Philosophy can shed light on mathematical modeling and the juxtaposition of modeling and empirical data. This paper explores three philosophical traditions of the structure of scientific theory—Syntactic, Semantic, and Pragmatic—to show that each illuminates mathematical modeling. The Pragmatic View identifies four critical functions of mathematical modeling: (1) unification of both models and data, (2) model fitting to data, (3) mechanism identification accounting for observation, and (4) prediction of future observations. Such facets are explored (...) using a recent exchange between two groups of mathematical modelers in plant biology. Scientific debate can arise from different modeling philosophies. (shrink)

Mathematical modeling offers a promising approach to improving mathematics education. This study aims to determine if the concept of metacognitive awareness in the learning process is associated with mathematical modeling. This study also considers the interaction effect of sex and academic year level on both variables. Focusing the study on preservice elementary teachers might address potential issues and targeted intervention in their preparation program concerning their ability to teach and guide young learners in modeling activities. (...) The research sample includes 140 preservice elementary teachers at Central Luzon State University, Philippines. Data collection used an adapted metacognitive awareness inventory and a validated researcher-made mathematical modeling competency test aligned with the K-12 mathematics curriculum in the Philippines. Results revealed that the preservice elementary teachers had a high metacognitive awareness and mathematical modeling competency, ranging from 22 to 31 out of 36 points. Besides, Factorial ANOVA indicates that academic year level positively affects both variables regardless of sex, and stepwise regression analysis unveiled that information management, declarative knowledge, and planning significantly predict 41.4% of the mathematical modeling competency variance. This suggests that developing metacognitive awareness supports preservice elementary teachers in performing modeling tasks that improve their competency level in mathematics. (shrink)

In the first part of this article we survey general similarities and differences between biological and social macroevolution. In the second (and main) part, we consider a concrete mathematical model capable of describing important features of both biological and social macroevolution. In mathematical models of historical macrodynamics, a hyperbolic pattern of world population growth arises from non-linear, second-order positive feedback between demographic growth and technological development. Based on diverse paleontological data and an analogy with macrosociological models, we suggest (...) that the hyperbolic character of biodiversity growth can be similarly accounted for by non-linear, second-order positive feedback between diversity growth and the complexity of community structure. We discuss how such positive feedback mechanisms can be modelled mathematically. (shrink)

This research aims at establishing the impact of information and communication technologies (ICT) on e-commerce development of industrial enterprises by means of economic and mathematical modelling. The goal was achieved using the following methods: theoretical generalization, analysis and synthesis (to critically analyse the scientific approaches of scientists regarding the expediency of using mathematical models in the context of enterprises’ e-commerce development), target, comparison and grouping (to reveal innovative methodological approach to assessing ICT impact on e-commerce development of industrial (...) enterprises), tabular, analytical and integral method (for summarizing the analysis results of enterprises readiness to implement ICT, ICT use in the activities of industrial enterprises of Ukraine and the analysis of e-commerce development), mathematical modelling (to build a regression model determining impact of changes in ICT use on the market share occupied by industrial enterprises), generalization (to determine promising directions of e-commerce developing of industrial enterprises). The implementation of a comprehensive approach to understanding the importance of ICT influence on e-commerce development of industrial enterprises will ensure acceleration of the digitalization of business processes, will contribute to the speed increase of enterprises response to customer requests, and increase the market share occupied by enterprises. A new vision of directions for developing e-commerce of industrial enterprises is suggested, which are determined by the need for enterprise rebranding, the development of e-commerce tools and technologies, the importance of outsourcing service automation and promotion of subscription trade. ICT is considered as integration factor that determines prospects for e-commerce development of industrial enterprises and contributes to increasing efficiency of online business management. Research results demonstrate that the use of economic and mathematical modelling is an important tool for assessing ICT impact, and its absence can negatively affect the accuracy and validity of online business management. (shrink)

The mental risk poses a high threat to the individuals, especially overseas demographic, including expatriates in comparison to the general Arab demographic. Since Arab countries are renowned for their multicultural environment with half of the population of students and faculties being international, this paper focuses on a comprehensive analysis of mental health problems such as depression, stress, anxiety, isolation, and other unfortunate conditions. The dataset is developed from a web-based survey. The detailed exploratory data analysis is conducted on the dataset (...) collected from Arab countries to study an individual’s mental health and indicative help-seeking pointers based on their responses to specific pre-defined questions in a multicultural society. The proposed model validates the claims mathematically and uses different machine learning classifiers to identify individuals who are either currently or previously diagnosed with depression or demonstrate unintentional “save our souls” (SOS) behaviors for an early prediction to prevent risks of danger in life going forward. The accuracy is measured by comparing with the classifiers using several visualization tools. This analysis provides the claims and authentic sources for further research in the multicultural public medical sector and decision-making rules by the government. (shrink)

In the first part of this article we survey general similarities and differences between biological and social macroevolution. In the second (and main) part, we consider a concrete mathematical model capable of describing important features of both biological and social macroevolution. In mathematical models of historical macrodynamics, a hyperbolic pattern of world population growth arises from non-linear, second-order positive feedback between demographic growth and technological development. This is more or less identical with the working of the collective learning (...) mechanism. Based on diverse paleontological data and an analogy with macrosociological models, we suggest that the hyperbolic character of biodiversity growth can be similarly accounted for by non-linear, second-order positive feedback between diversity growth and the complexity of community structure, suggesting the presence within the biosphere of a certain analogue of the collective learning mechanism. We discuss how such positive feedback mechanisms can be modelled mathematically. (shrink)

The classical (set-theoretic) concept of structure has become essential for every contemporary account of a scientific theory, but also for the metatheoretical accounts dealing with the adequacy of such theories and their methods. In the latter category of accounts, and in particular, the structural metamodels designed for the applicability of mathematics have struggled over the last decade to justify the use of mathematical models in sciences beyond their 'indispensability' in terms of either method or concepts/entities. In this paper, I (...) argue that these metamodels employ structures of different natures and epistemologies, and this diversity does pose a serious problem to the intended justification. (shrink)

