Mathematical Reality Research Articles

Social constructivism in mathematics? The promise and shortcomings of Julian Cole\u2019s institutional account

The core idea of social constructivism in mathematics is that mathematical entities are social constructs that exist in virtue of social practices, similar to more familiar social entities like institutions and money. Julian C. Cole has presented an institutional version of social constructivism about mathematics based on John Searle’s theory of the construction of the social reality. In this paper, I consider what merits social constructivism has and examine how well Cole’s institutional account meets the challenge of accounting for the characteristic features of mathematics, especially objectivity and applicability. I propose that in general social constructivism shows promise as an ontology of mathematics, because the view can agree with mathematical practice and it offers a way of understanding how mathematical entities can be real without conflicting with a scientific picture of reality. However, I argue that Cole’s specific theory does not provide an adequate social constructivist account of mathematics. His institutional account fails to sufficiently explain the objectivity and applicability of mathematics, because the explanations are weakened and limited by the three-level theoretical model underlying Cole’s account of the construction of mathematical reality and by the use of the Searlean institutional framework. The shortcomings of Cole’s theory give reason to suspect that the Searlean framework is not an optimal way to defend the view that mathematical reality is socially constructed.

واقع ممارسة معلمى الریاضیات بالمرحلة الثانویة لمهارات التواصل الریاضى الإلکترونى واتجاهاتهم نحو استخدامه فى التدریس The Reality of Mathematics Teacher Practice in the Secondary Stage of Electronic Mathematical Communication Skills and their Attitude towards its Use in Teaching

مستخلص البحث: هدف البحث إلى تعرف واقع ممارسة معلمى الرياضيات بالمرحلة الثانوية لمهارات التواصل الرياضى الإلکترونى، واتجاهاتهم نحو استخدامه فى التدريس، ولتحقيق الهدف من البحث قام الباحث بإعداد استبانة ممارسات التواصل الرياضى الإلکترونى، ومقياس الاتجاه نحو استخدام التواصل الرياضى الإلکترونى فى التدريس، واختيار عينة البحث، والتى تمثلت فى (35) معلما ومعلمة من معلمى الرياضيات بالمرحلة الثانوية، وأسفرت نتائج البحث عن: أن ممارسة معلمى الرياضيات بالمرحلة الثانوية لمهارت التواصل الرياضى الإلکترونى کانت متوسطة، واتجاههم نحو استخدامه فى تدريس الرياضيات کان متوسطا، وقدم البحث عددا من التوصيات والبحوث المقترحة. This study aimed at identifying the reality of mathematics teachers’ practices in the secondary stages of electronic Mathematical communication skills and their attitude towards its use in teaching. To achieve the aim of the study, the researcher prepared an Electronic Mathematical communication practices’ checklist, and a questionnaire of their attitudes towards using electronic Mathematical communication skills. Sample of the study consisted of (35) male and female mathematics teachers in the secondary stages. Results indicated that: Mathematics teachers in the secondary stage practice electronic Mathematical communication skills with an average level. Also, their attitudes towards using it in teaching mathematics was average .

Volume, Side-Area, and Force Direction of Berkovich and Cubecorner Indenters, Novel Important Insights

The iteration-free physical description of pyramidal indentations with closed mathematical equations is comprehensively described and extended for creating new insights in this important field of research and applications. All calculations are easily repeatable and should be programmed by instrument builders for even easier general use. Formulas for the volumes and side-areas of Berkovich and cubecorner as a function of depth are deduced and provided, as are the resulting forces and force directions. All of these allow for the detailed comparison of the different indenters on the mathematical reality. The pyramidal values differ remarkably from the ones of so-called “equivalent cones”. The worldwide use of such pseudo-cones is in severe error. The earlier claimed and used 3 times higher displaced volume with cube corner than with Berkovich is disproved. Both displace the same amount at the same applied force. The unprecedented mathematical results are experimentally confirmed for the physical indentation hardness and for the sharp-onset phase-transi- tions with calculated transition energy. The comparison of both indenters provides novel basic insights. Isotropic materials exhibit the same phase transition onset force, but the transition energy is larger with the cube corner, due to higher force and flatter force direction. This qualifies the cube corner for fracture toughness studies. Pile-up is not from the claimed “friction with the indenter”. Anisotropic materials with cleavage planes and channels undergo sliding along these under pressure, both to the surface and internally. Their volumes add to the depression volume. These volumes are essential for the exemplified pile-up management. Phase-transitions produce polymorph interfaces that are nucleation sites for cracks. Technical materials must be developed with onset forces higher than the highest thinkable stresses (at airliners, bridges, etc.). This requires urgent revision of ISO 14577-ASTM standards.

