The popular idea that mental calculation involves covert operations as counterparts to the scribblings, sayings or manipulations involved in classroom calculation is rejected by familiar arguments in Section I. Philosophers do not readily agree on an alternative account. Section II considers reasons why they are puzzled, reasons which encourage a return to the discredited position. The currently fashionable Causal or Functionalist view is criticised in Section III. Section IV reconsiders the stubborn fact that when someone calculates in their head (...) they do something at the time which explains their arithmetical successes or failures. Some dispositional descriptions are rejected as inadequate. Though a restrained dispositionalism is finally supported, it does not have the implication that nothing was going on in the silence. (shrink)

Mental arithmetic is characterised by a tendency to overestimate addition and to underestimate subtraction results: the operational momentum effect. Here, motivated by contentious explanations of this effect, we developed and tested an arithmetic heuristics and biases model that predicts reverse OM due to cognitive anchoring effects. Participants produced bi-directional lines with lengths corresponding to the results of arithmetic problems. In two experiments, we found regular OM with zero problems but reverse OM with non-zero problems. In a (...) third experiment, we tested the prediction of our model. Our results suggest the presence of at least three competing biases in mental arithmetic: a more-or-less heuristic, a sign-space association and an anchoring bias. We conclude that mental arithmetic exhibits shortcuts for decision-making similar to traditional domains of reasoning and problem-solving. (shrink)

The Heisenberg-Gabor uncertainty principle defines the limits of information resolution in both time and frequency domains. The limit of resolution discloses unique properties of a time series by frequency decomposition. However, classical methods such as Fourier analysis are limited by spectral leakage, particularly in longitudinal data with shifting periodicity or unequal intervals. Wavelet transformation provides a workable compromise by decomposing the signal in both time and frequency through translation and scaling of a basis function followed by correlation or convolution with (...) the original signal. This study aimed to compare the accuracy of predictive models in mental arithmetic in time and frequency domains. Analysis of the author's response time at mental arithmetic using a soroban was modeled for two periods, an initial period, and a return period both separated by an interval of 370 days. The median response times in seconds was longer for all tasks during the TI compared to the TR period, for addition [CTAdd 62 vs 50 s] and summation [CTSum 68 vs 57 s]. Response times were longer for errors regardless of the study period or task. There was an increasing phase difference for the addition and summation tasks during the TI period toward the end of the series 49.65o compared to the TR period where the phase difference between the two tasks was only 2.05o, indicating that both tasks are likely demonstrating similar learning rates during the latter study period. A comparison between time and time/frequency domain forecasts for an additional 100 tasks demonstrated higher accuracy of the maximum overlap discrete wavelet transform model, where the mean absolute percentage error ranged between 5.48 and 8.19% and that for the time domain models [autoregressive integrated moving average, generalized autoregressive conditional heteroscedasticity ] was 6.16–10.80%. (shrink)

Interactivity has been linked to better performance in problem solving, due in part to a more efficient allocation of attentional resources, a better distribution of cognitive load, but perhaps more important by enabling the reasoner to shape and reshape the physical problem presentation to promote the development of the problem solution. Interactivity in solving quotidian arithmetic problems involves gestures, pointing, and the recruitment of artefacts to facilitate computation and augment efficiency. In the experiment reported here, different types of interactivity (...) were examined with a series of mental arithmetic problems. Using a repeated-measures design, participants solved series of five 11-digit sums in four conditions that varied in the type of interactivity: no interactivity, pointing, pen and paper, and tokens. Performance in the four conditions was measured in terms of accuracy, calculation error, and efficiency. These quantitative analyses were supplemented by a detailed qualitative examination of a participant’s actions in the different conditions. The integration of artefacts, such as tokens or a pen, offered reasoners the opportunity to reconfigure the physical presentation of the problem, enacting different arithmetic strategies: the affordance landscape shifts as the problem trajectory is enacted through interactivity, and this generally produced better “mental” arithmetic performance. Participants also felt more positive about and better engaged with the task when they could reconfigure the problem presentation through interactivity. These findings underscore the importance of engineering task environments in the laboratory that offer a window on how problem solving unfolds through a coalition of mental and physical resources. (shrink)