Modelling of Generancy is a book on Indian philosophy. In this book I have tried to express various problems of philosophy in mathematical language. I think mathematics is a language. Everything can be expressed in this language. With the help of mathematics, the published issues are understandable to all. No one has any objection to this. In the realm of knowledge all terms or words are considered categories. This category is again of three types: substance, quality and action. In (...) another way, this category is divided into five parts. These five parts are: Purusa, prakrti, jagat, jnana and karma. These five subjects are known as panchatattva. A total of fifty-five words, including these five words, are discussed in detail in the panchatattva. Fifty-five letters of the Sanskrit alphabet have been used to identify these fifty-five words. I think the problems of the present world are the problems of philosophy and knowledge. If this philosophy or knowledge can be expressed in the language of mathematics, then it will be useful for everyone to understand. Everyone will be aware of their responsibilities. That is what has been tried in this book. (shrink)

This dissertation examines aspects of the interplay between computing and scientific practice. The appropriate foundational framework for such an endeavour is rather real computability than the classical computability theory. This is so because physical sciences, engineering, and applied mathematics mostly employ functions defined in continuous domains. But, contrary to the case of computation over natural numbers, there is no universally accepted framework for real computation; rather, there are two incompatible approaches --computable analysis and BSS model--, both claiming to formalise algorithmic (...) computation and to offer foundations for scientific computing. -/- The dissertation consists of three parts. In the first part, we examine what notion of 'algorithmic computation' underlies each approach and how it is respectively formalised. It is argued that the very existence of the two rival frameworks indicates that 'algorithm' is not one unique concept in mathematics, but it is used in more than one way. We test this hypothesis for consistency with mathematical practice as well as with key foundational works that aim to define the term. As a result, new connections between certain subfields of mathematics and computer science are drawn, and a distinction between 'algorithms' and 'effective procedures' is proposed. -/- In the second part, we focus on the second goal of the two rival approaches to real computation; namely, to provide foundations for scientific computing. We examine both frameworks in detail, what idealisations they employ, and how they relate to floating-point arithmetic systems used in real computers. We explore limitations and advantages of both frameworks, and answer questions about which one is preferable for computational modelling and which one for addressing general computability issues. -/- In the third part, analog computing and its relation to analogue (physical) modelling in science are investigated. Based on some paradigmatic cases of the former, a certain view about the nature of computation is defended, and the indispensable role of representation in it is emphasized and accounted for. We also propose a novel account of the distinction between analog and digital computation and, based on it, we compare analog computational modelling to physical modelling. It is concluded that the two practices, despite their apparent similarities, are orthogonal. (shrink)

Evolutionary anthropologists and archaeologists have been considerably successful in modelling the cumulative evolution of culture, of technological skills and knowledge in particular. Recently, one of these models has been introduced in the philosophy of science by De Cruz and De Smedt (Philos Stud 157:411–429, 2012), in an attempt to demonstrate that scientists may collectively come to hold more truth-approximating beliefs, despite the cognitive biases which they individually are known to be subject to. Here we identify a major shortcoming in that (...) attempt: De Cruz & De Smedt’s mathematical model makes one particularly strong tractability assumption that causes the model to largely miss its target (namely, truth accumulation in science), and that moreover conflicts with empirical observations. The second, more constructive part of the paper presents an alternative, agent-based model, which allows one to much better examine the conditions for scientific progress and decline. (shrink)

The purpose of this article is to establish a connection between modelling practices and interpretive approaches in quantum mechanics, taking as a starting point the literature on scientific representation. Different types of modalities play different roles in scientific representation. I postulate that the way theoretical structures are interpreted in this respect affects the way models are constructed. In quantum mechanics, this would be the case in particular of initial conditions and observables. I examine two formulations of quantum mechanics, the standard (...) wave-function formulation and the consistent histories formulation, and show that they correspond to opposite stances, which confirms my approach. Finally, I examine possible strategies for deciding between these stances. (shrink)

Compared with traditional statistics, only a few social scientists employ Bayesian analyses. The existing software programs for implementing Bayesian analyses such as OpenBUGS, WinBUGS, JAGS, and rstanarm can be daunting given that their complex computer codes involve a steep learning curve. In contrast, this paper introduces a new open software for implementing Bayesian network modelling and analysis: the bayesvl R package. The package aims at providing an intuitive gateway for beginners of Bayesian statistics to construct and analyse mathematical models (...) in social sciences... (shrink)

This study provides a basic introduction to agent-based modeling (ABM) as a powerful blend of classical and constructive mathematics, with a primary focus on its applicability for social science research. The typical goals of ABM social science researchers are discussed along with the culture-dish nature of their computer experiments. The applicability of ABM for science more generally is also considered, with special attention to physics. Finally, two distinct types of ABM applications are summarized in order to illustrate concretely the (...) duality of ABM: Real-world systems can not only be simulated with verisimilitude using ABM; they can also be efficiently and robustly designed and constructed on the basis of ABM principles. (shrink)

Just before the Scientific Revolution, there was a "Mathematical Revolution", heavily based on geometrical and machine diagrams. The "faculty of imagination" (now called scientific visualization) was developed to allow 3D understanding of planetary motion, human anatomy and the workings of machines. 1543 saw the publication of the heavily geometrical work of Copernicus and Vesalius, as well as the first Italian translation of Euclid.

