This chapter examines the pioneering work of Gottfried Wilhelm Leibniz (1646-1716) on various number systems, in particular binary, which he independently invented in the mid-to-late 1670s, and hexadecimal, which he invented in 1679. The chapter begins with the oft-debated question of who may have influenced Leibniz’s invention of binary, though as none of the proposed candidates is plausible I suggest a different hypothesis, that Leibniz initially developed binary notation as a tool to assist his investigations in mathematical problems that (...) were exercising him at the time, namely those concerning the divisibility of composite numbers, primality, and perfect numbers. The chapter then explores Leibniz’s development of binary, his little-known work on binary fractions and expansions, his use of binary as a symbol for creation in his philosophical-theology, and his response to the suggestion that there was a correlation between binary numeration and the hexagrams of the ancient Chinese divinatory text, the Yijing. The chapter then focuses on Leibniz’s work on other number systems, in particular his invention and exploration of hexadecimal as well as his work on duodecimal. The chapter concludes by revealing a hitherto unknown practical application of binary that Leibniz devised in the last year of his life. (shrink)

I argue that the human mind includes an innate domain-specific system for representing precise small numerical quantities. This theory contrasts with object-tracking theories and with domain-general theories that only make use of mental models. I argue that there is a good amount of evidence for innate representations of small numerical quantities and that such a domain-specific system has explanatory advantages when infants’ poor working memory is taken into account. I also show that the mental models approach requires previously (...) unnoticed domain-specific structure and consequently that there is no domain-general alternative to an innate domain-specific small number system. (shrink)

The approximate number system (ANS) enables organisms to represent the approximate number of items in an observed collection, quickly and independently of natural language. Recently, it has been proposed that the ANS goes beyond representing natural numbers by extracting and representing rational numbers (Clarke & Beck, 2021a). Prior work demonstrates that adults and children discriminate ratios in an approximate and ratio-dependent manner, consistent with the hallmarks of the ANS. Here, we use a well-known “connectedness illusion” to provide (...) evidence that these ratio-dependent ratio discriminations are (a) based on the perceived number of items in seen displays (and not just non-numerical confounds), (b) are not dependent on verbal working memory, or explicit counting routines, and (c) involve representations with a part-whole (or subset-superset) format, like a fraction, rather than a part-part format, like a ratio. These results vindicate key predictions of the hypothesis that the ANS represents rational numbers. (shrink)

All humans share a universal, evolutionarily ancient approximate number system (ANS) that estimates and combines the numbers of objects in sets with ratio-limited precision. Interindividual variability in the acuity of the ANS correlates with mathematical achievement, but the causes of this correlation have never been established. We acquired psychophysical measures of ANS acuity in child and adult members of an indigene group in the Amazon, the Mundurucú, who have a very restricted numerical lexicon and highly variable access to (...) mathematics education. By comparing Mundurucú subjects with and without access to schooling, we found that education significantly enhances the acuity with which sets of concrete objects are estimated. These results indicate that culture and education have an important effect on basic number perception. We hypothesize that symbolic and nonsymbolic numerical thinking mutually enhance one another over the course of mathematics instruction. (shrink)

This article accomplishes two goals. First, the paper clarifies Edmund Husserl’s investigation of the historical inception of the number system from his early works, Philosophy of Arithmetic and, “On the Logic of Signs (Semiotic)”. The article explores Husserl’s analysis of five historical developmental stages, which culminated in our ancestor’s ability to employ and enumerate with number signs. Second, the article reveals how Husserl’s conclusions about the history of the number system from his early works opens (...) up a fusion point with his investigations from his mature texts, The Crisis of the European Sciences and “The Origin of Geometry”. On the one hand, the essay shows that Husserl’s methodology was similar, as he sought in both his early and late writings to uncover the essence of the history of the formal sciences and was not executing mere intellectual history. On the other hand, the article discloses that Husserl’s insights from both time periods are strikingly analogous. Already in his early texts, Husserl saw that the sciences emerged from pre-theoretical experiences of the world and that the sciences are the result of a historical process, which involves the psychic activities of past individuals and the maintaining of discoveries over time by intersubjective communities. I conclude by showing how, in light of the analysis of this paper, we can rethink the evolution of Husserl’s philosophy. (shrink)

This Element, written for researchers and students in philosophy and the behavioral sciences, reviews and critically assesses extant work on number concepts in developmental psychology and cognitive science. It has four main aims. First, it characterizes the core commitments of mainstream number cognition research, including the commitment to representationalism, the hypothesis that there exist certain number-specific cognitive systems, and the key milestones in the development of number cognition. Second, it provides a taxonomy of influential views within (...) mainstream number cognition research, along with the central challenges these views face. Third, it identifies and critically assesses a series of core philosophical assumptions often adopted by number cognition researchers. Finally, the Element articulates and defends a novel version of pluralism about number concepts. (shrink)

On a now orthodox view, humans and many other animals possess a “number sense,” or approximate number system, that represents number. Recently, this orthodox view has been subject to numerous critiques that question whether the ANS genuinely represents number. We distinguish three lines of critique – the arguments from congruency, confounds, and imprecision – and show that none succeed. We then provide positive reasons to think that the ANS genuinely represents numbers, and not just non-numerical (...) confounds or exotic substitutes for number, such as “numerosities” or “quanticals,” as critics propose. In so doing, we raise a neglected question: numbers of what kind? Proponents of the orthodox view have been remarkably coy on this issue. But this is unsatisfactory since the predictions of the orthodox view, including the situations in which the ANS is expected to succeed or fail, turn on the kind of number being represented. In response, we propose that the ANS represents not only natural numbers, but also non-natural rational numbers. It does not represent irrational numbers, however, and thereby fails to represent the real numbers more generally. This distances our proposal from existing conjectures, refines our understanding of the ANS, and paves the way for future research. (shrink)