Mangarevan traditionally contained two numeration systems: a general one, which was highly regular, decimal, and extraordinarily extensive; and a specific one, which was restricted to specific objects, based on diverging counting units, and interspersed with binary steps. While most of these characteristics are shared by numeration systems in related languages in Oceania, the binary steps are unique. To account for these characteristics, this article draws on—and tries to integrate—insights from anthropology, archeology, linguistics, psychology, and cognitive science more generally. The analysis (...) of mental arithmetic with these systems reveals that both types of systems entailed cognitive advantages and served important functions in the cultural context of their application. How these findings speak to more general questions revolving around the theoretical models and evolutionary trajectory of numerical cognition will be discussed in the. (shrink)

Symbolic arithmetic is fundamental to science, technology and economics, but its acquisition by children typically requires years of effort, instruction and drill1,2. When adults perform mental arithmetic, they activate nonsymbolic, approximate number representations3,4, and their performance suffers if this nonsymbolic system is impaired5. Nonsymbolic number representations also allow adults, children, and even infants to add or subtract pairs of dot arrays and to compare the resulting sum or difference to a third array, provided that only approximate accuracy (...) is required6–10. Here we report that young children, who have mastered verbal counting and are on the threshold of arithmetic instruction, can build on their nonsymbolic number system to perform symbolic addition and subtraction11–15. Children across a broad socio-economic spectrum solved symbolic problems involving approximate addition or subtraction of large numbers, both in a laboratory test and in a school setting. Aspects of symbolic arithmetic therefore lie within the reach of children who have learned no algorithms for manipulating numerical symbols. Our findings help to delimit the sources of children’s difficulties learning symbolic arithmetic, and they suggest ways to enhance children’s engagement with formal mathematics. We presented children with approximate symbolic arithmetic problems in a format that parallels previous tests of non-symbolic arithmetic in preschool children8,9. In the first experiment, five- to six-year-old children were given problems such as ‘‘If you had twenty-four stickers and I gave you twenty-seven more, would you have more or less than thirty-five stickers?’’. Children performed well above chance (65.0%, t1952.77, P 5 0.012) without resorting to guessing or comparison strategies that could serve as alternatives to arithmetic. Children who have been taught no symbolic arithmetic therefore have some ability to perform symbolic addition problems. The children’s performance nevertheless fell short of performance on non-symbolic arithmetic tasks using equivalent addition problems with numbers presented as arrays of dots and with the addition operation conveyed by successive motions of the dots into a box (71.3% correct, F1,345 4.26, P 5 0.047)8.. (shrink)

People frequently gesture when problem-solving, particularly on tasks that require spatial transformation. Gesture often facilitates task performance by interacting with internal mental representations, but how this process works is not well understood. We investigated this question by exploring the case of mental abacus, a technique in which users not only imagine moving beads on an abacus to compute sums, but also produce movements in gestures that accompany the calculations. Because the content of MA is transparent and readily manipulated, (...) the task offers a unique window onto how gestures interface with mental representations. We find that the size and number of MA gestures reflect the length and difficulty of math problems. Also, by selectively interfering with aspects of gesture, we find that participants perform significantly worse on MA under motor interference, but that perceptual feedback is not critical for success on the task. We conclude that premotor processes involved in the planning of gestures are critical to mental representation in MA. (shrink)

Published in honor of Sergio Galvan, this collection concentrates on the application of logical and mathematical methods for the study of central issues in formal philosophy. The volume is subdivided into four sections, dedicated to logic and philosophy of logic, philosophy of mathematics, philosophy of science, metaphysics and philosophy of religion. The contributions adress, from a logical point of view, some of the main topics in these areas. The first two sections include formal treatments of: truth and paradoxes; definitions by (...) abstraction; the status of abstract objects, such as mathematical objects and universal concepts; and the structure of explicit knowledge. The last two sections include papers on classical problems in philosophy of science, such as the status of subjective probability, the notion of verisimilitude, the notion of approximation, and the theory of mind and mental causation, and specific issues in metaphysics and philosophy of religion, such as the ontology of species, actions, and intelligible worlds, and the logic of religious belonging. (shrink)

