 The resulting linguistic system of logic may in turn be studied mathematically, even independently of the mathematical activities and their recordings that it was originally abstracted from. Iterating the process, an infinite hierarchy arises of mathematical activities, their linguistic recordings, and the mathematical study of these recordings as linguistic objects independently of their original meaning. Brouwer describes this hierarchy (in more detail than we have done here) at the end of his dissertation (Brouwer 1907: 173ff), and criticizes Hilbert for not respecting it. Of particular interest is the distinction Brouwer makes between mathematics and “mathematics of the second order” (Brouwer 1907: 99n.1, 173 [1975: 61n.1, 94]), where one instance of the latter is the mathematical study of the language of the former in abstraction from its original meaning; this way, Brouwer made fully explicit the distinction between mathematics and (what became known as) metamathematics (e.g., Hilbert 1923: 153). Later, Brouwer claimed priority for this distinction, adding in a footnote that he had explained it to Hilbert in a series of conversations in 1909 (Brouwer 1928A2: 375 [Mancosu 1998: 44n.1]).