In this chapter, one considers finance at its very foundations, namely, at the place where assumptions are being made about the ways to measure the two key ingredients of finance: risk and return. It is well known that returns for a large class of assets display a number of stylized facts that cannot be squared with the traditional views of 1960s financial economics (normality and continuity assumptions, i.e. Brownian representation of market dynamics). Despite the empirical counterevidence, normality and continuity assumptions (...) were part and parcel of financial theory and practice, embedded in all financial practices and beliefs. Our aim is to build on this puzzle for extracting some clues revealing the use of one research strategy in academic community, model tinkering defined as a particular research habit. We choose to focus on one specific moment of the scientific controversies in academic finance: the ‘leptokurtic crisis’ opened by Mandelbrot in 1962. The profoundness of the crisis came from the angle of the Mandelbrot’s attack: not only he emphasized an empirical inadequacy of the Brownian representation, but also he argued for an inadequate grounding of this representation. We give some insights in this crisis and display the model tinkering strategies of the financial academic community in the 1970s and the 1980s. (shrink)

Much has been written about the free energy principle (FEP), and much misunderstood. The principle has traditionally been put forth as a theory of brain function or biological self-organisation. Critiques of the framework have focused on its lack of empirical support and a failure to generate concrete, falsifiable predictions. I take both positive and negative evaluations of the FEP thus far to have been largely in error, and appeal to a robust literature on scientific modelling to rectify the situation. A (...) prominent account of scientific modelling distinguishes between model structure and model construal. I propose that the FEP be reserved to designate a model structure, to which philosophers and scientists add various construals, leading to a plethora of models based on the formal structure of the FEP. An entailment of this position is that demands placed on the FEP that it be falsifiable or that it conform to some degree of biological realism rest on a category error. To this end, I deliver first an account of the phenomenon of model transfer and the breakdown between model structure and model construal. In the second section, I offer an overview of the formal elements of the framework, tracing their history of model transfer and illustrating how the formalism comes apart from any interpretation thereof. Next, I evaluate existing comprehensive critical assessments of the FEP, and hypothesise as to potential sources of existing confusions in the literature. In the final section, I distinguish between what I hold to be the FEP—taken to be a modelling language or modelling framework—and what I term “FEP models.”. (shrink)

The topics of modeling and information come together in at least two ways. Computational modeling and simulation play an increasingly important role in science, across disciplines from mathematics through physics to economics and political science. The philosophical questions at issue are questions as to what modeling and simulation are adding, altering, or amplifying in terms of scientific information. What changes with regard to information acquisition, theoretical development, or empirical confirmation with contemporary tools of computational modeling? In (...) this sense the title of this article is read in the following way: What kind of information is modeling information? What kind of information does modeling give us? (shrink)

This paper brings together Thompson's naive action explanation with interventionist modeling of causal structure to show how they work together to produce causal models that go beyond current modeling capabilities, when applied to specifically selected systems. By deploying well-justified assumptions about rationalization, we can strengthen existing causal modeling techniques' inferential power in cases where we take ourselves to be modeling causal systems that also involve actions. The internal connection between means and end exhibited in naive action (...) explanation has a modal strength like that of distinctively mathematical explanation, rather than that of causal explanation. Because it is stronger than causation, it can be treated as if it were merely causal in a causal model without thereby overextending the justification it can provide for inferences. This chapter introduces and demonstrate the usage of the Rationalization condition in causal modeling, where it is apt for the system(s) being modeled, and to provide the basics for incorporating R variables into systems of variables and R arrows into DAGs. Use of the Rationalization condition supplements causal analysis with action analysis where it is apt. (shrink)

The philosophy of mathematics has largely abandoned foundational studies, but is still fixated on theorem proving, logic and number theory, and on whether mathematical knowledge is certain. That is not what mathematics looks like to, say, a knot theorist or an industrial mathematical modeller. The "computer revolution" shows that mathematics is a much more direct study of the world, especially its structural aspects.

The present yearbook (which is the fourth in the series) is subtitled Trends & Cycles. It is devoted to cyclical and trend dynamics in society and nature; special attention is paid to economic and demographic aspects, in particular to the mathematical modeling of the Malthusian and post-Malthusian traps' dynamics. An increasingly important role is played by new directions in historical research that study long-term dynamic processes and quantitative changes. This kind of history can hardly develop without the application (...) of mathematical methods. There is a tendency to study history as a system of various processes, within which one can detect waves and cycles of different lengths – from a few years to several centuries, or even millennia. The contributions to this yearbook present a qualitative and quantitative analysis of global historical, political, economic and demographic processes, as well as their mathematical models. This issue of the yearbook consists of three main sections: (I) Long-Term Trends in Nature and Society; (II) Cyclical Processes in Pre-industrial Societies; (III) Contemporary History and Processes. We hope that this issue of the yearbook will be interesting and useful both for historians and mathematicians, as well as for all those dealing with various social and natural sciences. (shrink)

Recently, Bechtel and Abrahamsen have argued that mathematical models study the dynamics of mechanisms by recomposing the components and their operations into an appropriately organized system. We will study this claim through the practice of combinational modeling in circadian clock research. In combinational modeling, experiments on model organisms and mathematical/computational models are combined with a new type of model—a synthetic model. We argue that the strategy of recomposition is more complicated than what Bechtel and Abrahamsen indicate. (...) Moreover, synthetic modeling as a kind of material recomposition strategy also points beyond the mechanistic paradigm. (shrink)

There are no universally adopted answers to the natural questions about scientific concepts: What are they? What is their structure? What are their functions? How many kinds of them are there? Do they change? Ironically, most if not all scientific monographs or articles mention concepts, but the scientific studies of scientific concepts are rare in occurrence. It is well known that the necessary stage of any scientific study is constructing the model of objects in question. Many years logical modeling (...) was dominant in the concept studies. Last decades, concepts came to be regarded as the subject of mathematical modeling. However, different authors take different features of concepts as independent variables of their models. Our objective is to characterize informally the spectra of relevant variables for the modeling of scientific concepts. (shrink)

The epistemic value of models has traditionally been approached from a representational perspective. This paper argues that the artifactual approach evades the problem of accounting for representation and better accommodates the modal dimension of modeling. From an artifactual perspective, models are viewed as erotetic vehicles constrained by their construction and available representational tools. The modal dimension of modeling is approached through two case studies. The first portrays mathematical modeling in economics, while the other discusses the (...) class='Hi'>modeling practice of synthetic biology, which exploits and combines models in various modes and media. Neither model intends to represent any actual target system. Rather, they are constructed to study possible mechanisms through the construction of a model system with built-in dependencies. (shrink)

