Due to how fast life moves these days, most parents forget to keep an eye on their children’s development and math skills as early as 4 years old. The role of child care is very important to enhance quality assurance practices among staff for the development of future leaders. The main objective of this study is to determine the strength of the relationships between each element and arithmetic proficiency among Malaysian TASKA children. This study is significant to identify the (...) primary factors that influence children’s math skills. A questionnaire was utilized to collect data. From 376 TASKAs registered in Malaysia, only 103 of the 458 selected centers provide care and instruction for children aged 36 to 48 months. Between the ages of 36 and 46 months, language, communication, and early reading skills account for the majority of a child’ s mathematical ability. Therefore, an effective training module should be built on these characteristics. The upcoming study will compare all unregistered TASKA facilities to the facilities that have been registered throughout Malaysia. (shrink)

The aim of this research was to x-ray some causes of poor performance of pupils in primary school mathematics. Specifically, the study examined the use of instructional materials and pupils’ academic performance in mathematics; parents’ socio-economic background and pupils’ academic performance in mathematics; compared the performance of private and public primary school pupils in mathematics; examined ways in which teachers contribute to pupils’ poor performance in mathematics. The study employed a correlational and quasi- experimental research (...) designs. A simple random sampling technique was used to select a sample of 270 pupils and 45 teachers drawn from nine primary schools in Akamkpa local government area of Cross River State. A questionnaire and a mathematics achievement test (MAT) were instruments used for data collection. The collected data was analysed using a simple percentage (%) and arithmetic mean (X). The findings of the study revealed that the use of instructional materials adequately led to pupils’ poor performance in mathematics; parents’ socio-economic status contributed to the pupils’ performance in mathematics; pupils in private primary school perform better than their colleagues in public schools in mathematics; and teachers contribute to the poor performance of pupils in mathematics. Based on this results, conclusions and recommendations were made. (shrink)

Quality education needs quality teachers to achieve success. Thus, this study determined the factors related to the teachers’ performance in delivering the K to 12 Curriculum in the Narra del Sur district, Palawan, Philippines. A descriptive-correlational research design was employed, with a sample of 132 randomly selected public elementary teachers. The study used frequency counts and percentages, arithmetic mean and standard deviation, and Spearman’s rho to analyze and draw conclusions from the data. The findings revealed a correlation between (...) the respondents’ age and their utilization of teaching materials and ICT. The number of pupils taught was correlated with teachers’ commitment, while number of pupils taught, position, highest educational attainment, and length of service were related to the respondents’ performance. The main problem encountered by the respondents was excessive workloads, paperwork, and reports. To solve the problem teachers encountered in delivering the K to 12 Curriculum, the respondents preferred preparing a work plan and allocating time to daily and weekly tasks to avoid excessive workloads. These results may aid policymakers and curriculum developers in the Department of Education to make relevant enhancements to the factors that affect teachers’ performance in curriculum delivery. (shrink)

Much research supports the existence of an Approximate Number System (ANS) that is recruited by infants, children, adults, and non-human animals to generate coarse, non-symbolic representations of number. This system supports simple arithmetic operations such as addition, subtraction, and ordering of amounts. The current study tests whether an intuition of a more complex calculation, division, exists in an indigene group in the Amazon, the Mundurucu, whose language includes no words for large numbers. Mundurucu children were presented with a video (...) event depicting a division transformation of halving, in which pairs of objects turned into single objects, reducing the array's numerical magnitude. Then they were tested on their ability to calculate the outcome of this division transformation with other large-number arrays. The Mundurucu children effected this transformation even when non-numerical variables were controlled, performed above chance levels on the very first set of test trials, and exhibited performance similar to urban children who had access to precise number words and a surrounding symbolic culture. We conclude that a halving calculation is part of the suite of intuitive operations supported by the ANS. (shrink)

An interpretation of Wittgenstein’s much criticized remarks on Gödel’s First Incompleteness Theorem is provided in the light of paraconsistent arithmetic: in taking Gödel’s proof as a paradoxical derivation, Wittgenstein was drawing the consequences of his deliberate rejection of the standard distinction between theory and metatheory. The reasoning behind the proof of the truth of the Gödel sentence is then performed within the formal system itself, which turns out to be inconsistent. It is shown that the features of paraconsistent arithmetics (...) match with some intuitions underlying Wittgenstein’s philosophy of mathematics, such as its strict finitism and the insistence on the decidability of any mathematical question. (shrink)