In his [1937, 1938], Paul Dirac proposed his “Large Number Hypothesis” (LNH), as a speculative law, based upon what we will call the “Large Number Coincidences” (LNC’s), which are essentially “coincidences” in the ratios of about six large dimensionless numbers in physics. Dirac’s LNH postulates that these numerical coincidences reflect a deeper set of law-like relations, pointing to a revolutionary theory of cosmology. This led to substantial work, including the development of Dirac’s later [1969/74] cosmology, and other alternative (...) cosmologies, such as the Brans-Dicke modification of GTR, and to extensive empirical tests. We may refer to the generic hypothesis of “Large Number Relations” (LNR’s), as the proposal that there are lawlike relations of some kind between the dimensionless numbers, not necessarily those proposed in Dirac’s early LNH. Such relations would have a profound effect on our concepts of physics, but they remain shrouded in mystery. Although Dirac’s specific proposals for LNR theories have been largely rejected, the subject retains interest, especially among cosmologists seeking to test possible variations in fundamental constants, and to explain dark energy or the cosmological constant. In the first two sections here we briefly summarize the basic concepts of LNR’s. We then introduce an alternative LNR theory, using a systematic formalism to express variable transformations between conventional measurement variables and the true variables of the theory. We demonstrate some consistency results and review the evidence for changes in the gravitational constant G. The theory adopted in the strongest tests of Ġ/G, by the Lunar Laser Ranging (LLR) experiments, assumes: Ġ/G = 3(dr/dt)/r – 2(dP/dt)/P – (dm/dt)/m, as a fundamental relationship. Experimental measurements show the RHS to be close to zero, so it is inferred that significant changes in G are ruled out. However when the relation is derived in our alternative theory it gives: Ġ/G = 3(dr/dt)/r – 2(dP/dt)/P – (dm/dt)/m – (dR/dt)/R. The extra final term (which is the Hubble constant) is not taken into account in conventional derivations. This means the LLR experiments are consistent with our LNR theory (and others), and they do not really test for a changing value of G at all. This failure to transform predictions of LNR theories correctly is a serious conceptual flaw in current experiment and theory. (shrink)

In our target article, we argued that the number sense represents natural and rational numbers. Here, we respond to the 26 commentaries we received, highlighting new directions for empirical and theoretical research. We discuss two background assumptions, arguments against the number sense, whether the approximate number system represents numbers or numerosities, and why the ANS represents rational numbers.

The aim of this paper is to explore and understand the Albanian Party System. The analysis will cover the period from the collapse of the communist regime in 1991 until 2014. It will try to investigate what forces drive the battle of the parties, what cleavages 'divide' society and consequently the party system as well as which are the parties that count the most. in order to assess this, the paper will focus on the parliamentary parties and will (...) relay on various theoretical concepts of the contemporary literature on political parties. First it will draw on Sartori's concept of 'relevance' in order to understand whether all the parliamentary parties over the course of twenty four years of pluralism are important or not and what weight do they have. Then it will use the Laakso - Taagepera index of the Effective Number of Parties to understand how power is distributed within the parliament. (shrink)

On a now orthodox view, humans and many other animals are endowed with a “number sense”, or approximate number system (ANS), that represents number. Recently, this orthodox view has been subject to numerous critiques, with critics maintaining either that numerical content is absent altogether, or else that some primitive analog of number (‘numerosity’) is represented as opposed to number itself. We distinguish three arguments for these claims – the arguments from congruency, confounds, and imprecision (...) – and show that none succeed. We then highlight positive reasons for thinking that the ANS genuinely represents numbers. The upshot is that proponents of the orthodox view should not feel troubled by recent critiques of their position. (shrink)

Do scientific theories limit human knowledge? In other words, are there physical variables hidden by essence forever? We argue for negative answers and illustrate our point on chaotic classical dynamical systems. We emphasize parallels with quantum theory and conclude that the common real numbers are, de facto, the hidden variables of classical physics. Consequently, real numbers should not be considered as ``physically real" and classical mechanics, like quantum physics, is indeterministic.

Numbers are concepts whose content, structure, and organization are influenced by the material forms used to represent and manipulate them. Indeed, as argued here, it is the inclusion of multiple forms (distributed objects, fingers, single- and two-dimensional forms like pebbles and abaci, and written notations) that is the mechanism of numerical elaboration. Further, variety in employed forms explains at least part of the synchronic and diachronic variability that exists between and within cultural number systems. Material forms also impart characteristics (...) like linearity that may persist in the form of knowledge and behaviors, ultimately yielding numerical concepts that are irreducible to and functionally independent of any particular form. Material devices used to represent and manipulate numbers also interact with language in ways that reinforce or contrast different aspects of numerical cognition. Not only does this interaction potentially explain some of the unique aspects of numerical language, it suggests that the two are complementary but ultimately distinct means of accessing numerical intuitions and insights. The potential inclusion of materiality in contemporary research in numerical cognition is advocated, both for its explanatory power, as well as its influence on psychological, behavioral, and linguistic aspects of numerical cognition. (shrink)

