English mathematician George Boole (1815–1864) is considered a principal inventor of mathematical logic and a major predecessor of computer science. Historians typically emphasize his recasting Aristotelian logic in algebraic notation and applying the computational techniques of mathematical analysis to the laws governing deduction. By emphasizing Boole’s place on the trajectory from ancient to modern logic, however, such a narrative reduces his project to its novel theoretical claims and obscures its practical, moral stakes. In practice, implementing his system meant translating life’s concepts into algebraic symbols. The fragmentary pedagogical materials preserved in Boole’s personal archive exhibit a system oriented toward concrete matters of moral and theological concern. Boole entwined exposition with a parade of practical applications, analyzing such constructs as God’s chief end in creation and the Jewish legal definition of clean beasts. Though he insisted his appropriation of mathematical symbols was not theoretically necessary, this choice allowed him to harness the existing computational expertise of numerate readers. Faced with the often complex definitions of the objects populating heaven and earth, mathematical logic provided a new way to write them down, a symbolic language that a properly trained reader already knew how to manipulate. Boole’s efforts toward a textbook display a conviction that readers would find theological applications an especially interesting and intelligible manifestation of his logic. The assumptions underlying his notions of practicality and accessibility reveal an epistemic context in which logic constituted an arena for working out the still unsettled relationship between theological tradition and modern mathematical science.   

Abstract: Addressing the perennial question “under what conditions can real objects, attributes or relations be represented by numbers?” Hermann von Helmholtz gave an answer in his 1887 lecture “Zählen und Messen” which diverged from the predominant Kantian understanding of quantity and number. Unlike Kant, Helmholtz defined numbers prior to quantities, and regarded the concepts of homogeneity, unit, equality and addition as not having to do with necessary stages of human cognition, but physically determined in specific experimental contexts. Furthermore, Helmholtz did not define measurability by reference to measurement of space, time and mass, which made his views different from his contemporaries, such as the neo-Kantian philosopher Hermann Cohen, the mathematician Paul Du Bois Reymond and the physicist James Clerk Maxwell. The current paper argues that Helmholtz’s epistemology in “Zählen und Messen” closely mirrored the practices of measurement in electricity and magnetism, and was shaped by his involvement in efforts to establish an international electrical standard leading up to the 1881 International Congress of Electricians. The divergence between practice and theory, the lengthy process of calibration leading up to the definition and manufacture of units, and the ambiguous role of the measurement of length, mass and time in defining electromagnetic standards, all played a part in Helmholtz’s 1887 article.

During the later half of the first millennium BCE, Babylonian astrologers utilized two separate micro-zodiac schemes that partitioned a single zodiacal sign in different ways. On the one hand, the ‘micro-zodiac of 13’ synchronized the movements of sun and moon, by linking changes in the sun’s micro-zodiacal signs to changes in the moon’s thirteen zodiacal signs of an ideal month. On the other hand, the ‘micro-zodiac of 12’ depicted its twelve micro-zodiac divisions as a microcosm of the twelve zodiacal signs in a more straightforward manner, employed calculations involving the simpler divisor 12 (rather than 13), and extended twelve-part time divisions to spatial dimensions of the sky. I explore the intellectual and cultural contexts giving rise to each scheme, as well as the implications of their adaptation as dodekatemoria (lit. “twelfth parts”) in Greek and Latin sources, as expressed particularly in horoscopes and writings by Marcus Manilius, Paul of Alexandria, and Vettius Valens.