In this paper, representational structures of arithmetical thinking, encoded in human minds, are described. On the basis of empirical research, it is possible to distinguish four types of mental number lines: the shortest mental number line, summation mental number lines, point-place mental number lines and mental lines of exact numbers. These structures may be treated as generative mechanisms of forming arithmetical representations underlying our numerical acts of reference towards cardinalities, ordinals and magnitudes. In the paper, (...) the theoretical framework for a formal model of mental arithmetical representations is constructed. Many competitive conceptions of the mental system responsible for our arithmetical thinking may be unified within the presented framework. The paradigm underlying our research may be interpreted philosophically as a neo-Kantian approach to modeling the mind’s representational structures. (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)

Published in honor of Sergio Galvan, this collection concentrates on the application of logical and mathematical methods for the study of central issues in formal philosophy. The volume is subdivided into four sections, dedicated to logic and philosophy of logic, philosophy of mathematics, philosophy of science, metaphysics and philosophy of religion. The contributions adress, from a logical point of view, some of the main topics in these areas. The first two sections include formal treatments of: truth and paradoxes; definitions by (...) abstraction; the status of abstract objects, such as mathematical objects and universal concepts; and the structure of explicit knowledge. The last two sections include papers on classical problems in philosophy of science, such as the status of subjective probability, the notion of verisimilitude, the notion of approximation, and the theory of mind and mental causation, and specific issues in metaphysics and philosophy of religion, such as the ontology of species, actions, and intelligible worlds, and the logic of religious belonging. (shrink)

Here we present a systematic plan to the experimental study of test–retest reliability in the multitasking domain, adopting the multitrait-multimethod approach to evaluate the psychometric properties of performance in Düker-type speeded multiple-act mental arithmetic. These form of tasks capacitate the experimental analysis of integrated multi-step processing by combining multiple mental operations in flexible ways in the service of the overarching goal of completing the task. A particular focus was on scoring methodology, particularly measures of response speed variability. (...) To this end, we present data of two experiments with regard to test–retest reliability, between-measures correlational structure, and stability. Finally, we compared participants with high versus low performance variability to assess ability-related differences in measurement precision, which is especially relevant in the applied fields of clinical neuropsychology. The participants performed two classic integrated multi-act arithmetic tasks, combining addition and verification and addition and comparison. The results revealed excellent test–retest reliability for the standard and the variability measures. The analysis of between-measures correlational structure revealed the typical pattern of convergent and discriminant relationships, and also, that absolute response speed variability was highly correlated with average speed, indicating that these measures mainly deliver redundant information. In contrast, speed-adjusted variability revealed discriminant validity being correlated to a much lesser degree with average speed, indicating that this measure delivers additional information not already provided by the speed measure. Furthermore, speed-adjusted variability was virtually unaffected by test–retest practice, which makes this measure interesting in situations with repeated testing. (shrink)

The concept of affordances is rapidly gaining traction in the philosophy of mind and cognitive sciences. Affordances are opportunities for action provided by the environment. An important open question is whether affordances can be used to explain mental action such as attention, counting, and imagination. In this paper, we critically discuss McClelland’s (‘The Mental Affordance Hypothesis’, 2020, Mind, 129(514), pp. 401–427) mental affordance hypothesis. While we agree that the affordance concept can be fruitfully employed to explain (...) class='Hi'>mental action, we argue that McClelland’s mental affordance hypothesis contain remnants of a Cartesian understanding of the mind. By discussing the theoretical framework of the affordance competition hypothesis, we sketch an alternative research program based on the principles of embodied cognition that evades the Cartesian worries. We show how paradigmatic mental acts, such as imagination, counting, and arithmetic, are dependent on sensorimotor interaction with an affording environment. Rather than make a clear distinction between bodily and mental action, the mental affordances highlight the embodied nature of our mental action. We think that in developing our alternative research program on mental affordances, we can maintain many of the excellent insights of McClelland’s account without reintroducing the very distinctions that affordances were supposed to overcome. (shrink)

