Theories of writing and mind have proposed that the uses of literacy give rise to a distinct repertoire of cognitive skills, attitudes, and concepts. This paper reconsiders the earliest lexical lists of the Ancient Near East as one type of evidence on writing and mind. Past and present conceptions of the lists are briefly reviewed. Early views cast the lists as reflecting a Sumerian mentality or a uniquely literate mode of thought, while recent accounts suggest they may simply be routine (...) scribal exercises. A view from the philosophy of science, on which lists are considered a sub-type of ordering system, suggests a way of aligning a scribal practice account with aspects of earlier views by articulating the nature of list entries and the intentions of the list makers. On this account, the Ancient Near Eastern lists can be seen both as uniquely literate and as uniquely informative on the role of writing in mind. (shrink)

Theories of writing and mind have proposed that the uses of literacy give rise to a distinct repertoire of cognitive skills, attitudes, and concepts. This paper reconsiders the earliest lexical lists of the Ancient Near East as one type of evidence on writing and mind. Past and present conceptions of the lists are briefly reviewed. Early views cast the lists as reflecting a Sumerian mentality or a uniquely literate mode of thought, while recent accounts suggest they may simply be routine (...) scribal exercises. A view from the philosophy of science, on which lists are considered a sub-type of ordering system, suggests a way of aligning a scribal practice account with aspects of earlier views by articulating the nature of list entries and the intentions of the list makers. On this account, the Ancient Near Eastern lists can be seen both as uniquely literate and as uniquely informative on the role of writing in mind. (shrink)

One main challenge of non-platonist philosophy of mathematics is to account for the apparent objectivity of mathematical knowledge. Cole and Feferman have proposed accounts that aim to explain objectivity through the intersubjectivity of mathematical knowledge. In this paper, focusing on arithmetic, I will argue that these accounts as such cannot explain the apparent objectivity of mathematical knowledge. However, with support from recent progress in the empirical study of the development of arithmetical cognition, a stronger argument can be provided. I will (...) show that since the development of arithmetic is (partly) determined by biologically evolved proto-arithmetical abilities, arithmetical knowledge can be understood as maximally intersubjective. This maximal intersubjectivity, I argue, can lead to the experience of objectivity, thus providing a solution to the problem of reconciling non-platonist philosophy of mathematics with the (apparent) objectivity of mathematical knowledge. (shrink)

This paper presents writing as an extended cognitive system comprised of brain, body, and the material form that is writing. Part I introduces the theoretical framework used for the analysis, Material Engagement Theory (MET), and the initial insights into writing systems gained by applying MET to Mesopotamian artifacts for numbers and writing. Part II discusses how writing as a material form has changed over time and why this material change reflects, accumulates, and distributes change in the behaviors and brains of (...) generations of writers. Part III explains why forms of writing used today are a visible form of language in being comprised of contrastive graphic features. Part IV argues against the idea that writing should be excluded from being considered as an extended cognitive system. On the contrary, considering writing from this perspective can provide new insights into the ways we use material forms—not just in writing but more broadly—to change our behaviors and brains, and their roles in intensifying and perpetuating those changes. (shrink)

Human cognition is extended and enacted. Drawing the boundaries of cognition to include the resources and attributes of the body and materiality allows an examination of how these components interact with the brain as a system, especially over cultural and evolutionary spans of time. Literacy and numeracy provide examples of multigenerational, incremental change in both psychological functioning and material forms. Though we think materiality, its central role in human cognition is often unappreciated, for reasons that include conceptual distribution over multiple (...) material forms, the unconscious transparency of cognitive activity in general, and the different temporalities of metaplastic change in neurons and cultural forms. (shrink)

Using a model of cognition as extended and enactive, we examine the role of materiality in making minds as exemplified by lithics and writing, forms associated with conceptual thought and meta-awareness of conceptual domains. We address ways in which brain functions may change in response to interactions with material forms, the attributes of material forms that may cause such change, and the spans of time required for neurofunctional reorganization. We also offer three hypotheses for investigating co-influence and change in cognition (...) and material culture. This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No. 785793. (shrink)

Drawing on the material culture of the Ancient Near East as interpreted through Material Engagement Theory, the journey of how material number becomes a conceptual number is traced to address questions of how a particular material form might generate a concept and how concepts might ultimately encompass multiple material forms so that they include but are irreducible to all of them together. Material forms incorporated into the cognitive system affect the content and structure of concepts through their agency and affordances, (...) the capabilities and constraints they provide as the material component of the extended, enactive mind. Material forms give concepts the tangibility that enables them to be literally grasped and manipulated. As they are distributed over multiple material forms, concepts effectively become independent of any of them, yielding the abstract irreducibility that makes a concept like number what it is. Finally, social aspects of material use—collaboration, ordinariness, and time—have important effects on the generation and distribution of concepts. (shrink)