Matemática e metafísica em Platão e Aristóteles

As diferenças entre os modos de conceber a realidade matemática estão no próprio centro da divergência entre as filosofias de Platão e de Aristóteles, figurando de maneira crucial no estágio das controvérsias metafísicas que separam os dois filósofos. Platão, argumenta-se, mantém-se fiel à tradição filosófica anterior, ao estabelecer uma hierarquia ontológica que espelha diretamente os graus de inteligibilidade dos objetos da ciência. Aristóteles subverte fundamentalmente essa tendência, ao subordinar a matemática à física e propor, no lugar da matemática, a lógica como modelo primário da razão dianoética.

Inštrumentálny realizmus ako možné východisko teoretickej reflexie vyučovania matematiky

The aim of the paper is to present an epistemological position called instrumental realism and to show how this position can be used as a tool for the analysis of different approaches to mathematics education. Instrumental realism is based on the conviction that any successful study of a particular segment of reality must on the one hand take into account the specific features of the particular segment of reality that we are studying. On the other hand, it must take into account the particular epistemic tools, by means of which we approach this segment of reality. In the case of physics, biology, or psychology, these two requirement do not seem controversial. Nevertheless, in the case of mathematics to speak about mathematical reality that we study or about the instruments that we use is much less obvious. Instrumental realism thus opens a new perspective on knowledge acquisition, that, as we believe, has many important consequences for mathematics education.

ВИВЧЕННЯ ЯКОСТІ МАТЕМАТИЧНОГО РОЗВИТКУ ДІТЕЙ СТАРШОГО ДОШКІЛЬНОГО ВІКУ

In the article the author highlights and argues some approaches of modern native and foreign scientists and the results of her own pedagogical experience to the study of the quality of logical and mathematical development of preschool age children. In the context of this issue, the structure and content of the diagnostic methodology, the construction, organization and maintenance of the process of controlling and assessing the quality of mathematical knowledge and children's skills are considered. The author characterizes the structure of the diagnostic methodology and distinguishes in it the relevant directions: 1) study of the level of educational achievements of the child, accumulation of actual data that will reflect the following components: motivational, informational and procedural; 2) study of the influence of organizational and pedagogical conditions created by the tutor during the conduct of classes in mathematics and other pedagogical activities, on the quality of mathematical development of children, through purposeful observation and analysis of its results; 3) study of the methodological guidance of mathematical development of children in pre-school institutions, through the analysis of pedagogical documentation. Therefore, the diagnostic methodology includes: monitoring the activity of children in situations where they independently and adequately use mathematical knowledge and appropriate skills; individual conversations with children and their performance of respective tasks; questionnaire of teachers and study of pedagogical documentation, which presents the ways of methodological guidance for the development of professional skills of teachers in pre-school institutions. The results of the research work of the author of the article indicate that the quality of mathematical development is a relative characteristic of both the educational process as a whole and the development of the individual child. An important characteristic of mathematical development of the baby is the elementary mathematical competence, which involves the presence of knowledge about the number, shape, size, space, time, as well as the ability to apply this knowledge in different situations in life; detection of children independence, self-esteem, self-control, positive attitude to mathematical reality, cognitive interest, etc. These indicators will give the teacher the opportunity to evaluate the quality of mathematical development of the child, to build on this basis, individual development programs for each baby. Key words: mathematical development, quality of mathematical knowledge and skills; diagnostic technique.

The Case of Quantum Mechanics Mathematizing Reality: The ‘Superposition’ of Mathematically Modeled and Mathematical Reality: Is There Any Room for Gravity?