The abstract basis of modern computation is the formal description of a finite state machine, the Universal Turing Machine, based on manipulation of integers and logic symbols. In this contribution to the discourse on the computer-brain analogy, we discuss the extent to which analog computing, as performed by the mammalian brain, is like and unlike the digital computing of Universal Turing Machines. We begin with ordinary reality being a permanent dialog between continuous and discontinuous worlds. So it is with computing, (...) which can be analog or digital, and is often mixed. The theory behind computers is essentially digital, but efficient simulations of phenomena can be performed by analog devices; indeed, any physical calculation requires implementation in the physical world and is therefore analog to some extent, despite being based on abstract logic and arithmetic. The mammalian brain, comprised of neuronal networks, functions as an analog device and has given rise to artificial neural networks that are implemented as digital algorithms but function as analog models would. Analog constructs compute with the implementation of a variety of feedback and feedforward loops. In contrast, digital algorithms allow the implementation of recursive processes that enable them to generate unparalleled emergent properties. We briefly illustrate how the cortical organization of neurons can integrate signals and make predictions analogically. While we conclude that brains are not digital computers, we speculate on the recent implementation of human writing in the brain as a possible digital path that slowly evolves the brain into a genuine (slow) Turing machine. (shrink)

In this paper, we deal with the computation of Lie derivatives, which are required, for example, in some numerical methods for the solution of differential equations. One common way for computing them is to use symbolic computation. Computer algebra software, however, might fail if the function is complicated, and cannot be even performed if an explicit formulation of the function is not available, but we have only an algorithm for its computation. An alternative way to address the problem is to (...) use automatic differentiation. In this case, we only need the implementation of the algorithm that evaluates the function in terms of its analytic expression in a programming language, but we cannot use this if we have only a compiled version of the function. In this paper, we present a novel approach for calculating the Lie derivative of a function, even in the case where its analytical expression is not available, that is based on the In finity Computer arithmetic. A comparison with symbolic and automatic differentiation shows the potentiality of the proposed technique. (shrink)

Hauser argues that his pocket calculator (Cal) has certain arithmetical abilities: it seems Cal calculates. That calculating is thinking seems equally untendentious. Yet these two claims together provide premises for a seemingly valid syllogism whose conclusion - Cal thinks - most would deny. He considers several ways to avoid this conclusion, and finds them mostly wanting. Either we ourselves can't be said to think or calculate if our calculation-like performances are judged by the standards proposed to rule out Cal; or (...) the standards- e.g., autonomy and self-consciousness- make it impossible to verify whether anything or anyone (save oneself) meets them. While appeals to the intentionality of thought or the unity of minds provide more credible lines of resistance, available accounts of intentionality and mental unity are insufficiently clear and warranted to provide very substantial arguments against Cal's title to be called a thinking thing. Indeed, considerations favoring granting that title are more formidable than is generally appreciated. Rapaport's comments suggest that, on a strong view of thinking, mere calculating is not thinking (and pocket calculators don't think), but on a weak, but unexciting, sense of thinking, pocket calculators do think. He closes with some observations on the implications of this conclusion. (shrink)

In this article I offer an explicating interpretation of the procedure of content recarving as described by Frege in §64 of the Foundations of Arithmetic. I argue that the procedure of content recarving may be interpreted as an operation that while restricting the subject matter of a sentence, performs a generalization on what the sentence says about its subject matter. The characterization of the recarving operation is given in the setting of Yablo’s theory of subject matter and it is (...) based on the relation of determination between properties. The main advantage of the proposal is its generality, for it is applicable not just to the case of abstraction principles. (shrink)

This book develops a new approach to plural arbitrary reference and examines mereology, including considering four theses on the alleged innocence of mereology. The authors have advanced the notion of plural arbitrary reference in terms of idealized plural acts of choice, performed by a suitable team of agents. In the first part of the book, readers will discover a revision of Boolosʼ interpretation of second order logic in terms of plural quantification and a sketched structuralist reconstruction of second-order arithmetic (...) based on the axiom of infinite, a la Dedekind, as the unique non-logical axiom. The work goes on to analyse the pros and cons of the new interpretation, also with respect to Linneboʼs objections to the thesis that second order logic is genuine logic. A theory of concepts that can be labelled as a theory of logical concepts is expounded. In the second part of the book, the authors consider grounding megethology on plural arbitrary reference and argue that the arguments for the ontological innocence of mereology are not conclusive and that – for a certain use of mereology – a thesis of innocence, similar to that of plural arbitrary reference, is defensible. The work proposes a virtual theory of mereology in which the role of individuals is played by plural choices of atoms. This considered work will appeal to scholars from branches of analytic philosophy, logic and the philosophy of mathematics in particular. (shrink)

The main objective of this PhD Thesis is to present a method of obtaining strong normalization via natural ordinal, which is applicable to natural deduction systems and typed lambda calculus. The method includes (a) the definition of a numerical assignment that associates each derivation (or lambda term) to a natural number and (b) the proof that this assignment decreases with reductions of maximal formulas (or redex). Besides, because the numerical assignment used coincide with the length of a specific sequence of (...) reduction - the worst reduction sequence - it is the lowest upper bound on the length of reduction sequences. The main commitment of the introduced method is that it is constructive and elementary, produced only through analyzing structural and combinatorial properties of derivations and lambda terms, without appeal to any sophisticated mathematical tool. Together with the exposition of the method, it is presented a comparative study of some articles in the literature that also get strong normalization by means we can identify with the natural ordinal methods. Among them we highlight Howard[1968], which performs an ordinal analysis of Godel’s Dialectica interpretation for intuitionistic first order arithmetic. We reveal a fact about this article not noted by the author himself: a syntactic proof of strong normalization theorem for the system of typified lambda calculus λ⊃ is a consequence of its results. This would be the first strong normalization proof in the literature. (written in Portuguese). (shrink)

