Numbers are concepts whose content, structure, and organization are influenced by the material forms used to represent and manipulate them. Indeed, as argued here, it is the inclusion of multiple forms (distributed objects, fingers, single- and two-dimensional forms like pebbles and abaci, and written notations) that is the mechanism of numerical elaboration. Further, variety in employed forms explains at least part of the synchronic and diachronic variability that exists between and within cultural number systems. Material forms also impart characteristics like (...) linearity that may persist in the form of knowledge and behaviors, ultimately yielding numerical concepts that are irreducible to and functionally independent of any particular form. Material devices used to represent and manipulate numbers also interact with language in ways that reinforce or contrast different aspects of numerical cognition. Not only does this interaction potentially explain some of the unique aspects of numerical language, it suggests that the two are complementary but ultimately distinct means of accessing numerical intuitions and insights. The potential inclusion of materiality in contemporary research in numerical cognition is advocated, both for its explanatory power, as well as its influence on psychological, behavioral, and linguistic aspects of numerical cognition. (shrink)

The basic human ability to treat quantitative information can be divided into two parts. With proto-arithmetical ability, based on the core cognitive abilities for subitizing and estimation, numerosities can be treated in a limited and/or approximate manner. With arithmetical ability, numerosities are processed (counted, operated on) systematically in a discrete, linear, and unbounded manner. In this paper, I study the theory of enculturation as presented by Menary (2015) as a possible explanation of how we make the move from the proto-arithmetical (...) ability to arithmetic proper. I argue that enculturation based on neural reuse provides a theoretically sound and fruitful framework for explaining this development. However, I show that a comprehensive explanation must be based on valid theoretical distinctions and involve several stages in the development of arithmetical knowledge. I provide an account that meets these challenges and thus leads to a better understanding of the subject of enculturation. (shrink)

Arithmetical cognition is the result of enculturation. On a personal level of analysis, enculturation is a process of structured cultural learning that leads to the acquisition of evolutionarily recent, socio-culturally shaped arithmetical practices. On a sub-personal level, enculturation is realized by learning driven plasticity and learning driven bodily adaptability, which leads to the emergence of new neural circuitry and bodily action patterns. While learning driven plasticity in the case of arithmetical practices is not consistent with modularist theories of mental architecture, (...) it can be enriched by the theory of neural reuse. According to neural reuse, cerebral regions are reused to contribute to multiple neural circuits in functionally constrained ways throughout ontogeny. By hypothesis, learning driven plasticity is complemented by learning driven bodily adaptability, which suggests that there is an interesting functional relationship between finger gnosis, finger counting, and arithmetical practices. The emerging perspective on enculturated arithmetical cognition will be complemented by considerations on associated developmental and acquired disorders, namely developmental dyscalculia and acquired acalculia. The upshot is that we need to take the cerebral, extra-cerebral bodily, and socio-cultural dimensions of enculturation into account in order to arrive at a better understanding of the phylogenetic and ontogenetic conditions of arithmetical cognition. (shrink)

On knowledge and practices: a manifesto -- The web of practices -- Agents and frameworks -- Complementarity in mathematics -- Ancient Greek mathematics: a role for diagrams -- Advanced math: the hypothetical conception -- Arithmetic certainty -- Mathematics developed: the case of the reals -- Objectivity in mathematical knowledge -- The problem of conceptual understanding.

Many of our cognitive capacities are the result of enculturation. Enculturation is the temporally extended transformative acquisition of cognitive practices in the cognitive niche. Cognitive practices are embodied and normatively constrained ways to interact with epistemic resources in the cognitive niche in order to complete a cognitive task. The emerging predictive processing perspective offers new functional principles and conceptual tools to account for the cerebral and extra-cerebral bodily components that give rise to cognitive practices. According to this emerging perspective, many (...) cases of perception, action, and cognition are realized by the on-going minimization of prediction error. Predictive processing provides us with a mechanistic perspective that helps investigate the functional details of the acquisition of cognitive practices. The argument of this paper is that research on enculturation and recent work on predictive processing are complementary. The main reason is that predictive processing operates at a sub-personal level and on a physiological time scale of explanation only. A complete account of enculturated cognition needs to take additional levels and temporal scales of explanation into account. This complementarity assumption leads to a new framework—enculturated predictive processing—that operates on multiple levels and temporal scales for the explanation of the enculturated predictive acquisition of cognitive practices. Enculturated predictive processing is committed to explanatory pluralism. That is, it subscribes to the idea that we need multiple perspectives and explanatory strategies to account for the complexity of enculturation. The upshot is that predictive processing needs to be complemented by additional considerations and conceptual tools to realize its full explanatory potential. (shrink)

