The attempts of theoretically solving the famous puzzle-dictum of physicist Eugene Wigner regarding the “unreasonable” effectiveness of mathematics as a problem of analytical philosophy, started at the end of the 19th century, are yet far from coming out with an acceptable theoretical solution. The theories developed for explaining the empirical “miracle” of applied mathematics vary in nature, foundation and solution, from denying the existence of a genuine problem to structural theories with an advanced level of mathematical formalism. Despite (...) this variation, methodologically fundamental questions like “Which is the adequate theoretical framework for solving Wigner’s conjecture?” and “Can the logico-mathematical formalism solve it and is it entitled to do it?” did not receive answers yet. The problem of the applicability of mathematics in the physical reality has been treated unitarily in some sense, with respect to the semantic-conceptual use of the constitutive terms, within both the structural and non-structural theories. This unity (of consistency) applied to both the referred objects and concepts per se and the aims of the investigations. For being able to make an objective study of the possible alternatives of the existent theories, a foundational approach of them is needed, including through semantic-conceptual distinctions which to weaken the unity of consistency. (shrink)

According to Steiner (1998), in contemporary physics new important discoveries are often obtained by means of strategies which rely on purely formal mathematical considerations. In such discoveries, mathematics seems to have a peculiar and controversial role, which apparently cannot be accounted for by means of standard methodological criteria. M. Gell-Mann and Y. Ne׳eman׳s prediction of the Ω− particle is usually considered a typical example of application of this kind of strategy. According to Bangu (2008), this prediction is apparently based (...) on the employment of a highly controversial principle—what he calls the “reification principle”. Bangu himself takes this principle to be methodologically unjustifiable, but still indispensable to make the prediction logically sound. In the present paper I will offer a new reconstruction of the reasoning that led to this prediction. By means of this reconstruction, I will show that we do not need to postulate any “reificatory” role of mathematics in contemporary physics and I will contextually clarify the representative and heuristic role of mathematics in science. (shrink)

Abstract In the new millennium there have been important empirical developments in the philosophy of mathematics. One of these is the so-called “Empirical Philosophy of Mathematics”(EPM) of Buldt, Löwe, Müller and Müller-Hill, which aims to complement the methodology of the philosophy of mathematics with empirical work. Among other things, this includes surveys of mathematicians, which EPM believes to give philosophically important results. In this paper I take a critical look at the sociological part of EPM as (...) a case study of ... (shrink)

We argue that the mathematization of science should be understood as a normative activity of advocating for a particular methodology with its own criteria for evaluating good research. As a case study, we examine the mathematization of taxonomic classification in systematic biology. We show how mathematization is a normative activity by contrasting its distinctive features in numerical taxonomy in the 1960s with an earlier reform advocated by Ernst Mayr starting in the 1940s. Both Mayr and the numerical taxonomists sought (...) to formalize the work of classification, but Mayr introduced a qualitative formalism based on human judgment for determining the taxonomic rank of populations, while the numerical taxonomists introduced a quantitative formalism based on automated procedures for computing classifications. The key contrast between Mayr and the numerical taxonomists is how they conceptualized the temporal structure of the workflow of classification, specifically where they allowed meta-level discourse about difficulties in producing the classification. (shrink)

Socialist buildings in the city of Kruja (Albania) date back after the Second World War between the years 1945-1990. These buildings were built during the time of the socialist Albanian dictatorship and the totalitarian communist regime. A questionnaire with 30 questions was conducted and 14 people were interviewed. The interviewed residents belong to a certain area of the city of Kruja. Based on the results obtained, diagrams have been conceived and mathematical regression models have been developed which will serve as (...) a base for drawing conclusions. The purpose of this study is to find a relationship that specifies the role of the inhabitants, the role of the dwellings, the physical-mechanical characteristics, and their energy efficiency, in order to improve the living conditions of the inhabitants. The independent variables interact with each other assuming positive or negative values depending on the probabilistic mathematical regression, giving different results based on the calculated data. The variables that co-exist among them are living space, quality of life, time spent at home, social interaction of residents, dwelling humidity level, dwelling orientation, satisfaction level of the inhabitants, dwelling improvements demand, and monthly electricity bills. (shrink)

In spite of the many efforts made to clarify von Neumann’s methodology of science, one crucial point seems to have been disregarded in recent literature: his closeness to Hilbert’s spirit. In this paper I shall claim that the scientific methodology adopted by von Neumann in his later foundational reflections originates in the attempt to revaluate Hilbert’s axiomatics in the light of Gödel’s incompleteness theorems. Indeed, axiomatics continues to be pursued by the Hungarian mathematician in the spirit of Hilbert’s (...) school. I shall argue this point by examining four basic ideas embraced by von Neumann in his foundational considerations: a) the conservative attitude to assume in mathematics; b) the role that mathematics and the axiomatic approach have to play in all that is science; c) the notion of success as an alternative methodological criterion to follow in scientific research; d) the empirical and, at the same time, abstract nature of mathematical thought. Once these four basic ideas have been accepted, Hilbert’s spirit in von Neumann’s methodology of science will become clear. (shrink)

The methodology of scientific research programmes is a collection of papers published over time expressing a radical review of Popper's demarcation criterion between science and non-science, leading to a new theory of scientific rationality. Volume I address aspects of the philosophy of science, and volume II contains works on the philosophy of mathematics. For a science historian, the reconstruction proposed by Lakatos is attractive and explains the evolution of science to a level that has not been achieved before. (...) The basic evaluation unit proposed by Lakatos is the research programme, and especially the dynamics of these programmes. DOI: 10.13140/RG.2.2.31605.01767. (shrink)