A new advanced systems theory concerning the emergent nature of the Social, Consciousness, and Life based on Mathematics and Physical Analogies is presented. This meta-theory concerns the distance between the emergent levels of these phenomena and their ultra-efficacious nature. The theory is based on the distinction between Systems and Meta-systems (organized Openscape environments). We first realize that we can understand the difference between the System and the Meta-system in terms of the relationship between a ‘Whole greater than the (...) sum of the parts’ and a ‘Whole less than the sum of its parts’, i.e., a whole full of holes (like a sponge) that provide niches for systems in the environment. Once we understand this distinction and clarify the nature of the unusual organization of the Meta-system, then it is possible to understand that there is a third possibility which is a whole exactly equal to the sum of its parts that is only supervenient like perfect numbers. In fact, there are three kinds of Special System corresponding to the perfect, amicable, and sociable aliquot numbers. These are all equal to the sum of their parts but with different degrees of differing and deferring in what Jacques Derrida calls “differance”. All other numbers are either excessive (systemic) or deficient (metasystemic) in this regard. The Special Systems are based on various mathematical analogies and some physical analogies. But the most important of the mathematical analogies are the hypercomplex algebras, which include the Complex Numbers, Quaternions, and Octonions, with the Sedenions corresponding to the Emergent Meta-system. However, other analogies are the Hopf fibrations between hyperspheres of various dimensions, nonorientable surfaces, soliton solutions, etc. These Special Systems have a long history within the tradition since they can be traced back to the imaginary cities of Plato. The Emergent Meta-system is a higher order global structure that includes the System with the three Special Systems as a cycle. An example of this from our tradition is in the Monadology of Gottfried Wilhelm von Leibniz. There is a conjunctive relationship between the System schema and the Special Systems that produce the Meta-system schema cycle. The Special Systems are a meta-model for the relationship between the emergent levels of Consciousness (Dissipative Ordering based on the theory of negative entropy of Prigogine), Living (Autopoietic Symbiotic based on the theory of Maturana and Varela), and Social (Reflexive based on the theory of John O’Malley and Barry Sandywell). These different special systems are related to the various existenitals identified by Martin Heidegger in Being and Time and various temporal reference frames identified by Richard M. Pico. We also relate the special systems to morphodynamic and teleodynamic systems of Terrence Deacon in Incomplete Nature to which we add sociodynamic systems to complete the series of Special Systems. (shrink)

An increasing number of regulations propose the notion of ‘AI audits’ as an enforcement mechanism for achieving transparency and accountability for artificial intelligence (AI) systems. Despite some converging norms around various forms of AI auditing, auditing for the purpose of compliance and assurance currently have little to no agreed upon practices, procedures, taxonomies, and standards. We propose the ‘criterion audit’ as an operationalizable compliance and assurance external audit framework. We model elements of this approach after financial auditing practices, and (...) argue that AI audits should similarly provide assurance to their stakeholders about AI organizations’ ability to govern their algorithms in ways that mitigate harms and uphold human values. We discuss the necessary conditions for the criterion audit, and provide a procedural blueprint for performing an audit engagement in practice. We illustrate how this framework can be adapted to current regulations by deriving the criteria on which ‘bias audits’ for hiring algorithms can be performed, as required by the recently effective New York City Local Law 144 of 2021. We conclude by offering critical discussion on the benefits, inherent limitations, and implementation challenges of applying practices of the more mature financial auditing industry to AI auditing where robust guardrails against quality assurance issues are only starting to emerge. Our discussion as informed by experiences in performing these audits in practice highlights the critical role that an audit ecosystem plays in ensuring the effectiveness of such methodology. (shrink)

In his highly perceptive, if underappreciated introduction to Wittgenstein’s Tractatus, Russell identifies a “lacuna” within Wittgenstein’s theory of number, relating specifically to the topic of transfinite number. The goal of this paper is two-fold. The first is to show that Russell’s concerns cannot be dismissed on the grounds that they are external to the Tractarian project, deriving, perhaps, from logicist ambitions harbored by Russell but not shared by Wittgenstein. The extensibility of Wittgenstein’s theory of number to the (...) case of transfinite cardinalities is, I shall argue, a desideratum generated by concerns intrinsic, and internal to Wittgenstein’s logical and semantic framework. Second, I aim to show that Wittgenstein’s theory of number as espoused in the Tractatus is consistent with Russell’s assessment, in that Wittgenstein meant to leave open the possibility that transfinite numbers could be generated within his system, but did not show explicitly how to construct them. To that end, I show how one could construct a transfinite number line using ingredients inherent in Wittgenstein’s system, and in accordance with his more general theories of number and of operations. (shrink)

In this study, the archaic counting systems of Mesopotamia as understood through the Neolithic tokens, numerical impressions, and proto-cuneiform notations were compared to the traditional number-words and counting methods of Polynesia as understood through contemporary and historical descriptions of vocabulary and behaviors. The comparison and associated analyses capitalized on the ability to understand well-known characteristics of Uruk-period numbers like object-specific counting, polyvalence, and context-dependence through historical observations of Polynesian counting methods and numerical language, evidence unavailable for ancient numbers. Similarities (...) between the two number systems were then used to argue that archaic Mesopotamian numbers, like those of Polynesia, were highly elaborated and would have served as cognitively efficient tools for mental calculation. Their differences also show the importance of material technologies like tokens, impressions, and notations to developing mathematics. This project has received funding from the European Union’s Horizon 2020 research and innovation program under grant agreement No. 785793. (shrink)

Physical systems can store information and their informational properties are governed by the laws of information. In particular, the amount of information that a physical system can convey is limited by the number of its degrees of freedom and their distinguishable states. Here we explore the properties of the physical systems with absolutely one degree of freedom. The central point in these systems is the tight limitation on their information capacity. Discussing the implications of this limitation we demonstrate (...) that such systems exhibit a number of features, such as randomness, no-cloning, and non-commutativity, which are peculiarities attributed to quantum mechanics (QM). After demonstrating many astonishing parallels to quantum behavior, we postulate an interpretation of quantum physics as the physics of systems with a single degree of freedom. We then show how a number of other quantum conundrum can be understood by considering the informational properties of the systems and also resolve the EPR paradox. In the present work, we assume that the formalism of the QM is correct and well-supported by experimental verification and concentrate on the interpretational aspects of the theory. (shrink)

