This study explores the integration of the educational philosophy of Science, Technology, Engineering, Arts, and Mathematics (STEAM) with deep learning technologies for the development of high school English course resources. By introducing deep learning models, particularly the Transformer architecture, the study enables automatic generation and personalized delivery of instructional content, thereby enhancing the intelligence and relevance of course resources. An innovative Transformer-based framework incorporating STEAM-related semantic features is designed and implemented. The framework is trained on a large-scale, preprocessed English text dataset collected from 300 high school students participating in STEAM-integrated English learning tasks, with data from 150 students forming the core corpus for model training. The model employs an encoder-decoder architecture and is optimized using the Adam optimizer to support the generation of high-quality content tailored to diverse teaching objectives. Through on-site teaching experiments conducted with the 300 high school students, the results demonstrate that, the model's performance is rigorously evaluated through metrics such as Bilingual Evaluation Understudy score (0.78), innovation index (0.75), and content diversity (0.70). Results reveal that the proposed model outperforms comparative models (Model A and Model B) in language generation accuracy, innovation, and content diversity, highlighting its superior capability in producing high-quality English teaching resources. Although the model demonstrates promising results, its implementation in resource-limited environments presents challenges, highlighting the need for further refinements. Nonetheless, these findings demonstrate the model's potential to revolutionize English teaching by enhancing resource quality and offering a more efficient, innovative approach to course development.

The traditional teaching methods are difficult to conform to students’ ways of thinking and hard to meet teachers’ teaching demands. This work studies the teaching philosophy of Sciences, Technology, Engineering, Arts, and Mathematics (STEAM), a practice-oriented interdisciplinary education concept. This work focuses on the psychology of high school students’ English learning, emphasizing their learning interests and English evaluation criteria. It also introduces the STEAM education concept through task planning projects. In addition, the n-gram English learning evaluation model is integrated with the STEAM education concept. The proposed model cultivates senior high school students’ innovative thinking and comprehensive innovation ability. The analysis indicates that students’ innovative thinking has been enhanced across five aspects: flexibility, independence, openness, sophistication, and critical thinking. Moreover, their innovation ability is promoted through independent planning, novel method design, rich content, and comprehensive innovation. Therefore, the proposed STEAM model can provide students with a way of thinking in innovative education.

This paper examines the philosophy of mathematical language, exploring how mathematics functions as both a formal symbolic system and a natural language for describing abstract structures. We investigate the nature of mathematical meaning, the relationship between mathematical symbols and their referents, the role of natural language in mathematical practice, and contemporary debates about mathematical realism, platonism, and formalism. Through an interdisciplinary approach drawing from philosophy of mathematics, linguistics, cognitive science, and mathematical practice, this research illuminates how mathematical language shapes our understanding of mathematical truth, proof, and knowledge.

Abstract We re-examine the old question to what extent mathematics may be compared with a game. Mainly inspired by Hilbert and Wittgenstein, our answer is that mathematics is something like a “rhododendron of language games”, where the rules are inferential. The pure side of mathematics is essentially formalist, where we propose that truth is not carried by theorems correspondencing to whatever independent reality and arrived at through proof, but is defined by correctness of rule-following (and as such is objective given these rules). Gödel’s theorems, which are often seen as a threat to formalist philosophies of mathematics, actually strengthen our concept of truth. The applied side of mathematics arises from two practices: first, the dual nature of axiomatization as taking from heuristic practices like physics and informal mathematics whilst giving proofs and logical analysis; and second, the ability of using the inferential role of theorems to make “surrogative” inferences about natural phenomena. Our framework is pluralist, combining various (non-referential) philosophies of mathematics.

This paper aims to examine the central role of Platonism in the philosophy of mathematics and evaluates a range of critiques that challenge its ontological and epistemological assumptions. Beginning with Platos theory of Forms and the indispensability argument, the discussion traces the development of realism through Gdel, Benacerraf, and Field, before turning to alternatives such as intuitionism, formalism, social constructivism, and structuralism. Particular attention is paid to how Benacerrafs identification and epistemological problems motivated later approaches, and how Fields fictionalism reshaped the indispensability debate. Recent perspectives, including practice-based accounts, explanatory indispensability, and conceptual structuralism, are incorporated to highlight how contemporary debates extend beyond classical disputes. By integrating both historical and modern treatments, the paper argues that while Platonism continues to provide an appealing account of mathematical objectivity, its explanatory and epistemological difficulties encourage a pluralist outlook, in which structural, social, and cognitive perspectives together illuminate the evolving nature of mathematics.

