We Have Always Been . . . Cyborgs

philosophy of mathematics suited well the ‘‘general emphasisin later nineteenth-century mathematics on revisiting foundationalquestions’’ (p. 61). Pure geometry as it emerged from Frege, Paschand Hilbert, provided a diagram-free, purely sentential notion ofproof, which has lasted unto our own day.In the second, and greater, part of the book, Greaves traces outthe interrelations between philosophy, most especially metaphysics,and logic. Again, he begins with the ancients. To my mind, Greaves’very concise presentation of the connections, both structural anddevelopmental, among Aristotle’s logic, metaphysics, and science isthe most elegant version of the task I have ever seen. That alone iswell worth the book’s price.A major reason that diagrams did not figure until relatively late inthe career of syllogistic logic lies in the fact that ‘‘Aristotle developedthe syllogistic in support of a specific conception about the scope anduses to which logic would be put, and that this conception naturallyentails that sentences of a certain limited grammatical form be theprincipal targets of the theory’’ (p. 88). That is, given Aristotle’smetaphysics – especially the view of the relation between predicationand properties from the Categories – and his view of what a science is,syllogistic can have only the form it does have, namely, categorical–sentential.Because of these Aristotelian intentions, aside from some earlyoddities such as the efforts of Lull and Bruno, serious use of diagramsfor reasoning (rather than just depiction) did not occur until Eulerand the rise of extensional logic. As Greaves presents the case, it wasEuler’s decision to let his circles represent extension rather than thetraditional intension of the terms that would ‘‘make possible an en-tirely new sort of diagram type for logic, characterized by the use ofregions to represent concepts’’ (p. 122).Neither Euler’s diagrams (which had only limited power) nor hisextensional interpretation made much of a dent in logic at the time.However, ‘‘by the time of Boole and Venn, it is clear that a signifi-cant, although comparatively small, group of logicians had embracedan extensional, class-based interpretation of the component conceptsin a categorical proposition’’ (p. 127).Greaves’ discussion of developments in logic during the nineteenthcentury is delightfully economical. He covers, in turn, the develop-ment of Boole’s symbolic logic, emphasising the expansions that itmade in ‘‘Aristotle’s vision of the scope of logic along two maindimensions’’, namely the expansion of types of propositional forms221DEEP THOUGHTS ABOUT DIAGRAMS