This article reexamines Vico’s early critique of Cartesian reasoning and of how the Cartesian method, which comes from epistemology, creates problems for the sciences once embedded into their methodologies and given a foundational role. The focus will be on De nostri temporis studiorum ratione (1709), where Vico argues against generalizing the Cartesian method and overemphasizing clarity and distinctness in the search for truth. To this end, Vico’s relation to Cartesianism is first carefully contextualized. Then, Vico is presented as a hylomorphist (...) when it comes to how scientific disciplines and their instruments interrelate, something key for understanding why he questions the “instrumentalization” of the Cartesian method in the first place. Afterwards, Vico’s account of the impossibility of maximal scientific progress when the Cartesian method is prioritized across the sciences is presented, as probabilistic reasoning and the importance of the imagination and memory take centerstage. Lastly, the article looks at Vico’s opposition to the geometrical method’s role in physics and his defense of visual thinking in mathematics. The hope throughout is to advance a picture of the early Vico as being not only a neo- Baconian pragmatist, but a hypermodern ”meta-Cartesian” thinker concerned with humanity’s optimizing its creative potential in order to achieve scientific advances. (shrink)

Imre Lakatos' views on the philosophy of mathematics are important and they have often been underappreciated. The most obvious lacuna in this respect is the lack of detailed discussion and analysis of his 1976a paper and its implications for the methodology of mathematics, particularly its implications with respect to argumentation and the matter of how truths are established in mathematics. The most important themes that run through his work on the philosophy of mathematics and which (...) culminate in the 1976a paper are (1) the (quasi-)empirical character of mathematics and (2) the rejection of axiomatic deductivism as the basis of mathematical knowledge. In this paper Lakatos' later views on the quasi-empirical nature of mathematical theories and methodology are examined and specific attention is paid to what this view implies about the nature of mathematical argumentation and its relation to the empirical sciences. (shrink)

We give an overview of Lakatos’ life, his philosophy of mathematics and science, as well as of this issue. Firstly, we briefly delineate Lakatos’ key contributions to philosophy: his anti-formalist philosophy of mathematics, and his methodology of scientific research programmes in the philosophy of science. Secondly, we outline the themes and structure of the masterclass Lakatos’ Undone Work – The Practical Turn and the Division of Philosophy of Mathematics and Philosophy of Science, which gave rise to (...) this special issue. Lastly, we provide a summary of the contributions to this issue. (shrink)

Willem Jacob ’s Gravesande is widely remembered as a leading advocate of Isaac Newton’s work. In the first half of the eighteenth century, ’s Gravesande was arguably Europe’s most important proponent of what would become known as Newtonian physics. ’s Gravesande himself minimally described this discipline, which he called «physica», as studying empirical regularities mathematically while avoiding hypotheses. Commentators have as yet not progressed much beyond this view of ’s Gravesande’s physics. Therefore, much of its precise nature, its methodology, (...) and its relation to Newton’s actual work remains unclear. This article discusses one particular methodological element that ’s Gravesande himself often stressed in detail, namely the use of mathematics in philosophy and physics. In doing so, it takes exception to the claim that mathematics played only a minor role in ’s Gravesande’s work, a view put forward in recent historiography. Besides that, this article casts new light on the interpretation of ’s Gravesande’s philosophical notion of «mathematical reasoning», a notion that has remained somewhat obscure thus far. (shrink)

A recently developed computational methodology for executing numerical calculations with infinities and infinitesimals is described in this paper. The approach developed has a pronounced applied character and is based on the principle “The part is less than the whole” introduced by the ancient Greeks. This principle is applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). The point of view on infinities and infinitesimals (and in general, on Mathematics) presented in (...) this paper uses strongly physical ideas emphasizing interrelations that hold between a mathematical object under observation and the tools used for this observation. It is shown how a new numeral system allowing one to express different infinite and infinitesimal quantities in a unique framework can be used for theoretical and computational purposes. Numerous examples dealing with infinite sets, divergent series, limits, and probability theory are given. (shrink)

Indispensability arguments are used as a way of working out what there is: our best science tells us what things there are. Some philosophers think that indispensability arguments can be used to show that we should be committed to the existence of mathematical objects (numbers, functions, sets). Do indispensability arguments also deliver conclusions about the modal properties of these mathematical entities? Colyvan (in Leng, Paseau, Potter (eds) Mathematical knowledge, OUP, Oxford, 109-122, 2007) and Hartry Field (Realism, mathematics and modality, (...) Blackwell, Oxford, 1989) each suggest that a consequence of the empirical methodology of indispensability arguments is that the resulting mathematical objects can only be said to exist (or not exist) contingently. Kristie Miller has argued that this line of thought doesn’t work (Miller in Erkenntnis, 77 (3), 335-359, 2012). Miller argues that indispensability arguments are in direct tension with contingentism about mathematical objects, and that they cannot tell us about the modal status of mathematical objects. I argue that Miller’s argument is crucially imprecise, and that the best way of making it clearer no longer shows that the indispensability strategy collapses or is unstable if it delivers contingentist conclusions about what there is. (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)

The goal of the article is to substantiate that despite the criticism the paradigm in economics will not change because of the axiomatic assumptions of value-free economics. How these assumptions work is demonstrated on the example of Gary Becker’s economic approach which is analyzed from the perspective of scientific research programme. The author indicates hard core of economic approach and the protective belt which makes hard core immune from any criticism. This immunity leads economists to believe that they are objective (...) scientists and, consequently, it results in epistemological hubris. Due to its tautological nature, economic approach is considered to be a degenerative programme. This conclusion is extended on value-free economics. In spite of these problems, many economists still believe in positive economics and they dismiss normative approaches. It has a negative influence on people. The conclusion of the article is that thanks to axiomatic assumptions economists do not have objective and ironclad methodology and they should accept normative values in their research. (shrink)

