Human languages vary in terms of which meanings they lexicalize, but this variation is constrained. It has been argued that languages are under two competing pressures: the pressure to be simple (e.g., to have a small lexicon) and to allow for informative (i.e., precise) communication, and that which meanings get lexicalized may be explained by languages finding a good way to trade off between these two pressures. However, in certain semantic domains, languages can reach very high levels of informativeness even (...) if they lexicalize very few meanings in that domain. This is due to productive morphosyntax and compositional semantics, which may allow for construction of meanings which are not lexicalized. Consider the semantic domain of natural numbers: many languages lexicalize few natural number meanings as monomorphemic expressions, but can precisely convey very many natural number meanings using morphosyntactically complex numerals. In such semantic domains, lexicon size is not in direct competition with informativeness. What explains which meanings are lexicalized in such semantic domains? We will propose that in such cases, languages need to solve a different kind of trade‐off problem: the trade‐off between the pressure to lexicalize as few meanings as possible (i.e, to minimize lexicon size) and the pressure to produce as morphosyntactically simple utterances as possible (i.e, to minimize average morphosyntactic complexity of utterances). To support this claim, we will present a case study of 128 natural languages' numeral systems, and show computationally that they achieve a near‐optimal trade‐off between lexicon size and average morphosyntactic complexity of numerals. This study in conjunction with previous work on communicative efficiency suggests that languages' lexicons are shaped by a trade‐off between not two but three pressures: be simple, be informative, and minimize average morphosyntactic complexity of utterances. (shrink)

We present an elementary three-pass algorithm for computing addition in Ostrowski numeration systems. When a is quadratic, addition in the Ostrowski numeration system based on a is recognizable by a finite automaton. We deduce that a subset of X⊆Nn is definable in, where Va is the function that maps a natural number x to the smallest denominator of a convergent of a that appears in the Ostrowski representation based on a of x with a nonzero coefficient if and only (...) if the set of Ostrowski representations of elements of X is recognizable by a finite automaton. The decidability of the theory of follows. (shrink)

Recentemente diversi studi hanno mostrato come la distanza tra i Grundlagen e le precedenti pubblicazioni di Hilbert non sia tanto abissale come ritenuto in passato, ma vi sia una significativa consequenzialità con la teoria dei campi numerici. Nel ribadire questa visione, si intende mostrare come i risultati ottenuti da Hilbert, in particolare sui teoremi di Pappo e di Desargues, siano conseguenza di una ricerca più ampia sulla possibilità di introdurre all’interno della geometria dei sistemi numerici atti a coordinatizzare il piano (...) o lo spazio. Nello sviluppo della ricerca hanno avuto un ruolo determinante la scoperta di Hurwitz delle uniche quattro algebre di divisione normate e le idee maturate, già agli inizi della sua carriera, sulla finitezza degli strumenti da utilizzare in geometria, in particolare la rinuncia al concetto di continuità e la limitazione a sistemi numerici ordinati e numerabili. Nelle conclusioni dei Grundlagen, Hilbert rimarca l’importanza di dimostrare l’impossibilità delle assunzioni, tra cui l’impossibilità di determinare, attraverso solo relazioni geometriche di incidenza, dei sistemi numerici ordinati che abbiano le stesse proprietà dei quaternioni e degli ottonioni rispetto alle operazioni dell’aritmetica. (shrink)

A numeral system is an infinite sequence of different closed normal -terms intended to code the integers in -calculus. Barendregt has shown that if we can represent, for a numeral system, the functions Successor, Predecessor, and Zero Test, then all total recursive functions can be represented. In this paper we prove the independancy of these three particular functions. We give at the end a conjecture on the number of unary functions necessary to represent all total recursive functions.

