In Die Grundlagen der Arithmetik, Frege attempted to introduce cardinalnumbers as logical objects by means of a second-order abstraction principlewhich is now widely known as ``Hume's Principle'' (HP): The number of Fsis identical with the number of Gs if and only if F and G are equinumerous.The attempt miscarried, because in its role as a contextual definition HP fails tofix uniquely the reference of the cardinality operator ``the number of Fs''. Thisproblem of referential indeterminacy is usually called ``the Julius Caesar (...) problem''.In this paper, Frege's treatment of the problem in Grundlagen is critically assessed. In particular, I try to shed new light on it by paying special attention to the framework of his logicism in which it appears embedded. I argue, among other things, that the Caesar problem, which is supposed to stem from Frege's tentative inductive definition of the natural numbers, is only spurious, not genuine; that the genuine Caesar problem deriving from HP is a purely semantic one and that the prospects of removing it by explicitly defining cardinal numbers as objects which are not classes are presumably poor for Frege. I conclude by rejecting two closely connected theses concerning Caesar put forward by Richard Heck: (i) that Frege could not abandon Axiom V because he could not solve the Julius Caesar problem without it; (ii) that (by his own lights) his logicist programme in Grundgesetze der Arithmetik failed because he could not overcome that problem. (shrink)

Both Frege's Grundgesetze, and Lagrange's treatises on analytical functions pursue a foundational purpose. Still, the former's program is not only crucially different from the latter's. It also depends on a different idea of what foundation of mathematics should be like . Despite this contrast, the notion of function plays similar roles in their respective programs. The purpose of my paper is emphasising this similarity. In doing it, I hope to contribute to a better understanding of Frege's logicism, especially in relation (...) to its crucial differences with a set-theoretic foundational perspective. This should also spread some light on a question arisen by J. Hintikka and G. Sandu in a widely discussed paper, namely whether Frege should or should not be credited with the notion of arbitrary function underlying our standard interpretation of second-order logic. In section 1, I account for Lagrange's notion of function. In section 2, I advance some remarks on connected historical matters. This will provide an appropriate framework for discussing the role played by the notion of function in Frege's Grundgesetze. Section 3 is devoted to this. Some concluding remarks will close the paper. (shrink)

This paper aims to answer the question of whether or not Frege's solution limited to value-ranges and truth-values proposed to resolve the "problem of indeterminacy of reference" in section 10 of Grundgesetze is a violation of his principle of complete determination, which states that a predicate must be defined to apply for all objects in general. Closely related to this doubt is the common allegation that Frege was unable to solve a persistent version of the Caesar problem for value-ranges. It (...) is argued that, in Frege’s standards of reducing arithmetic to logic, his solution to the indeterminacy does not give rise to any sort of Caesar problem in the book. (shrink)

Resumen Como es muy conocido, Frege afirmó que la expresión ‘el concepto caballo’ se refiere a un objeto y no a un concepto. En este artículo, en primer lugar, mostramos cómo hay algunos barruntos de esta paradoja en textos anteriores a 1891. En segundo lugar, revisamos algunos argumentos que defienden que con el término ‘el concepto caballo’ Frege se refería a la extensión del mencionado concepto. Por último, sostendremos que, aun cuando el concepto caballo sea dicha extensión, es muy poco (...) lo que sabemos de su naturaleza. Esto último se debe a la forma fragmentaria en que Frege introduce las extensiones de concepto en los Grundgesetze y en ensayos previos.As is well known, Frege stated that the expression ‘the concept horse’ refers to an object and not to a concept. In this article we firstly show how there are some hints of this paradox in texts prior to 1891. Secondly we review some arguments given in the literature to hold that with ‘the concept horse’ Frege referred to the extension of the mentioned concept. Finally, we argue that, even if the concept horse is such an extension, we still know very little about its nature, given the fragmentary way in which Frege introduces the extensions of concepts in Grundgesetze and in previous essays. (shrink)

Gottlob Frege sostuvo que las palabras-concepto sólo se refieren a conceptos no a objetos y que los términos singulares se refieren sólo a objetos no a conceptos. Estas tesis dan lugar a la paradoja del concepto, de acuerdo con la cual ‘el concepto F’ no se refiere a ningún concepto. Mark Textor sostiene que su principio del espejar explica el origen del problema. Sin embargo, argumento que su artículo no explica todas las consecuencias de las tesis de Frege y mi (...) supuesto semántico ontológico puede verse como una consecuencia del principio del espejar de Textor. (shrink)