A case study of quantum mechanics is investigated in the framework of the philosophical opposition “mathematical model – reality”. All classical science obeys the postulate about the fundamental difference of model and reality, and thus distinguishing epistemology from ontology fundamentally. The theorems about the absence of hidden variables in quantum mechanics imply for it to be “complete” (versus Einstein’s opinion). That consistent completeness (unlike arithmetic to set theory in the foundations of mathematics in Godel’s opinion) can be interpreted furthermore as the coincidence of model and reality. The paper discusses the option and fact of that coincidence it its base: the fundamental postulate formulated by Niels Bohr about what quantum mechanics studies (unlike all classical science). Quantum mechanics involves and develops further both identification and disjunctive distinction of the global space of the apparatus and the local space of the investigated quantum entity as complementary to each other. This results into the analogical complementarity of model and reality in quantum mechanics. The apparatus turns out to be both absolutely “transparent” and identically coinciding simultaneously with the reflected quantum reality. Thus, the coincidence of model and reality is postulated as necessary condition for cognition in quantum mechanics by Bohr’s postulate and further, embodied in its formalism of the separable complex Hilbert space, in turn, implying the theorems of the absence of hidden variables (or the equivalent to them “conservation of energy conservation” in quantum mechanics). What the apparatus and measured entity exchange cannot be energy (for the different exponents of energy), but quantum information (as a certain, unambiguously determined wave function) therefore a generalized law of conservation, from which the conservation of energy conservation is a corollary. Particularly, the local and global space (rigorously justified in the Standard model) share the complementarity isomorphic to that of model and reality in the foundation of quantum mechanics. On that background, one can think of the troubles of “quantum gravity” as fundamental, direct corollaries from the postulates of quantum mechanics. Gravity can be defined only as a relation or by a pair of non-orthogonal separable complex Hilbert space attachable whether to two “parts” or to a whole and its parts. On the contrary, all the three fundamental interactions in the Standard model are “flat” and only “properties”: they need only a single separable complex Hilbert space to be defined.

How to improve critical thinking skills and spatial reasoning with augmented reality in mathematics learning?

Critical thinking skills and spatial reasoning are important aspect in human life. On the other hand, currently, implementing augmented reality in many fields be a trend, including in education field. This study aims to explain how to improve critical thinking skills and spatial reasoning with augmented reality in mathematics learning. The method used in this research is literature review. The data obtained from scientific journals, conference proceedings, and other relevant sources. The results show that to improve spatial reasoning can be done by utilizing its features that can develop confidence of learners, facilitate student-centered learning, create problem solving, and create interactive learning. Besides that, spatial reasoning can be improved by designing dynamic objects with AR. Dynamic objects permit learners to view objects from different sides. Therefore, active learning can be promoted.

Компьютер как новая реальность математики. I. Personal account

In the last decades there was much ado about computer proofs, computer aided proofs, computer verified proofs, etc. It is obvious that the advent and proliferation of computers have drastically changed applications of mathematics. What one discusses much less, however, is how computers changed mathematics itself, and mathematicians’ stance in regard of mathematical reality, both as far as the possibilities to immediately observe it, and the apprehension of what we can hope to prove. I am recounting my personal experience of using computers as a mathematical tool, and the experience of such similar use in the works of my colleagues that I could observe at close range. This experience has radically changed my perception of many aspects of mathematics, how it functions, and especially, how it should be taught. This first introductory part consists mostly of reminiscences and some philosophical observations. Further parts describe several specific important advances in algebra and number theory, that would had been impossible without computers.

Groups, Special Functions and Rigged Hilbert Spaces

We show that Lie groups and their respective algebras, special functions and rigged Hilbert spaces are complementary concepts that coexist together in a common framework and that they are aspects of the same mathematical reality. Special functions serve as bases for infinite dimensional Hilbert spaces supporting linear unitary irreducible representations of a given Lie group. These representations are explicitly given by operators on the Hilbert space H and the generators of the Lie algebra are represented by unbounded self-adjoint operators. The action of these operators on elements of continuous bases is often considered. These continuous bases do not make sense as vectors in the Hilbert space; instead, they are functionals on the dual space, Φ × , of a rigged Hilbert space, Φ ⊂ H ⊂ Φ × . In fact, rigged Hilbert spaces are the structures in which both, discrete orthonormal and continuous bases may coexist. We define the space of test vectors Φ and a topology on it at our convenience, depending on the studied group. The generators of the Lie algebra can often be continuous operators on Φ with its own topology, so that they admit continuous extensions to the dual Φ × and, therefore, act on the elements of the continuous basis. We investigate this formalism for various examples of interest in quantum mechanics. In particular, we consider S O ( 2 ) and functions on the unit circle, S U ( 2 ) and associated Laguerre functions, Weyl–Heisenberg group and Hermite functions, S O ( 3 , 2 ) and spherical harmonics, s u ( 1 , 1 ) and Laguerre functions, s u ( 2 , 2 ) and algebraic Jacobi functions and, finally, s u ( 1 , 1 ) ⊕ s u ( 1 , 1 ) and Zernike functions on a circle.

