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)

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 ...

Intuitively, we often see absences. For example, if someone steals your laptop at a café, you may see its absence from your table. However, absence perception presents a paradox. On prevailing models of perception, we see only present objects and scenes (Marr, Gibson, Dretske). So, we cannot literally see something that is not present. This suggests that we never literally perceive absences; instead, we come to believe that something is absent cognitively on the basis of what we perceive. But this (...) cognitive explanation does not do justice to the phenomenology. Many experiences of absence possess immediate, perceptual qualities. One may further argue that the ability to detect certain absences confers strong adaptive advantage and therefore must be as primitive and fundamental to humans as seeing positive things. I argue that we can literally see absences; in addition to representing objects, perception represents absences of objects. I present a model of seeing absence based on visual expectations and a visual matching process. The phenomenon of seeing absence can thus serve as an adequacy-test for a theory of perceptual content. If experiences of absence are possible, then we have another reason (following Siegel) to reject the view that perceptual content is restricted to colors and shapes. Furthermore, if the proposed account is correct, then we have grounds for dissociating seeing absence from other imagery-based phenomena termed “perceptual presence-in-absence” (Noë, Macpherson). (shrink)

Carrie Jenkins presents a new account of arithmetical knowledge, which manages to respect three key intuitions: a priorism, mind-independence realism, and empiricism. Jenkins argues that arithmetic can be known through the examination of empirically grounded concepts, non-accidentally accurate representations of the mind-independent world.

Five-year-old children categorized as skilled versus unskilled counters were given verbal estimation and number word comprehension tasks with numerosities 20 – 120. Skilled counters showed a linear relation between number words and nonsymbolic numerosities. Unskilled counters showed the same linear relation for smaller numbers to which they could count, but not for larger number words. Further tasks indicated that unskilled counters failed even to correctly order large number words differing by a 2 : 1 ratio, whereas they performed well on (...) this task with smaller numbers, and performed well on a nonsymbolic ordering task with the same numerosities. These findings provide evidence that large, approximate numerosity representations become linked to number words around the time that children learn to count to those words reliably. (shrink)

Develops a structuralist understanding of mathematics, as an alternative to set- or type-theoretic foundations, that respects classical mathematical truth while ...

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 unique book by Stewart Shapiro looks at a range of philosophical issues and positions concerning mathematics in four comprehensive sections. Part I describes questions and issues about mathematics that have motivated philosophers since the beginning of intellectual history. Part II is an historical survey, discussing the role of mathematics in the thought of such philosophers as Plato, Aristotle, Kant, and Mill. Part III covers the three major positions held throughout the twentieth century: the idea that mathematics is logic (logicism), (...) the view that the essence of mathematics is the rule-governed manipulation of characters (formalism), and a revisionist philosophy that focuses on the mental activity of mathematics (intuitionism). Finally, Part IV brings the reader up-to-date with a look at contemporary developments within the discipline. This sweeping introductory guide to the philosophy of mathematics makes these fascinating concepts accessible to those with little background in either mathematics or philosophy. (shrink)

Do numbers, sets, and so forth, exist? What do mathematical statements mean? Are they literally true or false, or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years, Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted, we are confronted with the following powerful dilemma. The desired continuity between mathematical and, say, scientific language suggests realism, but realism in this context suggests seemingly intractable epistemic (...) problems. As a way out of this dilemma, Shapiro articulates a structuralist approach. On this view, the subject matter of arithmetic, for example, is not a fixed domain of numbers independent of each other, but rather is the natural number structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework, realism in mathematics can be preserved without troublesome epistemic consequences. Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an "object" and the Quinean nature of ontological commitment. Clear, compelling, and tautly argued, Shapiro's work, noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics, will be of deep interest to both philosophers and mathematicians. (shrink)

Moving beyond both realist and anti-realist accounts of mathematics, Shapiro articulates a "structuralist" approach, arguing that the subject matter of a mathematical theory is not a fixed domain of numbers that exist independent of each other, but rather is the natural structure, the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle.

Describes the various senses as sensory systems that are attuned to the environment. Develops the notion of rich sensory information that specifies the distal environment. Includes a discussion of affordances.

How do we get out knowledge of the natural numbers? Various philosophical accounts exist, but there has been comparatively little attention to psychological data on how the learning process actually takes place. I work through the psychological literature on number acquisition with the aim of characterising the acquisition stages in formal terms. In doing so, I argue that we need a combination of current neologicist accounts and accounts such as that of Parsons. In particular, I argue that we learn the (...) initial segment of the natural numbers on the basis of the Fregean definitions, but do not learn the natural number structure as a whole on the basis of Hume's principle. Therefore, we need to account for some of the consistency of our number concepts with the Dedekind-Peano axioms in other terms. (shrink)

How can we acquire a grasp of cardinal numbers, even the first very small positive cardinal numbers, given that they are abstract mathematical entities? That problem of cognitive access is the main focus of this paper. All the major rival views about the nature and existence of cardinal numbers face difficulties; and the view most consonant with our normal thought and talk about numbers, the view that cardinal numbers are sizes of sets, runs into the cognitive access problem. The source (...) of the problem is the plausible assumption that cognitive access to something requires causal contact with it. It is argued that this assumption is in fact wrong, and that in this and similar cases we should accept that a certain recognise-and-distinguish capacity is sufficient for cognitive access. We can then go on to solve the cognitive access problem, and thereby support the set-size view of cardinal numbers, by paying attention to empirical findings about basic number abilities. To this end some selected studies of infants, pre-school children and a trained chimpanzee are briefly discussed. (shrink)

This influential 1888 publication explained the real numbers, and their construction and properties, from first principles.

