Critical Studies/book Reviews

algebra. After the first page or so, group theory is not concerned with the features common to all models of the group axioms, but rather with relationships among different models of those axioms, which is to say, different groups, especially those relationships established by certain kinds of functions, called homomorphisms. This makes it difficult even to imagine what a structuralist interpretation of the classification theorem for finite simple groups, say, would look like, unless it were simply a structuralist interpretation of the background set theory in which the existence or non-existence of groups and homomorphisms with various properties is established, and the theorem just mentioned ultimately proved. Similar remarks apply to other branches of abstract algebra. Real analysis before Dedekind and Cantor. However appropriate the structuralist interpretation of real analysis may be for the period since the ‘arithmetization of analysis’ in the nineteenth century, there is quite direct evidence that mathematicians of earlier periods had a quite different interpretation in mind. Newton, for instance, tells us quite directly, in the first pages of his UniversalArithmetick, that real numbers are ‘abstracted ratios’ of magnitudes such as lengths. Thus there is quite direct evidence that on Newton’s understanding of them, theorems of real analysis were theorems about a specific structure, the objects of whose domain were ratios of magnitudes such as lengths. This was, in fact, the dominant understanding in the seventeenth and eighteenth centuries. Euclid’s geometry. Going back further in history, to Euclid, we find that, unlike Hilbert at a much later period, he does not take points, lines, and circles to be just any objects satisfying the axioms of geometry, but rather takes them to be objects of certain specific kinds with which his readers are expected to be familiar. Perhaps the clearest indication of this fact is provided by the feature of the Elements whose seeming pointlessness has so often puzzled those brought up on modern axiomatics: his offering of definitions that play absolutely no role whatsoever in the subsequent development of his theory, beginning with the notorious ‘A point is that which hath no part’. Business arithmetic. As a structuralist analysis takes theorems of real analysis to be theorems about all models of the complete-ordered-field axioms, so a structuralist analysis takes theorems of number theory to be theorems about all models of the Peano postulates. However, the derivation of the ‘cardinal’ properties of natural numbers from the Peano postulates is