In this paper we study numeral systems in the -calculus. With one exception, we assume that all numerals have normal form. We study the independence of the conditions of adequacy of numeral systems. We find that, to a great extent, they are mutually independent. We then consider particular examples of numeral systems, some of which display paradoxical properties. One of these systems furnishes a counterexample to a conjecture of Böhm. Next, we turn to (...) the approach of Curry, Hindley, and Seldin. We dwell with the general problem of obtaining their results with the additional requirement of nonconvertibility of numerals. In particular we solve a problem that they left open. Finally, we give the first example of an adequate unsolvable numeral system without a test for zero in the usual sense, thus solving a problem of Barendregt and Barendsen. (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)

We study extensions of Presburger arithmetic with a unary predicate R and we show that under certain conditions on R, R is sparse and the theory of $\langle\mathbb{N}, +, R\rangle$ is decidable. We axiomatize this theory and we show that in a reasonable language, it admits quantifier elimination. We obtain similar results for the structure $\langle\mathbb{Q},+, R\rangle$.

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)

The ‘bat and ball’ is one of the problems most frequently employed as a testbed for research on the dual-system hypothesis of reasoning. Frederick (J Econ Perspect 19:25–42, 2005) is the first to envisage the possibility that different numerical arrangements of the ‘bat and ball’ problem could lead to different dynamics of activation of the dual-system, and so to different performances of subjects in task accomplishment. This possibility has triggered a strand of research oriented to accomplish ‘sensitivity analyses’ of the (...) ‘bat and ball’ problem. The scope of this paper is to test experimentally the specific hypothesis that numerals are responsible for the selective activation of the two systems of reasoning in this task. In particular, we argue that their role goes beyond and cannot be reduced to that of numbers conceived as magnitudes. To test our hypothesis, we devise an experimental setting in which the role of numbers (as magnitudes) is rendered irrelevant. We find experimental results consistent with our hypothesis. We further provide a link between the literature on mathematical problem-solving and that on mathematical cognition research, in particular that branch labeled embodied mathematical cognition. (shrink)

Many researchers, including Clarke and Beck, describe the human numerical system as unitary. We offer an alternative view – the coexistence of several systems; namely, multiple systems existing in parallel, ready to be activated depending on the task/need. Based on this alternative view, we present an account for the representation of rational numbers.

A numerical classification was performed on 69 elements with 54 chemicaland physicochemical properties. The elements fell into clusters in closeaccord with the electron shell s-, p- andd-blocks. The f-block elements were not included forlack of sufficiently complete data. The successive periods ofs- and p-block elements appeared in an ovalconfiguration, with d-block elements lying to one side. Morethan three axes were required to give good representation of thevariation, although the interpretation of the higher axes is difficult.Only 15 properties were scorable for (...) the noble gases, but despite thepaucity of properties reflecting chemical reactivity, analysis of the 69elements on these properties still showed the major features seen fromthe full set. (shrink)

In philosophical studies regarding mathematical models of dynamical systems, instability due to sensitive dependence on initial conditions, on the one side, and instability due to sensitive dependence on model structure, on the other, have by now been extensively discussed. Yet there is a third kind of instability, which by contrast has thus far been rather overlooked, that is also a challenge for model predictions about dynamical systems. This is the numerical instability due to the employment of numerical methods (...) involving a discretization process, where discretization is required to solve the differential equations of dynamical systems on a computer. We argue that the criteria for numerical stability, as usually provided by numerical analysis textbooks, are insufficient, and, after mentioning the promising development of backward analysis, we discuss to what extent, in practice, numerical instability can be controlled or avoided. (shrink)

Knowledge reduction of information systems is one of the most important parts of rough set theory in real-world applications. Based on the connections between the rough set theory and the theory of topology, a kind of topological reduction of incomplete information systems is discussed. In this study, the topological reduction of incomplete information systems is characterized by belief and plausibility functions from evidence theory. First, we present that a topological space induced by a pair of approximation operators (...) in an incomplete information system is pseudo-discrete, which deduces a partition. Then, the topological reduction is characterized by the belief and plausibility function values of the sets in the partition. A topological reduction algorithm for computing the topological reducts in incomplete information systems is also proposed based on evidence theory, and its efficiency is examined by an example. Moreover, relationships among the concepts of topological reduct, classical reduct, belief reduct, and plausibility reduct of an incomplete information system are presented. (shrink)