This paper explains how mathematical computation can be constructed from weaker recursive patterns typical of natural languages. A thought experiment is used to describe the formalization of computational rules, or arithmetical axioms, using only orally-based natural language capabilities, and motivated by two accomplishments of ancient Indian mathematics and linguistics. One accomplishment is the expression of positional value using versified Sanskrit number words in addition to orthodox inscribed numerals. The second is Pāṇini’s invention, around the fifth century BCE, of a formal (...) grammar for spoken Sanskrit, expressed in oral verse extending ordinary Sanskrit, and using recursive methods rediscovered in the twentieth century. The Sanskrit positional number compounds and Pāṇini’s formal system are construed as linguistic grammaticalizations relying on tacit cognitive models of symbolic form. The thought experiment shows that universal computation can be constructed from natural language structure and skills, and shows why intentional capabilities needed for language use play a role in computation across all media. The evolution of writing and positional number systems in Mesopotamia is used to transfer the thought experiment of “oral arithmetic” to inscribed computation. The thought experiment and historical evidence combine to show how and why mathematical computation is a cognitive technology extending generic symbolic skills associated with language structure, usage, and change. (shrink)

Most of what is told in this paper has been told before by the same author, in a number of publications of various kinds, but this is the first time that all this material has been brought together and treated in a uniform way. Smaller errors in the earlier publications are corrected here without comment. It has been known since the 1920s that quadratic equations played a prominent role in Babylonian mathematics. See, most recently, Høyrup (Hist Sci 34:1–32, 1996, and (...) Lengths, widths, surfaces: a portrait of old Babylonian algebra and its kin. Springer, New York, 2002). What has not been known, however, is how quadratic equations came to play that role, since it is difficult to think of any practical use for quadratic equations in the life and work of a Babylonian scribe. One goal of the present paper is to show how the need to find solutions to quadratic equations actually arose in Mesopotamia not later than in the second half of the third millennium BC, and probably before that in connection with certain geometric division of property problems. This issue was brought up for the first time in Friberg (Cuneiform Digit Lib J 2009:3, 2009). In this connection, it is argued that the tool used for the first exact solution of a quadratic equation was either a clever use of the “conjugate rule” or a “completion of the square,” but that both methods ultimately depend on a certain division of a square, the same in both cases. Another, closely related goal of the paper is to discuss briefly certain of the most impressive achievements of anonymous Babylonian mathematicians in the first half of the second millennium BC, namely recursive geometric algorithms for the solution of various problems related to division of figures, more specifically trapezoidal fields. For an earlier, comprehensive (but less accessible) treatment of these issues, see Friberg (Amazing traces of a Babylonian origin in Greek mathematics. WorldScientific, Singapore 2007b, Ch. 11 and App. 1). (shrink)

The world’s diverse written numeral systems are affected by human cognition; in turn, written numeral systems affect mathematical cognition in social environments. The present study investigates the constraints on graphic numerical notation, treating it neither as a byproduct of lexical numeration, nor a mere adjunct to writing, but as a specific written modality with its own cognitive properties. Constraints do not refute the notion of infinite cultural variability; rather, they recognize the infinity of variability within defined limits, thus transcending the (...) universalist/particularist dichotomy. In place of strictly innatist perspectives on mathematical cognition, a model is proposed that invokes domain-specific and notationally-specific constraints to explain patterns in numerical notations. The analysis of exceptions to cross-cultural generalizations makes the study of near-universals highly productive theoretically. The cross-cultural study of patterns in written numbers thus provides a rich complement to the cognitive analysis of writing systems. (shrink)

The world’s diverse written numeral systems are affected by human cognition; in turn, written numeral systems affect mathematical cognition in social environments. The present study investigates the constraints on graphic numerical notation, treating it neither as a byproduct of lexical numeration, nor a mere adjunct to writing, but as a specific written modality with its own cognitive properties. Constraints do not refute the notion of infinite cultural variability; rather, they recognize the infinity of variability within defined limits, thus transcending the (...) universalist/particularist dichotomy. In place of strictly innatist perspectives on mathematical cognition, a model is proposed that invokes domain-specific and notationally-specific constraints to explain patterns in numerical notations. The analysis of exceptions to cross-cultural generalizations makes the study of near-universals highly productive theoretically. The cross-cultural study of patterns in written numbers thus provides a rich complement to the cognitive analysis of writing systems. (shrink)