This study aimed to identify the effect of procedural justice on organizational loyalty from the point of view of Faculty Staff at Palestine Technical University- Kadoorei. It also aimed to identify the differences in the views of the study sample on the study variables according to the years of service. In order to achieve this, the researchers used a questionnaire consisting of (22) paragraphs where the first area (10) paragraphs looking at procedural justice while the paragraphs of the second area (...) and the number of (12) paragraph in the field of organizational allegiance to Faculty Staff at the university, (105) questionnaires were distributed on the sample of the study, and after the process of distribution of the questionnaire was collected and coded and entered into the computer and processed statistically using the statistical program of social sciences (SPSS). One of the most important findings of the study was that the degree of procedural justice at the heads of departments at Palestine Technical University- Kadoorei, from the point of view of Faculty Staff was between the medium and large, where the average arithmetic (3.65). Respondents also showed a high level of organizational loyalty (3.84). The study also showed a statistically significant effect at the level of significance (α = 0.05) for procedural fairness in achieving organizational loyalty, and the absence of differences attributed to years of experience. In the light of the results of the previous study, the researchers recommended several recommendations, the most important of which is to increase the awareness of the workers on the principles of procedural justice, to encourage adherence to them and to indicate their importance in job performance by creating systems and methods that ensure commitment to justice by raising the ability of leaders to build new policies and visions Which would promote the work of institutions. (shrink)

The paper explores the idea that some singular judgements about the natural numbers are immune to error through misidentification by pursuing a comparison between arithmetic judgements and first-person judgements. By doing so, the first part of the paper offers a conciliatory resolution of the Coliva-Pryor dispute about so-called “de re” and “which-object” misidentification. The second part of the paper draws some lessons about what it takes to explain immunity to error through misidentification. The lessons are: First, the so-called Simple (...) Account of which-object immunity to error through misidentification to the effect that a judgement is immune to this kind of error just in case its grounds do not feature any identification component fails. Secondly, wh-immunity can be explained by a Reference-Fixing Account to the effect that a judgement is immune to this kind of error just in case its grounds are constituted by the facts whereby the reference of the concept of the object which the judgement concerns is fixed. Thirdly, a suitable revision of the Simple Account explains the de re immunity of those arithmetic judgements which are not wh-immune. These three lessons point towards the general conclusion that there is no unifying explanation of de re and wh-immunity. (shrink)

Philosophers of science since Nagel have been interested in the links between intertheoretic reduction and explanation, understanding and other forms of epistemic progress. Although intertheoretic reduction is widely agreed to occur in pure mathematics as well as empirical science, the relationship between reduction and explanation in the mathematical setting has rarely been investigated in a similarly serious way. This paper examines an important particular case: the reduction of arithmetic to set theory. I claim that the reduction is unexplanatory. In (...) defense of this claim, I offer evidence from mathematical practice, and I respond to contrary suggestions due to Steinhart, Maddy, Kitcher and Quine. I then show how, even if set-theoretic reductions are generally not explanatory, set theory can nevertheless serve as a legitimate foundation for mathematics. Finally, some implications of my thesis for philosophy of mathematics and philosophy of science are discussed. In particular, I suggest that some reductions in mathematics are probably explanatory, and I propose that differing standards of theory acceptance might account for the apparent lack of unexplanatory reductions in the empirical sciences. (shrink)

The study aims to identify the efficiency of the university education performance from the perspective of postgraduate and undergraduate students in international and Palestinian universities. The analytical descriptive approach was used for this purpose and the questionnaire was used as a main tool for data collection. The study community consists of: post graduate students, (23850) graduate students and (146355) undergraduate students. The sample of the study was 378 graduate students and 383 undergraduate students. The random stratified sample was used. (...) The Statistical Package of Social Sciences (SPSS) was also used for data analysis. The study reached a number of results, the most important of which are: The level of efficiency of educational performance in Palestinian and international universities from the point of view of postgraduate and undergraduate students was high. And that there are significant differences between the average views of the sample of the study on the efficiency of educational performance in Palestinian and international universities attributed to the University and to the benefit of international universities. The study concluded many recommendations, the most important of which is the necessity of continuing to develop e-learning strategies that affect the efficiency of educational performance and research commensurate with the university's position in the local and international community, which puts it on the best classification between local and international universities through e-learning. (shrink)