A Mathematical Review by John Corcoran, SUNY/Buffalo -/- Macbeth, Danielle Diagrammatic reasoning in Frege's Begriffsschrift. Synthese 186 (2012), no. 1, 289–314. ABSTRACT This review begins with two quotations from the paper: its abstract and the first paragraph of the conclusion. The point of the quotations is to make clear by the “give-them-enough-rope” strategy how murky, incompetent, and badly written the paper is. I know I am asking a lot, but I have to ask you to read the quoted passages—aloud if (...) possible. Don’t miss the silly attempt to recycle Kant’s quip “Concepts without intuitions are empty; intuitions without concepts are blind”. What the paper was aiming at includes the absurdity: “Proofs without definitions are empty; definitions without proofs are, if not blind, then dumb.” But the author even bollixed this. The editor didn’t even notice. The copy-editor missed it. And the author’s proof-reading did not catch it. In order not to torment you I will quote the sentence as it appears: “In a slogan: proofs without definitions are empty, merely the aimless manipulation of signs according to rules; and definitions without proofs are, if no blind, then dumb.”[sic] The rest of my review discusses the paper’s astounding misattribution to contemporary logicians of the information-theoretic approach. This approach was cruelly trashed by Quine in his 1970 Philosophy of Logic, and thereafter ignored by every text I know of. The paper under review attributes generally to modern philosophers and logicians views that were never espoused by any of the prominent logicians—such as Hilbert, Gödel, Tarski, Church, and Quine—apparently in an attempt to distance them from Frege: the focus of the article. On page 310 we find the following paragraph. “In our logics it is assumed that inference potential is given by truth-conditions. Hence, we think, deduction can be nothing more than a matter of making explicit information that is already contained in one’s premises. If the deduction is valid then the information contained in the conclusion must be contained already in the premises; if that information is not contained already in the premises […], then the argument cannot be valid.” Although the paper is meticulous in citing supporting literature for less questionable points, no references are given for this. In fact, the view that deduction is the making explicit of information that is only implicit in premises has not been espoused by any standard symbolic logic books. It has only recently been articulated by a small number of philosophical logicians from a younger generation, for example, in the prize-winning essay by J. Sagüillo, Methodological practice and complementary concepts of logical consequence: Tarski’s model-theoretic consequence and Corcoran’s information-theoretic consequence, History and Philosophy of Logic, 30 (2009), pp. 21–48. The paper omits definitions of key terms including ‘ampliative’, ‘explicatory’, ‘inference potential’, ‘truth-condition’, and ‘information’. The definition of prime number on page 292 is as follows: “To say that a number is prime is to say that it is not divisible without remainder by another number”. This would make one be the only prime number. The paper being reviewed had the benefit of two anonymous referees who contributed “very helpful comments on an earlier draft”. Could these anonymous referees have read the paper? -/- J. Corcoran, U of Buffalo, SUNY -/- PS By the way, if anyone has a paper that has been turned down by other journals, any journal that would publish something like this might be worth trying. (shrink)

Semantic analysis in early analytic philosophy belongs to a long tradition of adopting geometrical methodologies to the solution of philosophical problems. In particular, it adapts Descartes’ development of formalization as a mechanism of analytic representation, for its application in natural language semantics. This article aims to trace the mathematical roots of Frege, Russel and Carnap’s analytic method. Special attention is paid to the formal character of modern analysis introduced by Descartes. The goal is to identify the particular conception of “form” (...) developed by the analytic tradition, from Descartes to early analytic philosophy, and to determine its relation to similar notions, like ‘function’ and ‘syntax’. Finally, I focus on how Frege, Russell and Carnap’s methods of semantic analysis fit the general characterization of formal analysis previously developed. (shrink)

In the debate over whether mathematical facts, properties, or entities explain physical events (in what philosophers call “extra-mathematical” explanations), Aidan Lyon’s (2012) affirmative answer stands out for its employment of the program explanation (PE) methodology of Frank Jackson and Philip Pettit (1990). Juha Saatsi (2012; 2016) objects, however, that Lyon’s examples from the indispensabilist literature are (i) unsuitable for PE, (ii) nominalizable into non-mathematical terms, and (iii) mysterious about the explanatory relation alleged to obtain between the PE’s mathematical explanantia (...) and physical explananda. In this paper, I propose a counterexample to Saatsi’s objections. My counterexample is Frank Jackson’s (1998a) program explanation for color experience, which I argue needs recasting as an extra-mathematical PE due to its implicit reliance on reflectance, a property that suffers conceptual regress unless redefined with Fourier harmonics. Pace Saatsi, I argue that this recast example is an authoritative PE, non-nominalizable, and minimally esoteric. Important for the indispensability debate at large, moreover, is that my counterexample reifies Fourier harmonics without the Enhanced Indispensability Argument (an argument to which Lyon applies PE as a premise). Indispensabilists have long overlooked the conditionalization of a limited mathematical realism on property realism, and my counterexample to Saatsi exploits this conditionalization. (shrink)

