Six-month-old infants discriminate between large sets of objects on the basis of numerosity when other extraneous variables are controlled, provided that the sets to be discriminated differ by a large ratio (8 vs. 16 but not 8 vs. 12). The capacities to represent approximate numerosity found in adult animals and humans evidently develop in human infants prior to language and symbolic counting.

In mathematics education, students' difficulties with negative numbers are well known. To explain these difficulties, researchers traditionally refer to obstacles raised by the concept of NEGATIVE NUMBERS itself throughout its historical evolution. In order to improve our understanding, I propose to take into consideration another point of view, based on Vygotsky's principles, which define a strong relationship between signs such as language or symbols and cognitive development. I show how it is of great interest to consider students' difficulties with negatives (...) in the light of the use of symbols, in particular the minus sign. I describe the results of an empirical study about students' strategies in solving arithmetical equations with negatives. My analysis shows that the problems raised by the presence of negatives in the equations are attributable to students' very restricted and often inadequate use of the minus sign, which prevents them from finding a negative solution or making sense of it. (shrink)

Many experiments with infants suggest that they possess quantitative abilities, and many experimentalists believe that these abilities set the stage for later mathematics: natural numbers and arithmetic. However, the connection between these early and later skills is far from obvious. We evaluate two possible routes to mathematics and argue that neither is sufficient: (1) We first sketch what we think is the most likely model for infant abilities in this domain, and we examine proposals for extrapolating the natural number concept (...) from these beginnings. Proposals for arriving at natural number by (empirical) induction presuppose the mathematical concepts they seek to explain. Moreover, standard experimental tests for children's understanding of number terms do not necessarily tap these concepts. (2) True concepts of number do appear, however, when children are able to understand generalizations over all numbers; for example, the principle of additive commutativity (a+b=b+a). Theories of how children learn such principles usually rely on a process of mapping from physical object groupings. But both experimental results and theoretical considerations imply that direct mapping is insufficient for acquiring these principles. We suggest instead that children may arrive at natural numbers and arithmetic in a more top-down way, by constructing mathematical schemas. (shrink)

Medieval Arabic algebra is a good example of an artificial language.Yet despite its abstract, formal structure, its utility was restricted to problem solving. Geometry was the branch of mathematics used for expressing theories. While algebra was an art concerned with finding specific unknown numbers, geometry dealtwith generalmagnitudes.Algebra did possess the generosity needed to raise it to a more theoretical level—in the ninth century Abū Kāmil reinterpreted the algebraic unknown “thing” to prove a general result. But mathematicians had no motive to (...) rework their theories in algebraic form. Because it offered no advantage over geometry, algebra remained a practical art in both the Islamic world and in Europe until the scientific uphevals of the 17th–18th centuries. (shrink)

A view of the sources of mathematical knowledge is sketched which emphasizes the close connections between mathematical and empirical knowledge. A platonistic interpretation of mathematical discourse is adopted throughout. Two skeptical views are discussed and rejected. One of these, due to Maturana, is supposed to be based on biological considerations. The other, due to Dummett, is derived from a Wittgensteinian position in the philosophy of language. The paper ends with an elaboration of Gödel's analogy between the mathematician and the physicist.

Ancient Greek mathematics developed the original feature of being deductive mathematics. This article attempts to give a explanation f or this achievement. The focus is on the use of a fixed system of linguistic formulae in Greek mathematical texts. It is shown that the structure of this system was especially adapted for the easy computation of operations of substitution on such formulae, that is, of replacing one element in a fixed formula by another, and it is further argued that such (...) operations of substitution were the main logical tool required by Greek mathematical deduction. The conclusion explains why, assuming the validity of the description above, this historical level is the best explanatory level for the phenomenon of Greek mathematical deduction. (shrink)