One of the most important questions in the young field of numerical cognition studies is how humans bridge the gap between the quantity-related content produced by our evolutionarily ancient brains and the precise numerical content associated with numeration systems like Indo-Arabic numerals. This gap problem is the main focus of this paper. The aim here is to evaluate the extent to which cultural factors can help explain how we come to think about numbers beyond the subitizing range. To do this, (...) I summarize Clark’s notion of a difference maker in explaining complex causation that criss-crosses between mind and world and apply it to Menary’s Open MIND. MIND Group, Frankfurt, 2015a) discussion of mathematical cognition as a case of enculturation. I argue that while Menary’s views on enculturation can help explain what makes the difference between numerate and anumerate cultures, it cannot help specify what makes the difference between numerate and anumerate individuals. I argue that features of enculturation do not provide an account of innovation capable of explaining how individuals manage to improve and modify the practices of their cultural niche. This is because Menary’s construal of the role of enculturation in the development of mathematical cognition focuses mostly on the inheritance and transmission of practices, not on their origins, which involve individual-level understanding, rather than population-level practices and pressures. The upshot is that culture provides the necessary background conditions against which individuals can innovate. This role is crucial in the development of numerical abilities—crucial, but explanatorily limited. (shrink)

Menary OpenMIND, MIND Group, Frankfurt am Main, 2015) has argued that the development of our capacities for mathematical cognition can be explained in terms of enculturation. Our ancient systems for perceptually estimating numerical quantities are augmented and transformed by interacting with a culturally-enriched environment that provides scaffolds for the acquisition of cognitive practices, leading to the development of a discrete number system for representing number precisely. Numerals and the practices associated with numeral systems play a significant role in this process. (...) However, the details of the relationship between the ancient number system and the discrete number system remain unclear. This lack of clarity is exacerbated by the problem of symbolic estrangement and the fact that unique features of how numeral systems represent require our ancient number system to play a dual role. These issues highlight that Dehaene’s From monkey brain to human brain, MIT Press, Cambridge, pp 133–157, 2005) neuronal recycling hypothesis may be insufficient to explain the neural mechanisms underlying the process of enculturation. In order to explain mathematical enculturation, and enculturation more generally, it may be necessary to adopt Anderson’s :245–266, 2010; After phrenology: neural reuse and the interactive brain, MIT Press, Cambridge, 2014) theory of neural reuse. (shrink)

In this paper I study the development of arithmetical cognition with the focus on metaphorical thinking. In an approach developing on Lakoff and Núñez, I propose one particular conceptual metaphor, the Process → Object Metaphor, as a key element in understanding the development of mathematical thinking.

This Oxford Handbook covers the current state of the art in the philosophy of maths and logic in a comprehensive and accessible manner, giving the reader an overview of the major problems, positions, and battle lines. The 26 newly-commissioned chapters are by established experts in the field and contain both exposition and criticism as well as substantial development of their own positions. Select major positions are represented by two chapters - one supportive and one critical. The book includes a comprehensive (...) bibliography. (shrink)

Many of our cognitive capacities are the result of enculturation. Enculturation is the temporally extended transformative acquisition of cognitive practices in the cognitive niche. Cognitive practices are embodied and normatively constrained ways to interact with epistemic resources in the cognitive niche in order to complete a cognitive task. The emerging predictive processing perspective offers new functional principles and conceptual tools to account for the cerebral and extra-cerebral bodily components that give rise to cognitive practices. According to this emerging perspective, many (...) cases of perception, action, and cognition are realized by the on-going minimization of prediction error. Predictive processing provides us with a mechanistic perspective that helps investigate the functional details of the acquisition of cognitive practices. The argument of this paper is that research on enculturation and recent work on predictive processing are complementary. The main reason is that predictive processing operates at a sub-personal level and on a physiological time scale of explanation only. A complete account of enculturated cognition needs to take additional levels and temporal scales of explanation into account. This complementarity assumption leads to a new framework—enculturated predictive processing—that operates on multiple levels and temporal scales for the explanation of the enculturated predictive acquisition of cognitive practices. Enculturated predictive processing is committed to explanatory pluralism. That is, it subscribes to the idea that we need multiple perspectives and explanatory strategies to account for the complexity of enculturation. The upshot is that predictive processing needs to be complemented by additional considerations and conceptual tools to realize its full explanatory potential. (shrink)