This chapter argues that the standard conception of Spinoza as a fellow-travelling mechanical philosopher and proto-scientific naturalist is misleading. It argues, first, that Spinoza’s account of the proper method for the study of nature presented in the Theological-Political Treatise (TTP) points away from the one commonly associated with the mechanical philosophy. Moreover, throughout his works Spinoza’s views on the very possibility of knowledge of nature are decidedly sceptical (as specified below). Third, in the seventeenth-century debates over proper methods in the (...) sciences, Spinoza sided with those that criticized the aspirations of those (the physico-mathematicians, Galileo, Huygens, Wallis, Wren, etc) who thought the application of mathematics to nature was the way to make progress. In particular, he offers grounds for doubting their confidence in the significance of measurement as well as their piece-meal methodology (see section 2). Along the way, this chapter offers a new interpretation of common notions in the context of treating Spinoza’s account of motion (see section 3). (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 condensed summary of the adventures of ideas (1990-2020). Methodology of evolutionary-phenomenological constitution of Consciousness. Vector (BeVector) of Consciousness. Consciousness is a qualitative vector quantity. Vector of Consciousness as a synthesizing category, eidos-prototecton, intentional meta-observer. The development of the ideas of Pierre Teilhard de Chardin, Brentano, Husserl, Bergson, Florensky, Losev, Mamardashvili, Nalimov. Dialectic of Eidos and Logos. "Curve line" of the Consciousness Vector from space and time. The lower and upper sides of the "abyss of being". The existential tension (...) of being. Five reference “points” (existential-extremum) - world events in the evolution of Consciousness from “homo habilis” to “homo sapiens sapiens”. “Prometheus Effect". Inversion and reversion of Сonsciousness. "Open" and "closed" Сonsciousness. Consciousness and Self-Consciousness. Protogeometer: fall in the future, in the "EgoLand". The problem of justification/substantiation of mathematics (knowledge) is an ontological problem. The crisis of ontology and ontological limits of cognition. Ontological structure of space. The project of constructive dialectic ontology. The methodology of dialectical-ontological construction (modeling): conceptual-figure synthesis, total unification of matter, coincidence of ontological opposites, primordial Meta-Axioma and Superprinciple, vector (bivector) of the absolute state (states) of matter (absolute forms of existence). The ontological "celestial triangle" (Plato). Ontological invariants of the Universum and their ontological paths. Ontological and gnoseological dimensions of space. Triune (ontological) space of nine gnoseological dimensions: absolute rest (linear state, "Continuum")+ absolute motion (absolute vortex, "Discretuum") + absolute becoming (absolute wave, "Dis-Continuum"). Primordial generating structure: ontological framework, ontological carcass and ontological foundation. Ontological (structural, cosmic) memory - the semantic core of the conceptual construction of the Universum being as an eternal process of generation of new meanings and structures. Consciousness is an absolute (unconditional) attractor of meanings, a univalent phenomenon of ontological memory, which manifests itself at a certain level of the Universum being. Ontological space-MatterMemory-Ontological time (S-MM-T). Ontological Rhythm and Cycle. The nature and structure of ontological time. Natural (absolute) determinism. (shrink)

The aim of this paper is to investigate the current debate on the state of economics from a methodological perspective. We claim that the majority of contributions criticizing modern economics are not based on clear methodological principles and thus many of them are not correct. We show this with respect to such issues as the problem of realisticness of models and their assumptions, the role of mathematics in economics, the way we conceptualize the relation between economics (theory) and economy (...) (empiria), as well as the general problem of comparing theories. In doing so we use the research apparatus taken form the philosophy of science and also we benefit a lot from recent developments in the philosophy of economics. Finally, we show one of the paradoxes of that debate, namely that many critics of economics accuse economists of using the wrong language (mathematics) while they do not use proper language themselves while criticizing economics, namely the apparatus taken from the philosophy of science. (shrink)

This paper considers non-standard analysis and a recently introduced computational methodology based on the notion of ①. The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the ①-based methodology does not use any (...) of these notions and proposes a more physical treatment of mathematical objects separating the objects from tools used to study them. It both offers a possibility to create new numerical methods using infinities and infinitesimals in floating-point computations and allows one to study certain mathematical objects dealing with infinity more accurately than it is done traditionally. In these notes, we explain that even though both methodologies deal with infinities and infinitesimals, they are independent and represent two different philosophies of Mathematics that are not in a conflict. It is proved that texts :539–555, 2017; Gutman and Kutateladze in Sib Math J 49:835–841, 2008; Kutateladze in J Appl Ind Math 5:73–75, 2011) asserting that the ①-based methodology is a part of non-standard analysis unfortunately contain several logical fallacies. Their attempt to show that the ①-based methodology can be formalized within non-standard analysis is similar to trying to show that constructivism can be reduced to the traditional Mathematics. (shrink)

The critique of philosophical foundations of neoclassical economics is significant, because of its hegemony on economic education and research programs in Iran and worldwide academies. Due to an epistemological fallacy, methodological individualism plays a prominent role in the philosophy of economic; since the ontological aspects of economy are reduced to methodological considerations. Accordingly, critique of methodological individualism is regarded as the main entry for philosophical analysis of neoclassical economics. This article aims to analyze and appraise the methodological individualism from critical (...) realism’s point of view. Critical realism is taken as a stand for criticizing methodological individualism, because critical realism and neoclassical economics as a positive approach to economics, both share a realistic worldview. The critical realism’ critique of methodological individualism is bases on analyzing the concepts of formalism, syllogism, social atomism, isolation and reduction. From critical realism’ point of view because of unrealistic postulations of methodological individualism, neoclassical economics is turned to an abstract and deductive science such as mathematics with a low potentiality of explanation. (shrink)

What is so special and mysterious about the Continuum, this ancient, always topical, and alongside the concept of integers, most intuitively transparent and omnipresent conceptual and formal medium for mathematical constructions and the battle field of mathematical inquiries ? And why it resists the century long siege by best mathematical minds of all times committed to penetrate once and for all its set-theoretical enigma ? -/- The double-edged purpose of the present study is to save from the transfinite deadlock of (...) higher set theory the jewel of mathematical Continuum -- this genuine, even if mostly forgotten today raison d'etre of all set-theoretical enterprises to Infinity and beyond, from Georg Cantor to W. Hugh Woodin to Buzz Lightyear, by simultaneously exhibiting the limits and pitfalls of all old and new reductionist foundational approaches to mathematical truth: be it Cantor's or post-Cantorian Idealism, Brouwer's or post-Brouwerian Constructivism, Hilbert's or post-Hilbertian Formalism, Goedel's or post-Goedelian Platonism. -/- In the spirit of Zeno's paradoxes, but with the enormous historical advantage of hindsight, we claim that Cantor's set-theoretical methodology, powerful and reach in proof-theoretic and similar applications as it might be, is inherently limited by its epistemological framework of transfinite local causality, and neither can be held accountable for the properties of the Continuum already acquired through geometrical, analytical, and arithmetical studies, nor can it be used for an adequate, conceptually sensible, operationally workable, and axiomatically sustainable re-creation of the Continuum. -/- From a strictly mathematical point of view, this intrinsic limitation of the constative and explicative power of higher set theory finds its explanation in the identified in this study ultimate phenomenological obstacle to Cantor's transfinite construction, similar to topological obstacles in homotopy theory and theoretical physics: the entanglement capacity of the mathematical Continuum. (shrink)