Number morphology (e.g., singular vs. plural) is a part of the grammar that captures numerical information. Some languages have morphological Number values, which express few (paucal), two (dual), three (trial) and sometimes (possibly) four (quadral). Interestingly, the limit of the attested morphological Number values matches the limit of non‐verbal numerical cognition. The latter is based on two systems, one estimating approximate numerosities and the other computing exact numerosities up to three or four. We compared the literature on non‐verbal number (...) systems with data on Number morphology from 218 languages. Our observations suggest that non‐verbal numerical cognition is reflected as a core part of language. (shrink)

Menary OpenMIND, MIND Group, Frankfurt am Main, 2015) has argued that the development of our capacities for mathematical cognition can be explained in terms of enculturation. Our ancient systems for perceptually estimating numerical quantities are augmented and transformed by interacting with a culturally-enriched environment that provides scaffolds for the acquisition of cognitive practices, leading to the development of a discrete number system for representing number precisely. Numerals and the practices associated with numeral systems play a significant role (...) in this process. However, the details of the relationship between the ancient number system and the discrete number system remain unclear. This lack of clarity is exacerbated by the problem of symbolic estrangement and the fact that unique features of how numeral systems represent require our ancient number system to play a dual role. These issues highlight that Dehaene’s From monkey brain to human brain, MIT Press, Cambridge, pp 133–157, 2005) neuronal recycling hypothesis may be insufficient to explain the neural mechanisms underlying the process of enculturation. In order to explain mathematical enculturation, and enculturation more generally, it may be necessary to adopt Anderson’s :245–266, 2010; After phrenology: neural reuse and the interactive brain, MIT Press, Cambridge, 2014) theory of neural reuse. (shrink)

The idea that there is a “Number Sense” (Dehaene, 1997) or “Core Knowledge” of number ensconced in a modular processing system (Carey, 2009) has gained popularity as the study of numerical cognition has matured. However, these claims are generally made with little, if any, detailed examination of which modular properties are instantiated in numerical processing. In this article, I aim to rectify this situation by detailing the modular properties on display in numerical cognitive processing. In the process, I review literature (...) from across the cognitive sciences and describe how the evidence reported in these works supports the hypothesis that numerical cognitive processing is modular. I outline the properties that would suffice for deeming a certain processing system a modular processing system. Subsequently, I use behavioral, neuropsychological, philosophical, and anthropological evidence to show that the number module is domain specific, informationally encapsulated, neurally localizable, subject to specific pathological breakdowns, mandatory, fast, and inaccessible at the person level; in other words, I use the evidence to demonstrate that some of our numerical capacity is housed in modular casing. (shrink)

Mathematical cognition has become an interesting case study for wider theories of cognition. Menary :1–20, 2015) argues that arithmetical cognition not only shows that internalist theories of cognition are wrong, but that it also shows that the Hypothesis of Extended Cognition is right. I examine this argument in more detail, to see if arithmetical cognition can support such conclusions. Specifically, I look at how the use of numerals extends our arithmetical abilities from quantity-related innate systems to systems that (...) can deal with exact numbers of arbitrary size. I then argue that the system underlying our grasp of small numbers is an unhelpful case study for Menary; it doesn’t support an argument for externalism over internalism. The system for large numbers, on the other hand, clearly displays important interactions between public numeral systems and our cognitive processes. I argue that the large number system supports an argument against internalist theories of arithmetical cognition, but that one cannot conclude that the Hypothesis of Extended Cognition is correct. In other words, the large number case doesn’t decide between the Hypothesis of Extended Cognition and the Hypothesis of EMbedded Cognition. (shrink)

Based on the notion that time, space, and number are part of a generalized magnitude system, we assume that the dual-systems approach to temporal cognition also applies to numerical cognition. Referring to theoretical models of the development of numerical concepts, we propose that children's early skills in processing numbers can be described analogously to temporal updating and temporal reasoning.