Julian Cole argues that mathematical domains are the products of social construction. This view has an initial appeal in that it seems to salvage much that is good about traditional platonistic realism without taking on the ontological baggage. However, it also has problems. After a brief sketch of social constructivist theories and Cole’s philosophy of mathematics, I evaluate the arguments in favor of social constructivism. I also discuss two substantial problems with the theory. I argue that unless and until social (...) constructivists can address the two concerns, we have reason to be skeptical about social constructivism in the philosophy of mathematics. (shrink)

In behavioral ecology some authors regard the innateness concept as irretrievably confused whilst others take it to refer to adaptations. In cognitive psychology, however, whether traits are 'innate' is regarded as a significant question and is often the subject of heated debate. Several philosophers have tried to define innateness with the intention of making sense of its use in cognitive psychology. In contrast, I argue that the concept is irretrievably confused. The vernacular innateness concept represents a key aspect of 'folkbiology', (...) namely, the explanatory strategy that psychologists and cognitive anthropologists have labeled 'folk essentialism'. Folk essentialism is inimical to Darwinism, and both Darwin and the founders of the modern synthesis struggled to overcome this way of thinking about living systems. Because the vernacular concept of innateness is part of folkbiology, attempts to define it more adequately are unlikely to succeed, making it preferable to introduce new, neutral terms for the various, related notions that are needed to understand cognitive development. (shrink)

Susan Carey's account of Quinean bootstrapping has been heavily criticized. While it purports to explain how important new concepts are learned, many commentators complain that it is unclear just what bootstrapping is supposed to be or how it is supposed to work. Others allege that bootstrapping falls prey to the circularity challenge: it cannot explain how new concepts are learned without presupposing that learners already have those very concepts. Drawing on discussions of concept learning from the philosophical literature, this article (...) develops a detailed interpretation of bootstrapping that can answer the circularity challenge. The key to this interpretation is the recognition of computational constraints, both internal and external to the mind, which can endow empty symbols with new conceptual roles and thus new contents. (shrink)

The purpose of this paper is to apply Crispin Wright’s criteria and various axes of objectivity to mathematics. I test the criteria and the objectivity of mathematics against each other. Along the way, various issues concerning general logic and epistemology are encountered.

The renowned philosopher John Searle reveals the fundamental nature of social reality. What kinds of things are money, property, governments, nations, marriages, cocktail parties, and football games? Searle explains the key role played by language in the creation, constitution, and maintenance of social reality. We make statements about social facts that are completely objective, for example: Barack Obama is President of the United States, the piece of paper in my hand is a twenty-dollar bill, I got married in London, etc. (...) And yet these facts only exist because we think they exist. How is it possible that we can have factual objective knowledge of a reality that is created by subjective opinions? This is part of a much larger question: How can we give an account of ourselves, with our peculiar human traits DS mind, reason, freedom, society - in a world that we know independently consists of mindless, meaningless particles? How can we account for our social and mental existence in a realm of brute physical facts? In answering this question, Searle avoids postulating different realms of being, a mental and a physical, or worse yet, a mental, a physical, and a social. There is just one reality: Searle shows how the human reality fits into that one reality. Mind, language, and civilization are natural products of the basic facts of the physical world described by physics, chemistry and biology. Searle explains how language creates and maintains the elaborate structures of human social institutions. These institutions serve to create and distribute power relations that are pervasive and often invisible. These power relations motivate human actions in a way that provides the glue that holds human civilization together. Searle shows how this account illuminates human rationality, free will, political power, and human rights. Our social world is a world created and maintained by language. (shrink)

In this paper I develop a philosophical account of actual mathematical infinity that does not demand ontologically or epistemologically problematic assumptions. The account is based on a simple metaphor in which we think of indefinitely continuing processes as defining objects. It is shown that such a metaphor is valid in terms of mathematical practice, as well as in line with empirical data on arithmetical cognition.

"First published in this translation 1955; second edition (revised) 1974; reprinted with additional revisions 1987; reissued with new Further Reading 2003; reissued with new introduction 2007"--T.p. verso.