Background: With the coming of the Industrial Revolution, the levels of pollution grow significantly. This Technological development contributed to the worsening of shortness breath problems in great shape. especially in infants and children. There are many shortness breath diseases that infants and children face in their lives. Shortness of breath is one of a very serious symptom in children and infants and should never be ignored. Objectives: Along these lines, the main goal of this expert system is to help (...) physician in diagnosing and describe some common causes of shortness of breath in infants and children by diagnosing their cases through our expert system. Moreover, this system which we are presenting will give patient the appropriate diagnosis of disease and the treatment required. Methods: In this paper the strategy of the expert system for diagnosing a number of the existed shortness of breath in infants and children diseases such as (Asthma , Bronchiolitis, Viral Pneumonia, cough, Shortness of breath' dyspnea ', Epiglottitis, Croup, ABSCESS in the tonsil 'peritonsillar abscess', Bronchitis, Viral Bronchitis, Wheezing, sudden infant death syndrome 'SIDS' ) is introduced, an overview about the shortness of breath in children and infants diseases are delineated, the cause of diseases are sketched and the treatment of disease whenever conceivable is given . SL5 Object Expert System language was utilized for designing and implementing the proposed expert system. Results: The proposed shortness of breath in children and infants diseases diagnosis expert system was estimated by Medical students and they were satisfied with its result. Conclusions: The Proposed expert system is very useful for Respiratory physician, pediatrician, recently graduated physician, and for children's parents with shortness of breath problem. (shrink)

Developing earlier studies of the system of numbers in Mundurucu, this paper argues that the Mundurucu numeral system is far more complex than usually assumed. The Mundurucu numeral system provides indirect but insightful arguments for a modular approach to numbers and numerals. It is argued that distinct components must be distinguished, such as a system of representation of numbers in the format of internal magnitudes, a system of representation for individuals and sets, and one-to-one correspondences (...) between the numerosity expressed by the number and its metrics. It is shown that while many-number systems involve a compositionality of units, sets and sets composed of units, few-number languages, such as Mundurucu, do not have access to sets composed of units in the usual way. The nonconfigurational character of the Mundurucu language, which is related to a property for which we coin the term 'low compositionality power', accounts for this and explains the curious fact that Mundurucus make use of marked one-to-one correspondence strategies in order to overcome the limitations of the core system at the perceptual/motor interface of the language faculty. We develop an analysis of a particular construction, parallel numbers, which has not been studied before, elucidating the whole system. This analysis, we argue, sheds new light on classical philosophical, psychological and linguistic debates about numbers and numerals and their relation to language, and more particularly, sheds light on few-number languages. (shrink)

The term “Complex Systems Biology” was introduced a few years ago [Kaneko, 2006] and, although not yet of widespread use, it seems particularly well suited to indicate an approach to biology which is well rooted in complex systems science. Although broad generalizations are always dangerous, it is safe to state that mainstream biology has been largely dominated by a gene-centric view in the last decades, due to the success of molecular biology. So the one gene - one trait approch, which (...) has often proved to be effective, has been extended to cover even complex traits. This simplifying view has been appropriately criticized, and the movement called systems biology has taken off. Systems biology [Noble, 2006] emphasizes the presence of several feedback loops in biological systems, which severely limit the range of validity of explanations based upon linear causal chains (e.g. gene → behaviour). Mathematical modelling is one the favourite tools of systems biologists to analyze the possible effects of competing negative and positive feedback loops which can be observed at several levels (from molecules to organelles, cells, tissues, organs, organisms, ecosystems). Systems biology is by now a well-established field, as it can be inferred by the rapid growth in number of conferences and journals devoted to it, as well as by the existence of several grants and funded projects.Systems biology is mainly focused upon the description of specific biological items, like for example specific organisms, or specific organs in a class of animals, or specific genetic-metabolic circuits. It therefore leaves open the issue of the search for general principles of biological organization, which apply to all living beings or to at least to broad classes. We know indeed that there are some principles of this kind, biological evolution being the most famous one. The theory of cellular organization also qualifies as a general principle. But the main focus of biological research has been that of studying specific cases, with some reluctance to accept (and perhaps a limited interest for) broad generalizations. This may however change, and this is indeed the challenge of complex systems biology: looking for general principles in biological systems, in the spirit of complex systems science which searches for similar features and behaviours in various kinds of systems. The hope to find such general principles appears well founded, and I will show in Section 2 that there are indeed data which provide support to this claim. Besides data, there are also general ideas and models concerning the way in which biological systems work. The strategy, in this case, is that of introducing simplified models of biological organisms or processes, and to look for their generic properties: this term, borrowed from statistical physics, is used for those properties which are shared by a wide class of systems. In order to model these properties, the most effective approach has been so far that of using ensembles of systems, where each member can be different from another one, and to look for those properties which are widespread. This approach was introduced many years ago [Kauffman, 1971] in modelling gene regulatory networks. At that time one had very few information about the way in which the expression of a given gene affects that of other genes, apart from the fact that this influence is real and can be studied in few selected cases (like e.g. the lactose metabolism in E. coli). Today, after many years of triumphs of molecular biology, much more has been discovered, however the possibility of describing a complete map of gene-gene interactions in a moderately complex organism is still out of reach. Therefore the goal of fully describing a network of interacting genes in a real organism could not be (and still cannot be) achieved. But a different approach has proven very fruitful, that of asking what are the typical properties of such a set of interacting genes. Making some plausible hypotheses and introducing some simplifying assumptions, Kauffman was able to address some important problems. In particular, he drew attention to the fact that a dynamical system of interacting genes displays selforganizing properties which explain some key aspects of life, most notably the existence of a limited number of cellular types in every multicellular organism (these numbers are of the order of a few hundreds, while the number of theoretically possible types, absent interactions, would be much much larger than the number of protons in the universe). In section 3 I will describe the ensemble based approach in the context of gene regulatory networks, and I will show that it can describe some important experimental data. Finally, in section 4 I will discuss some methodological aspects. (shrink)

