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Hrotsvit of Gandersheim sees music and arithmetic as parts of philosophy. Presenting the motives for engaging in literary creativity and describing her writing process, she says that by glorifying God with her work, she incorporated threads and pieces of the ancient mantle of philosophy into it. The paper aims to provide insight into Hrotsvit’s knowledge of philosophy and the specificities and significance of her approach to the contents of the disciplines of arithmetic and music (in particular to musica humana and the harmony of a world) in her plays Paphnutius and Sapientia, to analyze different elements of her authorial self-perception and to emphasize the similarities between her and Augustine’s understanding of music, and the role of certain types of writing. The paper concludes that Hrotsvit incorporated her knowledge and understanding of ancient and medieval theories of music and numbers into her plays uniquely, showing originality in their interpretation and merging them with everyday moral life and education. In this way, the rarely treated philosophical elements of her oeuvre are presented and evaluated.

The architectonic is key for situating Kant’s understanding of science in the coming century. For Kant the faculty of reason turns to ideas to form a complete system. The coherence of the system rests on these ideas. In contrast to technical unity which can be abstracted a posteriori, architectonic ideas are the source of a priori unity for the system of reason because they connect our reasonable pursuit to essential human ends. Given Kant’s focus on mathematics, in the architectonic and his critical philosophy more generally, we must have some sense of the architectonic idea of mathematics. In this paper, I argue for the key principles of the architectonic idea of mathematics: 1) because mathematics is grounded in a priori intuition, it is a peculiarly human activity; 2) the method of mathematics is one of a priori construction, a method only mathematics can employ and: 3) the objects of mathematics are extensive magnitudes. Given these principles, we can use the architectonic idea to have some clarity about how mathematics has dealt with historical development.

The purpose of this paper is to introduce the idea of how to construct transfinite numbers and study transfinite arithmetic. The research method used is a literature review, which involves collecting various sources such as scientific papers and books related to Cantorian set theory, infinity, ordinal or transfinite arithmetic, as well as the connection between infinity and theology. The study also involves constructing the objects of study, namely ordinal numbers such as finite ordinals and transfinite ordinals, and examining their arithmetic properties. The results of this research include the methods of constructing both finite and transfinite ordinal numbers using two generation principles. Both finite and transfinite ordinal numbers are defined as well-orderings that are also transitive sets. Arithmetic of finite ordinal numbers is well-known, but the arithmetic of transfinite ordinal numbers will be introduced in this paper, including addition, multiplication, and exponentiation.

Stochastic circuits are used in applications that require low area and power consumption. The computing performed using these circuits is referred to as Stochastic computing (SC). The arithmetic operations in this computing can be realized using minimum logic circuits. The SC system allows a tradeoff of computational accuracy and area; thereby, the challenge in SC is improving the accuracy. The accuracy depends on the SC system’s stochastic number generator (SNG) part. SNGs provide the appropriate stochastic input required for stochastic computation. Hence we explore the accuracy in SC for various arithmetic operations performed using stochastic computing with the help of logic circuits. The contributions in this paper are; first, we have performed stochastic computing for arithmetic components using two different SNGs. The SNGs considered are Linear Feed-back Shift Register (LFSR) -based traditional stochastic number generators and S-box-based stochastic number generators. Second, the arithmetic components are implemented in a combinational circuit for algebraic expression in the stochastic domain using two different SNGs. Third, computational analysis for stochastic arithmetic components and the stochastic algebraic equation has been conducted. Finally, accuracy analysis and measurement are performed between LFSR-based computation and S-box-based computation. The novel aspect of this work is the use of S-box-based SNG in the development of stochastic computing in arithmetic components. Also, the implementation of stochastic computing in the combinational circuit using the developed basic arithmetic components, and exploration of accuracy with respect to stochastic number generators used is presented.

It has been consistently reported that deaf individuals experience mathematical difficulties compared to their hearing peers. However, the idea that deafness and early language deprivation might differently affect verbal (i.e., multiplication) vs. visuospatial (i.e., subtraction) arithmetic performances is still under debate. In the present paper, three groups of 21 adults (i.e., deaf signers, hearing signers, and hearing controls) were therefore asked to perform, as fast and as accurately as possible, subtraction and multiplication operations. No significant group effect was found for accuracy performances. However, reaction time results demonstrated that the deaf group performed both arithmetic operations slower than the hearing groups. This group difference was even more pronounced for multiplication problems than for subtraction problems. Weaker language-based phonological representations for retrieving multiplication facts, and sensitivity to interference are two hypotheses discussed to explain the observed dissociation.