The review of V.P. Troitsky's book “The Trail and the Way, or the Onomatodoxy” and its conception and main ideas are considered. In terms of content, the book is a philosophical diary, which the author wrote for several decades. In his ideas, the author reflects on the concepts of both Russian religious thinkers of the “Soloviev” line (primarily Vyach I. Ivanov, Fr.P. Florensky, and A.F. Losev) and contemporary researchers and philosophers (e.g., A.N. Parshin). The review notes that the genre diversity of the book (academic articles, memoirs, letters, artistic sketches) is subordinated to a certain goal – to introduce the reader into the chronotope of the thinker with subsequent involvement in the intellectual subjects he explores. The main philosophical motive of Troitsky's book is to reveal the symbolic basis of reality in its numerous “visible” forms, from mystical and metaphysical to everyday life. It is shown that the most important philosophical problem of the book is the problem of the symbol in the formulation of domestic symbol theorists. In this perspective, the author's ideas are considered in the review in the context of chronologically similar studies of Russian Symbolism (S.S. Khoruzh, V.V. Bibikhin, L.A. Gogotishvili). Separately, this review pays attention to the author's description of the relationship between sign, symbol, and myth. A significant part of the book is also devoted to the philosophy of mathematics and philosophy of informatics. It is shown that the author solves a number of contemporary problems in these fields using symbolist methodology.

This essay attempts to highlight the unified nature of the idea in Plato's philosophy based on several passages from his dialogues and to understand this in light of the presentations on Plato's “unwritten teachings.” Particular importance will be attached to the interpretation of Phaedo 100e-101e, where Plato understands the number as an idea. Plato denies that a number is the result of an operation (in this case, addition) and regards it as a unity, in the same sense that an idea is a unity that is not composed of partial ideas. To interpret this passage, we will draw on a little-noticed essay by Taylor (1927), which, following the philosophy of mathematics of his time, distinguishes between mathematical operation and mathematical definition and seeks to understand Plato's methods in light of this distinction. A comparison with a report by Aristotle from Metaphysics M and a testimony by Alexander of Aphrodisias on Plato's “unwritten doctrine” will shed clearer light on the significance of the unified character of ideas in Plato's philosophy. The aim of the analysis is to better understand the essence of the idea and its functionality in Plato's philosophy.

ABSTRACT According to a widespread opinion in the philosophy of mathematics, the objects of mathematics (numbers, sets, topological spaces, categories, etc.) are non-spatiotemporal and causally inert. In the case of sets, this standard opinion has been challenged by various philosophers who claim that some sets do have spatiotemporal location. In this paper I want to examine the thesis that sets are spatiotemporal and explore its implications. First, I will offer precise formulations of the thesis that impure sets are spatiotemporal in the framework of formal theories of location. It turns out that there are logically non-equivalent ways of doing this, and I will explore their differences and logical connections. I will discuss some puzzles that those formulations of the thesis give rise to in the presence of certain assumptions about mereology and location.

This research aims to examine the relationship between the perspective of fallibilism and mathematics learning in the 21st century. The method used is a literature review, with data sources consisting of 11 scientific articles published between 2021 and 2025. The articles were obtained through searches using the Publish or Perish tool and the Google Scholar database, with keywords "fallibilism", "philosophy of mathematics", and "21st-century mathematics education". The data collection process involved selecting articles based on relevance criteria. The collected data were then analyzed qualitatively using content analysis techniques, focusing on identifying main themes and the interconnection of concepts. The analysis results are presented in two main tables depicting the integration of philosophy, curriculum, and technology in mathematics learning, and active learning strategies that align with the principles of fallibilism. The study's findings indicate that fallibilism views mathematics as knowledge that develops socially and is open to revision. This perspective is in line with contemporary educational principles that emphasize the importance of critical thinking skills, reflective thinking, and mastery of numerical literacy. Additionally, technology-based learning approaches such as Project-Based Learning and Problem-Based Learning are considered suitable with the principles of fallibilism because they encourage the exploration of ideas, problem-solving, and active student participation.