Four perspectives on numerical origins are examined. The nativist model sees numbers as an aspect of numerosity, the biologically endowed ability to appreciate quantity that humans share with other species. The linguistic model sees numbers as a function of language. The embodied model sees numbers as conceptual metaphors informed by physical experience and expressed in language. Finally, the extended model sees numbers as conceptual outcomes of a cognitive system that includes material forms as constitutive components. If numerical origins are to (...) be found, each perspective must address one or more critical questions that will require working across discipline boundaries. (shrink)

To solve fractional delay differential equation systems, the Laguerre Wavelets Method is presented and coupled with the steps method in this article. Caputo fractional derivative is used in the proposed technique. The results show that the current procedure is accurate and reliable. Different nonlinear systems have been solved, and the results have been compared to the exact solution and different methods. Furthermore, it is clear from the figures that the LWM error converges quickly when compared to other approaches. (...) When compared with the exact solution to other approaches, it is clear that LWM is more accurate and gets closer to the exact solution faster. Moreover, on the basis of the novelty and scientific importance, the present method can be extended to solve other nonlinear fractional-order delay differential equations. (shrink)

In a recent paper in Astrophysics and Space Science Vol. 364 no. 11 (2019), S. Ershkov & D. Leschenko presented a new solving procedure for Euler-Poisson equations for solving momentum equations of the CR3BP near libration points for uniformly rotating planets having inclined orbits in the solar system with respect to the orbit of the Earth. The system of equations of the CR3BP has been explored with regard to the existence of an analytic way of presentation of the approximated solution (...) in the vicinity of libration points. A new and elegant ansatz has been suggested in their publication whereby, in solving, the momentum equation is reduced to a system two coupled Riccati ODEs. In this paper, we presented a numerical solution of such coupled Riccati ODEs using Mathematica software package. (shrink)

This paper is concerned with the combinator representation of numeral systems with logarithmic space complexity of symbols. The principle used is based on the lexicographic ordering of words over a finite alphabet.

David Lewis (1969) introduced sender-receiver games as a way of investigating how meaningful language might evolve from initially random signals. In this report I investigate the conditions under which Lewis signaling games evolve to perfect signaling systems under various learning dynamics. While the 2-state/2- term Lewis signaling game with basic urn learning always approaches a signaling system, I will show that with more than two states suboptimal pooling equilibria can evolve. Inhomogeneous state distributions increase the likelihood of pooling equilibria, (...) but learning strategies with negative reinforcement or certain sorts of mutation can decrease the likelihood of, and even eliminate, pooling equilibria. Both Moran and APR learning strategies (Bereby-Meyer and Erev 1998) are shown to promote successful convergence to signaling systems. A model is presented that illustrates how a language that codes state-act pairs in an order-based grammar might evolve in the context of a Lewis signaling game. The terms, grammar, and the corresponding partitions of the state space co-evolve to generate a language whose structure appears to reflect canonical natural kinds. The evolution of these apparent natural kinds, however, is entirely in service of the rewards that accompany successful distinctions between the sender and receiver. Any metaphysics grounded on the structure of a natural language that evolved in this way would track arbitrary, but pragmatically useful distinctions. (shrink)

Past and present societies world-wide have employed well over 100 distinct notational systems for representing natural numbers, some of which continue to play a crucial role in intellectual and cultural development today. The diversity of these notations has prompted the need for classificatory schemes, or typologies, to provide a systematic starting point for their discussion and appraisal. The present paper provides a general framework for assessing the efficacy of these typologies relative to certain desiderata, and it uses this framework (...) to discuss the two influential typologies of Zhang & Norman and Chrisomalis. Following this, a new typology is presented that takes as its starting point the principles by which numerical notations represent multipliers (the principles of cumulation and cipherization), and bases (those of integration, parsing, and positionality). Many different examples show that this new typology provides a more refined classification of numerical notations than the ones put forward previously. In addition, the framework provided here can be used to assess typologies not only of numerical notations, but also of many other domains. (shrink)

Four perspectives on numerical origins are examined. The nativist model sees numbers as an aspect of numerosity, the biologically endowed ability to appreciate quantity that humans share with other species. The linguistic model sees numbers as a function of language. The embodied model sees numbers as conceptual metaphors informed by physical experience and expressed in language. Finally, the extended model sees numbers as conceptual outcomes of a cognitive system that includes material forms as constitutive components. If numerical origins are to (...) be found, each perspective must address one or more critical questions that will require working across discipline boundaries. (shrink)