In this paper we direct attention to 5–6-year-olds’ learning of arithmetic skills through a thorough analysis of changes in the children’s ways of encountering and experiencing numbers. The foundation for our approach is phenomenographic, in that our object of analysis is differences in children’s ways of completing an arithmetic task, which are considered to be expressions of their ways of experiencing numbers and what is possible to do with numbers. A qualitative analysis of 103 children’s ways of encountering the task gives an outcome space of varying ways of experiencing numbers. This is further analyzed through the lens of variation theory of learning, explaining why differences occur and how observed changes over a prolonged period of time can shed light on how children learn the meaning of numbers, allowing them to solve arithmetic problems. The results show how observed changes are liberating new and powerful problem-solving strategies. Emanating from empirical research, the results of our study contribute to the theoretical understanding of young children’s learning of arithmetic skills, taking the starting point in the child’s lived experiences rather than cognitive processes. This approach to interpreting learning, we suggest, has pedagogical implications concerning what is fundamental to teach children for their further development in mathematics.

This paper analyzes Danielewski's poetics of numbers and digits in the first season of The Familiar. I argue that Danielewski's recent work signals a balance shift from the topological to the arithmetical. In this regard, numbers fulfill a crucial role (1) in the serial makeup of the volumes, which could be labelled as his exorithmetic; (2) in the plot of the novel itself, his endorithmetic; and (3) as a recursive device that conjoins the materiality of the novel with the numbers in the plot, his mesorithmetic. While Danielewski's exorithmetic provokes hypotheses about the voluminousness of the project, the staggering numbers in the world of The Familiar are related to Xanther's epilepsy, Dov's teachings, and Anwar's trauma.

The paper sketches and defends two instances of the strategy Let N’s be whatever they have to be to explain our knowledge of them—one in which N’s are natural numbers and one in which N’s are propositions. The former, which makes heavy use of Hume’s principle and plural quantification, grounds our initial knowledge of number in (a) our identification of objects as falling under various types, (b) our ability to count (i.e. to pair memorized numerals with individuated objects of one’s attention), (c) our (initially perceptual) recognition of plural properties (e.g. being three in number), and (d) our predication of those properties of pluralities that possess them (even though no individuals in the pluralities do). Given this foundation, one can use Fregean techniques to non-paradoxically generate more extensive arithmetical knowledge. The second instance of my metaphysics-in-the-service-of-epistemology identifies propositions (i.e. semantic contents of some sentences, objects of the attitudes, and bearers of truth, falsity, necessity, contingency, and apriority) with certain kinds of purely representational cognitive acts, operations, or states. In addition to providing natural solutions to traditionally un-addressed epistemic problems involving linguistic cognition and language use, I argue that this metaphysical conception of propositions expands the solution spaces of many of the most recalcitrant and long-standing problems in natural-language semantics and the philosophy of language.

The brain solves problems in structures termed "working memory." Between 2001 and 2010, cognitive experiments verified that at each step when solving a problem, working memory can hold only a few small elements of knowledge that are not well-memorized. One implication of this limit is that students must rely almost exclusively on the application of memorized facts and algorithms when solving mathematical or scientific calculations. Unfortunately, since 1990, K-12 math standards in most U.S. states assumed that with access to calculators and computers, memorization in math could be de-emphasized. As a result, many students have deficits in “automaticity” in the recall of math that is necessary for chemistry. This paper will include evidence that if math fundamentals are moved into memory as preparation for a chemistry topic, student success in first-year chemistry improves substantially.

The learning of mathematical content in the lower grades of primary school tends to take place through working on mathematical problems. Students, however, have varying degrees of success in solving these problems, which depends on the context of the problems. This paper therefore aims to assess student achievement in solving arithmetic problems depending on the context of the problems. The research sample consisted of fourth-grade students from three primary schools in the city of Kragujevac. 165 students were asked to solve a test comprising 12 problems divided into three groups of 4 problems each, given in three different ways - problems given in the context of a mathematical expression, problems given in the context of a mathematical sentence, and problems given in the context of a real-life situation. The results show that fourth-grade students achieved uneven results when solving the same mathematical problems given in different contexts. The results also suggest that student achievement depends on students' gender.