This paper explores Simon Stevin’s l’Arithmétique of 1585, where we find a novel understanding of the concept of number. I will discuss the dynamics between his practice and philosophy of mathematics, and put it in the context of his general epistemological attitude. Subsequently, I will take a close look at his justificational concerns, and at how these are reflected in his inductive, a postiori and structuralist approach to investigating the numerical field. I will argue that Stevin’s renewed conceptualisation of the (...) notion of number is a sort of “existential closure” of the numerical domain, founded upon the practice of his predecessors and contemporaries. Accordingly, I want to make clear that l’Aritmetique have to be read not as an ontological analysis or exploration of the numerical field, but as an explication of a mathematical ethos. In this sense, this article also intends to make a specific contribution to the broader issue of the “ethics of geometry.”. (shrink)

Concepts are highly theoretical entities. One cannot study them empirically without committing oneself to substantial preliminary assumptions. Among the competing theories of concepts and categorization developed by psychologists in the last thirty years, the implicit theoretical assumption that what falls under a concept is determined by description () has never been seriously challenged. I present a nondescriptionist theory of our most basic concepts, which include (1) stuffs (gold, milk), (2) real kinds (cat, chair), and (3) individuals (Mama, Bill Clinton, the (...) Empire State Building). On the basis of something important that all three have in common, our earliest and most basic concepts of substances are identical in structure. The membership of the category like that of is a natural unit in nature, to which the concept does something like pointing, and continues to point despite large changes in the properties the thinker represents the unit as having. For example, large changes can occur in the way a child identifies cats and the things it is willing to call without affecting the extension of its word The difficulty is to cash in the metaphor of in this context. Having substance concepts need not depend on knowing words, but language interacts with substance concepts, completely transforming the conceptual repertoire. I will discuss how public language plays a crucial role in both the acquisition of substance concepts and their completed structure. (shrink)

What is a concept? Philosophers have given many different answers to this question, reflecting a wide variety of approaches to the study of mind and language. Nonetheless, at the most general level, there are two dominant frameworks in contemporary philosophy. One proposes that concepts are mental representations, while the other proposes that they are abstract objects. This paper looks at the differences between these two approaches, the prospects for combining them, and the issues that are involved in the dispute. We (...) argue that powerful motivations have been offered in support of both frameworks. This suggests the possibility of combining the two. Unlike Frege, we hold that the resulting position is perfectly coherent and well worth considering. Nonetheless, we argue that it should be rejected along with the view that concepts are abstract objects. (shrink)

Mathematical theorems are cultural artifacts and may be interpreted much as works of art, literature, and tool-and-craft are interpreted. The Fundamental Theorem of the Calculus, the Central Limit Theorem of Statistics, and the Statistical Continuum Limit of field theories, all show how the world may be put together through the arithmetic addition of suitably prescribed parts (velocities, variances, and renormalizations and scaled blocks, respectively). In the limit — of smoothness, statistical independence, and large N — higher-order parts, such as accelerations, (...) are, for the most, part irrelevant, affirming that, in the end, most of the world's particulars may be averaged over (a very un-Scriptural point of view). (We work out all of this in technical detail, including a nice geometric picture of stochastic integration, and a method of calculating the variance of the sum of dependent random variables using renormalization group ideas.) These fundamental theorems affirm a culture that is additive, ahistorical, Cartesian, and continuist, sharing in what might be called a species of modern culture. We understand mathematical results as useful because, like many other such artifacts, they have been adapted to fit the world, and the world has been adapted to fit their capacities. Such cultural interpretation is in effect motivation for the mathematics, and might well be offered to students as a way of helping them understand what is going on at the blackboard. Philosophy of mathematics might want to pay more attention to the history and detailed technical features of sophisticated mathematics, as a balance to the usual concerns that arise in formalist or even Platonist positions. (shrink)