The goal of the research programme I describe in this article is a realist epistemology for arithmetic which respects arithmetic's special epistemic status (the status usually described as a prioricity) yet accommodates naturalistic concerns by remaining fundamentally empiricist. I argue that the central claims which would allow us to develop such an epistemology are (i) that arithmetical truths are known through an examination of our arithmetical concepts; (ii) that (at least our basic) arithmetical concepts are accurate mental (...) representations of elements of the arithmetical structure of the independent world; (iii) that (ii) obtains in virtue of the normal functioning of our sensory apparatus. The first of these claims protects arithmetic's special epistemic status relative, for example, to the laws of physics, the second preserves the independence of arithmetical truth, and the third ensures that we remain empiricists. Preliminaries Justifying and grounding concepts Cameras and filters An epistemology for arithmetic Concluding remarks. (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)

The paper presents a formal model of the system of number representations as a multiplicity of mental number axes with a hierarchical structure. The hierarchy is determined by the mind as it acquires successive types of mental number axes generated by virtue of some algebraic mechanisms. Three types of algebraic structures, responsible for functioning these mechanisms, are distinguished: BASAN-structures, CASAN-structures and CAPPAN-structures. A foundational order holds between these structures. CAPPAN-structures are derivative from CASAN-structures which are extensions of BASAN-structures. (...) The constructed formal model unifies two competitive conceptions of cognitive arithmetic: namely, the conception of the mental number line and the conception of parallel individuation. The paper is the continuation of a paper entitled Representational structures of arithmetical thinking, in which rich empirical evidence supporting the model is presented. The main result achieved in the present paper may be philosophically interpreted as an attempt to formalize the Kantian conception of the pure idea of time, understood as the a priori form of human arithmetical thinking. In this way, our theory may be comprehended as a result of applying the hard method of logical reconstruction of fundamental epistemological categories. (shrink)

Hutto and Myin have proposed an account of radically enactive (or embodied) cognition (REC) as an explanation of cognitive phenomena, one that does not include mental representations or mental content in basic minds. Recently, Zahidi and Myin have presented an account of arithmetical cognition that is consistent with the REC view. In this paper, I first evaluate the feasibility of that account by focusing on the evolutionarily developed proto-arithmetical abilities and whether empirical data on them support the radical (...) enactivist view. I argue that although more research is needed, it is at least possible to develop the REC position consistently with the state-of-the-art empirical research on the development of arithmetical cognition. After this, I move the focus to the question whether the radical enactivist account can explain the objectivity of arithmetical knowledge. Against the realist view suggested by Hutto, I argue that objectivity is best explained through analyzing the way universal proto-arithmetical abilities determine the development of arithmetical cognition. (shrink)

There is at present a lively debate in cognitive psychology concerning the origin of natural number concepts. At the center of this debate is the system of mental magnitudes, an innately given cognitive mechanism that represents cardinality and that performs a variety of arithmetical operations. Most participants in the debate argue that this system cannot be the sole source of natural number concepts, because they take it to represent cardinality approximately while natural number concepts are precise. In this paper, (...) I argue that the claim that mental magnitudes represent cardinality approximately overlooks the distinction between a magnitude and the increments that compose to form that magnitude. While magnitudes do indeed represent cardinality approximately, they are composed of a precise number of increments. I argue further that learning the number words and the counting routine may allow one to mark in memory the number of increments that composed to form a magnitude, thereby creating a precise representation of cardinality. (shrink)

We investigated the immediate and maintenance effects of mental-imagery-based mnemonic training on improving youths’ working memory, long-term memory, arithmetic and spatial abilities, and fluid intelligence. In Experiment 1, 26 Chinese participants aged 10–16 years were divided into an experimental group that received 8 days of mental-imagery-based mnemonic training and a no-contact control group. Participants completed pre-, post-, and three follow-up tests. In Experiment 2, 54 Chinese children, all 12 years old, were divided into experimental and control groups. (...) Participants completed pre-, post-, and follow-up tests. Results showed that the training significantly affected long-term memory-related task performance but no effects were observed on working memory, arithmetic or spatial ability, or fluid intelligence-related tasks. Moreover, the effect of the training on long-term memory lasted up to one year; the more frequently the training was used, the more effective it was. A content analysis of the feedback submitted by parents of participants in Experiment 2 three months after the training showed that the children used the strategy more for memorizing content such as Chinese and English, as well as for musical scores. Furthermore, there was also the possibility that the training improved abilities and academic performance such as concentration and math performance. Our results provide a basis for the further exploration of mental-imagery-based mnemonic training as a novel training modality. (shrink)