Abstract This paper aims to compare the neuropsychological and educational profiles of Brazilian children with dyscalculia (n = 8), dyslexia (n = 13) and without learning disabilities (n = 12). The neuropsychological profile was composed of: (a) intelligence - assessed by the Wechsler Intelligence Scale for Children (WISC-III); (b) attention - WISC-III Coding and Symbol Search subtests; (c) executive functions - Digit Span (backward order) and WISC-III Arithmetic subtests, Pseudoword Repetition Test for Brazilian Children, Stroop Test and Wisconsin Card Sorting Test; (d) memory - WISC-III Digit Span subtest (forward order) and Rey Complex Figures. The educational profile was composed of reading, writing and mathematics, assessed by the Academic Performance Test and the Arithmetic Test. It was found that the groups with dyscalculia and dyslexia did not differentiate in any of the neuropsychological abilities, only in the reading and writing abilities. Neuropsychological variables that could explain these results were discussed.

Solving arithmetic word problems requires constructing a situation model based on the problem text and translating that into a mathematical model. As such, word problem solving makes demands on students’ language comprehension and their domain-general cognitive resources. These demands may decrease when students get more experienced and use strategies that do not require fully understanding the situation presented in the problem. The current study aims to address this hypothesis. Students (N=444) from third to sixth grade solved a paper-and-pencil task with 48 mathematics problems, comprising symbolic arithmetic problems and standard word problems, as well as more complex word problems that involve two arithmetic steps or include irrelevant numerical information. Their performance was analyzed with multilevel logistic regression analyses. Results showed that within each grade, performance on the different problem types did not differ, suggesting that already in third-grade students seem helped nor hindered by presenting arithmetic problems in a story, even if that story contains irrelevant numerical information. Non-verbal reasoning was more important in standard word problems than in arithmetic problems in symbolic format in one-step arithmetic, and reading comprehension was more important in solving two-step arithmetic word problems than in one-step arithmetic word problems.

This paper considers whether incompatibilism, the view that negation is to be explained in terms of a primitive notion of incompatibility, and Fregeanism, the view that arithmetical truths are analytic according to Frege’s definition of that term in §3 of Foundations of Arithmetic, can both be upheld simultaneously. Both views are attractive on their own right, in particular for a certain empiricist mind-set. They promise to account for two philosophical puzzling phenomena: the problem of negative truth and the problem of epistemic access to numbers. For an incompatibilist, proofs of numerical non-identities must appeal to primitive incompatibilities. I argue that no analytic primitive incompatibilities are forthcoming. Hence incompatibilists cannot be Fregeans.

q-rung picture fuzzy sets can handle complex fuzzy and impression information by changing a parameter q based on the different hesitation degree, and Yager operator is a useful aggregation technology that can control the uncertainty of valuating data from some experts and thus get intensive information in the process of decision-making. Thus, in this paper, we develop specific types of operators, namely, q-rung picture fuzzy Yager weighted average, q-rung picture fuzzy Yager ordered weighted average, q-rung picture fuzzy Yager hybrid weighted average, q-rung picture fuzzy Yager weighted geometric, q-rung picture fuzzy Yager ordered weighted geometric and q-rung picture fuzzy Yager hybrid weighted geometric operators. We propose q-rung picture fuzzy Yager aggregation operators to handle multiple attribute decision-making problems in a modernize way. Moreover, we discuss the effect of parameter on the decision-making results. To demonstrate the superiority and advantage of our proposed method, a comparison with existing methods is presented.

There is very little research on cognitive outcomes and challenges for children with Apert syndrome. This paper describes the findings of a 2½-year longitudinal exploration of the development of arithmetic skills in 10 children with Apert syndrome, who were aged between 4 and 9 years at the beginning of the study. There is evidence to suggest that children with Apert syndrome underachieve in mathematics, especially in areas requiring competence and confidence arithmetic. This study explored the changing strategies the children used as they developed their arithmetic problem solving skills. Of particular interest were the roles of finger gnosis and finger mobility in supporting the development of these skills. A case study approach was adopted in order to explore the children's problem solving strategies in depth. Children with Apert syndrome are born with their fingers fused and undergo several operations with the aim of ensuring that they have as many functioning fingers as possible. Finger gnosis and finger mobility were both seen to support arithmetic problem solving strategies and skills and reduce the reliance on mental strategies alone. This study found that children with Apert syndrome are disadvantaged if they are not supported to develop their finger gnosis and finger mobility skills. The findings have implications for children with Apert syndrome, but also add to the literature on the role of finger gnosis and finger mobility in the development of skills in early number and arithmetic.