LAUTMAN ON PROBLEMS AS THE CONDITIONS OF EXISTENCE OF SOLUTIONS

Albert Lautman (1908–44) was a philosopher of mathematics whose views on mathematical reality and on the philosophy of mathematics parted with the dominant tendencies of mathematical epistemology of the time. Lautman considered the role of philosophy, and of the philosopher, in relation to mathematics to be quite specific. He writes that “in the development of mathematics, a reality is asserted that mathematical philosophy has as a function to recognize and describe” (Mathematics, Ideas and the Physical Real (London: Bloomsbury, 2011) 87). He goes on to characterise this reality as an “ideal reality” that “governs” the development of mathematics. The relation between mathematical problems as they arise in the historical development of mathematics and the solutions that are provided to these problems by mathematicians, in the form of new mathematical theories, definitions or axioms, are governed by what Lautman characterises as a dialectics of mathematics. The aim of this paper is to give an account of this Lautmanian dialectic and of how it can be understood to govern the development of solutions to mathematical problems.

A Season of Science: April to August 2017

Letter| November 01 2017 A Season of Science: April to August 2017 ABT Editor: William F. McComas William F. McComas ABT Editor Search for other works by this author on: This Site PubMed Google Scholar The American Biology Teacher (2017) 79 (9): 701. https://doi.org/10.1525/abt.2017.79.9.701 Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Facebook Twitter LinkedIn MailTo Tools Icon Tools Get Permissions Cite Icon Cite Search Site Citation William F. McComas; A Season of Science: April to August 2017. The American Biology Teacher 1 November 2017; 79 (9): 701. doi: https://doi.org/10.1525/abt.2017.79.9.701 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentThe American Biology Teacher Search It is hard to ignore the frequent attacks on science, its implications and conclusions. So, please forgive me if I join the chorus of worry and express my deep concern about the increasing decline in trust for science in this essay. We live in extraordinary times. It seems that all one needs to do to reject a personally disagreeable science conclusion is to cry out “fake news,” “alternative facts,” or exclaim “you have your opinions and I have mine!” Perhaps it is our strong democratic tradition that has allowed so many to become de facto science-deniers to think we can vote on everything (including the “truth”) and require that we must honor any opinion no matter how little empiricism it contains. Attacks on science and reason are not new. In 1887, Indiana considered establishing the value of pi as 3.2 until Purdue professor Clarence Waldo warned the legislature against voting... You do not currently have access to this content.

The state vector of a quantum system: Mathematical fiction or physical reality?

A version of an experiment with a correlated pair of entangled particles is considered. This experiment demonstrates an interesting effect of variations in the entangled photon polarization that shows the reality of all of the various superposition components and the corresponding state vector of the quantum system. The possible consequences of this are analyzed.

Mathematical Enargeia: The Rhetoric of Early Modern Mathematical Notation

This article proposes and explicates a rhetorical model for the function of notational writing in sixteenth- and seventeenth-century European mathematics. Drawing on enargeia's requirement that both author and reader contribute to the full realization of a text, mathematical enargeia enables the transformation of images of mathematical imagination resulting from an encounter with mathematical writing into further written acts of mathematical creation. Mathematical enargeia provides readers with an ability to understand a text as if they created it themselves. Within the period's dominant reading of classical geometry as a synthetic presentation that suppressed, hid, or obscured analytic mathematical reality, notational mathematics found favor as a rhetorically unmediated expression of mathematical truth. Consequently, mathematical enargeia creates an operational and presentational link between mathematics' past and its future.

Reinventing fractions and division as they are used in algebra: the power of preformal productions

In this paper, we explore algebra students’ mathematical realities around fractions and division, and the ways in which students reinvented mathematical productions involving fractions and division. We find that algebra students’ initial realities do not include the fraction-as-quotient sub-construct. This can be problematic because in algebra, quotients are almost always represented as fractions. In a design experiment, students progressively reinvented the fraction-as-quotient sub-construct. Analyzing this experiment, we find that a particular type of mathematical production, which we call preformal productions, played two meditational roles: (1) they mediated mathematical activity, and (2) they mediated the reinvention of more formal mathematical productions. We suggest that preformal productions may emerge even when they are not designed for, and we show how preformal productions embody historic classroom activity and social interaction.

Indefinite Divisibility

Some hold that the lesson of Russell’s paradox and its relatives is that mathematical reality does not form a ‘definite totality’ but rather is ‘indefinitely extensible’. There can always be more sets than there ever are. I argue that certain contact puzzles are analogous to Russell’s paradox this way: they similarly motivate a vision of physical reality as iteratively generated. In this picture, the divisions of the continuum into smaller parts are ‘potential’ rather than ‘actual’. Besides the intrinsic interest of this metaphysical picture, it has important consequences for the debate over absolute generality. It is often thought that ‘indefinite extensibility’ arguments at best make trouble for mathematical platonists; but the contact arguments show that nominalists face the same kind of difficulty, if they recognize even the metaphysical possibility of the picture I sketch.

