This paper studies Paul Cohen’s philosophy of mathematics and mathematical practice as expressed in his writing on set-theoretic consistency proofs using his method of forcing. Since Cohen did not consider himself a philosopher and was somewhat reluctant about philosophy, the analysis uses semiotic and literary textual methodologies rather than mainstream philosophical ones. Specifically, I follow some ideas of Lévi-Strauss’s structural semiotics and some literary narratological methodologies. I show how Cohen’s reflections and rhetoric attempt to bridge what he experiences as an (...) uncomfortable tension between reality and the formal by means of his notion of intuition. (shrink)
One of the distinguishing features of mathematics is the exceptional level of consensus among mathematicians. However, an analysis of what mathematicians agree on, how they achieve this agreement, and the relevant historical conditions is lacking. This paper is a programmatic intervention providing a preliminary analysis and outlining a research program in this direction.First, I review the process of ‘negotiation’ that yields agreement about the validity of proofs. This process most often does generate consensus, however, it may give rise to another (...) kind of disagreement: whether the original and new proof are effectively the same, leading to potential disagreement about the validity of the original proof.Second, I historicize the phenomenon of consensus. I show that in earlier European mathematics and other mathematical cultures, consensus about the validity of arguments was substantially weaker or conceived differently than it is today. This means that contemporary consensus about the validity of mathematical proofs should be explained by historical changes in mathematical practice.Finally, I conjecture what brought about the contemporary form of mathematical consensus. Since a sharp rise in consensus occurs around the turn of the 20th century, it makes sense to explain this consensus by the concurrent formalization of mathematics. However, this explanation has a major flaw: it explains a really existing phenomenon (consensus) by something that hardly ever happens (writing proofs in formal languages). I will therefore explain the ways in which formalization does enter mathematical practice so as to account for contemporary forms of mathematical consensus. (shrink)
"Roy Wagner is a one-of-a-kind anthropologist whose books provide intense intellectual stimulation. His way of connecting the world of New Guinea to the world of anthropology is unique and, well, mind-blowing.... He writes books that you actually want to and will read more than once."--Steven Feld, author of Sound and Sentiment "Wagner asks, daringly, what it would be like to imagine one of the most significant of human activities, the activity of description or representation, as a self-scaling phenomenon.... One begins (...) to glimpse a genuine 'alternative anthropology.'"--Marilyn Strathern, author of The Gender of the Gift. (shrink)
The purpose of this article is to analyse the mathematical practices leading to Rafael Bombelli’s L’algebra (1572). The context for the analysis is the Italian algebra practiced by abbacus masters and Renaissance mathematicians of the fourteenth to sixteenth centuries. We will focus here on the semiotic aspects of algebraic practices and on the organisation of knowledge. Our purpose is to show how symbols that stand for underdetermined meanings combine with shifting principles of organisation to change the character of algebra.
This introduction to the Common Knowledge symposium titled “Comparative Relativism” outlines a variety of intellectual contexts where placing the unlikely companion terms comparison and relativism in conjunction offers analytical purchase. If comparison, in the most general sense, involves the investigation of discrete contexts in order to elucidate their similarities and differences, then relativism, as a tendency, stance, or working method, usually involves the assumption that contexts exhibit, or may exhibit, radically different, incomparable, or incommensurable traits. Comparative studies are required to (...) treat their objects as alike, at least in some crucial respects; relativism indicates the limits of this practice. Jensen argues that this seeming paradox is productive, as he moves across contexts, from Lévi-Strauss's analysis of comparison as an anthropological method to Peter Galison's history of physics, and on to the anthropological, philosophical, and historical examples offered in symposium contributions by Barbara Herrnstein Smith, Eduardo Viveiros de Castro, Marilyn Strathern, and Isabelle Stengers. Comparative relativism is understood by some to imply that relativism comes in various kinds and that these have multiple uses, functions, and effects, varying widely in different personal, historical, and institutional contexts that can be compared and contrasted. Comparative relativism is taken by others to encourage a “comparison of comparisons,” in order to relativize what different peoples—say, Western academics and Amerindian shamans—compare things “for.” Jensen concludes that what is compared and relativized in this symposium are the methods of comparison and relativization themselves. He ventures that the contributors all hope that treating these terms in juxtaposition may allow for new configurations of inquiry. (shrink)
This article studies Ibn al-Haytham’s treatment of the common notions from Euclid’s Elements (usually referred to today as the axioms). We argue that Ibn al-Haytham initiated a new approach with regard to these foundational statements, rejecting their qualification as innate, self-evident, or primary. We suggest that Ibn al-Haytham’s engagement with experimental science, especially optics, led him to revise the framing of Euclidean common notions in a way that would fit his experimental approach.
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.
