] Kessler, G. 1980.- Frege, Mill and the Foundations of Arithmetic, Journal of Philosophy, LXXVII, 65-79.

[12] Lavine, S. 1994.- Understanding the Infinite, Cambridge MA : Harvard University Press. | MR | Zbl

[13] Leisenring, A.C. 1969.- Mathematical Logic and Hilbert's Epsilon Symbol, London : Macdonald. | Zbl

[14] Lewis, D.K. 1991.- Parts of Classes, Oxford : Blackwell. | Zbl

[15] Maddy, P. 1990a.- Physicalistic Platonism, in Irvine, A.D. (ed.), Physicalism in Mathematics, Dordrecht : Kluwer.

[16] Maddy, P. 1990b.- Realism in Mathematics, Oxford : Clarendon. | MR | Zbl

[17] Maddy, P. 1997.- Naturalism in Mathematics, Oxford : Clarendon. | MR | Zbl

[18] Maddy, P. 2000.- Does Mathematics need New Axioms ?, The Bulletin of Symbolic Logic, 6.4, 413-422. | MR

[19] Mayberry, J.P. 2000.- The Foundations of Mathematics in the Theory of Sets, Cambridge : C.U.P. | MR | Zbl

[20] Mycielski, J. 1981.- Analysis without Actual Infinity, Journal of Symbolic Logic, 46.3, 625-633. | MR | Zbl

[21] Quine, W.V.O. 1963.- Set Theory and its Logic, Cambridge MA : Belknap Press. | MR | Zbl

[22] Rodych, V. 2000.- Wittgenstein's Critique of Set Theory, The Southern Journal of Philosophy, XXXVIII, 281-319.

[23] Ryle, G. 1973.- The Concept of Mind, Harmondsworth : Penguin.

[24] Simons, P. 1982.- Against the Aggregate Theory of Number, Journal of Philosophy, LXXIX, 163-167

[25] Slater, B.H. 2000.- Concept and Object in Frege, Minerva, 4.. See : http://www.ul.ie/~philos/vol4/frege.html

[26] Tiles, M. 1989.- The Philosophy of Set Theory, Oxford : Blackwell. | MR | Zbl

[27] Wright, C. 1983.- Frege's Conception of Numbers as Objects, Aberdeen : Aberdeen University Press. | MR | Zbl