Strong nativist views about numerical concepts claim that human beings have at least some innate precise numerical representations. Weak nativist views claim only that humans, like other animals, possess an innate system for representing approximate numerical quantity. We present a new strong nativist model of the origins of numerical concepts and defend the strong nativist approach against recent cross-cultural studies that have been interpreted to show that precise numerical concepts are dependent on language and that they are restricted to (...) speakers of languages with the right kind of structure. (shrink)

Theories of number concepts often suppose that the natural numbers are acquired as children learn to count and as they draw an induction based on their interpretation of the first few count words. In a bold critique of this general approach, Rips, Asmuth, Bloomfield [Rips, L., Asmuth, J. & Bloomfield, A.. Giving the boot to the bootstrap: How not to learn the natural numbers. Cognition, 101, B51–B60.] argue that such an inductive inference is consistent with a representational system (...) that clearly does not express the natural numbers and that possession of the natural numbers requires further principles that make the inductive inference superfluous. We argue that their critique is unsuccessful. Provided that children have access to a suitable initial system of representation, the sort of inductive inference that Rips et al. call into question can in fact facilitate the acquisition of larger integer concepts without the addition of any further principles. (shrink)

This paper demonstrates that Edmund Husserl’s frequently overlooked 1890 manuscript, “On the Logic of Signs,” when closely investigated, reveals itself to be the hermeneutical touchstone for his seminal 1891 Philosophy of Arithmetic. As the former comprises Husserl’s earliest attempt to account for all of the different kinds of signitive experience, his conclusions there can be directly applied to the latter, which is focused on one particular type of sign; namely, number signs. Husserl’s 1890 descriptions of motivating and replacing signs (...) will be respectively employed to clarify his 1891 understanding of the authentic and inauthentic presentations of numbers via number signs. Moreover, his schematic classification of replacement-signs in Semiotic will illuminate the reasons why he believed the number system to be necessary for the operation of replacing number signs. (shrink)

The relationship between abstract formal procedures and the activities of actual physical systems has proved to be surprisingly subtle and controversial, and there are a number of competing accounts of when a physical system can be properly said to implement a mathematical formalism and hence perform a computation. I defend an account wherein computational descriptions of physical systems are high-level normative interpretations motivated by our pragmatic concerns. Furthermore, the criteria of utility and success vary according to our diverse (...) purposes and pragmatic goals. Hence there is no independent or uniform fact to the matter, and I advance the ‘anti-realist’ conclusion that computational descriptions of physical systems are not founded upon deep ontological distinctions, but rather upon interest-relative human conventions. Hence physical computation is a ‘conventional’ rather than a ‘natural’ kind. (shrink)

Donors to global health programs and policymakers within national health systems have to make difficult decisions about how to allocate scarce health care resources. Principled ways to make these decisions all make some use of summary measures of health, which provide a common measure of the value (or disvalue) of morbidity and mortality. They thereby allow comparisons between health interventions with different effects on the patterns of death and ill health within a population. The construction of a summary measure of (...) health requires that a number be assigned to the harm of death. But the harm of death is currently a matter of debate: different philosophical theories assign very different values to the harm of death at different ages. This chapter considers how we should assign numbers to the harm of deaths at different ages in the face of uncertainty and disagreement. (shrink)

In his early philosophy as well as in his middle period, Wittgenstein holds a purely syntactic view of logic and mathematics. However, his syntactic foundation of logic and mathematics is opposed to the axiomatic approach of modern mathematical logic. The object of Wittgenstein’s approach is not the representation of mathematical properties within a logical axiomatic system, but their representation by a symbolism that identifies the properties in question by its syntactic features. It rests on his distinction of descriptions and (...) operations; its aim is to reduce mathematics to operations. This paper illustrates Wittgenstein’s approach by examining his discussion of irrational numbers. (shrink)

William Lane Craig has argued that there cannot be actual infinities because inverse operations are not well-defined for infinities. I point out that, in fact, there are mathematical systems in which inverse operations for infinities are well-defined. In particular, the theory introduced in John Conway's *On Numbers and Games* yields a well-defined field that includes all of Cantor's transfinite numbers.

The boundless nature of the natural numbers imposes paradoxically a high formal bound to the use of standard artificial computer programs for solving conceptually challenged problems in number theory. In the context of the new cognitive foundations for mathematics' and physics' program immersed in the setting of artificial mathematical intelligence, we proposed a refined numerical system, called the physical numbers, preserving most of the essential intuitions of the natural numbers. Even more, this new numerical structure additionally possesses the (...) property of being a bounded object allowing us to work with quite similar axioms like the classic Peano axioms, but in a finite environment and with an enriched physical dimension. Finally, we present several enlightening examples and we conclude that the physical numbers provide a natural formal setting for approaching classic problems in number theory from a more hybrid perspective, i.e. with a potential participation in the generation of solutions, not only of working mathematicians but also of more sophisticated artificial interactive (mathematical) intelligent agents (within the context of cognitive-computational metamathematics or artificial mathematical intelligence) and more controlled by physically-inspired principles. Finally, this paper addresses a highly new, bottom-up and paradigm-shifting approach to the foundations of physics: One should refine and improve the conceptual formal setting of the language that we use for understanding physical phenomena (e.g. the mathematical grounding concepts and theories) for being able to understand better more subtle and complex physical phenomena. (shrink)

