Concepts of Proof in Mathematics, Philosophy, and Computer Science

The article discusses the problem of teaching computer science, mathematics and logic in modern education, in particular, in a secondary general education school as inherited from the problems of the existing relationships between the sciences of computer science and mathematics, as well as generated by the prevailing forms and practice of teaching in the system of general education. The methodological aspects of teaching computer science, mathematics and logic are considered inherent in them and applied to them in a differentiated form: - a meaningful approach inherent in computer science and applied in mathematics and logic; - a formal approach inherent in mathematics and used in computer science and logic; - a socio-cultural approach characteristic of the triad ”computer science-mathematics-logic”. Against the background of the existing correspondence of the selected approaches (or the measure of this correspondence) to each local and independent element of the triad, it is assumed that there is an identification of the regular relations between the subject and the subject of education, the principles of organizing the processes of implementing methodological knowledge based on these regularities. For our study, it is important that the methodology and methodology of each differentiated system of the triad should contain everything that is ”necessary and sufficient” for the functioning of the educational system of the corresponding level. In this aspect, the article notes that mathematics studies form, computer science - form and content. The study of objects only in form leads to formalism in the information sphere (which is unacceptable), therefore, possible integration must be combined with the necessary differentiation. Since the processes of the development of knowledge of science, its methodology and teachingmethods are interrelated and interdependent, it is unconditionally interesting to reveal the potential (possibilities) of intersubject and metasubject connections and relations of the triad under consideration as close, but not homogeneous educational systems that have their own subject. specificity of the content, methods and forms of training. In this perspective, possible ways of solving the problems that have arisen are proposed in the context of changes in the school curricula of the content of information and mathematics education under the influence of the requirements of the new federal state educational standards and the structure of subject teaching in computer science and mathematics, ensuring their efficiency and effectiveness.

Based on the analogy between mathematical proofs and computer programs this paper illustrates how it is possible to apply methodologies from computer science, used in the development and presentation of computer programs, for the presentation of mathematical theorems and proofs. The following three methodologies are discussed: stepwise refinement, abstraction, and program verification.

This paper investigates mathematics trainee teachers’ conceptions of proof writing in algebra at Copperbelt College of Education in Zambia. The ability to construct proofs seems to be an important skill for all mathematicians and those who have interest in mathematics education. However, literature shows that proof is widely acknowledged to be one of the most difficult aspects in the mathematics curriculum. The sample comprised of 35 male and 15 female third year mathematics trainee teachers. Data was collected by means of a self-administered questionnaire, a written diagnostic test and oral interviews. The three instruments were used to provide multiple views to triangulate the data. The results show that students seem to understand proof as a verification and confirmation tool. Many of them applied empirical approaches in their written proofs. They did not have sufficient knowledge of the right set of accepted statements such as axioms and could not identify which statements required proofs in mathematics. These findings suggest that students’ conceptions of proof were underdeveloped hence affecting their ability to produce valid proofs. These findings will assist in guiding curricula designers and teachers of proof related knowledge. DOI: 10.5901/mjss.2014.v5n14p331