We present data and argument to show that in Tetris - a real-time interactive video game - certain cognitive and perceptual problems are more quickly, easily, and reliably solved by performing actions in the world rather than by performing computational actions in the head alone. We have found that some translations and rotations are best understood as using the world to improve cognition. These actions are not used to implement a plan, or to implement a reaction; they are used to (...) change the world in order to simplify the problem-solving task. Thus, we distinguish pragmatic actions ñ actions performed to bring one physically closer to a goal - from epistemic actions - actions performed to uncover information that is hidden or hard to compute mentally. To illustrate the need for epistemic actions, we first develop a standard information-processing model of Tetris-cognition, and show that it cannot explain performance data from human players of the game - even when we relax the assumption of fully sequential processing. Standard models disregard many actions taken by players because they appear unmotivated or superfluous. However, we describe many such actions that are actually taken by players that are far from superfluous, and that play valuable roles in improving human performance. We argue that traditional accounts are limited because they regard action as having a single function: to change the world. By recognizing a second function of action - an epistemic function - we can explain many of the actions that a traditional model cannot. Although, our argument is supported by numerous examples specifically from Tetris, we outline how the one category of epistemic action can be incorporated into theories of action more generally. (shrink)

Experimental studies indicate that nonhuman animals and infants represent numerosities above three or four approximately and that their mental number line is logarithmic rather than linear. In contrast, human children from most cultures gradually acquire the capacity to denote exact cardinal values. To explain this difference, I take an extended mind perspective, arguing that the distinctly human ability to use external representations as a complement for internal cognitive operations enables us to represent natural numbers. Reviewing neuroscientific, developmental, and anthropological evidence, (...) I argue that the use of external media that represent natural numbers (like number words, body parts, tokens or numerals) influences the functional architecture of the brain, which suggests a two-way traffic between the brain and cultural public representations. (shrink)

Where does the mind stop and the rest of the world begin? The question invites two standard replies. Some accept the demarcations of skin and skull, and say that what is outside the body is outside the mind. Others are impressed by arguments suggesting that the meaning of our words "just ain't in the head", and hold that this externalism about meaning carries over into an externalism about mind. We propose to pursue a third position. We advocate a very different (...) sort of externalism: an _active externalism_, based on the active role of the environment in driving cognitive processes. (shrink)

What is the relation between the material, conventional symbol structures that we encounter in the spoken and written word, and human thought? A common assumption, that structures a wide variety of otherwise competing views, is that the way in which these material, conventional symbol-structures do their work is by being translated into some kind of content-matching inner code. One alternative to this view is the tempting but thoroughly elusive idea that we somehow think in some natural language (such as English). (...) In the present treatment I explore a third option, which I shall call the "complementarity" view of language. According to this third view the actual symbol structures of a given language add cognitive value by complementing (without being replicated by) the more basic modes of operation and representation endemic to the biological brain. The "cognitive bonus" that language brings is, on this model, not to be cashed out either via the ultimately mysterious notion of "thinking in a given natural language" or via some process of exhaustive translation into another inner code. Instead, we should try to think in terms of a kind of coordination dynamics in which the forms and structures of a language qua material symbol system play a key and irreducible role. Understanding language as a complementary cognitive resource is, I argue, an important part of the much larger project (sometimes glossed in terms of the "extended mind") of understanding human cognition as essentially and multiply hybrid: as involving a complex interplay between internal biological resources and external non-biological resources. (shrink)

At their extremes, the modernization of ancient mathematical texts (absolute presentism) leaves nothing of the source and the refusal to modernize (absolute antiquarism) changes nothing. The extremes exist only as tendencies. This paper attempts to justify the admissibility of broad modernization of mathematical sources (presentism) in the context of a socio-cultural (non-fundamentalist) philosophy of mathematics.

An alarming number of philosophers and cognitive scientists have argued that mind extends beyond the brain and body. This book evaluates these arguments and suggests that, typically, it does not. A timely and relevant study that exposes the need to develop a more sophisticated theory of cognition, while pointing to a bold new direction in exploring the nature of cognition Articulates and defends the “mark of the cognitive”, a common sense theory used to distinguish between cognitive and non-cognitive processes Challenges (...) the current popularity of extended cognition theory through critical analysis and by pointing out fallacies and shortcoming in the literature Stimulates discussions that will advance debate about the nature of cognition in the cognitive sciences. (shrink)

A critique of the hypothesis of extended cognition.