Stereotypes of social construction suggest that the existence of social constructs is accidental and that such constructs have arbitrary and subjective features. In this paper, I explore a conception of social construction according to which it consists in the collective imposition of function onto reality and show that, according to this conception, these stereotypes are incorrect. In particular, I argue that the collective imposition of function onto reality is typically non-accidental and that the products of such imposition frequently have non-arbitrary (...) and objective features. These conclusions are interesting in and of themselves since they debunk important aspects of our socially constructed conception of social construction. Yet, additionally, they have important implications for the viability of mathematical social constructivism since resistance to such constructivism is frequently grounded in the observation that mathematics is non-accidental, non-arbitrary, and objective. As a secondary focus, I explore these implications in this paper. (shrink)

Recent years have seen an explosion of empirical data concerning arithmetical cognition. In this paper that data is taken to be philosophically important and an outline for an empirically feasible epistemological theory of arithmetic is presented. The epistemological theory is based on the empirically well-supported hypothesis that our arithmetical ability is built on a protoarithmetical ability to categorize observations in terms of quantities that we have already as infants and share with many nonhuman animals. It is argued here that arithmetical (...) knowledge developed in such a way cannot be totally conceptual in the sense relevant to the philosophy of arithmetic, but neither can arithmetic understood to be empirical. Rather, we need to develop a contextual a priori notion of arithmetical knowledge that preserves the special mathematical characteristics without ignoring the roots of arithmetical cognition. Such a contextual a priori theory is shown not to require any ontologically problematic assumptions, in addition to fitting well within a standard framework of general epistemology. (shrink)

In Reading in the Brain, Stanislas Dehaene presents a compelling account of how the brain learns to read. Central to this account is his neuronal recycling hypothesis: neural circuitry is capable of being ‘recycled’ or converted to a different function that is cultural in nature. The original function of the circuitry is not entirely lost and constrains what the brain can learn. It is argued that the neural niche co-evolves with the environmental niche in a way that does not undermine (...) the core ideas of neuronal recycling, but which is quite different from the models of cognitive and cultural evolution provided by evolutionary psychology and epidemiology. Dehaene contrasts neuronal recycling with a naïve model of the brain as a general learning device that is unconstrained in what it can learn. Consequently a tension develops in Dehaene's account of the role of plasticity in the acquisition of language. It is argued that the functional and structural changes in the brain that Dehaene documents in great detail are driven by learning and that this learning-driven plasticity does not commit us to a naïve model of the brain. (shrink)

Intersubjectivity refers to the variety of possible relations between perspectives. It is indispensable for understanding human social behaviour. While theoretical work on intersubjectivity is relatively sophisticated, methodological approaches to studying intersubjectivity lag behind. Most methodologies assume that individuals are the unit of analysis. In order to research intersubjectivity, however, methodologies are needed that take relationships as the unit of analysis. The first aim of this article is to review existing methodologies for studying intersubjectivity. Four methodological approaches are reviewed: comparative self-report, (...) observing behaviour, analysing talk and ethnographic engagement. The second aim of the article is to introduce and contribute to the development of a dialogical method of analysis. The dialogical approach enables the study of intersubjectivity at different levels, as both implicit and explicit, and both within and between individuals and groups. The article concludes with suggestions for using the proposed method for researching intersubjectivity both within individuals and between individuals and groups. (shrink)

Platonism about mathematics (or mathematical platonism) isthe metaphysical view that there are abstract mathematical objectswhose existence is independent of us and our language, thought, andpractices. Just as electrons and planets exist independently of us, sodo numbers and sets. And just as statements about electrons and planetsare made true or false by the objects with which they are concerned andthese objects' perfectly objective properties, so are statements aboutnumbers and sets. Mathematical truths are therefore discovered, notinvented., Existence. There are mathematical objects.

According to Carey (2009), humans construct new concepts by abstracting structural relations among sets of partly unspecified symbols, and then analogically mapping those symbol structures onto the target domain. Using the development of integer concepts as an example, I give reasons to doubt this account and to consider other ways in which language and symbol learning foster conceptual development.