My aim in this thesis is to reveal the true nature of mathematical reasoning and to show the necessity of pictorial representations in mathematics. In order to support my thesis I argue that using only the formal language, without fully integrating visual reasoning in mathematical communication, mathematics operates insufficiently and it can never reach its full potential. First of all, I provide a short historical approach to philosophy of mathematics as well as a summary of current state of visualization in mathematics. Then, I bring forth a complete picture of philosophy of mathematics and mathematical objects, which consistently allows visual reasoning, intuitions and constructions in mathematical communication. This interpretation makes it easy to see the pictorial representations as a natural part of mathematical reasoning. Moreover, I lay out the medium for visual reasoning in mathematics: Space. After providing several definitions of space I conclude that Kant’s characterization is the finest one as a base for mathematical reasoning, in particular for visual reasoning. Furthermore, I bring out the method of visual reasoning: Synthetic a priori. In order to show the link between visualization and synthetic a priori I provide hitherto definitions of visualization and argue that these lead up to Kantian characterization of constructions and synthetic a priori. Then, I provide several interpretations to show that Kant’s characterization of mathematics has elements valuable for modern mathematics that modern logic cannot capture. Finally, I offer three main arguments leading to the main thesis “pictorial repre- sentations/ diagrams are necessary parts of mathematical reasoning”: (1) There are formal systems with diagrams. Hence, not all uses of diagrams are synthetic. (2) There are necessary non-formal uses of diagrams in mathematical proofs. (3) There is a strong link between visualization and Kantian philosophy of mathematics. As a result, I claim that the real capacity of mathematics is not in turning it into a machine language but integrating and appreciating the aesthetic part of it as well. I also indicate the necessity of having a theory of diagrammatics, which should be set up as a branch of mathematics.

هدف البحث الحالي التعرف على فاعلية نموذج البنية المعرفية لتدريس الرياضيات في تنمية مهارات التواصل الرياضي لدى تلاميذ المرحلة الإعدادية، ولتحقيق أهداف البحث استخدم الباحث المنهج شبه التجريبي، وتکونت أدوات الدراسة من اختبار في مهارات التواصل الرياضي في مهارات (الکتابة الرياضية، التمثيل الرياضي، القراءة الرياضية)، وتم تطبيق البحث على عينة اختيرت بالطريقة العشوائية حيث بلغ إجمالي العينة (113) تلميذ من تلاميذ الصف الأول الإعدادي من مدرسة منشأة البدوي الجديدة للتعليم الأساسي (ع) مرکز طلخا محافظة الدقهلية، تکونت المجموعة التجريبية من (57) تلميذ تم تدريسهم باستخدام نموذج البنية المعرفية لتدريس الرياضيات، والمجموعة الضابطة من (56) تلميذ تم تدريسهم بالطريقة التقليدية، وقد توصلت نتائج البحث إلى النتائج التالية: - يوجد فرق دال احصائيا عند مستوى(α ≤ 0.05) بين متوسطي درجات المجموعة التجريبية التي درست وفقا لنموذج البنية المعرفية لتدريس الرياضيات والمجموعة الضابطة التي درست بالطريقة التقليدية في التطبيق البعدي في اختبار مهارات التواصل الرياضي لصالح المجموعة التجريبية. - يوجد فرق دال احصائيا عند مستوى(α ≤ 0.05) بين متوسطي درجات المجموعة التجريبية في التطبيقين (القبلي والبعدي) لاختبار مهارات التواصل الرياضي لصالح التطبيق البعدي. وفي ضوء النتائج قدم الباحث مجموعة من التوصيات کان من أهمها ضرورة استخدام نموذج البنية المعرفية لتدريس الرياضيات کمدخل لتنمية مهارات التواصل الرياضي. (الکلمات المفتاحية: نموذج البنية المعرفية لتدريس الرياضيات، مهارات التواصل الرياضي) Abstract The purpose of the current research was to identify the effectiveness of the cognitive structure model to teach mathematics on development of skills of mathematical communication among the preparatory school students. To achieve the research objectives, the researcher used the quasi-experimental approach. The study tools consisted of a test in mathematical communication skills in written communication skills, mathematical representation and mathematical reading, and enabling the application of research on a randomly chosen sample with a total of (113) students from the first preparatory grade students from Menshat Badawi El-Gedida School for Basic Education, the experimental group consisted of (57) students were taught using the cognitive structure of mathematics teaching model, and the control group (56) students were taught using the traditional teaching technique. The study reached to the following results: - There was a statistically significant difference (α ≤ 0.05) between the average scores of posttest of mathematical communication between the experimental group that was taught using cognitive structure model of mathematics teaching and the control group that was taught using the traditional method in favor of the experimental group. - There was a statistically significant difference (α ≤ 0.05) between the mean scores of the experimental group in the two applications (pre and post) to test mathematical communication skills in favor of post application. In the light of the obtained results, the researcher suggested a set of recommendations, the most important of which was the necessity of using the cognitive structure model in mathematics teaching aiming to the development of mathematical communication skills. (Keywords: cognitive structure model for mathematics teaching, mathematical communication skills)