The study ascertained mathematical problem-solving skills and self-efficacy as correlates of Pre-service NCE Mathematics Teachers’ Performance in South-East, Nigeria. Seven research questions guided the study while seven hypotheses were tested at 0.05 level of significance. Correlation research design was used for the study. The population of the study was 197 pre-service NCE Mathematics teachers in South-East, Nigeria. All the population of 197 was studied as sample because, it is small and manageable. Mathematics Problem-Solving Skill Test (MPSST) and Pre-Service Teachers’ (...) Self-Efficacy Questionnaire (PSTSEQ) were used as instruments for data collection. Both instruments were validated by three experts in science Education while reliabilities of MPSST and PSTSEQ were established using Cronbach Alpha method and coefficient values of 0.82 and 0.80 respectively were obtained which were considered high enough to confirm the instruments as reliable. Data were collected and analyzed using Pearson product moment correlation coefficient. The findings revealed that a high positive and significant relationship existed between mathematical problem-solving skills scores and academic performance scores and between self-efficacy scores and academic performance scores of pre-service NCE mathematics teachers. It was recommended among others that Mathematics lecturers should make conscious effort towards improving the mathematical problem-solving skills and self-efficacy of the pre-service NCE mathematics teachers through modeling, conditioning, and behaviour modification for improved academic performance in Mathematics. (shrink)

The present Yearbook (which is the fourth in the series) is subtitled Trends & Cycles. Already ancient historians (see, e.g., the second Chapter of Book VI of Polybius' Histories) described rather well the cyclical component of historical dynamics, whereas new interesting analyses of such dynamics also appeared in the Medieval and Early Modern periods (see, e.g., Ibn Khaldūn 1958 [1377], or Machiavelli 1996 [1531] 1). This is not surprising as the cyclical dynamics was dominant in the agrarian social systems. With (...) modernization, the trend dynamics became much more pronounced and these are trends to which the students of modern societies pay more attention. Note that the term trend – as regards its contents and application – is tightly connected with a formal mathematical analysis. Trends may be described by various equations – linear, exponential, power-law, etc. On the other hand, the cliodynamic research has demonstrated that the cyclical historical dynamics can be also modeled mathematically in a rather effective way (see, e.g., Usher 1989; Chu and Lee 1994; Turchin 2003, 2005a, 2005b; Turchin and Korotayev 2006; Turchin and Nefedov 2009; Nefedov 2004; Korotayev and Komarova 2004; Korotayev, Malkov, and Khaltourina 2006; Korotayev and Khaltourina 2006; Korotayev 2007; Grinin 2007), whereas the trend and cycle components of historical dynamics turn out to be of equal importance. (shrink)

In the first part of this article we survey general similarities and differences between biological and social macroevolution. In the second (and main) part, we consider a concrete mathematical model capable of describing important features of both biological and social macroevolution. In mathematical models of historical macrodynamics, a hyperbolic pattern of world population growth arises from non-linear, second-order positive feedback between demographic growth and technological development. Based on diverse paleontological data and an analogy with macrosociological models, we suggest (...) that the hyperbolic character of biodiversity growth can be similarly accounted for by non-linear, second-order positive feedback between diversity growth and the complexity of community structure. We discuss how such positive feedback mechanisms can be modelled mathematically. (shrink)

Games of chance are developed in their physical consumer-ready form on the basis of mathematical models, which stand as the premises of their existence and represent their physical processes. There is a prevalence of statistical and probabilistic models in the interest of all parties involved in the study of gambling – researchers, game producers and operators, and players – while functional models are of interest more to math-inclined players than problem-gambling researchers. In this paper I present a structural analysis (...) of the knowledge attached to mathematical models of games of chance and the act of modeling, arguing that such knowledge holds potential in the prevention and cognitive treatment of excessive gambling, and I propose further research in this direction. (shrink)

Many biological processes and objects can be described by fractals. The paper uses a new type of objects – blinking fractals – that are not covered by traditional theories considering dynamics of self-similarity processes. It is shown that both traditional and blinking fractals can be successfully studied by a recent approach allowing one to work numerically with infinite and infinitesimal numbers. It is shown that blinking fractals can be applied for modeling complex processes of growth of biological systems including (...) their season changes. The new approach allows one to give various quantitative characteristics of the obtained blinking fractals models of biological systems. (shrink)

While the predominant focus of the philosophical literature on scientific modeling has been on single-scale models, most systems in nature exhibit complex multiscale behavior, requiring new modeling methods. This challenge of modeling phenomena across a vast range of spatial and temporal scales has been called the tyranny of scales problem. Drawing on research in the geosciences, I synthesize and analyze a number of strategies for taming this tyranny in the context of conceptual, physical, and mathematical (...) class='Hi'>modeling. This includes several strategies that can be deployed in physical modeling, even when strict dynamical scaling fails. In all cases, I argue that having an adequate conceptual model—given both the nature of the system and the particular purpose of the model—is essential. I draw a distinction between depiction and representation, and use this research in the geosciences to advance a number of debates in the philosophy of modeling. (shrink)

