 Frege does not extract a unified, consistent position from the work of Heine and Thomae, and much of his criticism is devoted to showing that they inconsistently slip into modes of thought which are only appropriate for ‘contentful’ arithmetic, which Frege takes to be a body of truths expressed by utterances in which numerical expressions designate abstract referents independent of the mind (or at least any particular individual’s mind). Heine and Thomae talk of mathematical domains and structures, of prohibitions on what may be uttered (e.g. against writing ‘\(3\div 0\)’ which is deemed, in some special sense, meaningless), of numbers being greater than or less than one another (rather than physical marks being larger or smaller, darker or lighter)—all things which make no sense if arithmetic is a theory of marks and their physical properties or is just a body of transformations of referent-less symbols. (Heine however reserves, with Kronecker, a special place for arithmetic, treating it in non-formalist fashion; this position may thus be more coherent than Thomae’s. See Simons, 2009 especially 293–6.)