Is calculation possible without language? Or is the human ability for arithmetic dependent on the language faculty? To clarify the relation between language and arithmetic, we studied numerical cognition in speakers of Mundurukú, an Amazonian language with a very small lexicon of number words. Although the Mundurukú lack words for numbers beyond 5, they are able to compare and add large approximate numbers that are far beyond their naming range. However, they fail in exact arithmetic with numbers (...) larger than 4 or 5. Our results imply a distinction between a nonverbal system of number approximation and a language-based counting system for exact number and arithmetic. (shrink)

The subject of this article is Modal-Epistemic Arithmetic (MEA), a theory introduced by Horsten to interpret Epistemic Arithmetic (EA), which in turn was introduced by Shapiro to interpret Heyting Arithmetic. I will show how to interpret MEA in EA such that one can prove that the interpretation of EA is MEA is faithful. Moreover, I will show that one can get rid of a particular Platonist assumption. Then I will discuss models for MEA in light of the (...) problems of logical omniscience and logical competence. Awareness models, impossible worlds models and syntactical models have been introduced to deal with the first problem. Certain conditions on the accessibility relations are needed to deal with the second problem. I go on to argue that those models are subject to the problem of quantifying in, for which I will provide a solution. (shrink)

SEMANTIC ARITHMETIC: A PREFACE John Corcoran Abstract Number theory, or pure arithmetic, concerns the natural numbers themselves, not the notation used, and in particular not the numerals. String theory, or pure syntax, concems the numerals as strings of «uninterpreted» characters without regard to the numbe~s they may be used to denote. Number theory is purely arithmetic; string theory is purely syntactical... in so far as the universe of discourse alone is considered. Semantic arithmetic is a broad (...) subject which begins when numerals are mentioned (not just used) and mentioned as names of numbers (not just as syntactic objects). Semantic arithmetic leads to many fascinating and surprising algorithms and decision procedures; it reveals in a vivid way the experiential import of mathematical propositions and the predictive power of mathematical knowledge; it provides an interesting perspective for philosophical, historical, and pedagogical studies of the growth of scientific knowledge and of the role metalinguistic discourse in scientific thought. (shrink)

An arithmetic theory of oppositions is devised by comparing expressions, Boolean bitstrings, and integers. This leads to a set of correspondences between three domains of investigation, namely: logic, geometry, and arithmetic. The structural properties of each area are investigated in turn, before justifying the procedure as a whole. Io finish, I show how this helps to improve the logical calculus of oppositions, through the consideration of corresponding operations between integers.

The revival of the philosophy of mathematics in the 60s following its post-1931 slump left us with two conflicting positions on arithmetic’s ontological relationship to set theory. W.V. Quine’s view, presented in 'Word and Object' (1960), was that numbers are sets. The opposing view was advanced in another milestone of twentieth-century philosophy of mathematics, Paul Benacerraf’s 'What Numbers Could Not Be' (1965): one of the things numbers could not be, it explained, was sets; the other thing numbers could not (...) be, even more dramatically, was objects. Curiously, although Benacerraf’s article appeared in the heyday of Quine’s influence, it declined to engage the Quinean position squarely, even seemed to think it was not its business to do so. Despite that, in my experience, most philosophers believe that Benacerraf’s article put paid to the reductionist view that numbers are sets (though perhaps not the view that numbers are objects). My chapter will attempt to overturn this orthodoxy. (shrink)

Orthodoxy holds that there is a determinate fact of the matter about every arithmetical claim. Little argument has been supplied in favour of orthodoxy, and work of Field, Warren and Waxman, and others suggests that the presumption in its favour is unjustified. This paper supports orthodoxy by establishing the determinacy of arithmetic in a well-motivated modal plural logic. Recasting this result in higher-order logic reveals that even the nominalist who thinks that there are only finitely many things should think (...) that there is some sense in which arithmetic is true and determinate. (shrink)

The paper introduces and utilizes a few new concepts: “nonstandard Peano arithmetic”, “complementary Peano arithmetic”, “Hilbert arithmetic”. They identify the foundations of both mathematics and physics demonstrating the equivalence of the newly introduced Hilbert arithmetic and the separable complex Hilbert space of quantum mechanics in turn underlying physics and all the world. That new both mathematical and physical ground can be recognized as information complemented and generalized by quantum information. A few fundamental mathematical problems of the (...) present such as Fermat’s last theorem, four-color theorem as well as its new-formulated generalization as “four-letter theorem”, Poincaré’s conjecture, “P vs NP” are considered over again, from and within the new-founding conceptual reference frame of information, as illustrations. Simple or crucially simplifying solutions and proofs are demonstrated. The link between the consistent completeness of the system mathematics-physics on the ground of information and all the great mathematical problems of the present (rather than the enumerated ones) is suggested. (shrink)

(Goodsell, Journal of Philosophical Logic, 51(1), 127-150 2022) establishes the noncontingency of sentences of first-order arithmetic, in a plausible higher-order modal logic. Here, the same result is derived using significantly weaker assumptions. Most notably, the assumption of rigid comprehension—that every property is coextensive with a modally rigid one—is weakened to the assumption that the Boolean algebra of properties under necessitation is countably complete. The results are generalized to extensions of the language of arithmetic, and are applied to answer (...) a question posed by Bacon and Dorr (2024). (shrink)