The thesis deals with the concept of truth and the paradoxes of truth. Philosophical theories usually consider the concept of truth from a wider perspective. They are concerned with questions such as - Is there any connection between the truth and the world? And, if there is - What is the nature of the connection? Contrary to these theories, this analysis is of a logical nature. It deals with the internal semantic structure of language, the mutual semantic connection of sentences, (...) above all the connection of sentences that speak about the truth of other sentences and sentences whose truth they speak about. Truth paradoxes show that there is a problem in our basic understanding of the language meaning and they are a test for any proposed solution. It is important to make a distinction between the normative and analytical aspect of the solution. The former tries to ensure that paradoxes will not emerge. The latter tries to explain paradoxes. Of course, the practical aspect of the solution is also important. It tries to ensure a good framework for logical foundations of knowledge, for related problems in Artificial Intelligence and for the analysis of the natural language. Tarski’s analysis emphasized the T-scheme as the basic intuitive principle for the concept of truth, but it also showed its inconsistency with the classical logic. Tarski’s solution is to preserve the classical logic and to restrict the scheme: we can talk about the truth of sentences of a language only inside another essentially richer metalanguage. This solution is in harmony with the idea of reflexivity of thinking and it has become very fertile for mathematics and science in general. But it has normative nature | truth paradoxes are avoided in a way that in such frame we cannot even express paradoxical sentences. It is also too restrictive because, for the same reason we cannot express a situation in which there is a circular reference of some sentences to other sentences, no matter how common and harmless such a situation may be. Kripke showed that there is no natural restriction to the T-scheme and we have to accept it. But then we must also accept the riskiness of sentences | the possibility that under some circumstances a sentence does not have the classical truth value but it is undetermined. This leads to languages with three-valued semantics. Kripke did not give any definite model, but he gave a theoretical frame for investigations of various models | each fixed point in each three-valued semantics can be a model for the concept of truth. The solutions also have normative nature | we can express the paradoxical sentences, but we escape a contradiction by declaring them undetermined. Such a solution could become an analytical solution only if we provide the analysis that would show in a substantial way that it is the solution that models the concept of truth. Kripke took some steps in the direction of finding an analytical solution. He preferred the strong Kleene three-valued semantics for which he wrote it was "appropriate" but did not explain why it was appropriate. One reason for such a choice is probably that Kripke finds paradoxical sentences meaningful. This eliminates the weak Kleene three valued semantics which corresponds to the idea that paradoxical sentences are meaningless, and thus indeterminate. Another reason could be that the strong Kleene three valued semantics has the so-called investigative interpretation. According to this interpretation, this semantics corresponds to the classical determination of truth, whereby all sentences that do not have an already determined value are temporarily considered indeterminate. When we determine the truth value of these sentences, then we can also determine the truth value of the sentences that are composed of them. Kripke supplemented this investigative interpretation with an intuition about learning the concept of truth. That intuition deals with how we can teach someone who is a competent user of an initial language (without the predicate of truth T) to use sentences that contain the predicate T. That person knows which sentences of the initial language are true and which are not. We give her the rule to assign the T attribute to the former and deny that attribute to the latter. In that way, some new sentences that contain the predicate of truth, and which were indeterminate until then, become determinate. So the person gets a new set of true and false sentences with which he continues the procedure. This intuition leads directly to the smallest fixed point of strong Kleene semantics as an analytically acceptable model for the logical notion of truth. However, since this process is usually saturated only on some transfinite ordinal, this intuition, by climbing on ordinals, increasingly becomes a metaphor. This thesis is an attempt to give an analytical solution to truth paradoxes. It gives an analysis of why and how some sentences lack the classical truth value. The starting point is basic intuition according to which paradoxical sentences are meaningful (because we understand what they are talking about well, moreover we use it for determining their truth values), but they witness the failure of the classical procedure of determining their truth value in some "extreme" circumstances. Paradoxes emerge because the classical procedure of the truth value determination does not always give a classically supposed (and expected) answer. The analysis shows that such an assumption is an unjustified generalization from common situations to all situations. We can accept the classical procedure of the truth value determination and consequently the internal semantic structure of the language, but we must reject the universality of the exterior assumption of a successful ending of the procedure. The consciousness of this transforms paradoxes to normal situations inherent to the classical procedure. Some sentences, although meaningful, when we evaluate them according to the classical truth conditions, the classical conditions do not assign them a unique value. We can assign to them the third value, \undetermined", as a sign of definitive failure of the classical procedure. An analysis of the propagation of the failure in the structure of sentences gives exactly the strong Kleene three-valued semantics, not as an investigative procedure, as it occurs in Kripke, but as the classical truth determination procedure accompanied by the propagation of its own failure. An analysis of the circularities in the determination of the classical truth value gives the criterion of when the classical procedure succeeds and when it fails, when the sentences will have the classical truth value and when they will not. It turns out that the truth values of sentences thus obtained give exactly the largest intrinsic fixed point of the strong Kleene three-valued semantics. In that way, the argumentation is given for that choice among all fixed points of all monotone three-valued semantics for the model of the logical concept of truth. An immediate mathematical description of the fixed point is given, too. It has also been shown how this language can be semantically completed to the classical language which in many respects appears a natural completion of the process of thinking about the truth values of the sentences of a given language. Thus the final model is a language that has one interpretation and two systems of sentence truth evaluation, primary and final evaluation. The language through the symbol T speaks of its primary truth valuation, which is precisely the largest intrinsic fixed point of the strong Kleene three valued semantics. Its final truth valuation is the semantic completion of the first in such a way that all sentences that are not true in the primary valuation are false in the final valuation. (shrink)

Templates travel because they offer a tractable format that can be used for model-building in a variety of domains. It is often because of this quality that a particular template is chosen. But one cannot assume that there are always templates ready to model a new phenomenon, and moreover, templates have also been designed at some point. A critical aspect of this designing process is the choice of the mathematical objects with which one hopes to capture this phenomenon. This (...) means that one has to assess which of the properties of different kinds of mathematical objects and which combinations of them are most appropriate. The selection of these objects is based on whether their properties enable the model to perform its function. This selection of mathematical objects is similar to the selection of materials in mechanical design. (shrink)

Models are indispensable tools of scientific inquiry, and one of their main uses is to improve our understanding of the phenomena they represent. How do models accomplish this? And what does this tell us about the nature of understanding? While much recent work has aimed at answering these questions, philosophers' focus has been squarely on models in empirical science. I aim to show that pure mathematics also deserves a seat at the table. I begin by presenting two cases: Cramér’s random (...) model of the prime numbers and the function field model of the integers. These cases show that mathematicians, like empirical scientists, rely on unrealistic models to gain understanding of complex phenomena. They also have important implications for some much-discussed theses about scientific understanding. First, modeling practices in mathematics confirm that one can gain understanding without obtaining an explanation. Second, these cases undermine the popular thesis that unrealistic models confer understanding by imparting counterfactual knowledge. (shrink)

This paper examines two parallel discussions of scientific modeling which have invoked experimentation in addressing the role of models in scientific inquiry. One side discusses the experimental character of models, whereas the other focuses on their exploratory uses. Although both relate modeling to experimentation, they do so differently. The former has considered the similarities and differences between models and experiments, addressing, in particular, the epistemic value of materiality. By contrast, the focus on exploratory modeling has highlighted the (...) various kinds of exploratory functions of models in the early stages of inquiry. These two perspectives on modeling are discussed through a case study in the field of synthetic biology. The research practice in question explores biological control by making use of an ensemble of different epistemic means: mathematical models and simulations, synthetic genetic circuits and intracellular measuring devices, and finally electronic circuits. We argue that the study of exploratory modeling should trace the ways different epistemic means, in different materialities, are being combined over time. Finally, the epistemic status of such novel investigative objects as synthetic genetic circuits is evaluated, with the conclusion that they can function as both experiments and models. (shrink)