We present a number of puzzles arising for the interpretation of modified numerals. Following Büring and others we assume that the main difference between comparative and superlative modifiers is that only the latter convey disjunctive meanings. We further argue that the inference patterns triggered by disjunction and superlative modifiers are hard to capture in existing semantic and pragmatic analyses of these phenomena (neo-Gricean or grammatical alike), and we propose a novel account of these inferences in the framework of bilateral state-based (...) modal logic defining a first order extension of Aloni (Semant Pragmat 15:5-EA, 2022. https://doi.org/10.3765/sp.15.5)’s. (shrink)

An interpretive system for the IBM 704 computer permitting interpretation of the genetic pattern of a numeric symbioorganism as a game strategy has been developed. Selection for best performance in a simple game has been applied in a preliminary experiment. An objective method to measure the quality of a game played is described. The results presented in the article show a small but significant improvement of game quality during a period of 2300 generations.The general characteristics of the phenomena observed are (...) similar to those of preceding evolution experiments . However, a startling infection process caused by a parasite whose behaviour was influenced by the game competition is described. The parasite never developed an independent game strategy to any degree of efficiency and was entirely dependent on its host organism for game competitions with, and transmission of the infection to, uninfected hosts.The consequences of substituting the reproduction and mutation rules used in the preceding evolution experiments are discussed. Particularly, the use of rules applying the reproduction pattern ofDNA-molecules is considered, and the theoretic aspects of symbiogenetic evolution experiments based onDNA-norm are discussed. A few inferences concerning the origin and history of terrestrial life suggested by the results of the symbiogenesis experiments are presented.Ein interpretierendes System für die IBM 704 Rechenmaschine, dass Interpretation des genetischen Modells von einem numerischen Symbio-organismus wie eine Spiel Strategie, erlaubt, wurde entwickelt. Eine Forführung für Fähigkeit in einem einfachen Spiel wurde angewandt in einem preliminarisches Experiment. Eine objektive Methode zur Messung der Qualität des betreffenden Spiels wurde beschrieben. Die in dem Artikel angegebenen Resultate zeigen eine kleine, aber bedeutende Verbesserung der Spiel-Qualität während einr Periode von 2300 Generationen.Die allgemeinen Kennziechen von den beobachteten Phänomenen sind denen von vorhergehenden Evolutions-Experimenten ähnlich . Jedoch, ein überraschender Infektions-Prozess, verursacht durch einen Parasit, wessen Verhalten beinflusst wurde durch die Spiel-Konkurrenz, wurde beschrieben. Der Parasit entwickelte niemals eine unabhängige Spiel-Strategie zu irgendeinem Masze von Fähigkeit und war ganz abhängig von seinem Gastherr-Organismus für Spiel-Konkurrenz mit, und Überbringung von der Infektion auf, uninfektierte Gäste.Die Folgerungen von dem Ersetzen der Fortpflanzung und Mutations-Regeln , angewandt in vorhergehenden Evolutions-Experimente wurden besprochen. Besonders der Gebrauch von Regeln welche das FortpflanzungsModell von DNA-Molekülen anwenden wurde betrachtet, und die theorethischen Aspekte von Symbiogenetischen Evolutions-Experimenten basiert auf DNA-Norm wurden besprochen. Einige Schlussfolgerungen in Bezug auf die Entstehung und Geschichte von dem Erdenleben suggeriert durch die Resultate von den Symbiogenese-Experimenten wurden gezeigt. (shrink)

Number systems differ cross-culturally in characteristics like how high counting extends and which number is used as a productive base. Some of this variability can be linked to the way the hand is used in counting. The linkage shows that devices like the hand used as external representations of number have the potential to influence numerical structure and organization, as well as aspects of numerical language. These matters suggest that cross-cultural variability may be, at least in part, a matter (...) of whether devices are used in counting, which ones are used, and how they are used. (shrink)

Learning programs that generalize from real‐world examples will have to deal with many different kinds of data. Continuous numeric data can cause problems for algorithms that search for examples with identical property values. These problems can be surmounted by categorizing the numeric data. However, this process has problems of its own. In this paper, we look at the need for categorizing numeric data and several methods for doing so. We concentrate on the use of generalization‐based memory, a memory organization where (...) actual examples are stored along with generalizations, which leads to a generalization‐based categorization algorithm. We also consider how to use a number heuristic, looking for gaps. These methods have been implemented in the UNIMEM computer system. Examples are presented of these algorithms categorizing data about the states of the United States. (shrink)

We agree that the approximate number system truly represents number. We endorse the authors' conclusions on the arguments from confounds, congruency, and imprecision, although we disagree with many claims along the way. Here, we discuss some complications with the meanings that undergird theories in numerical cognition, and with the language we use to communicate those theories.