Many of our questions about religion, says renowned anthropologist Pascal Boyer, are no longer mysteries. We are beginning to know how to answer questions such as "Why do people have religion?" Using findings from anthropology, cognitive science, linguistics, and evolutionary biology, Religion Explained shows how this aspect of human consciousness is increasingly admissible to coherent, naturalistic explanation. This brilliant and controversial book gives readers the first scientific explanation for what religious feeling is really about, what it consists of, and where (...) it comes from. (shrink)

Extends the ideas of social constructivism to the philosophy of mathematics, developing a powerful critique of traditional absolutist conceptions of mathematics, and proposing a reconceptualization of the philosophy of mathematics.

'I want to begin by declaring that I regard scientific knowledge as the most important kind of knowledge we have', writes Sir Karl Popper in the opening essay of this book, which collects his meditations on the real improvements science has wrought in society, in politics and in the arts in the course of the twentieth century. His subjects range from the beginnings of scientific speculation in classical Greece to the destructive effects of twentieth century totalitarianism, from major figures of (...) the Enlightenment such as Kant and Voltaire to the role of science and self-criticism in the arts. The essays offer striking new insights into the mind of one of the greatest twentieth century philosophers. (shrink)

The operation of developing a concept is a common procedure in mathematics and in natural science, but has traditionally seemed much less possible to philosophers and, especially, logicians. Meir Buzaglo's innovative study proposes a way of expanding logic to include the stretching of concepts, while modifying the principles which block this possibility. He offers stimulating discussions of the idea of conceptual expansion as a normative process, and of the relation of conceptual expansion to truth, meaning, reference, ontology and paradox, and (...) analyzes the views of Kant, Wittgenstein, Godel, and others, paying especially close attention to Frege. His book will be of interest to a wide range of readers, from philosophers to logicians, mathematicians, linguists, and cognitive scientists. (shrink)

At a purely instrumental level, quantum theory is all about multiplication, addition and taking mod squares of complex numbers called probability amplitudes. The rules for combining amplitudes are deceptively simple. When two or more events are independent you multiply their respective probability amplitudes and when they are mutually exclusive you add them. Whenever you want to calculate probabilities you take mod squares of respective amplitudes. That’s it. If you are prepared to ignore the explanatory power of the theory (which you (...) should not) the rest is just a set of convenient mathematical tools developed for the purpose of book-keeping of amplitudes. Thus we tabulate amplitudes into state vectors and unitary matrices, and place them in Hilbert spaces. (shrink)

This paper examines some of the methods animals and humans have of adapting their environment. Because there are limits on how many different tasks a creature can be designed to do well in, creatures with the capacity to redesign their environments have an adaptive advantage over those who can only passively adapt to existing environmental structures. To clarify environmental redesign I rely on the formal notion of a task environment as a directed graph where the nodes are states and the (...) links are actions. One natural form of redesign is to change the topology of this graph structure so as to increase the likelihood of task success or to reduce its expected cost, measured in physical terms. This may be done by eliminating initial states hence eliminating choice points; by changing the action repertoire; by changing the consequence function; and lastly, by adding choice points. Another major method for adapting the environment is to change its cognitive congeniality. Such changes leave the state space formally intact but reduce the number and cost of mental operations needed for task success; they reliably increase the speed, accuracy or robustness of performance. The last section of the paper describes several of these epistemic or complementary actions found in human performance. (shrink)

To analyze the task of mental arithmetic with external representations in different number systems we model algorithms for addition and multiplication with Arabic and Roman numerals. This demonstrates that Roman numerals are not only informationally equivalent to Arabic ones but also computationally similar—a claim that is widely disputed. An analysis of our models' elementary processing steps reveals intricate tradeoffs between problem representation, algorithm, and interactive resources. Our simulations allow for a more nuanced view of the received wisdom on Roman numerals. (...) While symbolic computation with Roman numerals requires fewer internal resources than with Arabic ones, the large number of needed symbols inflates the number of external processing steps. (shrink)