Textual and historical subtleties aside, let's call the idea that numbers are properties (or classes) of equinumerous sets ‘the Fregean thesis.’ In a recent paper, Palle Yourgrau claims to have found a decisive refutation of this thesis. More surprising still, he claims in addition that the essence of this refutation is found in the Grundlagen itself – the very masterpiece in which Frege first proffered his thesis. My intention in this note is to evaluate these claims, and along the way to shed some light on relevant passages of the Grundlagen. I will argue that Yourgrau does not make his case.

The arguments with which we are concerned are found in the last three sections of Yourgrau's paper. A pervasive difficulty in these sections is that it is not clear exactly what Yourgrau is arguing against. The stated object of his attack is the Fregean thesis (d. 581-6), a thesis about what numbers are; however, instead of a frontal assault, his strategy is to embark on a foray into the ill-defined issue of what it is that numbers number (586ff.), where, roughly speaking, a number n numbers an object x just in case n can be legitimately assigned to x. The reason for this shift in emphasis appears to be rooted in a misconception. As we’ll see in more detail shortly, Yourgrau's argument against the Fregean thesis is based on an extension of a well known argument of Frege's found in §§22-3 of the Grundlagen, which Glenn Kessler has tagged the ‘relativity argument,’ (henceforth ‘RA’). According to Yourgrau, this is an argument ‘to the effect that what is literally numbered cannot simply be concrete objects’ (586). This is incorrect. Frege himself clarifies the point of the argument in §21 with the following preface:

In this paper, a method for design and optimization of folded reflect-array antennas is proposed based on particleswarm optimization (PSO). In addition to such a powerful optimization algorithm, two further requirements have to be fulfilled. The first one is a good and fast algorithm for the exact prognosis of the far-field radiation diagram, resulting from a specific element configuration on the reflector. Additionally, a good choice of the fitness function for the evaluation of the resulting radiation diagrams is necessary. In both, reflect-array-related aspects such as phase truncation, reflection losses, and cell discontinuities have to be considered. Antenna optimization based on this technique is presented in this paper at the example of two 77 GHz folded reflect-array antennas. The efficiency of this approach is demonstrated with these examples, and the results are verified by measurement, showing an excellent agreement with the specifications of the diagram masks. The implemented tool, including a realistic antenna diagram preview, allows the investigation of the design parameters’ influence on the antenna performance, such as illumination amplitude, high substrate losses, and phase truncation.

Professor Grosholz packs a lot into her interesting and suggestive paper “Formal Unities and Real Individuals” (Grosholz 1990b). In the limited space available I can comment briefly on its several parts, or direct more substantive comments at a single issue. I will opt for the latter; specifically, I want to address her critique of mathematical structuralism, as found especially in the writings of Michael Resnik.

I begin with a brief, hence necessarily caricatured, summary of Resnik’s influential view. According to structuralism, the subject matter of a mathematical theory is a given pattern, or structure, and the objects of the theory are intrinsically unstructured points, or positions, within that pattern. Mathematical objects thus have no identity, and no intrinsic features, outside of the patterns in which they occur. Hence, they cannot be given in isolation but only in their role within an antecedently given pattern, and are distinguishable from one another only in virtue of the relations they bear to one another in the pattern (see, e.g., Resnik (1981)).