Mathematicians tend to think of themselves as scientists investigating the features of real mathematical things, and the wildly successful application of mathematics in the physical sciences reinforces this picture of mathematics as an objective study. For philosophers, however, this realism about mathematics raises serious questions: What are mathematical things? Where are they? How do we know about them? Offering a scrupulously fair treatment of both mathematical and philosophical concerns, Penelope Maddy here delineates and defends a novel version of mathematical realism. (...) She answers the traditional questions and poses a challenging new one, refocusing philosophical attention on the pressing foundational issues of contemporary mathematics. (shrink)

Foundations of Set Theory discusses the reconstruction undergone by set theory in the hands of Brouwer, Russell, and Zermelo. Only in the axiomatic foundations, however, have there been such extensive, almost revolutionary, developments. This book tries to avoid a detailed discussion of those topics which would have required heavy technical machinery, while describing the major results obtained in their treatment if these results could be stated in relatively non-technical terms. This book comprises five chapters and begins with a discussion of (...) the antinomies that led to the reconstruction of set theory as it was known before. It then moves to the axiomatic foundations of set theory, including a discussion of the basic notions of equality and extensionality and axioms of comprehension and infinity. The next chapters discuss type-theoretical approaches, including the ideal calculus, the theory of types, and Quine's mathematical logic and new foundations; intuitionistic conceptions of mathematics and its constructive character; and metamathematical and semantical approaches, such as the Hilbert program. This book will be of interest to mathematicians, logicians, and statisticians. (shrink)

What is the nature of number systems and arithmetic that we use in science for quantification, analysis, and modeling? I argue that number concepts and arithmetic are neither hardwired in the brain, nor do they exist out there in the universe. Innate subitizing and early cognitive preconditions for number— which we share with many other species—cannot provide the foundations for the precision, richness, and range of number concepts and simple arithmetic, let alone that of more complex mathematical concepts. Numbers and (...) arithmetic, and mathematics in general, have unique features—precision, objectivity, rigor, generalizability, stability, symbolizability, and applicability to the real world—that must be accounted for. They are sophisticated concepts that developed culturally only in recent human history. I suggest that numbers and arithmetic are realized through precise combinations of non-mathematical everyday cognitive mechanisms that make human imagination and abstraction possible. One such mechanism, conceptual metaphor, is a neurally instantiated inference-preserving cross-domain mapping that allows the conceptualization of abstract entities in terms of grounded bodily experience. I analyze how the inferential organization of the properties and “laws” of arithmetic emerge metaphorically from everyday meaningful actions. Numbers and arithmetic are thus—outside of natural selection—the product of the biologically constrained interaction of individuals with the appropriate cultural and historical phenotypic variation supported by language, writing systems, and education. (shrink)

The eclipse riddle -- Seeing surfaces -- The disappearing act -- Spinning shadows -- Berkeley's shadow -- Para-reflections -- Para-refractions : shadowgrams and the black drop -- Goethe's colored shadows -- Filtows -- Holes in the light -- Black and blue -- Seeing in black and white -- We see in the dark -- Hearing silence.

This massive two-volume reference presents a comprehensive selection of the most important works on the foundations of mathematics. While the volumes include important forerunners like Berkeley, MacLaurin, and D'Alembert, as well as such followers as Hilbert and Bourbaki, their emphasis is on the mathematical and philosophical developments of the nineteenth century. Besides reproducing reliable English translations of classics works by Bolzano, Riemann, Hamilton, Dedekind, and Poincare, William Ewald also includes selections from Gauss, Cantor, Kronecker, and Zermelo, all translated here for (...) the first time. (shrink)

For the modern set theorist the empty set Ø, the singleton {a}, and the ordered pair 〈x, y〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building locks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the complexities of Power Set, Replacement, and Choice (...) are broached in the formal elaboration of the ‘set of’f {} operation. So it is surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano. In the development of modern mathematical logic out of the turbulence of 19th century logic, the emergence of the empty set, the singleton, and the ordered pair as clear and elementary set-theoretic concepts serves as amotif that reflects and illuminates larger and more significant developments in mathematical logic: the shift from the intensional to the extensional viewpoint, the development of type distinctions, the logical vs. the iterative conception of set, and the emergence of various concepts and principles as distinctively set-theoretic rather than purely logical. Here there is a loose analogy with Tarski's recursive definition of truth for formal languages: The mathematical interest lies mainly in the procedure of recursion and the attendant formal semantics in model theory, whereas the philosophical interest lies mainly in the basis of the recursion, truth and meaning at the level of basic predication. Circling back to the beginning, we shall see how central the empty set, the singleton, and the ordered pair were, after all. (shrink)

Neurons in the primate parieto-frontal network represent the number of visual items in a collection, but it is unknown whether this system encodes empty sets as conveying null quantity. We recorded from the ventral intraparietal area (VIP) and the prefrontal cortex (PFC) of monkeys performing a matching task including empty sets and countable numerosities as stimuli. VIP neurons encoded empty sets predominantly as a distinct category from numerosities. In contrast, PFC neurons represented empty sets more similarly to numerosity one than (...) to larger numerosities, exhibiting numerical distance and size effects. Moreover, prefrontal neurons represented empty sets abstractly and irrespective of stimulus variations. Compared to VIP, the activity of numerosity neurons in PFC correlated better with the behavioral outcome of empty-set trials. Our results suggest a hierarchy in the processing from VIP to PFC, along which empty sets are steadily detached from visual properties and gradually positioned in a numerical continuum. These findings elucidate how the brain transforms the absence of countable items, nothing, into an abstract quantitative category, zero. (shrink)