Plastic waste is a significant environmental problem for almost all countries; therefore, protecting the environment from the problem is crucial. The most sensible solution to these problems is substituting the natural aggregates with substantial plastic waste in various building materials. This study aimed to optimise the mixed design ratio of cement brick containing plastic waste as aggregate replacement. Plastic cement brick mixtures were prepared by the incorporation of four different types of plastic waste such as polyethylene terephthalate (PET), high-density polyethylene, (...) low-density polyethylene and polypropylene into cement bricks with different cement contents (150, 300 and 450 g) and plastic replacement percentages (0, 3 and 6%). Compressive strength and water absorption of the plastic cement bricks were analysed using a statistical model through the response surface methodology. It revealed the optimum cement brick mixed design is C3-1% PET with the compressive strength of 27.50 MPa and water absorption of 1.16%. The optimised plastic cement brick also satisfied the general ASTM C62-17 requirements for building bricks despite the higher porosity observed by the scanning electron microscopy. The results from Fourier transform infrared spectroscopy analysis also showed that the addition of the plastic waste into cement brick was unlikely to modify the chemical compound within the cement brick mixtures. Thus, the proposed mathematical model can predict the required hardened properties of plastic cement bricks and could lead to greater utilisation of plastic waste in building materials. (shrink)

It is discerned what light can bring the recent historical reconstructions of maxwellian optics and electromagnetism unification on the following philosophical/methodological questions. I. Why should one believe that Nature is ultimately simple and that unified theories are more likely to be true? II. What does it mean to say that a theory is unified? III. Why theory unification should be an epistemic virtue? To answer the questions posed genesis and development of Maxwellian electrodynamics are elucidated. It is enunciated that the (...) Maxwellian Revolution is a far more complicated phenomenon than it may be seen in the light of Kuhnian and Lakatosian epistemological models. Correspondingly it is maintained that maxwellian electrodynamics was elaborated in the course of the old pre-maxwellian programmes’ reconciliation: the electrodynamics of Ampére-Weber, the wave theory of Young-Fresnel and Faraday’s programme. To compare the different theoretical schemes springing from the different language games James Maxwell had constructed a peculiar neutral language. Initially it had encompassed the incompressible fluid models; eventually – the vortices ones. The three programmes’ encounter engendered the construction of the hybrid theory at first with an irregular set of theoretical schemes. However, step by step, on revealing and gradual eliminating the contradictions between the programmes involved, the hybrid set is “put into order” (Maxwell’s term). A hierarchy of theoretical schemes starting from ingenious crossbreeds (the displacement current) and up to usual hybrids is set up. After the displacement current construction the interpenetration of the pre-maxwellian programmes begins that marks the commencement of theoretical schemes of optics, electricity and magnetism real unification. Maxwell’s programme surpassed that of Ampére-Weber because it did absorb the ideas of the Ampére-Weber programme, as well as the presuppositions of the programmes of Young-Fresnel and Faraday properly co-ordinating them with each other. But the opposite statement is not true. The Ampére-Weber programme did not assimilate the propositions of the Maxwellian programme. Maxwell’s victory over his rivals became possible because the gist of Maxwell’s unification strategy was formed by Kantian epistemology looked in the light of William Whewell and such representatives of Scottish Enlightenment as Thomas Reid and Sir William Hamilton. Maxwell did put forward as basic synthetic principles the ideas that radically differed from that of Ampére-Weber approach by their open, flexible and contra-ontological, genuinly epistemological, Kantian character. For Maxwell, ether was not the ultimate building block of physical reality, from which all the charges and fields should be constructed. “Action at a distance”, “incompressible fluid”, “molecular vortices”, etc. were contrived analogies for Maxwell, capable only to direct the researcher at the “right” mathematical relations. Key words: J.C. Maxwell, unification of optics and electromagnetism, I. Kant, T. Reid, W. Hamilton . (shrink)

Open peer commentary on the article “Constructivist Model Building: Empirical Examples From Mathematics Education” by Catherine Ulrich, Erik S. Tillema, Amy J. Hackenberg & Anderson Norton. Upshot: I argue that radical constructivism poses a series of deep methodological constraints on educational research. We focus on the work of Ulrich et al. to illustrate the practical implications of these constraints.

Francis Ysidro Edgeworth’s unduly neglected monograph New and Old Methods of Ethics (1877) advances a highly sophisticated and mathematized account of social well-being in the utilitarian tradition of his 19th-century contemporaries. This article illustrates how his usage of the ‘calculus of variations’ was combined with findings from empirical psychology and economic theory to construct a consequentialist axiological framework. A conclusion is drawn that Edgeworth is a methodological predecessor to several important methods, ideas, and issues that continue to be discussed in (...) contemporary social well-being studies. (shrink)

Gödel argued that intuition has an important role to play in mathematical epistemology, and despite the infamy of his own position, this opinion still has much to recommend it. Intuitions and folk platitudes play a central role in philosophical enquiry too, and have recently been elevated to a central position in one project for understanding philosophical methodology: the so-called ‘Canberra Plan’. This philosophical role for intuitions suggests an analogous epistemology for some fundamental parts of mathematics, which casts a (...) number of themes in recent philosophy of mathematics (concerning a priority and fictionalism, for example) in revealing new light. (shrink)