The paper considers a generalization of Peano arithmetic, Hilbert arithmetic as the basis of the world in a Pythagorean manner. Hilbert arithmetic unifies the foundations of mathematics (Peano arithmetic and set theory), foundations of physics (quantum mechanics and information), and philosophical transcendentalism (Husserl’s phenomenology) into a formal theory and mathematical structure literally following Husserl’s tracе of “philosophy as a rigorous science”. In the pathway to that objective, Hilbert arithmetic identifies by itself information related to finite (...) sets and series and quantum information referring to infinite one as both appearing in three “hypostases”: correspondingly, mathematical, physical and ontological, each of which is able to generate a relevant science and area of cognition. Scientific transcendentalism is a falsifiable counterpart of philosophical transcendentalism. The underlying concept of the totality can be interpreted accordingly also mathematically, as consistent completeness, and physically, as the universe defined not empirically or experimentally, but as that ultimate wholeness containing its externality into itself. (shrink)

The systems of arithmetic discussed in this work are non-elementary theories. In this paper, natural numbers are characterized axiomatically in two di erent ways. We begin by recalling the classical set P of axioms of Peano’s arithmetic of natural numbers proposed in 1889 (including such primitive notions as: set of natural numbers, zero, successor of natural number) and compare it with the set W of axioms of this arithmetic (including the primitive notions like: set of natural numbers (...) and relation of inequality) proposed by Witold Wilkosz, a Polish logician, philosopher and mathematician, in 1932. The axioms W are those of ordered sets without largest element, in which every non-empty set has a least element, and every set bounded from above has a greatest element. We show that P and W are equivalent and also that the systems of arithmetic based on W or on P, are categorical and consistent. There follows a set of intuitive axioms PI of integers arithmetic, modelled on P and proposed by B. Iwanuś, as well as a set of axioms WI of this arithmetic, modelled on the W axioms, PI and WI being also equivalent, categorical and consistent. We also discuss the problem of independence of sets of axioms, which were dealt with earlier. (shrink)

A language in which we can express arithmetic and which contains its own satisfaction predicate (in the style of Kripke's theory of truth) can be formulated using just two nonlogical primitives: (the successor function) and Sat (a satisfaction predicate).

Does the human mind resemble the machines that can behave like it? Biologically inspired machine-learning systems approach “human-level” accuracy in an astounding variety of domains, and even predict human brain activity—raising the exciting possibility that such systems represent the world like we do. However, even seemingly intelligent machines fail in strange and “unhumanlike” ways, threatening their status as models of our minds. How can we know when human–machine behavioral differences reflect deep disparities in their underlying capacities, vs. when such failures (...) are only superficial or peripheral? This article draws on a foundational insight from cognitive science—the distinction between performance and competence—to encourage “species-fair” comparisons between humans and machines. The performance/competence distinction urges us to consider whether the failure of a system to behave as ideally hypothesized, or the failure of one creature to behave like another, arises not because the system lacks the relevant knowledge or internal capacities (“competence”), but instead because of superficial constraints on demonstrating that knowledge (“performance”). I argue that this distinction has been neglected by research comparing human and machine behavior, and that it should be essential to any such comparison. Focusing on the domain of image classification, I identify three factors contributing to the species-fairness of human–machine comparisons, extracted from recent work that equates such constraints. Species-fair comparisons level the playing field between natural and artificial intelligence, so that we can separate more superficial differences from those that may be deep and enduring. (shrink)

Over the years many mathematicians have voiced a preference for proofs that stay “close” to the statements being proved, avoiding “foreign”, “extraneous”, or “remote” considerations. Such proofs have come to be known as “pure”. Purity issues have arisen repeatedly in the practice of arithmetic; a famous instance is the question of complex-analytic considerations in the proof of the prime number theorem. This article surveys several such issues, and discusses ways in which logical considerations shed light on these issues.

The subject of the first section is Carnapian modal logic. One of the things I will do there is to prove that certain description principles, viz. the ''self-predication principles'', i.e. the principles according to which a descriptive term satisfies its own descriptive condition, are theorems and that others are not. The second section will be devoted to Carnapian modal arithmetic. I will prove that, if the arithmetical theory contains the standard weak principle of induction, modal truth collapses to truth. (...) Then I will propose a different formulation of Carnapian modal arithmetic and establish that it is free of collapse. Noteworthy is that one can retain the standard strong principle of induction. I will occupy myself in the third section with Carnapian epistemic logic and arithmetic. Here too it is claimed that the standard weak principle of induction is invalid and that the alternative principle is valid. In the fourth and last section I will get back to the self-predication principles and I will point to some of the consequences if one adds them to Carnapian Epistemic arithmetic. The interaction of self-predication principles and the strong principle of induction results in a collapse of de re knowability. (shrink)