Two families of mathematical methods lie at the heart of investigating the hierarchical structure of genetic variation in Homo sapiens: /diversity partitioning/, which assesses genetic variation within and among pre-determined groups, and /clustering analysis/, which simultaneously produces clusters and assigns individuals to these “unsupervised” cluster classifications. While mathematically consistent, these two methodologies are understood by many to ground diametrically opposed claims about the reality of human races. Moreover, modeling results are sensitive to assumptions such as preexisting theoretical commitments (...) to certain linguistic, anthropological, and geographic human groups. Thus, models can be perniciously reified. That is, they can be conflated and confused with the world. This fact belies standard realist and antirealist interpretations of “race,” and supports a pluralist conventionalist interpretation. (shrink)

A holobiont is a composite organism consisting of a host together with its microbiome, such as a coral with its zooxanthellae. To explain the often intimate integration between hosts and their microbiomes, some investigators contend that selection operates on holobionts as a unit and view the microbiome’s genes as extending the host’s nuclear genome to jointly comprise a hologenome. Because vertical transmission of microbiomes is uncommon, other investigators contend that holobiont selection cannot be effective because a holobiont’s microbiome is an (...) acquired condition rather than an inherited trait. This disagreement invites a simple mathematical model to see how holobiont selection might operate and to assess its plausibility as an evolutionary force. This paper presents two variants of such a model. In one variant, juvenile hosts obtain microbiomes from their parents (vertical transmission). In the other variant, microbiomes of juvenile hosts are assembled from source pools containing the combined microbiomes of all parents (horizontal transmission). According to both variants, holobiont selection indeed causes evolutionary change in holobiont traits. Therefore, holobiont selection is plausibly an effective evolutionary force with either mode of microbiome transmission. The modeling employs two distinct concepts of inheritance, depending on the mode of microbiome transmission: collective inheritance whereby juveniles inherit a sample of the collected genomes from all parents, as contrasted with lineal inheritance whereby juveniles inherit the genomes from only their own parents. A distinction between collective and lineal inheritance also features in theories of multilevel selection. (shrink)

Fundamental Science is undergoing an acute conceptual-paradigmatic crisis of philosophical foundations, manifested as a crisis of understanding, crisis of interpretation and representation, “loss of certainty”, “trouble with physics”, and a methodological crisis. Fundamental Science rested in the "first-beginning", "first-structure", in "cogito ergo sum". The modern crisis is not only a crisis of the philosophical foundations of Fundamental Science, but there is a comprehensive crisis of knowledge, transforming by the beginning of the 21st century into a planetary existential crisis, which has (...) exacerbated the question of the existence of Humanity and life on Earth. Due to the unsolved problem of justification of Mathematics, paradigm problems in Computational mathematics have arisen. It's time to return ↔ Into Dialectics. The solution to the problem of the foundations of Mathematics, and therefore knowledge in general, is the solution to the problem of modeling (constructing) the ontological basis of knowledge - the ontological model of the primordial generating process. The idea and model of the primordial generating process, its ontological structure directs thinking to the need for the introduction of superconcept → ontological (cosmic, structural) memory, concept-attractor, supercategory, substantial semantic core of the scientific picture of the world of the nuclear-ecological-information age. Model of basic Ideality→ “Space-MatterMemory-Time” [S-MM-T]. (shrink)

At the beginning of the COVID-19 pandemic, high hopes were put on digital contact tracing, using mobile phone apps to record and immediately notify contacts when a user reports as infected. Such apps can now be downloaded in many countries, but as second waves of COVID-19 are raging, these apps are playing a less important role than anticipated. We argue that this is because most countries have opted for app configurations that cannot provide a means of rapidly informing users of (...) likely infections while avoiding too many false positive reports. Mathematical modelling suggests that differently configured apps have the potential to do this. These require, however, that some pseudonymised data be stored on a central server, which privacy advocates have cautioned against. We contend that their influential arguments are subject to two fallacies. First, they have tended to one-sidedly focus on the risks that centralised data storage entails for privacy, while paying insufficient attention to the fact that inefficient contact tracing involves ethical risks too. Second, while the envisioned system does entail risks of breaches, such risks are also present in decentralised systems, which have been falsely presented as ‘privacy preserving by design’. When these points are understood, it becomes clear that we must rethink our approach to digital contact tracing in our fight against COVID-19. There are no data in this work. (shrink)

Mathematical models are a well established tool in most natural sciences. Although models have been neglected by the philosophy of science for a long time, their epistemological status as a link between theory and reality is now fairly well understood. However, regarding the epistemological status of mathematical models in the social sciences, there still exists a considerable unclarity. In my paper I argue that this results from specific challenges that mathematical models and especially computer simulations face in (...) the social sciences. The most important difference between the social sciences and the natural sciences with respect to modeling is that in the social sciences powerful and well confirmed background theories (like Newtonian mechanics, quantum mechanics or the theory of relativity in physics) do not exist in the social sciences. Therefore, an epistemology of models that is formed on the role model of physics may not be appropriate for the social sciences. I discuss the challenges that modeling faces in the social sciences and point out their epistemological consequences. The most important consequences are that greater emphasis must be placed on empirical validation than on theoretical validation and that the relevance of purely theoretical simulations is strongly limited. (shrink)

This paper provides a method for characterizing space events using the framework of conceptual spaces. We focus specifically on estimating and ranking the likelihood of collisions between space objects. The objective is to design an approach for anticipatory decision support for space operators who can take preventive actions on the basis of assessments of relative risk. To make this possible our approach draws on the fusion of both hard and soft data within a single decision support framework. Contextual data is (...) also taken into account, for example data about space weather effects, by drawing on the Space Domain Ontologies, a large system of ontologies designed to support all aspects of space situational awareness. The framework is coupled with a mathematical programming scheme that frames a mathematically optimal approach for decision support, providing a quantitative basis for ranking potential for collision across multiple satellite pairs. The goal is to provide the broadest possible information foundation for critical assessments of collision likelihood. (shrink)

