The paper undertakes three interdisciplinary tasks. The first one consists in constructing a formal model of the basic arithmetic competence, that is, the competence sufficient for solving simple arithmetic story-tasks which do not require any mathematical mastery knowledge about laws, definitions and theorems. The second task is to present a generalized arithmetic theory, called the arithmetic of indexed numbers (INA). All models of the development of counting abilities presuppose the common assumption that our simple, folk arithmetic encoded linguistically in the (...) mind is based on the linear number representation. This classical conception is rejected and a competitive hypothesis is formulated according to which the basic mature representational system of cognitive arithmetic is a structure composed of many numerical axes which possess a common constituent, namely, the numeral zero. Arithmetic of indexed numbers is just a formal tool for modelling the basic mature arithmetic competence. The third task is to develop a standpoint called temporal pluralism, which is motivated by neo-Kantian philosophy of arithmetic. (shrink)

The main aim of this study was to analyze the patterns of changes in Approximate Number Sense precision from grade 1 to grade 9 in a sample of Russian schoolchildren. To fulfill this aim, the data from a longitudinal study of two cohorts of children were used. The first cohort was assessed at grades 1–5, and the second cohort was assessed at grades 5–9. ANS precision was assessed by accuracy and reaction time in a non-symbolic comparison test. The patterns of (...) change were estimated via mixed-effect growth models. The results revealed that in the first cohort, the average accuracy increased from grade 1 to grade 5 following a non-linear pattern and that the rate of growth slowed after grade 3. The non-linear pattern of changes in the second cohort indicated that accuracy started to increase from grade 7 to grade 9, while there were no changes from grade 5 to grade 7. However, the RT in the non-symbolic comparison test decreased evenly from grade 1 to grade 7, and the rate of processing non-symbolic information tended to stabilize from grade 7 to grade 9. Moreover, the changes in the rate of processing non-symbolic information were not explained by the changes in general processing speed. The results also demonstrated that accuracy and RT were positively correlated across all grades. These results indicate that accuracy and the rate of non-symbolic processing reflect two different processes, namely, the maturation and development of a non-symbolic representation system. (shrink)

ABSTRACT I argue against both neuropsychological and cognitive accounts of our grasp of numbers. I show that despite the points of divergence between these two accounts, they face analogous problems. Both presuppose too much about what they purport to explain to be informative, and also characterize our grasp of numbers in a way that is absurd in the light of what we already know from the point of view of mathematical practice. Then I offer a positive methodological proposal about the (...) role that cognitive science should play in the philosophy of mathematics. (shrink)

Savant syndrome and prime numbers Oliver Sacks reported that a pair of autistic twins had extraordinary number abilities and that they spontaneously generated huge prime numbers. Such abilities could contradict our understanding of human abilities. Sacks' report attracted widespread attention, and several researchers speculated theoretically. Unfortunately, most of the explanations in the literature are wrong. Here a correct explanation on prime number identification is provided. Fermat's little theorem is implemented in spreadsheet. Also, twenty years after the report, questionable aspects were (...) found in it. Extreme abilities became dubious. One possibility for the less extreme abilities is incomplete trial division. (shrink)

How can we explain the intentional nature of an expert’s actions, performed without immediate and conscious control, relying instead on automatic cognitive processes? How can we account for the differences and similarities with a novice’s performance of the same actions? Can a naturalist explanation of intentional expert action be in line with a philosophical concept of intentional action? Answering these and related questions in a positive sense, this dissertation develops a three-step argument. Part I considers different methods of explanations in (...) cognitive neuroscience (Bennett & Hacker’s philosophical, conceptual analysis; Marr’s three levels of explanation; Neural Correlates of Consciousness research; mechanistic explanation), defending ‘mechanistic explanation’ as a method that provides the necessary tools for integrating interdisciplinary insights into human action. Furthermore, a dynamic, explanatory mechanism allows the assessment of the impact of learning and development on expert action in a valuable way that other methods don’t. Part II continues by scrutinizing several cognitive neuroscientific theories of learning and development (neuroconstructivism; dual-processing theories; simulation theory; extended mind/cognition hypothesis), arguing for the complex interactions between different types of processing and different action representations involved in expert action performances. Moreover, according to our discussion of a particular ‘simulation theory’ these interactions can be influenced in several ways with the use of language, allowing an agent to configure a specific action representation for performance at a later stage. The results of Parts I and II are then applied in Part III to a parallel discussion of philosophical analyses of intentional action (discussing i.a. Frankfurt, Bratman, Pacherie and Ricoeur) and cognitive neuroscientific insights in it. Both approaches are found to converge in emphasizing the importance for an expert to develop stable patterns of actions that comply maximally not only with his intentions, but also with his motor expertise and with situational conditions. Consequently, his actions – automatic, or not – rely on this ‘sculpted space of actions’. (shrink)

The changing conditions for the accumulation and transmission of knowledge in the age of multimedia networks make it inevitable that old philosophical problems become formulated in a new light. Above all, the problem of the unity of knowledge is once again a topical issue. The situation-dependent acquisition of knowledge that is made possible by mobile learning transcends the boundaries of traditional disciplines, linking the domains of text, diagram, and picture. Database integration and multimedia search become central problems in the epistemology (...) of the 21st century, while handheld devices are emerging as vital technologies for supporting collaborative learning. -- Kristóf Nyíri has published widely on Wittgenstein, Austrian intellectual history, and the philosophy of communication. He directs the interdisciplinary research program COMMUNICATIONS IN THE 21ST CENTURY, conducted jointly by the Institute for Philosophical Research of the Hungarian Academy of Sciences and Westel Mobile Telecommunications, Budapest. (shrink)

The purpose of this paper is to look at the problem of rule-following—notably discussed by Kripke (Wittgenstein on rules and private language, 1982) and Wittgenstein (Philosophical investigations, 1953)—from the perspective of the study of generics. Generics are sentences that express generalizations that tolerate exceptions. I first suggest that meaning ascriptions be viewed as habitual sentences, which are a sub-set of generics. I then seek a proper semantic analysis for habitually construed meaning sentences. The quantificational approach is rejected, due to its (...) persistent difficulties. Instead, a cognitive approach is adopted, where psychological considerations of meaning attributors play a crucial role. This account is then compared with the picture of meaning offered by Kripke and Wittgenstein, respectively. I show how this fresh way of conceiving of meaning sentences respects some of their insights while avoiding some of the drawbacks, and serves to improve the framework in which the current debate and inquiry about rule-following are conducted. (shrink)

Neurosciences and cognitive sciences provide us with myriad empirical findings that shed light on hypothesised primitive numerical processes in the brain and in the mind. Yet, the hypotheses on which the experiments are based, and hence the results, depend strongly on sophisticated abstract models used to describe and explain neural data or cognitive representations that supposedly are the empirical roots of primary arithmetical activity. I will question the foundational role of such models. I will even cast doubt upon the search (...) for a general and unified philosophical foundation of an empirical science. First, it seems to me hard to draw a global and coherent view from the innumerable and piecemeal neuropsychological experiments and their variable, and sometimes uneasily compatible or fully divergent interpretations. Secondly, I think that the aim of empirical research is to describe dynamical processes, establishing correlations between different sets of data, without meaning to fix an origin or to point to a cause, let alone to a ground. From the very scientific and philosophical point of view it is essential to distinguish between explanations, which provide correlations or, at best, causal mechanisms, and grounding, which involves a claim to some form of determinism. (shrink)