In this paper, we investigate the numerical solution of the coupled fractional massive Thirring equation with the aid of He’s fractional complex transform. This study plays a significant aspect in the field of quantum physics, weakly nonlinear thrilling waves, and nonlinear optics. The main advantage of FCT is that it converts the fractional differential equation into its traditional parts and is also capable to handle the fractional order, whereas the homotopy perturbation method is employed to tackle the nonlinear terms in (...) the coupled fractional massive Thirring equation. An example is illustrated to present the efficiency and validity of the two-scale theory. The solutions are obtained in the form of series with simple and easy computations which confirm that the present approach is good in agreement and is easy to implement for such type of complex systems in science and engineering. (shrink)

A Theory of Magnitude suggests that space, time, and quantities are processed through a generalized magnitude system. ATOM posits that task-irrelevant magnitudes interfere with the processing of task-relevant magnitudes as all the magnitudes are processed by a common system. Many behavioral and neuroimaging studies have found support in favor of a common magnitude processing system. However, it is largely unknown whether such cross-domain monotonic mapping arises from a change in the accuracy of the magnitude judgments or results from changes in (...) precision of the processing of magnitude. Therefore, in the present study, we examined whether large numerical magnitude affects temporal accuracy or temporal precision, or both. In other words, whether numerical magnitudes change our temporal experience or simply bias duration judgments. The temporal discrimination paradigm was used to present numerical magnitudes across varied durations. We estimated temporal accuracy and precision for each numerical magnitude. The results revealed that temporal accuracy for large numerical magnitude was significantly lower than that of small and identical magnitudes. This implies that the temporal duration was overestimated for large numerical magnitude compared to small and identical numerical magnitude, in line with ATOM’s prediction. However, no influence of numerical magnitude was observed on temporal precision. The findings of the present study suggest that task-irrelevant numerical magnitude selectively affects the accuracy of processing of duration but not duration discrimination itself. Further, we argue that numerical magnitude may not directly affect temporal processing but could influence via attentional mechanisms. (shrink)

The emergence of systems biology raises many fascinating questions: What does it mean to take a systems approach to problems in biology? To what extent is the use of mathematical and computational modelling changing the life sciences? How does the availability of big data influence research practices? What are the major challenges for biomedical research in the years to come? This book addresses such questions of relevance not only to philosophers and biologists but also to readers interested in (...) the broader implications of systems biology for science and society. The book features reflections and original work by experts from across the disciplines including systems biologists, philosophers, and interdisciplinary scholars investigating the social and educational aspects of systems biology. In response to the same set of questions, the experts develop and defend their personal perspectives on the distinctive character of systems biology and the challenges that lie ahead. Readers are invited to engage with different views on the questions addressed, and may explore numerous themes relating to the philosophy of systems biology. This edited work will appeal to scholars and all levels, from undergraduates to researchers, and to those interested in a variety of scholarly approaches such as systems biology, mathematical and computational modelling, cell and molecular biology, genomics, systems theory, and of course, philosophy of biology. (shrink)

Our epistemic access to mathematical objects, like numbers, is mediated through our external representations of them, like numerals. Nevertheless, the role of formal notations and, in particular, of the internal structure of these notations has not received much attention in philosophy of mathematics and cognitive science. While systems of number words and of numerals are often treated alike, I argue that they have crucial structural differences, and that one has to understand how the external representation works in order to (...) form an advanced conception of numbers. The relation between external representations and mathematical cognition will be discussed by drawing on experimental results from cognitive science and mathematics education, and I argue for the constitutive role of external representations for the development of an advanced conception of numbers. (shrink)