This paper extracts some of the main theses in the philosophy of mathematics from my book, The Construction of Logical Space. I show that there are important limits to the availability of nominalistic paraphrase functions for mathematical languages, and suggest a way around the problem by developing a method for specifying nominalistic contents without corresponding nominalistic paraphrases. Although much of the material in this paper is drawn from the book — and from an earlier paper — I hope the present (...) discussion will earn its keep by motivating the ideas in a new way, and by suggesting further applications. (shrink)

A uniquely integrative work, this volume provides a much needed compilation of primary source material to researchers from basic neuroscience, psychology, developmental science, neuroimaging, neuropsychology and theoretical biology. * The ...

Is calculation possible without language? Or is the human ability for arithmetic dependent on the language faculty? To clarify the relation between language and arithmetic, we studied numerical cognition in speakers of Mundurukú, an Amazonian language with a very small lexicon of number words. Although the Mundurukú lack words for numbers beyond 5, they are able to compare and add large approximate numbers that are far beyond their naming range. However, they fail in exact arithmetic with numbers larger than 4 (...) or 5. Our results imply a distinction between a nonverbal system of number approximation and a language-based counting system for exact number and arithmetic. (shrink)

I contend that mathematical domains are freestanding institutional entities that, at least typically, are introduced to serve representational functions. In this paper, I outline an account of institutional reality and a supporting metaontological perspective that clarify the content of this thesis. I also argue that a philosophy of mathematics that has this thesis as its central tenet can account for the objectivity, necessity, and atemporality of mathematics.

One of the most fundamental questions in the philosophy of mathematics concerns the relation between truth and formal proof. The position according to which the two concepts are the same is called deflationism, and the opposing viewpoint substantialism. In an important result of mathematical logic, Kurt Gödel proved in his first incompleteness theorem that all consistent formal systems containing arithmetic include sentences that can neither be proved nor disproved within that system. However, such undecidable Gödel sentences can be established to (...) be true once we expand the formal system with Alfred Tarski s semantical theory of truth, as shown by Stewart Shapiro and Jeffrey Ketland in their semantical arguments for the substantiality of truth. According to them, in Gödel sentences we have an explicit case of true but unprovable sentences, and hence deflationism is refuted. -/- Against that, Neil Tennant has shown that instead of Tarskian truth we can expand the formal system with a soundness principle, according to which all provable sentences are assertable, and the assertability of Gödel sentences follows. This way, the relevant question is not whether we can establish the truth of Gödel sentences, but whether Tarskian truth is a more plausible expansion than a soundness principle. In this work I will argue that this problem is best approached once we think of mathematics as the full human phenomenon, and not just consisting of formal systems. When pre-formal mathematical thinking is included in our account, we see that Tarskian truth is in fact not an expansion at all. I claim that what proof is to formal mathematics, truth is to pre-formal thinking, and the Tarskian account of semantical truth mirrors this relation accurately. -/- However, the introduction of pre-formal mathematics is vulnerable to the deflationist counterargument that while existing in practice, pre-formal thinking could still be philosophically superfluous if it does not refer to anything objective. Against this, I argue that all truly deflationist philosophical theories lead to arbitrariness of mathematics. In all other philosophical accounts of mathematics there is room for a reference of the pre-formal mathematics, and the expansion of Tarkian truth can be made naturally. Hence, if we reject the arbitrariness of mathematics, I argue in this work, we must accept the substantiality of truth. Related subjects such as neo-Fregeanism will also be covered, and shown not to change the need for Tarskian truth. -/- The only remaining route for the deflationist is to change the underlying logic so that our formal languages can include their own truth predicates, which Tarski showed to be impossible for classical first-order languages. With such logics we would have no need to expand the formal systems, and the above argument would fail. From the alternative approaches, in this work I focus mostly on the Independence Friendly (IF) logic of Jaakko Hintikka and Gabriel Sandu. Hintikka has claimed that an IF language can include its own adequate truth predicate. I argue that while this is indeed the case, we cannot recognize the truth predicate as such within the same IF language, and the need for Tarskian truth remains. In addition to IF logic, also second-order logic and Saul Kripke s approach using Kleenean logic will be shown to fail in a similar fashion. (shrink)

