This article discusses the concept of mathematical metaphor as a tool for analyzing the formation of mathematical knowledge. It reflects on the work of Lakoff and Núñez as a reference point against which to rearticulate a richer notion of mathematical metaphor that can account for actual mathematical evolution. To reach its goal this article analyzes historical case studies, draws on cognitive research, and applies lessons from the history of metaphors in philosophy as analyzed by Derrida and de Man.

ABSTRACT In a series of papers read to the Cambridge Philosophical Society through the 1820s, the Cambridge mathematician George Peacock laid the foundation for a natural history of arithmetic that would tell a story of human progress from counting to modern arithmetic. The trajectory of that history, Peacock argued, established algebraic analysis as a form of universal reasoning that used empirically warranted operations of mind to think with symbols on paper. The science of counting would suggest arithmetic, arithmetic would suggest (...) arithmetical algebra, and, finally, arithmetical algebra would suggest symbolic algebra. This philosophy of suggestion provided the foundation for Peacock's “principle of equivalent forms,” which justified the practice of nineteenth-century English symbolic algebra. Peacock's philosophy of suggestion owed a considerable debt to the early Cambridge Philosophical Society culture of natural history. The aim of this essay is to show how that culture of natural history was constitutively significant to the practice of nineteenth-century English algebra. (shrink)

ABSTRACT In a series of papers read to the Cambridge Philosophical Society through the 1820s, the Cambridge mathematician George Peacock laid the foundation for a natural history of arithmetic that would tell a story of human progress from counting to modern arithmetic. The trajectory of that history, Peacock argued, established algebraic analysis as a form of universal reasoning that used empirically warranted operations of mind to think with symbols on paper. The science of counting would suggest arithmetic, arithmetic would suggest (...) arithmetical algebra, and, finally, arithmetical algebra would suggest symbolic algebra. This philosophy of suggestion provided the foundation for Peacock's “principle of equivalent forms,” which justified the practice of nineteenth-century English symbolic algebra. Peacock's philosophy of suggestion owed a considerable debt to the early Cambridge Philosophical Society culture of natural history. The aim of this essay is to show how that culture of natural history was constitutively significant to the practice of nineteenth-century English algebra. (shrink)

I have always been a philosopher at heart. I write history of science and history of its philosophy primarily as a philosopher wary of his abstractions and broad conceptualizations. But that has not always been the case. Lakatos famously portrayed history of science as the testing ground for theories of scientific rationality. But he did so along the crudest Hegelian lines that did injury both to Hegel and to the history and methodology of science. Since science is ultimately rational, he (...) argued, rival methodologies can prove their mettle by competing for whose tendentiously reconstructed account of the history of science renders more of it rational! My own approach to the relationship between history and philosophy of science started out perhaps a little more open-mindedly than Lakatos’s, but in a manner no less crude. Over the years the relationship between the history I wrote and the philosophy to which I was committed took on a firmer and more reciprocal shape. It did so in the course of a process that I now realize exemplified the philosophical position it eventually yielded. I would like to trace that development in the following pages and reflect as best I can on where it has lead and left me. (shrink)

Scholars in both digital humanities and media studies have noted an apparent disconnect between computation and the interpretive methods of the humanities. Alan Liu has argued that literary scholars employing digital methods encounter a “meaning problem” due to the difficulty of reconciling algorithmic methods with interpretive ones. Conversely, the media scholar Friedrich Kittler has questioned the adequacy of hermeneutics as a means of studying computers. This paper argues that that this disconnect results from a set of contingent decisions made in (...) both humanistic and mathematical disciplines in the first half of the nineteenth century that delineated, with implications that continue to resonate in the present day, which aspects of human activity would come to be formalized in algorithms and which would not. I begin with a discussion of Nicolas de Condorcet, who attempted, at the height of the 1789 revolution, to turn algebra into a universal language; his work, I argue, exemplifies the form of algorithmic thinking that existed before the Romantic turn. Next, I discuss William Wordsworth’s arguments about the relationship of poetry and science. While Wordsworth is sometimes viewed as a critic of science, I argue that his polemic is specifically targeted at highly politicized projects like Condorcet’s that sought to supplant existing modes of thought with scientific rationality. Finally, I demonstrate the importance of Romantic thought for George Boole, creator of the logic system that would eventually form the basis of digital electronics. The reason Boole was able to succeed where Condorcet had failed, I argue, was that Romantic notions of culture enabled him to reconcile a mechanical view of mathematical reasoning with an organic view of the development of meaning—a dichotomy that remains a key assumption of computer interfaces in the twenty-first century. (shrink)