In this paper a Knowledge-Based System (KBS) for determining the appropriate students major according to his/her preferences for sophomore student enrolled in the Faculty of Engineering and Information Technology in Al-Azhar University of Gaza was developed and tested. A set of predefined criterions that is taken into consideration before a sophomore student can select a major is outlined. Such criterion as high school score, score of subject such as Math I, Math II, Electrical Circuit I, and Electronics I taken (...) during the student freshman year, number of credits passed, student cumulative grade point average of freshman year, among others, were then used as input data to KBS. KBS was designed and developed using Simpler Level Five (SL5) Object expert system language. KBS was tested on three generation of sophomore students from the Faculty of Engineering and Information Technology of the Al-Azhar University, Gaza. The results of the evaluation show that the KBS is able to correctly determine the appropriate students major without errors. (shrink)

The Unified Medical Language System and the Gene Ontology are among the most widely used terminology resources in the biomedical domain. However, when we evaluate them in the light of simple principles for wellconstructed ontologies we find a number of characteristic inadequacies. Employing the theory of granular partitions, a new approach to the understanding of ontologies and of the relationships ontologies bear to instances in reality, we provide an application of this theory in relation to an example drawn (...) from the context of the pathophysiology of hypertension. This exercise is designed to demonstrate how, by taking ontological principles into account we can create more realistic biomedical ontologies which will also bring advantages in terms of efficiency and robustness of associated software applications. (shrink)

Physical systems without time and dynamics have been considered. The principle of how to construct spacetime in a physical system without time and dynamics has been proposed. It has been found what can be objects in such a spacetime, and what can be an interaction between such objects. Within the framework of the considered class of systems, answers to the following problems of philosophy and physics have been found: the nature of consciousness and the connection of body and consciousness (...) (mind-body problem), the nature of time, the anthropic principle and the problem of fine-tuning the universe, the effectiveness of mathematics in describing physical phenomena, the limits of knowledge. There are a number of indications that our Universe is not based on one of the systems of the class considered. The considered class of systems makes it possible to find answers to questions that cannot be answered for our Universe. This shows that it is fundamentally possible to find answers to these questions for our Universe as well. (shrink)

It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can’t contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant. Moreover, a better terminology for the so-called real numbers is “random numbers”, as their series of bits are truly random. I propose an alternative classical mechanics, which is (...) empirically equivalent to classical mechanics, but uses only finite-information numbers. This alternative classical mechanics is non-deterministic, despite the use of deterministic equations, in a way similar to quantum theory. Interestingly, both alternative classical mechanics and quantum theories can be supplemented by additional variables in such a way that the supplemented theory is deterministic. Most physicists straightforwardly supplement classical theory with real numbers to which they attribute physical existence, while most physicists reject Bohmian mechanics as supplemented quantum theory, arguing that Bohmian positions have no physical reality. (shrink)

In the last 20 years, a stream of research emerged under the label of „complex problem solving“ (CPS). This research was intended to describe the way people deal with complex, dynamic, and intransparent situations. Complex computer-simulated scenarios were as stimulus material in psychological experiments. This line of research lead to subtle insights into the way how people deal with complexity and uncertainty. Besides these knowledge-rich, realistic, intransparent, complex, dynamic scenarios with many variables, a second line of research used more simple, (...) knowledge-lean scenarios with a low number of variables („minimal complex systems“, MCS) that have been proposed recently in problem-solving research for the purpose of educational assessment. In both cases, the idea behind the use of microworlds is to increase validity of problem solving tasks by presenting interactive environments that can be explored and controlled by participants while pursuing certain action goals. The main argument presented here is: both types of systems - CPS and MCS – can only be dealt with successfully if causal dependencies between input and output variables are identified and used for system control. System knowledge is necessary for control and intervention. But CPS and MCS differ in their way of how causal dependencies are identified and how the mental model is constructed; therefore, they cannot be compared directly to each other with respect to the cognitive processes that are necessary for solving the tasks. Knowledge-poor MCS tasks address only a small fraction of the cognitive processes and structures needed for knowledge-rich CPS situations. (shrink)

Organisations are computing systems. The university’s sports centre is a computing system for managing sports teams and facilities. The tenure committee is a computing system for assigning tenure status. Despite an increasing number of publications in group ontology, the computational nature of organisations has not been recognised. The present paper is the first in this debate to propose a theory of organisations as groups structured for computing. I begin by describing the current situation in group ontology and (...) by spelling out the thesis in more detail. I then present the example of a sports centre to illustrate why one might intuitively think of organisations as computing systems. To substantiate the thesis, I introduce Piccinini’s restrictive analysis of physical computation. As I show, organisations meet all criteria for being computing systems. Organisations are structured groups with the function of manipulating medium-independent vehicles according to rules. Furthermore, I argue for the modal claim that this is a necessary feature of organisations. Having sketched the computational account of organisations, I compare it to other proposals in the literature. (shrink)

What are numbers, and where do they come from? A novel answer to these timeless questions is proposed by cognitive archaeologist Karenleigh A. Overmann, based on her groundbreaking study of material devices used for counting in the Ancient Near East—fingers, tallies, tokens, and numerical notations—as interpreted through the latest neuropsychological insights into human numeracy and literacy. The result, a unique synthesis of interdisciplinary data, outlines how number concepts would have been realized in a pristine original condition to develop into (...) one of the ancient world’s greatest mathematical traditions, a foundation for mathematical thinking today. In this view, numbers are abstract from their inception and materially bound at their most elaborated. The research updates historical work on Neolithic tokens and interpretations of Mesopotamian numbers, challenging several longstanding assumptions about numbers in the process. The insights generated are also applied to the role of materiality in human cognition more generally, including how concepts become distributed across and independent of the material forms used for their representation and manipulation; how societies comprised of average individuals use material structures to create elaborated systems of numeracy and literacy; and the differences between thinking through and thinking about materiality. (shrink)

In this paper, we introduce for the first time the neutrosophic quadruple numbers (of the form a + bT + cI + dF) and the refined neutrosophic quadruple numbers. Then we define an absorbance law, based on a prevalence order, both of them in order to multiply the neutrosophic components T, I, F, or their sub-components tJ, Ik, Fl and thus to construct the multiplication of neutrosophic quadruple numbers.