Assalamu’alaikum Wr. Wb. and Greetings.I am pleased and honored to welcome you at the First International Conference on Computer Science and Applied Mathematic (ICCSAM 2018), which was held this year by AMIK and STIKOM Tunas Bangsa Pematangsiantar in collaboration with the Indonesian Computer and Information Professional Association (IPKIN), Indonesian Mathematical Society (INDOMS) and Bank Muamalat. As Chair of the Foundation, I would like to convey a greeting Welcome to Parapat City Lake Toba to presenters from various countries and especially those I respect for the presence of scientists in the field of Computer and Mathematical Sciences and industry practitioners in the 2018 ICCSAM event.I see that this activity is designed to enhance the exchange of knowledge and new discoveries in computer science and mathematics and related fields in industry. I hope that scientists in the field of computer science and mathematics and those who work in the industry can share knowledge and work together as a team that has mutual relations with each other. I am very pleased to say that the theme of this conference, “Advancing Computability Innovation” is very much in line with the objectives of AMIK and STIKOM conferences and missions, namely “to become a Science and Technology based Study Program that meets industrial needs and functions as a research center in Information Data Science. The conference was held in response to output and aoutcame which had a significant contribution in the field of computer science and mathematics as an inseparable unit of science towards the development of local and global industries. It is our happiness and honor to welcome distinguished professors present to convey their expertise at this conference.I hope this meeting will enable the development of productive dialogue between participants from various countries. They also provide invaluable opportunities for networking among participants, institutions and industries. I also hope that diversity in these fields can reveal more opportunities for researchers and practitioners from all over the world to start a lot of research related to the industry in the future.I would like to congratulate AMIK and STIKOM Tunas Bangsa for starting this conference with a synergistic contribution from dedicated partners namely the USU Faculty of Mathematics and Natural Sciences, USU’s Faculty of Computer and Information Technology, the Indonesian Computer and Informatics Professional Association (IPKIN), Indonesian Mathematical Society (INDOMS), Bank Muamalat, Malaysia Pahang University (UMP), University of Essex, United Kingdom, Ankara, Turkey and Institute of Applied Mathematics Middle East Technical University to all delegates for their full support, cooperation and contribution to the 2018 ICCSAM. I also want to thank the Organizing Committee for their perseverance and extraordinary efforts. Various sponsors are also rewarded for their contributions. I really hope that all participants will get a pleasant stay here at NIAGARA HOTEL and bring back unforgettable experiences and valuable knowledge from this conference.Thanks.H. Ahmad Ridwansyah Putra

