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)

Our only channel of information about the world is the impact of external forces on our sensory surfaces. So says science itself. There is no clairvoyance. How, then, can we have parlayed this meager sensory input into a full-blown scientific theory of the world? This is itself a scientific question. The pursuit of it, with free use of scientific theory, is what I call naturalized epistemology. The Roots of Reference falls within that domain. Its more specific concern, within that domain, (...) is reference to concrete and abstract objects: what such reference consists in, and how we achieve it. (shrink)

Charles Parsons examines the notion of object, with the aim to navigate between nominalism, denying that distinctively mathematical objects exist, and forms of Platonism that postulate a transcendent realm of such objects. He introduces the central mathematical notion of structure and defends a version of the structuralist view of mathematical objects, according to which their existence is relative to a structure and they have no more of a 'nature' than that confers on them. Parsons also analyzes the concept of intuition (...) and presents a conception of it distantly inspired by that of Kant, which describes a basic kind of access to abstract objects and an element of a first conception of the infinite. (shrink)

Reprint of the original, first published in 1869.

Until recently, cognitive science focused on such mental functions as problem solving, grammar, and pattern-the functions in which the human mind most closely resembles a computer. But humans are more than computers: we invent new meanings, imagine wildly, and even have ideas that have never existed before. Today the cutting edge of cognitive science addresses precisely these mysterious, creative aspects of the mind.The Way We Think is a landmark analysis of the imaginative nature of the mind. Conceptual blending is already (...) widely known in research laboratories throughout the world; this book, written to be accessible to both lay readers and interested scientists, is its definitive statement. Gilles Fauconnier and Mark Turner show that conceptual blending is the root of the cognitively modern human mind, and that conceptual blends themselves are continually combined and reblended to create the rich mental fabric in which we live.The Way We Think shows how this blending operates; how it is affected by (and gives rise to) language, identity, culture, and invention; and how we imagine what could be and what might have been. The result is a bold and exciting new view of how the mind works. (shrink)

In this entry, Frege's logic is introduced and described in some detail. It is shown how the Dedekind-Peano axioms for number theory can be derived from a consistent fragment of Frege's logic, with Hume's Principle replacing Basic Law V.

The following values have no corresponding Zotero field: PB - Lawrence Erlbaum Associates Inc Mahwah, NJ.

Concepts are the elementary units of reason and linguistic meaning. They are conventional and relatively stable. As such, they must somehow be the result of neural activity in the brain. The questions are: Where? and How? A common philosophical position is that all concepts—even concepts about action and perception—are symbolic and abstract, and therefore must be implemented outside the brain’s sensory-motor system. We will argue against this position using (1) neuroscientific evidence; (2) results from neural computation; and (3) results about (...) the nature of concepts from cognitive linguistics. We will propose that the sensory-motor system has the right kind of structure to characterise both sensory-motor and more abstract concepts. Central to this picture are the neural theory of language and the theory of cogs, according to which, brain structures in the sensory-motor regions are exploited to characterise the so-called “abstract” concepts that constitute the meanings of grammatical constructions and general inference patterns. (shrink)

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)

The Pirahã language challenges simplistic application of Hockett’s nearly universally accepted design features of human language by showing that some of these features may be culturally constrained. In particular, Pirahã culture constrains communication to nonabstract subjects which fall within the immediate experience of interlocutors. This constraint explains a number of very surprising features of Pirahã grammar and culture: the absence of numbers of any kind or a concept of counting and of any terms for quantification, the absence of color terms, (...) the absence of embedding, the simplest pronoun inventory known, the absence of "relative tenses," the simplest kinship system yet documented, the absence of creation myths and fiction, the absence of any individual or collective memory of more than two generations past, the absence of drawing or other art and one of the simplest material cultures documented, and the fact that the Pirahã are monolingual after more than 200 years of regular contact with Brazilians and the Tupi-Guarani-speaking Kawahiv. (shrink)

Seven studies explored the empirical basis for claims that infants represent cardinal values of small sets of objects. Many studies investigating numerical ability did not properly control for continuous stimulus properties such as surface area, volume, contour length, or dimensions that correlate with these properties. Experiment 1 extended the standard habituation/dishabituation paradigm to a 1 vs 2 comparison with three-dimensional objects and confirmed that when number and total front surface area are confounded, infants discriminate the arrays. Experiment 2 revealed that (...) infants dishabituated to a change in front surface area but not to a change in number when the two variables were pitted against each other. Experiments 3 through 5 revealed no sensitivity to number when front surface area was controlled, and Experiments 6 and 7 extended this pattern of findings to the Wynn (1992) transformation task. Infants’ lack of a response to number, combined with their demonstrated sensitivity to one or more dimensions of continuous extent, supports the hypothesis that the representations subserving object-based attention, rather than those subserving enumeration, underlie performance in the above tasks. (shrink)

The idea is that, in a wide range of contexts, utterances of the sentences in (a) in each case will communicate the assumption in (b) in each case (or something closely akin to it, there being a certain amount of contextually governed variation in the speaker's propositional attitude and so the scope of the negation). These scalar inferences are taken to be one kind of (generalized) conversational implicature. As is the case with pragmatic inference quite generally, these inferences are defeasible (...) (cancellable), which distinguishes them from entailments, and they are nondetachable, which distinguishes them from conventional implicatures. The core idea is that the choice of a weaker element from a scale of elements ordered in terms of semantic strength (that is, numbers of entailments) tends to implicate that, as far as the speaker knows, none of the stronger elements in the scale holds in this instance. The pattern is quite clear in (1) and (2), where the weak/strong alternatives are some/all and five/six respectively. In the case of (3), the stronger expression must be intelligent and good-hearted which entails intelligent; what Y's utterance implicates is that Mary does not have the two properties: intelligence and good-heartedness, so that, given the proposition expressed (Mary is intelligent) it follows, deductively, that she is not good-hearted, in Y's opinion. The example in (4) involves a scale inversion due to the negation, so that the weak/strong alternatives are not necessarily/not possibly; the negation which the scalar inference generates creates a double negation, which is eliminated giving possibly. (shrink)

Mathematical investigation, when done well, can confer understanding. This bare observation shouldn’t be controversial; where obstacles appear is rather in the effort to engage this observation with epistemology. The complexity of the issue of course precludes addressing it tout court in one paper, and I’ll just be laying some early foundations here. To this end I’ll narrow the field in two ways. First, I’ll address a specific account of explanation and understanding that applies naturally to mathematical reasoning: the view proposed (...) by Philip Kitcher and Michael Friedman of explanation or understanding as involving the unification of theories that had antecedently appeared heterogeneous. For the second narrowing, I’ll take up one specific feature (among many) of theories and their basic concepts that is sometimes taken to make the theories and concepts preferred: in some fields, for some problems, what is counted as understanding a problem may involve finding a way to represent the problem so that it (or some aspect of it) can be visualized. The final section develops a case study which exemplifies the way that this consideration – the potential for visualizability – can rationally inform decisions as to what the proper framework and axioms should be. The discussion of unification (in sections 3 and 4) leads to a mathematical analogue of Goodman’s problem of identifying a principled basis for distinguishing grue and green. Just as there is a philosophical issue about how we arrive at the predicates we should use when making empirical predictions, so too there is an issue about what properties best support many kinds of mathematical reasoning that are especially valuable to us. The issue becomes pressing via an examination of some physical and mathematical cases that make it seem unlikely that treatments of unification can be as straightforward as the philosophical literature has hoped. Though unification accounts have a grain of truth (since a phenomenon (or cluster of phenomena) called “unification” is in fact important in many cases) we are far from an analysis of what “unification” is.. (shrink)