In the first section of this paper we show that i Π1 ≡ W⌝⌝lΠ1 and that a Kripke model which decides bounded formulas forces iΠ1 if and only if the union of the worlds in any path in it satisflies IΠ1. In particular, the union of the worlds in any path of a Kripke model of HA models IΠ1. In the second section of the paper, we show that for equivalence of forcing and satisfaction of Πm-formulas in a linear Kripke (...) model deciding Δ0-formulas it is necessary and sufficient that the model be Σm-elementary. This implies that if a linear Kripke model forces PEMprenex, then it forces PEM. We also show that, for each n ≥ 1, iΦn does not prove ℋ(IΠn's are Burr's fragments of HA. (shrink)

The basic human ability to treat quantitative information can be divided into two parts. With proto-arithmetical ability, based on the core cognitive abilities for subitizing and estimation, numerosities can be treated in a limited and/or approximate manner. With arithmetical ability, numerosities are processed (counted, operated on) systematically in a discrete, linear, and unbounded manner. In this paper, I study the theory of enculturation as presented by Menary (2015) as a possible explanation of how we make the move from the proto-arithmetical (...) ability to arithmetic proper. I argue that enculturation based on neural reuse provides a theoretically sound and fruitful framework for explaining this development. However, I show that a comprehensive explanation must be based on valid theoretical distinctions and involve several stages in the development of arithmetical knowledge. I provide an account that meets these challenges and thus leads to a better understanding of the subject of enculturation. (shrink)

The primary purpose of this note is to demonstrate that predicative Frege arithmetic naturally interprets certain weak but non-trivial arithmetical theories. It will take almost as long to explain what this means and why it matters as it will to prove the results.

This present collection of (translations of) reviews is intended to help obtain a more balanced picture of the reception and impact of Edmund Husserl’s first book, the 1891 Philosophy of Arithmetic. One of the insights to be gained from this non-exhaustive collection of reviews is that the Philosophy of Arithmetic had a much more widespread reception than hitherto assumed: in the present collection alone there already are fourteen, all published between 1891 and 1895. Three of the reviews appeared (...) in mathematical journals (Jahrbuch über die Fortschritte der Mathematik, Zeitschrift für Mathematik und Physik, and Zeitschrift für mathema- tischen und naturwissenschaftlichen Unterricht), three were published in English journals (The Philosophical Review, The Monist, Mind), two were written by other members of the School of Brentano (Franz Hillebrand and Alois Höfler). Some of the reviews and notices appear to be very superficial, consisting merely of para- phrases (often without references) and lists of topics taken from the table of con- tents, presenting barely acceptable summaries. Others, among which Höfler might be the most significant, engage much more deeply with the topics and problems that Husserl addresses, analyzing his approach in the context of the mathematics of his time and the School of Brentano. (shrink)

Aims: The purpose of this study is to identify the factors associated with the academic performance of Grade 10 students during their First Quarter of school as well as the significant correlation between the factors and student academic performance in Mathematics. -/- Study Design: Descriptive Correlation Design. -/- Place and Duration of Study: The study was conducted at Agusan National High School in Cagayan de Or City's East 1 District during the school year: 2022 – 2023. -/- Methodology: (...) The respondents were Two hundred thirty-one (231) students in Grade 10 at Agusan National High School in Cagayan de Oro City's East 1 District. This study used a researcher-made questionnaire that underwent validity and reliability testing and the academic performance of the students. -/- Results: The results showed that students agree with routines related to mastering Mathematics at a high level. Furthermore, there is no correlation between student study habits and Mathematics performance in terms of self-confidence, but there is a substantial positive association between the student's study habits and performance in terms of attitude. The study habits and learning techniques used by students were found to be important determinants of how well they performed in Mathematics. The researcher strongly suggested using the enhancement plan in teaching Mathematics to Junior High school students. -/- Conclusion: Students have a very positive attitude toward their study habits when learning Mathematics. Students felt that studying and learning Mathematics was essential. Students' success and academic growth depend heavily on their Mathematics performance. It is essential for pupils to master and comprehend its concepts. Students' study habits in terms of attitudes have an impact on how they learn Mathematics. (shrink)

Neo-Fregean logicists claim that Hume’s Principle (HP) may be taken as an implicit definition of cardinal number, true simply by fiat. A long-standing problem for neo-Fregean logicism is that HP is not deductively conservative over pure axiomatic second-order logic. This seems to preclude HP from being true by fiat. In this paper, we study Richard Kimberly Heck’s Two-Sorted Frege Arithmetic (2FA), a variation on HP which has been thought to be deductively conservative over second-order logic. We show that it (...) isn’t. In fact, 2FA is not conservative over n-th order logic, for all $n \geq 2$. It follows that in the usual one-sorted setting, HP is not deductively Field-conservative over second- or higher-order logic. (shrink)

We argue that, under the usual assumptions for sufficiently strong arithmetical theories that are subject to Gödel’s First Incompleteness Theorem, one cannot, without impropriety, talk about *the* Gödel sentence of the theory. The reason is that, without violating the requirements of Gödel’s theorem, there could be a true sentence and a false one each of which is provably equivalent to its own unprovability in the theory if the theory is unsound.