Forecasting on Energy Consumption in China Based on Logistic-Markov

Logistic model has a simple and apparent reality in mathematics, it can well characterize the feedback mechanism of energy consumption and economic growth. Based on the Logistic model, this paper proposes a new method for solving Logistic. With China’s total energy consumption in 2000-2013 as raw data, it uses Logistic model to predict our country’s total energy consumption. And then, it improves the predicted results with Markov chain. The results show that: the average relative error is 4.71 percent if it uses Logistic model, while the average relative error is only 2.45 percent if it is improved. The prediction accuracy is improved of 2.26 percent. Therefore, this paper uses Logistic-Markov method to predict energy consumption of 2014-2020. Finally, it analyzes energy consumption trends by predicting, and proposes the corresponding energy conservation measures.

A Rational Belief: The Method of Discovery in the Complex Variable

The importance of mathematics in the context of the scientific and technological development of humanity is determined by the possibility of creating mathematical models of the objects studied under the different branches of Science and Technology. The arithmetisation process that took place during the nineteenth century consisted of the quest to discover a new mathematical reality in which the validity of logic would stand as something essential and central. Nevertheless, in contrast to this process, the development of mathematical analysis within a framework that largely involves intuition and geometry is a fact that cannot go unnoticed amongst the mathematics community, as we shall show in this paper through the research made by Bernhard Riemann on complex variables.

Neurostimulation options for failed back surgery syndrome: The need for rational and objective measurements. Proposal of an international clinical network using an integrated database and health economic analysis: The proback network

IntroductionIn the context of failed back surgery syndrome (FBSS) treatment, the current practice in neurostimulation varies from center-to-center and most clinical decisions are based on an individual diagnosis. Neurostimulation evaluation tools and pain relief assessement are of major concern, as they now constitute one of the main biases of clinical trials. Moreover, the proliferation of technological devices, in a fertile and unsatisfied market, fosters and only furthers the confusion. There are three options available to apply scientific debates to our daily neurostimulation practice: intentional ignorance, standardized evidence-based practice or alternative data mining approach. ObjectiveIn view of the impossibility of conducting multiple randomized clinical trials comparing various devices, one by one, the proposed concept would be to redefine the indications and the respective roles of the various spinal cord and peripheral nerve stimulation devices with large-scale computational modeling/data mining approach, by conducting a multicenter prospective database registry, supported by a clinician's global network called “PROBACK”. MethodsWe chose to specifically analyze 6 parameters: device coverage performance/coverage selectivity/persistence of the long-term electrical response (technical criteria) and comparative mapping of patient pain relief/persistence of the long-term clinical response/safety and complications occurrence (clinical criteria). Two types of analysis will be performed: immediate analysis (including cost analysis) and computational analysis, i.e. demonstration of the robustness of certain correlations of variables, in order to extract response predictors. Discussion/conclusionBy creating an international prospective database, the purpose of the PROBACK project was to set up a process of extraction and comparative analysis of data derived from the selection, implantation and follow-up of FBSS patients candidates for implanted neurostimulation. This evaluation strategy should help to change the opinions of each implanter and each health system towards a more rational decision-making approach subtended by mathematical reality.

Improving the accuracy of geological model by using seismic forward and inverse techniques

Abstract Sequential Gaussian simulation method is taken as an example to analyze the characteristics and defects of stochastic simulation methods, and a modeling strategy is proposed that is based on the results of seismic inversion to improve modeling accuracy. Stochastic simulation can only achieve “mathematical reality” by recovery of parameter's macro statistical regularity through examples, and seismic forward model can verify whether the simulation results deviate from the “geological reality”. Seismic forward modeling can verify the reliability of the geological model, and the results of geological modeling constrained by seismic inversion can improve the accuracy of the model to achieve the “geological reality”. For braided river delta, a modeling strategy constrained by seismic inversion results in the condition of “multi-levels” and “multi-conditions” is put forward. For lithofacies modeling, division of lithofacies in single well acts as the first variable; in the stage of exploration and development, lithology probability and lithology body act as the second variable respectively, and then establish facies models. For property modeling, take lithofacies model and the seismic impedance as the first variable and the second variable respectively to create property model, in the condition of horizontal and vertical impedance constraints. The proposed modeling strategy maintains the statistical regularities of input data, keep a better consistency with seismic data, and match with dynamic field production.