Recent research has suggested that the Pirahã, an Amazonian tribe with a number-less language, are able to match quantities > 3 if the matching task does not require recall or spatial transposition. This finding contravenes previous work among the Pirahã. In this study, we re-tested the Pirahãs’ performance in the crucial one-to-one matching task utilized in the two previous studies on their numerical cognition, as well as in control tasks requiring recall and mental transposition. We also conducted a novel quantity (...) recognition task. Speakers were unable to consistently match quantities > 3, even when no recall or transposition was involved. We provide a plausible motivation for the disparate results previously obtained among the Pirahã. Our findings are consistent with the suggestion that the exact recognition of quantities > 3 requires number terminology. (shrink)

Ambitious and elegant, this book builds a bridge between evolutionary theory and cultural psychology. Michael Tomasello is one of the very few people to have done systematic research on the cognitive capacities of both nonhuman primates and human children. The Cultural Origins of Human Cognition identifies what the differences are, and suggests where they might have come from. -/- Tomasello argues that the roots of the human capacity for symbol-based culture, and the kind of psychological development that takes place within (...) it, are based in a cluster of uniquely human cognitive capacities that emerge early in human ontogeny. These include capacities for sharing attention with other persons; for understanding that others have intentions of their own; and for imitating, not just what someone else does, but what someone else has intended to do. In his discussions of language, symbolic representation, and cognitive development, Tomasello describes with authority and ingenuity the "ratchet effect" of these capacities working over evolutionary and historical time to create the kind of cultural artifacts and settings within which each new generation of children develops. He also proposes a novel hypothesis, based on processes of social cognition and cultural evolution, about what makes the cognitive representations of humans different from those of other primates. -/- Lucid, erudite, and passionate, The Cultural Origins of Human Cognition will be essential reading for developmental psychology, animal behavior, and cultural psychology. (from the HUP website). (shrink)

I discuss a puzzle that shows there is a need to develop a new metaphysical interpretation of mathematical theories, because all well-known interpretations conflict with important aspects of mathematical activities. The new interpretation, I argue, must authenticate the ontological commitments of mathematical theories without curtailing mathematicians' freedom and authority to creatively introduce mathematical ontology during mathematical problem-solving. Further, I argue that these two constraints are best met by a metaphysical interpretation of mathematics that takes mathematical entities to be constitutively constructed (...) by human activity in a manner similar to the constitutive construction of the US Supreme Court by certain legal and political activities. Finally, I outline some of the philosophical merits of metaphysical interpretations of mathematical theories of this type. (shrink)

Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. In this survey article, the view is clarified and distinguished from some related views, and arguments for and against the view are discussed.

Science Without Numbers caused a stir in 1980, with its bold nominalist approach to the philosophy of mathematics and science. It has been unavailable for twenty years and is now reissued in a revised edition with a substantial new preface presenting the author's current views and responses to the issues raised in subsequent debate.

In The Construction of Social Reality, John Searle argues that there are two kinds of facts--some that are independent of human observers, and some that require..

Key words: the continuum, structuralism, conceptual structuralism, basic structural conceptions, Euclidean geometry, Hilbertian geometry, the real number system, settheoretical conceptions, phenomenological conceptions, foundational conceptions, physical conceptions.

Only human beings have a rich conceptual repertoire with concepts like tort, entropy, Abelian group, mannerism, icon and deconstruction. How have humans constructed these concepts? And once they have been constructed by adults, how do children acquire them? While primarily focusing on the second question, in The Origin of Concepts , Susan Carey shows that the answers to both overlap substantially. Carey begins by characterizing the innate starting point for conceptual development, namely systems of core cognition. Representations of core cognition (...) are the output of dedicated input analyzers, as with perceptual representations, but these core representations differ from perceptual representations in having more abstract contents and richer functional roles. Carey argues that the key to understanding cognitive development lies in recognizing conceptual discontinuities in which new representational systems emerge that have more expressive power than core cognition and are also incommensurate with core cognition and other earlier representational systems. Finally, Carey fleshes out Quinian bootstrapping, a learning mechanism that has been repeatedly sketched in the literature on the history and philosophy of science. She demonstrates that Quinian bootstrapping is a major mechanism in the construction of new representational resources over the course of childrens cognitive development. Carey shows how developmental cognitive science resolves aspects of long-standing philosophical debates about the existence, nature, content, and format of innate knowledge. She also shows that understanding the processes of conceptual development in children illuminates the historical process by which concepts are constructed, and transforms the way we think about philosophical problems about the nature of concepts and the relations between language and thought. (shrink)

This book not only outlines the indispensability argument in considerable detail but also defends it against various challenges.