The paper considers a new type of objects – blinking fractals – that are not covered by traditional theories studying dynamics of self-similarity processes. It is shown that the new approach allows one to give various quantitative characteristics of the newly introduced and traditional fractals using infinite and infinitesimal numbers proposed recently. In this connection, the problem of the mathematical modelling of continuity is discussed in detail. A strong advantage of the introduced computational paradigm consists of its well-marked numerical (...) character and its own instrument – Infinity Computer – able to execute operations with infinite and infinitesimal numbers. (shrink)

Much work in MKM depends on the application of formal logic to mathematics. However, much mathematical knowledge is informal. Luckily, formal logic only represents one tradition in logic, specifically the modeling of inference in terms of logical form. Many inferences cannot be captured in this manner. The study of such inferences is still within the domain of logic, and is sometimes called informal logic. This paper explores some of the benefits informal logic may have for the management of (...) informal mathematical knowledge. (shrink)

(THIS PAPER NEEDS A CORRECTION) Satisfaction is a complex concept which has a key role in each individual’s everyday life and impacts their behavior. Abraham Maslow (1943) suggested a framework [1] to study human motivation, which was a starting point towards developing the quality of life(QOL) theory. On that article, he described a hierarchy of human needs, that is generally consist of fundamental needs which are required for human survival, and environment dependent ones, like society, safety and etc. In this (...) study, we are trying to represent a tangible mathematical model for assessing the sense of need for water in terms of the available source of water. On this very subject, M. Joseph Sirgy [2] states that ”From a human developmental perspective, QOL goals can be defined as satisfaction of human developmental needs in a community or society”. Accordingly, in the first section we propose a model for the water needs of an individual. In the second section, we continue the discussion by checking the behavior of the model on a given population, and each time by assuming that we have two out of three involved factors, we try to suggest an estimation for the other one. (shrink)

Following Marr’s famous three-level distinction between explanations in cognitive science, it is often accepted that focus on modeling cognitive tasks should be on the computational level rather than the algorithmic level. When it comes to mathematical problem solving, this approach suggests that the complexity of the task of solving a problem can be characterized by the computational complexity of that problem. In this paper, I argue that human cognizers use heuristic and didactic tools and thus engage in cognitive (...) processes that make their problem solving algorithms computationally suboptimal, in contrast with the optimal algorithms studied in the computational approach. Therefore, in order to accurately model the human cognitive tasks involved in mathematical problem solving, we need to expand our methodology to also include aspects relevant to the algorithmic level. This allows us to study algorithms that are cognitively optimal for human problem solvers. Since problem solving methods are not universal, I propose that they should be studied in the framework of enculturation, which can explain the expected cultural variance in the humanly optimal algorithms. While mathematical problem solving is used as the case study, the considerations in this paper concern modeling of cognitive tasks in general. (shrink)

2nd edition. Many-valued logics are those logics that have more than the two classical truth values, to wit, true and false; in fact, they can have from three to infinitely many truth values. This property, together with truth-functionality, provides a powerful formalism to reason in settings where classical logic—as well as other non-classical logics—is of no avail. Indeed, originally motivated by philosophical concerns, these logics soon proved relevant for a plethora of applications ranging from switching theory to cognitive modeling, (...) and they are today in more demand than ever, due to the realization that inconsistency and vagueness in knowledge bases and information processes are not only inevitable and acceptable, but also perhaps welcome. The main modern applications of (any) logic are to be found in the digital computer, and we thus require the practical knowledge how to computerize—which also means automate—decisions (i.e. reasoning) in many-valued logics. This, in turn, necessitates a mathematical foundation for these logics. This book provides both these mathematical foundation and practical knowledge in a rigorous, yet accessible, text, while at the same time situating these logics in the context of the satisfiability problem (SAT) and automated deduction. The main text is complemented with a large selection of exercises, a plus for the reader wishing to not only learn about, but also do something with, many-valued logics. (shrink)