The world’s diverse written numeral systems are affected by human cognition; in turn, written numeral systems affect mathematical cognition in social environments. The present study investigates the constraints on graphic numerical notation, treating it neither as a byproduct of lexical numeration, nor a mere adjunct to writing, but as a specific written modality with its own cognitive properties. Constraints do not refute the notion of infinite cultural variability; rather, they recognize the infinity of variability within defined (...) limits, thus transcending the universalist/particularist dichotomy. In place of strictly innatist perspectives on mathematical cognition, a model is proposed that invokes domain-specific and notationally-specific constraints to explain patterns in numerical notations. The analysis of exceptions to cross-cultural generalizations makes the study of near-universals highly productive theoretically. The cross-cultural study of patterns in written numbers thus provides a rich complement to the cognitive analysis of writing systems. (shrink)

The world’s diverse written numeral systems are affected by human cognition; in turn, written numeral systems affect mathematical cognition in social environments. The present study investigates the constraints on graphic numerical notation, treating it neither as a byproduct of lexical numeration, nor a mere adjunct to writing, but as a specific written modality with its own cognitive properties. Constraints do not refute the notion of infinite cultural variability; rather, they recognize the infinity of variability within defined (...) limits, thus transcending the universalist/particularist dichotomy. In place of strictly innatist perspectives on mathematical cognition, a model is proposed that invokes domain-specific and notationally-specific constraints to explain patterns in numerical notations. The analysis of exceptions to cross-cultural generalizations makes the study of near-universals highly productive theoretically. The cross-cultural study of patterns in written numbers thus provides a rich complement to the cognitive analysis of writing systems. (shrink)

In this article, we investigated a deterministic model of pneumonia-meningitis coinfection. Employing the Atangana–Baleanu fractional derivative operator in the Caputo framework, we analyze a seven-component approach based on ordinary differential equations. Furthermore, the invariant domain, disease-free as well as endemic equilibria, and the validity of the model’s potential results are all investigated. According to controller design evaluation and modelling, the modulation technique devised is effective in diminishing the proportion of incidences in various compartments. A fundamental reproducing value is generated by (...) exploiting the next generation matrix to assess the properties of the equilibrium. The system’s reliability is further evaluated. Sensitivity analysis is used to classify the impact of each component on the spread or prevention of illness. Using simulation studies, the impacts of providing therapy have been determined. Additionally, modelling the appropriate configuration demonstrated that lowering the fractional order from 1 necessitates a rapid initiation of the specified control technique at the largest intensity achievable and retaining it for the bulk of the pandemic’s duration. (shrink)

This article presents research on assemblages among humans and computational systems in which physical and virtual autonomous processes occur in order to create artworks allowing the emergence of mixed sensory set-ups. It begins with triadic relationships computer, physical objects and participants aimed at co-relations among bands of bots (virtual and physical) with groups of humans (interactors). The bots have a representation of the virtual world: physical bots live on a flat surface (Abbot 1991), a projection of the 3D environment (...) where the virtual bots live. The artwork also explores the metaphor of higher dimensions being understood through their projections in lower dimensions. Its goal is to weave interaction and autonomous processes, looking for poetic results integrating art, science and technology; results with aesthetic, poetic and functional qualities; and results exploring emergency and agency for design, construction and set-up of expressive systems. For these symbioses to happen the assemblage of environments with aesthetic and poetic appeal leads one to think about mixed cognitive processes that provoke emergent behaviours. I am interested in the behaviours emerged in such inter-relationships. Therefore, to achieve this goal, I have created a Java framework that allows me to program interfaces weaving such behaviors1 and also I have programmed a set of sketches using Processing2 to test the hypothesis. The main inquiries of such research are as follows: What kind of artwork could mix people with autonomous computational processes without becoming invasive? and Is such a dichotomy possible? To answer these questions, I came up with the idea of playing with swarms and flock algorithms mixing virtual and physical domains. Will this create the mixed behaviour I am looking for? Will the participants contribute with their actions, emotions and affects while the computer develops its own autonomous processes? (shrink)