In this paper, we propose a calibration estimator of population mean in stratified sampling using the known mean and variance information from multi-auxiliary variables. The problem of determining the optimum calibrated weights is formulated as a Nonlinear Programming Problem (NLPP) that is solved using the Lagrange multiplier technique. Numerical example with real data is presented to illustrate the computational details of the proposed estimator. A comparison study is also carried out using real and simulated data to evaluate the performance and the usefulness of the proposed estimator. The study reveals that the proposed estimator with multi-auxiliary information is more efficient estimator of the population mean as it provides least estimated variance and highest gain in relative efficiency (RE). References Jean Claude Deville and Carl Erik Sarndal. Calibration estimators in survey sampling. Journal of the American statistical Association, 87(418):376–382, 1992. doi:10.1080/01621459.1992.10475217 . Victor M Estevao and Carl Erik Sarndal. Survey estimates by calibration on complex auxiliary information. International Statistical Review, 74(2):127–147, 2006. doi:110.1111/j.1751-5823.2006.tb00165.x Patrick J Farrell and Sarjinder Singh. Model-assisted higher-order calibration of estimators of variance. Australian and New Zealand Journal of Statistics, 47(3):375–383, 2005. doi:10.1111/j.1467-842X.2005.00402.x Wolfram Research, Inc. Mathematica, Version 11.3. Champaign, IL, 2018. Jong Min Kim, Engin A Sungur, and Tae Young Heo. Calibration approach estimators in stratified sampling. Statistics and probability letters, 77(1):99–103, 2007. doi:10.1016/j.spl.2006.05.015 Phillip S Kott. Using calibration weighting to adjust for nonresponse and coverage errors. Survey Methodology, 32(2):133, 2006. Dinesh K Rao. Mathematical programing in stratified random sampling. PhD thesis, School of Computing, Information and Mathematical Sciences, The University of the South Pacific, Fiji, February 2017. Dinesh K. Rao, Tokaua. Tekabu, and Mohammad G M Khan. New calibration estimators in stratified sampling. In Proceedings of Asia-Pacific World Congress on Computer Science and Engineering, pages 66–70. IEEE, 2016. Gurmindar K Singh, Dinesh K Rao, and Mohammed GM Khan. Calibration estimator of population mean in stratified random sampling. In Proceedings of Asia-Pacific World Congress on Computer Science and Engineering (APWC on CSE), pages 1–5. IEEE, 2014. Sarjindar Singh, Stephen Horn, and Frank Yu. Estimation of variance of general regression estimator: Higher level calibration approach. Survey Methodology, 24:41–50, 1998. Sarjinder Singh. Advanced Sampling Theory With Applications: How Michael Selected Amy, volume I and II. Kluwer Academic Publishers, Netherlands, 2003. Sarjinder Singh. On the calibration of design weights using a displacement function. Metrika, 75(1):85–107, 2012. doi:10.1007/s00184-010-0316-6 Sarjinder Singh, Stephen Horn, Sadeq Chowdhury, and Frank Yu. Theory and methods: Calibration of the estimators of variance. Australian and New Zealand Journal of Statistics, 41(2):199–212, 1999. doi:10.1111/1467-842X.00074 D S Tracy, S Singh, and R Arnab. Note on calibration in stratified and double sampling. Survey Methodology, 29(1):99–104, 2003. Changbao Wu and Randy R Sitter. A model-calibration approach to using complete auxiliary information from survey data. Journal of the American Statistical Association, 96(453):185–193, 2001. doi:10.1198/016214501750333054

We present results from a classroom-based intervention designed to help a class of grade 10 students (14–15 years old) learn proof while studying trigonometry in a dynamic geometry software environment. We analysed some students’ solutions to conjecture-and-proof problems that let them gain experience in stating conjectures and developing proofs. Grounded on a conception of proof that includes both empirical and deductive mathematical argumentations, we show the trajectories of some students progressing from developing basic empirical proofs towards developing deductive proofs and understanding the role of conjectures and proofs in mathematics. Our analysis of students’ solutions is based on networking Boero et al.’s construct of cognitive unity of theorems, Pedemonte’s structural and referential analysis of conjectures and proofs, and Balacheff and Margolinas’ cK¢ model, while using Toulmin schemes to represent students’ productions. This combination has allowed us to identify several emerging types of cognitive unity/rupture, corresponding to different ways of solving conjecture-and-proof problems. We also show that some types of cognitive unity/rupture seem to induce students to produce deductive proofs, whereas other types seem to induce them to produce empirical proofs.

This report reviews the research activities in the Mathematics and Computer Science Division at Argonne National Laboratory for the period September 1989 through February 1991. The body of the report gives a brief look at the MCS staff and the research facilities and then discusses the diverse research projects carried out in the division. Projects funded by non-DOE sources are also discussed, and new technology transfer activities are described. Further information on staff, visitors, workshops, and seminars is found in the appendixes.