The research team designed and evaluated a mobile game to promote rapid retrieval of arithmetic facts among a group of children aged 7–8 years (n = 97). The design of the game was based on principles drawn from research literature in mathematical cognition, game-based learning, and game design. The game trains basic number knowledge within a motivating context. It tested an implication of theory of automatization of arithmetic facts that training of recognition of multiples of single-digit numbers should lead to greater fluency in solving multiplication and division problems. A quasi-experimental design was employed to test whether the game improves retrieval of arithmetic facts. Children played the game in their classrooms for 20 min a day for 2 weeks. Comparisons between pre- and post-tests showed that the game playing group outperformed controls with a medium to large effect size (>0.6). These results suggest an improvement in arithmetic fluency equivalent to around 7 months' progress and provide rare empirical evidence supporting transfer of game-based training to a pencil-and-paper test. The findings are consistent with a connectionist theory of arithmetic skill, by showing that improved recognition of multiples contributes to multiplication and division skill. The theoretical and practical implications of these findings are discussed.

Wu introduced the interval range of fuzzy sets. Based on this, he defined a kind of arithmetic of fuzzy sets using a gradual number and gradual sets. From the point of view of soft computing, this definition provides a new way of handling the arithmetic operations of fuzzy sets. The interval range is an important characterization of a fuzzy set. The interval range is also useful for analyses and applications of arithmetic. In this paper, we present general conclusions on crucial problems related to interval ranges of fuzzy sets induced by this arithmetic. These conclusions indicate that the corresponding conclusions in previous works should be modified: firstly, we give properties of the arithmetic and the composites of finite arithmetic. Then, we discuss the relationship between the domain of a gradual set and the range of its induced fuzzy set, and the relationship between the domain of a gradual set and the interval range of its induced fuzzy set. Based on the above results, we present the relationship between the intersection of the interval ranges of a group of fuzzy sets and the interval ranges of their resulting fuzzy sets obtained by compositions of finite arithmetic. Furthermore, we construct examples to show that even under conditions stronger than in previous work, there are still various possibilities in the relationship between the intersection of interval ranges of a group of fuzzy sets and the ranges of their resulted fuzzy sets, and there are still various possibilities in the relationship between the intersection of the interval ranges of a group of fuzzy sets and the interval ranges of their resulting fuzzy sets.

Floating-point division is indispensable and becoming increasingly important in many modern applications. To improve speed performance of floating-point division in actual microprocessors, this paper proposes a low-latency architecture with a multi-precision architecture for floating-point division which will meet the IEEE-754 standard. There are three parts in the floating-point division design: pre-configuration, mantissa division, and quotient normalization. In the part of mantissa division, based on the fast division algorithm, a Predict–Correct algorithm is employed which brings about more partial quotient bits per cycle without consuming too much circuit area. Detailed analysis is presented to support the guaranteed accuracy per cycle with no restriction to specific parameters. In the synthesis using TSMC, 90 nm standard cell library, the results show that the proposed architecture has ≈63.6% latency, ≈30.23% total time (latency × period), ≈31.8% total energy (power × latency × period), and ≈44.6% efficient average energy (power × latency × period/efficient length) overhead over the latest floating-point division structure. In terms of latency, the proposed division architecture is much faster than several classic processors.

This paper starts with an explanation of how the logicist research program can be approached within the framework of Leśniewski’s systems. One nice feature of the system is that Hume’s Principle is derivable in it from an explicit definition of natural numbers. I generalize this result to show that all predicative abstraction principles corresponding to second-level relations, which are provably equivalence relations, are provable. However, the system fails, despite being much neater than the construction of Principia Mathematica (PM). One of the key reasons is that, just as in the case of the system of PM, without the assumption that infinitely many objects exist, (renderings of) most of the standard axioms of Peano Arithmetic are not derivable in the system. I prove that introducing modal quantifiers meant to capture the intuitions behind potential infinity results in the (renderings of) axioms of Peano Arithmetic (PA) being valid in all relational models (i.e. Kripke-style models, to be defined later on) of the extended language. The second, historical part of the paper contains a user-friendly description of Leśniewski’s own arithmetic and a brief investigation into its properties.