Whereas recent commentators have suggested that consumer demand, typecasting and marketing lead performers to maintain continuities across films, I argue that cinema has historically made it difficult to subtract performers from roles, leading to relatively constant comportment, and that casting, marketing and audience preference are not only causes but also effects of this. I do so using thought experiments and empirical experiments, for example, by pondering why people say they see Jesus in paintings of him and rarely mention models, but (...) stress that they see actors when encountering photographic stills of performers portraying him. Arguably, this relates to what photographs have historically come to mean, and these meanings would make it difficult for audiences to subtract and not see the actor. Based on such thinking, along with what filmmakers have said and done, and adding to classic accounts of Cavell, Santayana and others, I build the case that cinematic media invite performers to play themselves. (shrink)

We take an argument of Gödel's from his ground‐breaking 1931 paper, generalize it, and examine its validity. The argument in question is this: "the sentence G says about itself that it is not provable, and G is indeed not provable; therefore, G is true".

A speculative investigation of how Frege's logical views change between Begriffsschrift and Grundgesetze and how this might have affected the formal development of logicism.

Second-order Peano Arithmetic minus the Successor Axiom is developed from first principles through Quadratic Reciprocity and a proof of self-consistency. This paper combines 4 other papers of the author in a self-contained exposition.

Hilbert arithmetic in a wide sense, including Hilbert arithmetic in a narrow sense consisting by two dual and anti-isometric Peano arithmetics, on the one hand, and the qubit Hilbert space (originating for the standard separable complex Hilbert space of quantum mechanics), on the other hand, allows for an arithmetic version of Gentzen’s cut elimination and quantum measurement to be described uniformy as two processes occurring accordingly in those two branches. A philosophical reflection also justifying that unity by (...) quantum neo-Pythagoreanism links it to the opposition of propositional logic, to which Gentzen’s cut rule refers immediately, on the one hand, and the linguistic and mathematical theory of metaphor therefore sharing the same structure borrowed from Hilbert arithmetic in a wide sense. An example by hermeneutical circle modeled as a dual pair of a syllogism (accomplishable also by a Turing machine) and a relevant metaphor (being a formal and logical mistake and thus fundamentally inaccessible to any Turing machine) visualizes human understanding corresponding also to Gentzen’s cut elimination and the Gödel dichotomy about the relation of arithmetic to set theory: either incompleteness or contradiction. The metaphor as the complementing “half” of any understanding of hermeneutical circle is what allows for that Gödel-like incompleteness to be overcome in human thought. (shrink)

El objetivo del presente artículo de reflexión es hacer evidente la relación entre la categoría de performatividad en Judith Butler y el arte de la performance como una aproximación a la construcción de la memoria. Por un lado, el arte de la performance ofrece un problema en torno al registro, al archivo y a la memoria de la acción artística al considerar que el acto es irrepetible, único y fugaz. Esta discusión está vinculada con la reperformación como alternativa (...) de memoria que conserva la naturaleza de esta prácticaartística. Por otro lado, de acuerdo con Butler, la identidad es un efecto de una serie de performances regulados, esto revela cómo la performatividad sugiere un tipo de identidad no esencial donde la memoria está siempre abierta a nuevas inscripciones cuyo soporte está en la reiteración de actos discontinuos. En consecuencia, la repetición de los actos que propone Butler permite comprender que en la reperformance existe otra forma para abordar una memoria restaurada. (shrink)

A natural problem from elementary arithmetic which is so strongly undecidable that it is not even Trial and Error decidable (in other words, not decidable in the limit) is presented. As a corollary, a natural, elementary arithmetical property which makes a difference between intuitionistic and classical theories is isolated.

The Turing machine halting problem can be explained by several factors, including arithmetic logic irreversibility and memory erasure, which contribute to computational uncertainty due to information loss during computation. Essentially, this means that an algorithm can only preserve information about an input, rather than generate new information. This uncertainty arises from characteristics such as arithmetic logical irreversibility, Landauer's principle, and memory erasure, which ultimately lead to a loss of information and an increase in entropy. To measure this uncertainty (...) and loss of information, the concept of arithmetic logical entropy can be used. The Turing machine and its equivalent, general recursive functions can be understood through the λ calculus and the Turing/Church thesis. However, there are certain recursive functions that cannot be fully understood or predicted by other algorithms due to the loss of information during logical-arithmetic operations. In other words, the behaviour of these functions cannot be completely determined at every stage of the computation due to a lack of information in their definition. While there are some cases where the behaviour of recursive functions is highly predictable, the lack of information in most cases makes it impossible for algorithms to determine if a program will halt or not. This inability to predict the outcome of the computation is the essence of the halting problem of the Turing machine. Even in cases where more information is available about the program, it is still difficult to determine with certainty if the program will halt or not. This also highlights the importance of the Turing oracle machine, which introduces information from outside the computation to compensate for the lack of information and ultimately decide the result of the computation. (shrink)

