Crítica, Revista Hispanoamericana de Filosofía, Volume 23, number 68, agosto 1991
Números, objetos y estructuras
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Francisco Rodríguez Consuegra
Instituto S. Vilaseca
McMaster University


Abstract: Introduction. This paper is principally a critical exposition of the
celebrated article by Benacerraf, indicating briefly íts antecedents,
emphasizing its accomplishments, problems and basic insufficiencies,
followed by an evaluation of the main criticisms to which Benacerraf's article has been subjected, as well as a study of the historical
framework in which a new global criticism is meaningful. The paper
ends with the examination of a possible connection with the structuralist
philosophy of mathematics, which is in part inspired by the
work of Benacerraf.

Mathematical reduction according to Benacerraf (sections 2 and
3). It is here shown that the fundamental "objective" antecedents
to Benacerraf's work are Quine, along with Parsons, and the nominalism
of Goddard; and sorne important differences are also pointed
out. Discussed is Benacerraf's rejection of the identification of
numbers and objects, and its substitution by progressions in the
framework of the typically Quinean argument of set polymorphism,
as well as his difficult theory of identity, all of which without a clearly
relativist ontological contexto His reduction of numbers to positions
in a progression is situated in the now old debate between the cardinal
and the ordinal, and is a step in the direction of the nascent
structuralism, although it lacks sufficient justification.

Some criticisms [sections 4 and 5). An evaluative study is made of
the criticisms that seem to me most accurate, or the most revealing
of underlying problems. Reviewed are the most relevant among such
criticisms in the literature: Steiner, Resnik, Maddy, Wright, and Hale
along with others of the enormous quantity of articles discussing this
topic that have appeared over the last twenty years, Common lines
are traced out, and some possible defenses of Benaeerraf are indicated,
although again the weaknesses of his position are pointed out,
weaknesses stemming from its unresolved problems (the historieal
framework, the ill-defined ontology, the nascent structuralism, etc.).

Essential criticisms (section 6). Beginning with the problem of
counting, the axis of Benaeerraf's work, an historical excursion is
presented, in which it is shown that the problem pointed out above
(cardinal versus ordinal) can be seen as the center of the indicated
difficulties. The theory of Dedekind-Peano is compared with that of
Cantor, and the epistemological and constructive advantages of the latter are noted. It is shown how positions very similar to Benacerraf's
were already held by Cassirer and Weyl (without mentioning
Berkeley!); meanwhile, the Cantorian approach of Couturat and RusseU
is shown to be superior, at least from the point of view of a global
coneeption. Finally, the eonneetion between eonstructions and polymorphism
---a problem shared by logic, mathematics and physics---is
pointed out.

The structuralist tendency and platonism (section 7). The antecedents
of the structuralism of Resnik and Shapiro are traced to Benacerraf
himself ---and the historical trace is further extended back to
the ordinalists, Bourbaki and Quine--- in the hope of shedding light
on the basic problem: the supposed antithesis between terms and
relations (already familiar in Bradley and Russell). Further, I suggest
and examine a parallelism with the relativism of mathematical entities
such as this appears after the limitations of (at least first order)
axiomatization. The paper ends making a connection of the subject
with the theory of categories, which, surprisingly, still has not been
eonsidered by the strueturalist, despite the fact that it is quite clearly
a natural extension of the structuralist point of view.

[Traducción de Raúl Orayen y Mark Rollins]

Keywords:

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