The rapid growth in the amount of data in the digital world leads to the need for data compression, and so forth, reducing the number of bits needed to represent a text file, an image, audio, or video content. Compressing data saves storage capacity and speeds up data transmission. In this paper, we focus on the text compression and provide a comparison of algorithms (in particular, entropy-based arithmetic and dictionary-based Lempel–Ziv–Welch (LZW) methods) for text compression in different languages (Croatian, Finnish, Hungarian, Czech, Italian, French, German, and English). The main goal is to answer a question: ”How does the language of a text affect the compression ratio?” The results indicated that the compression ratio is affected by the size of the language alphabet, and size or type of the text. For example, The European Green Deal was compressed by 75.79%, 76.17%, 77.33%, 76.84%, 73.25%, 74.63%, 75.14%, and 74.51% using the LZW algorithm, and by 72.54%, 71.47%, 72.87%, 73.43%, 69.62%, 69.94%, 72.42% and 72% using the arithmetic algorithm for the English, German, French, Italian, Czech, Hungarian, Finnish, and Croatian versions, respectively.

In fuzzy set theory, t-norms and t-conorms are fundamental binary operators. Yager proposed respective parametric families of both t-norms and t-conorms. In this paper, we apply these operators for the analysis of Pythagorean fuzzy sets. For this purpose, we introduce six families of aggregation operators named Pythagorean fuzzy Yager weighted averaging aggregation, Pythagorean fuzzy Yager ordered weighted averaging aggregation, Pythagorean fuzzy Yager hybrid weighted averaging aggregation, Pythagorean fuzzy Yager weighted geometric aggregation, Pythagorean fuzzy Yager ordered weighted geometric aggregation and Pythagorean fuzzy Yager hybrid weighted geometric aggregation. These tools inherit the operational advantages of the Yager parametric families. They enable us to study two multi-attribute decision-making problems. Ultimately we can choose the best option by comparison of the aggregate outputs through score values. We show this procedure with two practical fully developed examples.

A dynamical model of an animal’s nervous system, or synthetic nervous system (SNS), is a potentially transformational control method. Due to increasingly detailed data on the connectivity and dynamics of both mammalian and insect nervous systems, controlling a legged robot with an SNS is largely a problem of parameter tuning. Our approach to this problem is to design functional subnetworks that perform specific operations, and then assemble them into larger models of the nervous system. In this paper, we present networks that perform addition, subtraction, multiplication, division, differentiation, and integration of incoming signals. Parameters are set within each subnetwork to produce the desired output by utilizing the operating range of neural activity, R, the gain of the operation, k, and bounds based on biological values. The assembly of large networks from functional subnetworks underpins our recent results with MantisBot.

The Genetic Algorithm (GA) is an evolutionary algorithms and technique based on natural selections of individuals called chromosomes. In this paper, a method for solving Knapsack problem via GA (Genetic Algorithm) is presented. We compared six different crossovers: Crossover single point, Crossover Two point, Crossover Scattered, Crossover Heuristic, Crossover Arithmetic and Crossover Intermediate. Three different dimensions of knapsack problems are used to test the convergence of knapsack problem. Based on our experimental results, two point crossovers (TP) emerged the best result to solve knapsack problem. Keywords: Genetic Algorithm, Crossover, Heuristic, Arithmetic, Intermediate, Evolutionary Algorithm

The use of real numbers in a program can introduce differences between the expected and the actual behavior of the program, due to the finite representation of these numbers. Therefore, one may want to define programs using real numbers such that this difference vanishes. This paper defines a program transformation for a certain class of programs that improves the accuracy of the computations on real number representations by removing the square root and division operations from the original program in order to enable exact computation with addition, multiplication and subtraction. This transformation is meant to be used on embedded systems, therefore the produced programs have to respect constraints relative to this kind of code. In order to ensure that the transformation is correct, i.e. preserves the semantics, we also aim at specifying and proving this transformation using the PVS proof assistant.

The paper concerns the concept of number (arithmos), important for dialectical method of later Plato. It becomes clear that the arithmos in Plato’s dialectics should be understood as a concrete operation, a sort of tekhne, such as counting, enumeration, compilation of a comprehensive and systematic list, etc., rather then the theoretical number of abstract arithmetic. The author analyses a series of grammatical, musical and rhetorical examples, supplied by Plato in the Philebus and other dialogues, and traces the usage of arithmos and similar words in the earlier tradition, particularly, in Homer, Hesiod, the classical tragedy, and Herodotus.