On 28-29 April 2000, the Board on Mathematical Sciences and the Computer Science and Telecommunications Board of the National Research Council cosponsored a workshop on the interface between the mathematical sciences and three areas of computer science: network traffic modeling, computer vision, and data mining. The workshop's goal was to identify some open research questions in the mathematical sciences whose solutions could contribute to important unsolved problems in those computer science disciplines. The 48 participants engaged in lively discussions and identified at least 36 research challenges of a mathematical, statistical, or probabilistic nature.

Games provide mathematical models for interaction. Numerous tasks in computer science can be formulated in game-theoretic terms. This fresh and intuitive way of thinking through complex issues reveals underlying algorithmic questions and clarifies the relationships between different domains. This collection of lectures, by specialists in the field, provides an excellent introduction to various aspects of game theory relevant for applications in computer science that concern program design, synthesis, verification, testing and design of multi-agent or distributed systems. Originally devised for a Spring School organised by the GAMES Networking Programme in 2009, these lectures have since been revised and expanded, and range from tutorials concerning fundamental notions and methods to more advanced presentations of current research topics. This volume is a valuable guide to current research on game-based methods in computer science for undergraduate and graduate students. It will also interest researchers working in mathematical logic, computer science and game theory.

The design and initial implementation of a project-based directed study course in mathematical research and communication is described. The intention of the course is to expose students to the foundation of mathematical research through inquiry-based learning. Students are asked to approach an exercise in topology as though it is an open problem. At the end of the semester, students are also required to prepare and deliver a presentation on their work from the semester.

The lecture “Music, Mathematics and Computer Science” is subdivided into two parts. The fi rst part, “Music, Mathematics and Computer Science: History of Interaction” (see ICONI journal, No. 3, 2020) examines the processes of interconnection and interpenetration of various fi elds of music, mathematics and computer science, spanning the period from Ancient Times to the turn of the 20th and 21st centuries. The second part of the lecture: “Music, Mathematics and Computer Science: Particular Features of Functioning of Computer-Musical Technologies” is devoted to examining various aspects of developing and applying computer-musical technologies in contemporary musical practice, including musical composition, performance and the sphere of music education

This report reviews the research activities in the Mathematics and Computer Science Division at Argonne National Laboratory for the period January 1988 - August 1989. The body of the report gives a brief look at the MCS staff and the research facilities, and discusses various projects carried out in two major areas of research: analytical and numerical methods and advanced computing concepts. Projects funded by non-DOE sources are also discussed, and new technology transfer activities are described. Further information on division staff, visitors, workshops, and seminars is found in the appendices.

There are various logic-based approaches to modelling concurrent programming. The use of logic for system development concerns both the specification step and the study of the operational aspects (through proof analysis and construction) of such development. This means that we have to consider different logical systems for different uses of logic. Moreover, we can also consider concurrency from both points of view: we want to reason and to specify systems where some concurrency aspects are involved, but also to have some operational interpretation of concurrency within logic (focusing on the concept of proof). For this purpose, it seems clear that we have to identify and to study, on the one hand, the role and the treatment of objects (representation, inheritance, modularity, communication, and so on) in this context, and, on the other hand, the interaction of work on proofs and concurrency with the Object-Oriented Programming paradigm.The ECOOP’96 workshop on ‘Proof Theory of Concurrent Object-Oriented Programming’ took place in Linz, Austria, in July 1996. Its objective was to provide an integrated forum for the presentation of research and the exchange of ideas and experiences in the topics concerned with proofs, concurrency and object-oriented programming (specification, proof development, and so on). The call for papers for this Special Issue of Mathematical Structures in Computer Science can be considered to have been a natural scientific continuation of the workshop.The papers selected for this Special Issue address some of the topics discussed in the workshop. They present different alternative frameworks that are effectively based on a proof-theoretic approach. They illustrate, from different points of view, the interest, potentialities and difficulties of dealing with such an approach in the design of object-based concurrent systems.