Most of us are familiar with the phenomenology of mental effort accompanying cognitively demanding tasks, like focusing on the next chess move or performing lengthy mental arithmetic. In this paper, I argue that phenomenology of mental effort poses a novel counterexample to tracking intentionalism, the view that phenomenal consciousness is a matter of tracking features of one’s environment in a certain way. I argue that an increase in the phenomenology of mental effort does not accompany (...) a change in any of the following candidate representational contents: representation of externally presented features, e.g. brightness, contrast, and so on representation of task difficulty, representation of the possibility of error, representation of trying to achieve some state of affairs, representation of bodily changes like muscle tension, or representation of change in cognitive resource availability and lost opportunity cost. While tracking intentionalism about some phenomenal experiences like pains might obtain, it does not seem to obtain for all phenomenal experiences. This puts the intentionalist into an uncomfortable position of trying to explain why some phenomenal experiences have representational content and not others. Since many believe that tracking intentionalism or something like it provides the best chance of naturalizing consciousness, these arguments deserve detailed consideration. (shrink)

The mental representation of brief temporal durations, when assessed in standard laboratory conditions, is highly accurate. Here we show that adding or subtracting temporal durations systematically results in strong and opposite biases, namely over-estimation for addition and under-estimation for subtraction. The difference with respect to a baseline temporal reproduction task changed across durations in an operation-specific way and survived correcting for the effect due to operation sign alone, indexing a reliable signature of arithmetic processing on time representation. A (...) second experiment replicated these findings with a different set of stimuli. This novel behavioral marker conceptually mirrors in the time domain the representational momentum found with motion, whereby the estimated spatial position of a visual target is displaced in the direction of motion itself. This momentum effect in temporal arithmetic suggests a striking analogy between time processing and visuospatial processing, which might index the presence of common computational principles. (shrink)

The feeling of error (FOE) is the subjective experience that something went wrong during a reasoning or calculation task. The main goal of the present study was to assess the accuracy of the FOE in the context of mental mathematical calculation. We used the number bisection task (NBT) to evoke this metacognitive feeling and assessed it by asking participants if they felt they have committed an error after solving the task. In the NBT participants have to determine whether the (...) number presented in the middle of a triplet corresponds to the arithmetic mean of the two outer numbers (e.g., 07_16_25) with a Yes/No answer. Our results show that FOE reports were strongly correlated with arithmetic errors and numerical properties of the NBT, suggesting that the FOE accurately represents the error. This finding indicates that even very fast metacognitive feelings are reliable when it comes to evaluating one’s own mental performance. Moreover, our results suggest that the occurrence of FOEs is determined by the fluency with which each triplet was solved and the post-decision evaluation processes that occurred after the NBT was solved. Additionally, we asked participants to report their confidence in the given answer for the cases where they did not report FOEs. Participants reported less confidence for the (objectively) incorrect answers than for the (objectively) correct ones, suggesting that in cases where they did not have a conscious FOE they still were able to implicitly detect their errors. Remarkably, confidence was also determined by the fluency of the NBT. (shrink)

In the book, I argue that the mind can be explained computationally because it is itself computational—whether it engages in mental arithmetic, parses natural language, or processes the auditory signals that allow us to experience music. All these capacities arise from complex information-processing operations of the mind. By analyzing the state of the art in cognitive science, I develop an account of computational explanation used to explain the capacities in question.

The domain of numbers provides a paradigmatic case for investigating interactions of culture, language, and cognition: Numerical competencies are considered a core domain of knowledge, and yet the development of specifically human abilities presupposes cultural and linguistic input by way of counting sequences. These sequences constitute systems with distinct structural properties, the cross-linguistic variability of which has implications for number representation and processing. Such representational effects are scrutinized for two types of verbal numeration systems—general and object-specific ones—that were in parallel (...) use in several Oceanic languages. The analysis indicates that the object-specific systems outperform the general systems with respect to counting and mental arithmetic, largely due to their regular and more compact representation. What these findings reveal on cognitive diversity, how the conjectures involved speak to more general issues in cognitive science, and how the approach taken here might help to bridge the gap between anthropology and other cognitive sciences is discussed in the conclusion. (shrink)