This paper explores the semiotic status of algebraic variables. To do that we build on a structuralist and post-structuralist train of thought going from Mauss and L vi-Strauss to Baudrillard and Derrida. We import these authors' semiotic thinking from the register of indigenous concepts (such as mana), and apply it to the register of algebra via a concrete case study of generating functions. The purpose of this experiment is to provide a philosophical language that can explore the openness of mathematical (...) signs to reinterpretation, and bridge some barriers between philosophy of mathematics and critical approaches to knowledge. (shrink)
This article interprets Józef Maria Hoëné Wronski’s (1776–1853) use of actual infinities in his mathematical work. The interpretation places this usage, which undermined Wronski’s acceptance as a mathematician, in his contemporary mathematical and philosophical context and in the context of his own sociopolitical-philosophical project.
S(zp,zp) performs an innovative analysis of one of modern logic's most celebrated cornerstones: the proof of Gödel's first incompleteness theorem. The book applies the semiotic theories of French post- structuralists such as Julia Kristeva, Jacques Derrida and Gilles Deleuze to shed new light on a fundamental question: how do mathematical signs produce meaning and make sense? S(zp,zp) analyses the text of the proof of Gödel's result, and shows that mathematical language, like other forms of language, enjoys the full complexity of (...) language as a process, with its embodied genesis, constitutive paradoxical forces and unbounded shifts of meaning. These effects do not infringe on the logico-mathematical validity of Gödel's proof. Rather, they belong to a mathematical unconscious that enables the successful function of mathematical texts for a variety of different readers. S(zp,zp) breaks new ground by synthesising mathematical logic and post-structural semiotics into a new form of philosophical fabric, and offers an original way of bridging the gap between the "two cultures". (shrink)
This note opens with brief evaluations of classical foundationalist endeavors – those of Frege, Russell, Brouwer and Hilbert. From there we proceed to some pluralist approaches to foundations, focusing on Putnam and Wittgenstein, making a note of what enables their pluralism. Then, I bring up approaches that find foundations potentially harmful, as expressed by Rav and Lakatos. I conclude with a brief discussion of a late medieval Indian case study in order to show what an “unfounded” mathematics could look like. (...) The general purpose is to re-evaluate the desiderata of foundational programs in mathematics. (shrink)
ArgumentThis paper argues for the viability of a different philosophical point of view concerning classical Greek geometry. It reviews Reviel Netz's interpretation of classical Greek geometry and offers a Deleuzian, post-structural alternative. Deleuze's notion of haptic vision is imported from its art history context to propose an analysis of Greek geometric practices that serves as counterpoint to their linear modular cognitive narration by Netz. Our interpretation highlights the relation between embodied practices, noisy material constraints, and operational codes. Furthermore, it sheds (...) some new light on the distinctness and clarity of Greek mathematical conceptual divisions. (shrink)
A few years ago, a manuscript by Jost Bürgi (1552–1632) was brought to scholarly attention, which included an ingenious sine calculation method. The purpose of this paper is to discuss two aspects of this manuscript. First, we wish to improve the current understanding of Bürgi’s method of sine calculation, especially with respect to the calculation of sines at a resolution of 1 min. Second, we wish to suggest a possible transfer of knowledge between India’s Kerala School of mathematical astronomy and (...) Bürgi. The evidence for the latter seems to be stronger than the evidence for other available case studies, but still revolves mainly around analogies, and can therefore not be considered as conclusive proof of transmission. We also append a translation of the relevant chapter of Bürgi’s treatise. (shrink)
This essay asks: Is “culture” the subject of a communication among anthropologists, or are anthropologists subjects to a communication among cultures? Put more simply, is there only one culture, comprised of multiplex variations recovered from various parts of an ever-changing world, or are there a great many, all of them variations on a single theoretical insight, which anthropologists have made up in secret and carefully keep as a secret from themselves? (Why not? the author asks, adding that such is exactly (...) how the modern state operates.) Is it possible, he further asks, for a memory to have an independent existence, untroubled by the people who constantly keep occurring to it? It is of course absurd to ascribe agency to what amount to mere figures of speech, metaphors, or enigmatic perceptual cues, “but if a metaphor could not think, as an agency in and of itself, then neither could we.” (Memories would have to be metaphors; otherwise how could we forget them?) Finally, the essays asks: What is more natural than the agency of the one you see in the mirror, that steals your act of looking, but only to view itself? (shrink)
This paper will apply post-structural semiotic theories to study the texts of Gödel's first incompleteness theorem. I will study the texts’ own articulations of concepts of ‘meaning’, analyze the mechanisms they use to sustain their senses of validity, and point out how the texts depend (without losing their mathematical rigor) on sustaining some shifts of meaning. I will demonstrate that the texts manifest semiotic effects, which we usually associate with poetry and everyday speech. I will conclude with an analysis of (...) how the picture I paint relates to an ethics of mathematical production. (shrink)
We present the notion of finite high-order Gowers games, and prove a statement parallel to Gowers's Combinatorial Lemma for these games. We derive ‘quantitative’ versions of the original Gowers Combinatorial Lemma and of Gowers's Dichotomy, which place them in the context of the recently introduced infinite dimensional asymptotic theory for Banach spaces.