The number of independent messages a physical system can carry is limited by the number of its adjustable properties. In particular, systems that have only one adjustable property cannot carry more than a single message at a time. We demonstrate this is the case for the single photons in the double-slit experiment, and the root of the fundamental limit on measuring the complementary aspect of the photons. Next, we analyze the other ‘quantal’ behavior of the systems with (...) a single adjustable property, such as noncommutativity and no-cloning. Finally, we formulate a mathematical theory to describe the dynamics of such systems and derive the standard Hilbert-space formalism of quantum mechanics. Our derivation demonstrates the physical foundation of the quantum theory. (shrink)

A history of the construction of number has been in line with the process of recognition about the properties of geometry. Natural number representing countability is exhibited on a straight line and the completeness of real number is also originated from the continuous property of the number line. Complex number on a plane off the number line is established and thereafter, the whole number system is completed. When the process of constructing a (...) number with geometric features is investigated from different perspectives, it provides a new insight into the fundamental principle of number, which leads to a novel methodology in mathematics. (shrink)

The First Hilbert problem is studied in this paper by applying two instruments: a new methodology distinguishing between mathematical objects and mathematical languages used to describe these objects; and a new numeral system allowing one to express different infinite numbers and to use these numbers for measuring infinite sets. Several counting systems are taken into consideration. It is emphasized in the paper that different mathematical languages can describe mathematical objects (in particular, sets and the number of their elements) (...) with different accuracies. The traditional and the new approaches are compared and discussed. (shrink)

Systems biologists often distance themselves from reductionist approaches and formulate their aim as understanding living systems “as a whole.” Yet, it is often unclear what kind of reductionism they have in mind, and in what sense their methodologies would offer a superior approach. To address these questions, we distinguish between two types of reductionism which we call “modular reductionism” and “bottom-up reductionism.” Much knowledge in molecular biology has been gained by decomposing living systems into functional modules or through detailed studies (...) of molecular processes. We ask whether systems biology provides novel ways to recompose these findings in the context of the system as a whole via computational simulations. As an example of computational integration of modules, we analyze the first whole-cell model of the bacterium M. genitalium. Secondly, we examine the attempt to recompose processes across different spatial scales via multi-scale cardiac models. Although these models rely on a number of idealizations and simplifying assumptions as well, we argue that they provide insight into the limitations of reductionist approaches. Whole-cell models can be used to discover properties arising at the interfaces of dynamically coupled processes within a biological system, thereby making more apparent what is lost through decomposition. Similarly, multi-scale modeling highlights the relevance of macroscale parameters and models and challenges the view that living systems can be understood “bottom-up.” Specifically, we point out that system-level properties constrain lower-scale processes. Thus, large-scale modeling reveals how living systems at the same time are more and less than the sum of the parts. (shrink)

Humans possess two nonverbal systems capable of representing numbers, both limited in their representational power: the first one represents numbers in an approximate fashion, and the second one conveys information about small numbers only. Conception of exact large numbers has therefore been thought to arise from the manipulation of exact numerical symbols. Here, we focus on two fundamental properties of the exact numbers as prerequisites to the concept of EXACT NUMBERS : the fact that all numbers can be generated by (...) a successor function and the fact that equality between numbers can be defined in an exact fashion. We discuss some recent findings assessing how speakers of Munduruc (an Amazonian language), and young Western children (3-4 years old) understand these fundamental properties of numbers. (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)

Abstract: Heart attacks, or myocardial infarctions, are a leading cause of mortality worldwide. Early prediction and accurate analysis of potential risk factors play a crucial role in preventing heart attacks and improving patient outcomes. In this study, we conduct a comprehensive review of datasets related to heart attack analysis and prediction. We begin by examining the various types of datasets available for heart attack research, encompassing clinical, demographic, and physiological data. These datasets originate from diverse sources, including hospitals, research institutions, (...) and public health agencies. Our analysis aims to identify common features, data quality, and potential biases in these datasets. Next, we explore the predictive modeling techniques employed in heart attack prediction. Machine learning algorithms, such as decision trees, support vector machines, and deep neural networks, have gained prominence in predicting heart attacks. We discuss the strengths and limitations of these methods and highlight recent advancements in predictive modeling. Furthermore, we delve into the critical risk factors associated with heart attacks. Factors such as age, gender, hypertension, diabetes, and cholesterol levels are examined for their significance in predicting cardiac events. We also investigate the role of lifestyle factors, including smoking, diet, and physical activity, in heart attack risk assessment. Additionally, this review addresses the importance of data preprocessing and feature engineering in improving prediction accuracy. Feature selection methods, missing data handling, and data scaling techniques are discussed to enhance the robustness of heart attack prediction models. In conclusion, this comprehensive dataset review provides valuable insights into the state of heart attack analysis and prediction. It serves as a resource for researchers and healthcare professionals seeking to better understand the datasets available for heart attack research and the methods employed for accurate prediction. Ultimately, our efforts in dataset analysis and predictive modeling contribute to the advancement of preventive cardiology and the reduction of heart attack-related morbidity and mortality. (shrink)