Why would we want to develop artificial human-like arithmetical intelligence, when computers already outperform humans in arithmetical calculations? Aside from arithmetic consisting of much more than mere calculations, one suggested reason is that AI research can help us explain the development of human arithmetical cognition. Here I argue that this question needs to be studied already in the context of basic, non-symbolic, numerical cognition. Analyzing recent machine learning research on artificial neural networks, I show how AI studies could potentially (...) shed light on the development of human numerical abilities, from the proto-arithmetical abilities of subitizing and estimating to counting procedures. Although the current results are far from conclusive and much more work is needed, I argue that AI research should be included in the interdisciplinary toolbox when we try to explain the development and character of numerical cognition and arithmetical intelligence. This makes it relevant also for the epistemology of mathematics. (shrink)

Recent years have seen an explosion of empirical data concerning arithmetical cognition. In this paper that data is taken to be philosophically important and an outline for an empirically feasible epistemological theory of arithmetic is presented. The epistemological theory is based on the empirically well-supported hypothesis that our arithmetical ability is built on a protoarithmetical ability to categorize observations in terms of quantities that we have already as infants and share with many nonhuman animals. It is argued here that (...) arithmetical knowledge developed in such a way cannot be totally conceptual in the sense relevant to the philosophy of arithmetic, but neither can arithmetic understood to be empirical. Rather, we need to develop a contextual a priori notion of arithmetical knowledge that preserves the special mathematical characteristics without ignoring the roots of arithmetical cognition. Such a contextual a priori theory is shown not to require any ontologically problematic assumptions, in addition to fitting well within a standard framework of general epistemology. (shrink)

Øystein Linnebo has recently shown that the existence of successors cannot be proven in predicative Frege arithmetic, using Frege’s definitions of arithmetical notions. By contrast, it is shown here that the existence of successor can be proven in ramified predicative Frege arithmetic.

According to Streeck and Vogl, the neoliberalization of the state has been the result of political-economic developments that render the state dependent on financial markets. However, they do not explain the discursive shifts that would have been required for demoting the state to the role of an agent to bondholders. I propose to explain this shift via the performative effect of neoliberal agency theory. In 1976, Michael Jensen and William Meckling claimed that corporate managers are agents to shareholding principals, which (...) implied that their main task was the procurement of shareholder value. Agency theory subsequently prescribes a series of measures to ensure the alignment of principal and agent interests in corporations. The diffusion of agency theory, however, moved beyond corporate governance to reconfigure the state. Due to its reliance on capital markets, the state supposedly likewise becomes an agent of the investment public and should procure bondholder value. (shrink)

The aim of the research is to identify the relationship between the performance criteria and the achievement of the objectives of supervision which is represented in the performance of the job at the Islamic University in Gaza Strip. To achieve the objectives of the research, the researchers used the descriptive analytical approach to collect information. The questionnaire consisted of (22) paragraphs distributed to three categories of employees of the Islamic University (senior management, faculty members, their assistants and members (...) of the administrative board). A random sample of 314 employees was selected, 276 responses were retrieved with a return rate of 88.1%. The SPSS program was used to enter, process, and analyze the data. The results of the study showed a positive relationship between the performance criteria and the achievement of the control objectives represented by the job performance in the Islamic University from the point of view of the members (senior management, faculty and their assistants and the administrative board). The researchers also recommended a number of recommendations, including the provision of an appropriate level of control system components today through the continuous updating and development of performance standards and the need to provide the necessary physical and financial resources to continue the development and achievement within the university. Expand the development of technology in the various activities of the university through the construction of a complete and integrated system to support the control systems in the university to suit its size. The researchers also recommended the follow-up, review of the performance standards and work to modify them in line with the mission of the university and the goals that the university seeks to reach. (shrink)

This paper aims to determine knowledge management (KM) factors which have strong impact on high performance. Also, the study aims to compare KMM between intermediate colleges. This study was applied on three intermediate colleges in Gaza strip, Palestine. Asian productivity organization model was applied to measure KMM. Second dimension which assess high performance was developed by the authors. The controlled sample was 190. Several statistical tools were used for data analysis and hypotheses testing, including reliability correlation using Cronbach’s (...) alpha, “ANOVA”, simple linear regression and step wise regression and LSD test. The overall findings of the current study show that maturity level is in the second level. Findings also support the main hypothesis and its sub- hypotheses. The most important factors effecting high performance are the processes, KM leadership, people and KM outcomes. In addition, there are differences in high performance for college (PTC). Furthermore, the current study is unique by the virtue of its nature, scope and way of implied investigation, as it is the first comparative study between intermediate colleges in Palestine that explores the status of KMM using the Asian productivity model. Research limitations was that the survey findings were based on intermediate colleges in Gaza Strip. (shrink)