This research article was championed as a way of providing discourses pertaining to the concept of "Critical Realism (CR)" approach, which is amongst many othe forms of competing postmodern philosophical concepts for the engagement of dialogical discourses in the area of established econonetric methodologies for effective policy prescription in the economic science discipline. On the the whole, there is no doubt surrounding the value of empirical endeavours in econometrics to address real world economic problems, but equally so, the heavy weighted (...) use and reliance on mathematical contents as a way of justifying its scientific base seemed to be loosing traction of the intended focus of economics when it comes to confronting real world problems in the domain of social interaction. In this vein, the construction of mixed methods discourse(s), which favour that of CR philosophy is hereby suggested in this article as a way forward in confronting with issues raised by critics of mainstream economics and other professionals in the postmodern era. (shrink)

Penelope Maddy’s Second Philosophy is one of the most well-known ap- proaches in recent philosophy of mathematics. She applies her second-philosophical method to analyze mathematical methodology by reconstructing historical cases in a setting of means-ends relations. However, outside of Maddy’s own work, this kind of methodological analysis has not yet been extensively used and analyzed. In the present work, we will make a first step in this direction. We develop a general framework that allows us to clarify the (...) procedure and aims of the Second Philosopher’s investigation into set-theoretic methodology; pro- vides a platform to analyze the Second Philosopher’s methods themselves; and can be applied to further questions in the philosophy of set theory. (shrink)

High grade chromite ore has been decreased constantly due to the importance of chromium element in industrial uses such as metallurgical, chemical, and refractory industries. Therefore, beneficiation of low grade has been more significant .Response Surface Methodology (RSM) is combination of statistical and mathematical methods used for modeling and analyzing problems. In this study, the Central Composite Design (CCD) was applied by using Design-Expert (version 6.0.5) for modeling and optimizing the effect of operating variables on the performance of gravity (...) separation via pilot plant shaking table for chromite ore. Three operating variables were studied, namely feed rate, tilt angle, and flow rate during the tests. The sample under study is low grade chromite ore, containing (30.21%, Cr2O3). The mathematical model equations of ANOVA model revealed that the grade of concentrate is more sensetive for feed rate (g/min) compered to water flow rate (l/min).whereas , recovery of concentrate is more sensetive for tilt angle compered to water flow rate (l/min) and feed rate (g/min). Optimized responses for the beneficiation process were found at concentrate with 48.52% Cr2O3 in with 83.09% recovery and it was achieved at water flow rate 15.33 l/min, tilt angle 2.16 ≈ 2.00 degree, and feed rate of 195.38 g/min. (shrink)

Extensive scientific, philosophical and artistic activities were carried out in the Islamic World’s various science and civilization centers during the early Middle Ages. In these centers, noteworthy works of mathematics, astronomy, geography, medicine, pharmacology, optics, botany, chemistry and other fields of science, which would later determine improvement paths for these fields, were created. Abu al-Izz Ismail ibn al-Razzaz al-Jazari (12th-13th centuries), was a magnificent Muslim scientist known for his work named The Book of Knowledge of Ingenious Mechanical Devices (Kitab (...) fi ma‘rifati’l-hiyal al-handasiyya) and he was born in geography where many scientists were trained. He contributed to the development of science and civilization, the joint legacy of humanity and demonstrated his true capability and success in the field of ilm al-hiyal (the scientific field which addresses the information and practices regarding mechanical technology and engineering in Islamic science history). From this perspective, it is evident that al-Jazari had a wide field of interest, made his mark in the age he lived in with his brave and innovative ideas, and was a wise engineer. al-Jazari’s detailed descriptions and explanations of water clocks, oil lamp clocks and water pumping machines, as well as their construction, were astounding. In addition, al-Jazari’s studies on metal casting techniques, cone valves, wheel balance and the real-world use of aspiration tubes are important in contemporary engineering studies. When all these are considered, it becomes clear that his designation as the “Leonardo da Vinci of the Islamic World” is not a coincidence. Many studies on al-Jazari’s scientific legacy have been conducted in various parts of the world. There are various manuscripts of his aforementioned book. Based on these manuscripts, the work has been -fully or partially- translated into many languages, including Turkish. The German and English translations served as a basis for studies on al-Jazari in other languages. In recent years, studies conducted on al-Jazari in Türkiye (books, articles, domestic and international symposiums, congresses etc.) have demonstrated his brilliance and competence as a scientist. Furthermore, studies on al-Jazari are still ongoing in Türkiye. As for Russia, the scientific legacy of the scientists raised in the Islamic World such as Avicenna, al-Farabi, Abu Bakr al-Razi, Abu Rayhan al-Biruni, Abu al-Qasim al-Zahrawi, Muhammad ibn Musa al-Khwarizmi, al-Battani, al-Kashi, Abd al-Rahman al-Khazini, Nasir al-Din al-Tusi was researched within the scope of a special plan and program. Depending on the relevant studies, al-Jazari and his work drew attention as well. A small portion of the aforementioned Russian studies is known in Türkiye. It is safe to assume that al-Jazari studies in Russia are virtually unknown in Türkiye. The present study primarily aims to contribute to the literature on the scientific legacy of al-Jazari, a Middle Age Islamic scientist best known for his famous work in the mechanical field. More specifically, the present study aims to: 1) to identify the studies on al-Jazari in Russia, and in the former USSR countries, where academic studies are still in Russian after the Soviet Union period, to create a chronological biography and to set forth their compilation; in other words, to create a bibliography which includes Russian studies on al-Jazari; 2) to provide information on the structure and content of the studies in question, to detect the methodological issues encountered in these works, and make solution recommendations to solve these issues; 3) to contribute to continuing and future studies in Türkiye on al-Jazari in terms of source and material index. In conclusion, it should be stated that comparative reading was conducted centered on The Book of Knowledge of Ingenious Mechanical Devices (Kitab fi ma‘rifati’l-hiyal al-handasiyya). It should also be noted that Russian popular internet publications (which are not based on academic research and are created/reproduced for commercial and personal purposes) were excluded from the assessment. (shrink)