This inquiry seeks to understand how the original form of writing in Mesopotamia—the small pictures and conventions of protocuneiform—became cuneiform, a script that could not be read without acquiring the neurological and behavioral reorganizations understood today as literacy. The process is described as involving small neurological and behavioral changes realized, accumulated, and distributed to new users through interactions with and concomitant incremental changes in the material form of writing. A related inquiry focuses on why and how numerical notations differ from (...) other written signs. Crucially, numerical signs instantiate their meaning, a representational mode that contrasts with the signification used to represent non-numerical language and which makes numerical notations contiguous with their unwritten precursors, technologies like fingers, tallies, and counters. Instantiation is related to the perceptual system for quantity; this so-called number sense influences the function and form of numerical signs. Reading is then discussed as a cognitive activity that necessarily involves a material form, a plausible example of extended cognition. Because numerical notations share function and often form with precursor technologies, if the former participate in extended cognition, the latter likely do as well. In conjunction with the contiguity between numerical notations and their unwritten precursors, this complicates the idea that (all) writing is (just) language. Finally, potential follow-on research is suggested. (shrink)

This paper discusses the relationship between Kantian idealism and Marxian materialism. Part I examines the reasons this relationship is misconstrued to be predominantly a matter of practical philosophy and turns to the neglected works of Alfred Sohn-Rethel and Richard Seaford to outline the importance of money for understanding Kant’s theoretical work. Part II considers an objection that Kant confuses the commodity form for the transcendental object of experience. I am ultimately concerned with defusing the accusation that the identity of the (...) commodity with a Kantian thing renders Kantian idealism “bourgeois” in the pejorative sense. Money is implicated, historically and conceptually, in the very intelligibility of metaphysics. In that case, Kantian idealism and Marxian materialism are two sides of the same coin. (shrink)

Complex systems like literacy and numeracy emerge through multigenerational interactions of brains, behaviors, and material forms. In such systems, material forms – writing for language and notations for numbers – become increasingly refined to elicit specific behavioral and psychological responses in newly indoctrinated individuals. These material forms, however, differ fundamentally in things like semiotic function: language signifies, while numbers instantiate. This makes writing for language able to represent the meanings and sounds of particular languages, while notations for numbers are semantically (...) meaningful without phonetic specification. This representational distinction is associated with neurofunctional and behavioral differences in what neural activity and behaviors like handwriting contribute to literacy and numeracy. In turn, neurofunctional and behavioral differences place written representations for language and numbers under different pressures that influence the forms they take and how those forms change over time as they are transmitted across languages and cultures. (shrink)

I argue for the Wittgensteinian thesis that mathematical statements are expressions of norms, rather than descriptions of the world. An expression of a norm is a statement like a promise or a New Year's resolution, which says that someone is committed or entitled to a certain line of action. A expression of a norm is not a mere description of a regularity of human behavior, nor is it merely a descriptive statement which happens to entail a norms. The view can (...) be thought of as a sort of logicism for the logical expressivist---a person who believes that the purpose of logical language is to make explicit commitments and entitlements that are implicit in ordinary practice. The thesis that mathematical statements are expression of norms is a kind of logicism, not because it says that mathematics can be reduced to logic, but because it says that mathematical statements play the same role as logical statements. ;I contrast my position with two sets of views, an empiricist view, which says that mathematical knowledge is acquired and justified through experience, and a cluster of nativist and apriorist views, which say that mathematical knowledge is either hardwired into the human brain, or justified a priori, or both. To develop the empiricist view, I look at the work of Kitcher and Mill, arguing that although their ideas can withstand the criticisms brought against empiricism by Frege and others, they cannot reply to a version of the critique brought by Wittgenstein in the Remarks on the Foundations of Mathematics. To develop the nativist and apriorist views, I look at the work of contemporary developmental psychologists, like Gelman and Gallistel and Karen Wynn, as well as the work of philosophers who advocate the existence of a mathematical intuition, such as Kant, Husserl, and Parsons. After clarifying the definitions of "innate" and "a priori," I argue that the mechanisms proposed by the nativists cannot bring knowledge, and the existence of the mechanisms proposed by the apriorists is not supported by the arguments they give. (shrink)