Números, objetos y estructuras
Authors: Rodríguez Consuegra Francisco 
Published: March 2020

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 paperends 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]