The purpose of this study was to determine whether the application of learning brain-friendly through the whole brain teaching a positive effect on the character of creative students, to study the response of the students, and to determine whether the students' response to the application of learning brain-friendly through the whole brain teaching positively correlated with the character of creative students in mathematics. The research method used is quantitative. The instruments used are student questionnaire responses related to the application (...) of brain-friendly learning through the whole brain teaching and observation sheet on student creativity in the learning of mathematics after the implementation of this method. Results of research with correlation analysis show that the greater significance of the alpha value (5%), which means accepting and rejecting H0 Ha which means students' response to the application of brain-friendly learning through the whole brain teaching is not positively correlated with the creative character of students in the learning of mathematics. The average score of the students' response to this methodology very well categorized in the amount of 85%. The results of the observation of the creative character of the students after the implementation of this method, the average score of 68% were categorized quite creative. (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)

In this chapter we will argue, firstly, that Bacon’s engages in a pecu-liar form of mathematization of nature that develops a quali-quantitative methodology of measurement. Secondly, we will show that medicine is one of the disciplines where that dual way of measurement is practiced. In the first section of the chapter, we will expose the ontology involved in the Baconian proposal of measurement of nature. The second section will address the place that mixed mathematics occupies in Bacon’s scheme (...) of scientific branches and will suggest that a proper advancement of medicine can generate a new branch of mixed mathematics. The next section will reconstruct Bacon’s approach to measurement and expose its quali-quantitative import. In the last section, we will show some examples of medicine in which this quali-quantitative measurement is applied. (shrink)

While there are satisfactory answers to the question “How should we teach children mathematics?”, there are no satisfactory answers to the question “What mathematics should we teach children?”. This paper provides an answer to the last question for preschool children (early childhood), although the answer is also applicable to older children. This answer, together with an appropriate methodology on how to teach mathematics, gives a clear conception of the place of mathematics in the children’s world (...) and our role in helping children develop their mathematical abilities. Briefly, children’s mathematics consists of the world of children’s internal activities that they eventually purposefully organize in order to understand and control the outside world and organize their overall activities in it. We need to support a child in mathematical activities that she does spontaneously and in which she shows interest, and we need to teach her mathematics that she is interested in developing through these activities. In doing so, we must be fully aware that the child’s mathematics is part of the child’s world of internal activities and is not outside of it. We help the child develop mathematical abilities by developing them in the context of her world and not outside of it. From the point of view of this conception, the standards established today are limiting and too focused on numbers and geometric figures: these topics are too prominent and elaborated, and other mathematical contents are subordinated to them. Adhering to the standards, we drastically limit the mathematics of the child’s world, hamper the correct mathematical development of a child, and we can turn her away from mathematics. (shrink)

This chapter tries to answer the following question: How should we conceive of the method of mathematics, if we take a naturalist stance? The problem arises since mathematical knowledge is regarded as the paradigm of certain knowledge, because mathematics is based on the axiomatic method. Moreover, natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some authors (...) tried to naturalize mathematics by relying on evolutionism. But several difficulties arise when we try to do this. This chapter suggests that, in order to naturalize mathematics, it is better to take the method of mathematics to be the analytic method, rather than the axiomatic method, and thus conceive of mathematical knowledge as plausible knowledge. (shrink)

The influence of religious beliefs to several leading mathematicians in early Soviet years, especially among members of the Moscow Mathematical Society, had drawn the attention of militant Soviet marxists, as well as Soviet authorities. The issue has also drawn significant attention from scholars in the post-Soviet period. According to the currently prevailing interpretation, reported purges against Moscow mathematicians due to their religious inclination are the focal point of the relevant history. However, I maintain that historical data arguably offer reasons to (...) cast reasonable doubts on this interpretation. In this paper, by reviewing the relevant literature, I raise some methodological and philosophical concerns, in an attempt to contribute to a better understanding of the issue. I maintain that an efficient line of reasoning is to discuss issues in the context of their making, taking into consideration the specific features of each era’s culture. Thus, by focusing on P.A. Nekrasov’s case, I attempt to point to an alternative interpretation, in which the different treatment of religious inclined mathematicians by Soviet authorities is explained in the context of the ideological confrontation between two contrasting worldviews, as part of the ongoing class war in the several phases of Soviet history. (shrink)

In 1901 Russell had envisaged the new analytic philosophy as uniquely systematic, borrowing the methods of science and mathematics. A century later, have Russell’s hopes become reality? David Lewis is often celebrated as a great systematic metaphysician, his influence proof that we live in a heyday of systematic philosophy. But, we argue, this common belief is misguided: Lewis was not a systematic philosopher, and he didn’t want to be. Although some aspects of his philosophy are systematic, mainly his pluriverse (...) of possible worlds and its many applications, that systematicity was due to the influence of his teacher Quine, who really was an heir to Russell. Drawing upon Lewis’s posthumous papers and his correspondence as well as the published record, we show that Lewis’s non- Quinean influences, including G.E. Moore and D.M. Armstrong, led Lewis to an anti- systematic methodology which leaves each philosopher’s views and starting points to his or her own personal conscience. (shrink)