The paper explicates the stages of the author’s philosophical evolution in the light of Kopnin’s ideas and heritage. Starting from Kopnin’s understanding of dialectical materialism, the author has stated that category transformations of physics has opened from conceptualization of immutability to mutability and then to interaction, evolvement and emergence. He has connected the problem of physical cognition universals with an elaboration of the specific system of tools and methods of identifying, individuating and distinguishing objects from a scientific theory domain. The (...) role of vacuum conception and the idea of existence (actual and potential, observable and nonobservable, virtual and hidden) types were analyzed. In collaboration with S.Crymski heuristic and regulative functions of categories of substance, world as a whole as well as postulates of relativity and absoluteness, and anthropic and self-development principles were singled out. Elaborating Kopnin’s view of scientific theories as a practically effective and relatively true mapping of their domains, the author in collaboration with M. Burgin have originated the unified structure-nominative reconstruction (model) of scientific theory as a knowledge system. According to it, every scientific knowledge system includes hierarchically organized and complex subsystems that partially and separately have been studied by standard, structuralist, operationalist, problem-solving, axiological and other directions of the current philosophy of science. 1) The logico-linguistic subsystem represents and normalizes by means of different, including mathematical, languages and normalizes and logical calculi the knowledge available on objects under study. 2) The model-representing subsystem comprises peculiar to the knowledge system ways of their modeling and understanding. 3) The pragmatic-procedural subsystem contains general and unique to the knowledge system operations, methods, procedures, algorithms and programs. 4) From the viewpoint of the problem-heuristic subsystem, the knowledge system is a unique way of setting and resolving questions, problems, puzzles and tasks of cognition of objects into question. It also includes various heuristics and estimations (truth, consistency, beauty, efficacy, adequacy, heuristicity etc) of components and structures of the knowledge system. 5) The subsystem of links fixes interrelations between above-mentioned components, structures and subsystems of the knowledge system. The structure-nominative reconstruction has been used in the philosophical and comparative case-studies of mathematical, physical, economic, legal, political, pedagogical, social, and sociological theories. It has enlarged the collection of knowledge structures, connected, for instance, with a multitude of theoreticity levels and with an application of numerous mathematical languages. It has deepened the comprehension of relations between the main directions of current philosophy of science. They are interpreted as dealing mainly with isolated subsystems of scientific theory. This reconstruction has disclosed a variety of undetected knowledge structures, associated also, for instance, with principles of symmetry and supersymmetry and with laws of various levels and degrees. In cooperation with the physicist Olexander Gabovich the modified structure-nominative reconstruction is in the processes of development and justification. Ideas and concepts were also in the center of Kopnin’s cognitive activity. The author has suggested and elaborated the triplet model of concepts. According to it, any scientific concept is a dependent on cognitive situation, dynamical, multifunctional state of scientist’s thinking, and available knowledge system. A concept is modeled as being consisted from three interrelated structures. 1) The concept base characterizes objects falling under a concept as well as their properties and relations. In terms of volume and content the logical modeling reveals partially only the concept base. 2) The concept representing part includes structures and means (names, statements, abstract properties, quantitative values of object properties and relations, mathematical equations and their systems, theoretical models etc.) of object representation in the appropriate knowledge system. 3) The linkage unites a structures and procedures that connect components from the abovementioned structures. The partial cases of the triplet model are logical, information, two-tired, standard, exemplar, prototype, knowledge-dependent and other concept models. It has introduced the triplet classification that comprises several hundreds of concept types. Different kinds of fuzziness are distinguished. Even the most precise and exact concepts are fuzzy in some triplet aspect. The notions of relations between real scientific concepts are essentially extended. For example, the definition and strict analysis of such relations between concepts as formalization, quantification, mathematization, generalization, fuzzification, and various kinds of identity are proposed. The concepts «PLANET» and «ELEMENTARY PARTICLE» and some of their metamorphoses were analyzed in triplet terms. The Kopnin’s methodology and epistemology of cognition was being used for creating conception of the philosophy of law as elaborating of understanding, justification, estimating and criticizing legal system. The basic information on the major directions in current Western philosophy of law (legal realism, feminism, criticism, postmodernism, economical analysis of law etc.) is firstly introduced to the Ukrainian audience. The classification of more than fifty directions in modern legal philosophy is suggested. Some results of historical, linguistic, scientometric and philosophic-legal studies of the present state of Ukrainian academic science are given. (shrink)

Throughout this paper, we are trying to show how and why our Mathematical frame-work seems inappropriate to solve problems in Theory of Computation. More exactly, the concept of turning back in time in paradoxes causes inconsistency in modeling of the concept of Time in some semantic situations. As we see in the first chapter, by introducing a version of “Unexpected Hanging Paradox”,first we attempt to open a new explanation for some paradoxes. In the second step, by applying this (...) paradox, it is demonstrated that any formalized system for the Theory of Computation based on Classical Logic and Turing Model of Computation leads us to a contradiction. We conclude that our mathematical frame work is inappropriate for Theory of Computation. Furthermore, the result provides us a reason that many problems in Complexity Theory resist to be solved.(This work is completed in 2017 -5- 2, it is in vixra in 2017-5-14, presented in Unilog 2018, Vichy). (shrink)

By the end of his life Plato had rearranged the theory of ideas into his teaching about ideal numbers, but no written records have been left. The Ideal mathematics of Plato is present in all his dialogues. It can be clearly grasped in relation to the effective use of mathematical modelling. Many problems of mathematical modelling were laid in the foundation of the method by cutting the three-level idealism of Plato to the single-level “ideism” of Aristotle. For a (...) long time, the real, ideal numbers of Plato’s Ideal mathematics eliminates many mathematical problems, extends the capabilities of modelling, and improves mathematics. (shrink)

Global processes significantly affect the mobility of the population. In the context of geopolitical transformation, globalization and quarantine restrictions of Covid-19, it is important to predict the development of the migration movement of countries that are developing. Therefore, the article is aimed at modelling migration changes according to alternative scenarios using the example of Ukraine. The theoretical and methodological basis of the research is formed by a number of scientific works of leading scientists from different countries, statistical information on migration (...) processes and socio-economic indicators of Ukraine’s development, economic, mathematical and scenario methods. In the course of the study, the main factors were identified that more affect the migration processes of Ukraine, taking into account the trends in the impact of Covid-19 on them. These include population size, life expectancy, GDP per capita, average monthly wages, and the volume of remittances from individuals to Ukraine. With the help of correlation-regression analysis, a multivariate econometric model of migration growth (reduction) has been built. This made it possible to study the absolute and relative influence of factors on the magnitude of the migration increase (decrease), determine the potential reserves for its increase (decrease), evaluate them using a comparative analysis and carry out predictive calculations of the volume of migration increase (decrease) in Ukraine. (shrink)

Jobst Langrebe’s and Barry Smith’s book Why Machines Will Never Rule the World argues that artificial general intelligence (AGI) will never be realized. Drawing on theories of complexity they argue that it is not only technically, but mathematically impossible to realize AGI. The book is the result of cooperation between a philosopher and a mathematician. In addition to a thorough treatment of mathematical modelling of complex systems the book addresses many fundamental philosophical questions. The authors show that philosophy is (...) still relevant for questions of information technology in general and artificial intelligence in particular. This paper endorses Landgrebe’s and Smith’s arguments that artificial general intelligence cannot be realized, but not their conclusion that machines will never rule the world. It is not only a question of what technology can do. An equally important question is what technology does to us. Machines may not take over the world in a literal sense, but they may have many negative effects. Some of the most serious can be placed under the category of the “degeneration effect”. (shrink)

This year’s topic is “Genomics and Philosophy of Race.” Different researchers might work on distinct subsets of the six thematic clusters below, which are neither mutually exclusive nor collectively exhaustive: (1) Concepts of ‘Race’; (2) Mathematical Modeling of Human History and Population Structure; (3) Data and Technologies of Human Genomics; (4) Biological Reality of Race; (5) Racialized Selves in a Global Context; (6) Pragmatic Consequences of ‘Race Talk’ among Biologists.