Epistemic questions have become an important area of debate within democratic theory. Epistemic democrats have revived epistemic justification of democracy, while social scientific research has speared a significant debate on voter knowledge. An area which has received less attention, however, is the epistemic case for markets. Market advocates have developed a number of epistemic critiques of democracy which suggest that most goods are better provided by markets than democratic institutions. Despite representing important challenges to democracy, these critiques have gone (...) without reply as democratic theorists have tended to exclude markets from consideration. This article responds to these critiques and argues that there are good epistemic grounds for granting a much greater role to democracy than its market critics have claimed. It argues that there is a broad range of goods, including important ethical goods, which are better provided by democracy than markets due to the particular epistemic burdens they create. (shrink)

Issues of sociopolitical systems’ stability and risks of their destabi-lization in process of political transformations belong to the most important ones as regards the social development perspectives, as has been shown again by the recent events in Ukraine. In this re-spect it appears necessary to note that the transition to democracy may pose a serious threat to the stability of respective sociopolitical systems. This article studies the issue of democratization of countries within globalization context, it points to the unreasonably high (...) economic and social costs of a rapid transition to democracy as a result of revolutions or of similar large-scale events for the countries unprepared for it. The authors believe that in a number of cases the authoritarian regimes turn out to be more effective in economic and social terms in comparison with emerging democracies especially of the revolutionary type, which are often incapable to insure social order and may have a swing to authoritarianism. Effective authoritarian regimes can also be a suitable form of a transition to efficient and stable democracy. The article investigates various correlations between revolutionary events and possibilities of establishing democracy in a society on the basis of the historical and contemporary examples as well as the recent events in Egypt. The authors demonstrate that one should take into account a country’s degree of sociopolitical and cultural preparedness for democratic institutions. In case of favorable background, revolutions can proceed smoothly (“velvet revolutions”) with efficient outcomes. On the contrary, democracy is established with much difficulty, throwbacks, return to totalitarianism, and with outbreaks of violence and military takeovers in the countries with high illiteracy rate and rural population share, with low female status, with widespread religious fundamental ideology, where a substantial part of the population hardly ever hears of democracy while the liberal intellectuals idealize this form, where the opposing parties are not willing to respect the rules of democratic game when defeated at elections. (shrink)

It is usual to identify initial conditions of classical dynamical systems with mathematical real numbers. However, almost all real numbers contain an infinite amount of information. I argue that a finite volume of space can’t contain more than a finite amount of information, hence that the mathematical real numbers are not physically relevant. Moreover, a better terminology for the so-called real numbers is “random numbers”, as their series of bits are truly random. I propose an alternative classical mechanics, which is (...) empirically equivalent to classical mechanics, but uses only finite-information numbers. This alternative classical mechanics is non-deterministic, despite the use of deterministic equations, in a way similar to quantum theory. Interestingly, both alternative classical mechanics and quantum theories can be supplemented by additional variables in such a way that the supplemented theory is deterministic. Most physicists straightforwardly supplement classical theory with real numbers to which they attribute physical existence, while most physicists reject Bohmian mechanics as supplemented quantum theory, arguing that Bohmian positions have no physical reality. (shrink)

The stability of socio-political systems and the risks of destabilisation in the process of political transformation are among the most important issues of social development; the transition to democracy may pose a serious threat to the stability of a respective socio-political system. This article studies the issue of democratisation. It highlights the high economic and social costs of a rapid transition to democracy for countries unprepared for it—democracy resulting from revolutions or similar large-scale events. The authors believe that in (...) a number of cases authoritarian regimes turn out to be more effective in economic and social terms than emerging democracies, especially those of a revolutionary type, which are often incapable of ensuring social order and may have a swing to authoritarianism. Effective authoritarian regimes can also be a suitable form of transition to an efficient and stable democracy. Using historical and contemporary examples, particularly the recent events in Egypt, the article investigates various correlations between revolutionary events and the possibility of establishing democracy in a society. (shrink)

The article considers the problem of the system model of family counseling, in particular, the analysis of the family as a social system, as a complex of elements and their properties, which are in dynamic connections and relationships. The analysis of the theory of systems and the description of the principles of family counseling is carried out. Particular attention is paid to highlighting the main provisions of the individual (“adlerian”) psychology in counseling the family. -/- Currently among specialists (...) there is a high interest in the provision of psychological assistance to a family in crisis. This is largely due to the fact that over the past decades, the institution of the family in this country is experiencing an increase in destructive tendencies. This is evidenced by an increase in the number of appeals to school psychologists, psychological counseling, psychological services and centers from both the individual members of the family and families in general. -/- The psychological help to the family is positioned as a relatively new field of practice for a psychologist. The practice of counseling, including family, is largely determined by the theoretical skills of the counselor, primarily as (s)he understands the personality, determination of behavior, the source of family problems, and the possibility of change. Today, practitioners prefer the integrative method, which is a system approach. -/- The purpose of this article is to analyze the theory of systems as the basis of family counseling. The system model of family counseling is considered to be one of the youngest and most widespread models that received their recognition at the end of the twentieth century. (shrink)

The paper studies a cluster of systems for fully disquotational truth based on the restriction of initial sequents. Unlike well-known alternative approaches, such systems display both a simple and intuitive model theory and remarkable proof-theoretic properties. We start by showing that, due to a strong form of invertibility of the truth rules, cut is eliminable in the systems via a standard strategy supplemented by a suitable measure of the number of applications of truth rules to formulas in derivations. Next, (...) we notice that cut remains eliminable when suitable arithmetical axioms are added to the system. Finally, we establish a direct link between cut-free derivability in infinitary formulations of the systems considered and fixed-point semantics. Noticeably, unlike what happens with other background logics, such links are established without imposing any restriction to the premisses of the truth rules. (shrink)