Over the past two decades, gamblers have begun taking mathematics into account more seriously than ever before. While probability theory is the only rigorous theory modeling the uncertainty, even though in idealized conditions, numerical probabilities are viewed not only as mere mathematical information, but also as a decision-making criterion, especially in gambling. This book presents the mathematics underlying the major games of chance and provides a precise account of the odds associated with all gaming events. It begins by (...) explaining in simple terms the meaning of the concept of probability for the layman and goes on to become an enlightening journey through the mathematics of chance, randomness and risk. It then continues with the basics of discrete probability, combinatorics and counting arguments for those interested in the supporting mathematics. These mathematic sections may be skipped by readers who do not have a minimal background in mathematics; these readers can skip directly to the Guide to Numerical Results to pick the odds and recommendations they need for the desired gaming situation. Doing so is possible due to the organization of that chapter, in which the results are listed at the end of each section, mostly in the form of tables. The chapter titled The Mathematics of Games of Chance presents these games not only as a good application field for probability theory, but also in terms of human actions where probability-based strategies can be tried to achieve favorable results. Through suggestive examples, the reader can see what are the experiments, events and probability fields in games of chance and how probability calculus works there. The main portion of this work is a collection of probability results for each type of game. Each game s section is packed with formulas and tables. Each section also contains a description of the game, a classification of the gaming events and the applicable probability calculations. The primary goal of this work is to allow the reader to quickly find the odds for a specific gaming situation, in order to improve his or her betting/gaming decisions. Every type of gaming event is tabulated in a logical, consistent and comprehensive manner. The complete methodology and complete or partial calculations are shown to teach players how to calculate probability for any situation, for every stage of the game for any game. Here, readers can find the real odds, returned by precise mathematical formulas and not by partial simulations that most software uses. Collections of odds are presented, as well as strategic recommendations based on those odds, where necessary, for each type of gaming situation. The book contains much new and original material that has not been published previously and provides great coverage of probabilities for the following games of chance: Dice, Slots, Roulette, Baccarat, Blackjack, Texas Hold em Poker, Lottery and Sport Bets. Most of games of chance are predisposed to probability-based decisions. This is why the approach is not an exclusively statistical one, but analytical: every gaming event is taken as an individual applied probability problem to solve. A special chapter defines the probability-based strategy and mathematically shows why such strategy is theoretically optimal.". (shrink)

How should one conceive of the method of mathematics, if one takes a naturalist stance? Mathematical knowledge is regarded as the paradigm of certain knowledge, since mathematics is based on the axiomatic method. Natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some naturalists try to naturalize mathematics relying on Darwinism. But several difficulties arise when (...) one tries to naturalize in this way the traditional view of mathematics, according to which mathematical knowledge is certain and the method of mathematics is the axiomatic method. This paper suggests that, in order to naturalize mathematics through Darwinism, it is better to take the method of mathematics not to be the axiomatic method. (shrink)

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

The paper investigates how the mathematical languages used to describe and to observe automatic computations influence the accuracy of the obtained results. In particular, we focus our attention on Single and Multi-tape Turing machines which are described and observed through the lens of a new mathematical language which is strongly based on three methodological ideas borrowed from Physics and applied to Mathematics, namely: the distinction between the object (we speak here about a mathematical object) of an observation and the (...) instrument used for this observation; interrelations holding between the object and the tool used for the observation; the accuracy of the observation determined by the tool. Results of the observation executed by the traditional and new languages are compared and discussed. (shrink)

Reverse mathematics studies which subsystems of second order arithmetic are equivalent to key theorems of ordinary, non-set-theoretic mathematics. The main philosophical application of reverse mathematics proposed thus far is foundational analysis, which explores the limits of different foundations for mathematics in a formally precise manner. This paper gives a detailed account of the motivations and methodology of foundational analysis, which have heretofore been largely left implicit in the practice. It then shows how this account can (...) be fruitfully applied in the evaluation of major foundational approaches by a careful examination of two case studies: a partial realization of Hilbert’s program due to Simpson [1988], and predicativism in the extended form due to Feferman and Schütte. -/- Shore [2010, 2013] proposes that equivalences in reverse mathematics be proved in the same way as inequivalences, namely by considering only omega-models of the systems in question. Shore refers to this approach as computational reverse mathematics. This paper shows that despite some attractive features, computational reverse mathematics is inappropriate for foundational analysis, for two major reasons. Firstly, the computable entailment relation employed in computational reverse mathematics does not preserve justification for the foundational programs above. Secondly, computable entailment is a Pi-1-1 complete relation, and hence employing it commits one to theoretical resources which outstrip those available within any foundational approach that is proof-theoretically weaker than Pi-1-1-CA0. (shrink)

The sudden shift in the education system during the pandemic and its subsequent evolution during the post-pandemic era have been pivotal in fostering significant educational development and growth. However, this paradigm shift has not been without challenges. This paper aims to investigate the challenges faced by 68 mathematics teachers in four public elementary schools in the Philippines. The respondents were purposively selected to answer the study’s instrument. Using a descriptive survey research methodology, this study explored the five domains (...) in teaching mathematics, namely planning lessons, executing lessons, assessment, mathematical knowledge, and using technology tools. This paper revealed that novice, experienced, and highly experienced teachers alike had moderate difficulty enriching mathematical knowledge and utilizing technology in teaching the subject matter. Moreover, teachers with more than five years’ experience had moderate difficulty in the five domains of teaching mathematics in the post-pandemic era. This finding emphasizes the widespread impact of the education shift for teachers of all levels of experience. The paper further emphasizes the importance of supporting experienced teachers during this transformative phase of formulating innovative learning activities and adapting to reimagined pedagogical concepts. Consequently, this study underscores the need for additional training interventions to enhance their mathematical knowledge and use of technology as an integral teaching tool. (shrink)

The paper discusses the possibility of constructing economic models using the methodology of model construction in classical mechanics. At the same time, unlike the "econophysical" approach, the properties of economic models are derived without involvement of any equivalent physical properties, but with account of the types of symmetry existing in the economic system. It has been shown that at this approach practically all known mechanical variables have their "economic twins". The variational principle is formulated on the basis of formal (...) mathematical construction without involving the subjective factor common to the majority of models in economics. The dynamics of interaction of two companies has been studies in details, on the basis of which we can proceed to modeling of more complex and realistic economic systems. Prediction of the possibility of constructing economic theory on the basis of primary principles analogously to physics has been made. (shrink)

One of the key features of modern mathematics is the adoption of the abstract method. Our goal in this paper is to propose an explication of that method that is rooted in the history of the subject.