The philosophy of mathematics plays a crucial role in explaining the origins and status of mathematical knowledge. In the context of mathematics education, epistemological understanding is increasingly relevant, given the persistent dominance of absolutist views that regard mathematics as a certain and final body of knowledge. However, approaches such as empiricism and fallibilism offer alternative perspectives that are more reflective and dynamic. This study aims to explore the philosophical ideas of empiricism and fallibilism in mathematics and analyze their implications for mathematics education. The research employs a systematic literature review with a thematic analysis approach to identify key themes from each perspective and their potential contributions to educational practice. The findings reveal that empiricism emphasizes the role of concrete experience in the formation of mathematical knowledge, while fallibilism highlights the open and non-absolute nature of mathematical truth. The integration of these two approaches has the potential to support a contextual learning environment that embraces error and fosters critical thinking. Therefore, this study contributes theoretically to the development of mathematics education that is more humanistic, reflective, and centered on students' understanding.

This essay deals with one of the most basic questions that concern the philosophy of mathematics, which has come to be known as the Continuum Hypothesis. First put forth by Georg Cantor in 1878, the Continuum Hypothesis is a postulate on whether there exists an infinite set of real numbers whose cardinality lies strictly between that of the natural numbers and that of the real numbers themselves. The independence of Continuum Hypothesis from the standard axiomatic system of Zermelo-Fraenkel set theory with the Axiom of Choice was shown by Kurt Gödel and Paul Cohen; the independence has given rise to much interesting philosophical debate about the nature of mathematical truth. This essay argues from a Platonist perspective, maintaining that Continuum Hypothesis must have a determinate truth value, independent of the limitations of formal systems. The essay contrasts this view with formalism, which sees mathematical truths as dependent on the choice of axioms. By drawing historical analogies and examining both Platonist and formalist viewpoints, the paper advocates for the pursuit of new axioms and alternative frameworks—such as large cardinal and forcing axioms—that might ultimately resolve the Continuum Hypothesis. The discussion highlights the broader implications of Continuum Hypothesis for understanding the nature of infinity, the completeness of mathematical systems, and the foundations of mathematics itself.

As an academic discipline, mathematics is more than a set of computing mechanisms; it is argued by scholars that the quality of mathematics education is in decline today in human society. Historically, mathematics and philosophy had a close interrelationship enabling the understanding of various natural phenomena. According to Platonism of pure mathematics, the dynamics of natural phenomena and human societies follow invariant and absolute mathematical principles. This article presents a socio-philosophical argument that the concept of society within the natural space can be classified into natural society and synthetic society based on the concept of mathematical purifications, and mathematics education has a role in it. The existence of logical inversions of various forms in the synthetic societies are analysed and corrective roles of mathematics education are explained. The Pythagorean philosophy of mathematics in building an improved as well as just society is an appropriate solution that calls for a relook into mathematics education in order to reduce utilitarian distortion in mathematics education today and to promote intellectualism as well as harmony while reforming mathematics education. Contribution: This article presents detailed analysis of the existing critical issues related to mathematics education today in human society and it compares various social forms in nature, including human, in light of the concepts of socio-philosophy. It is illustrated that these two aspects are interrelated in view of developments in education and evolution of human society. The historical perspectives of mathematics education forming a free and harmonious society following the Pythagorean philosophy of mathematics and society are presented, which can be a solution to the multidimensional problems in human society and in mathematics education today.

Motivated by the articles on Gender Studies, in particular, Gender differences in Mathematics Learning, a deep search is made. In that context an innovative paper titled GENDER PARITY IN MATHEMATICS EDUCATION IN INDIA published with a peer reviewed journal viz., International Research Journal of Modernization in Engineering Technology and Science is identified and it was authored by Ankur Nandi, Amit Bagdi, Anita Chatterjee, Barsha Ghosh, Supriya Dutta, Soumen Roy, Dr. Tapash Das and Dr. Lutful Haque. This paper is a review article on Gender differences in Mathematics Education in India. This study investigates gender parity in mathematics education at the higher education level (Bachelor of Mathematics, Master of Philosophy in Mathematics, and Doctor of Philosophy in Mathematics) in India. The study employs qualitative methods and documentary analysis. Documentary analysis addresses the goals of the study by reviewing the available reports from the Department of Higher Education, Government of India's All- India Survey on Higher Education. It was found that there was a notable increase in the percentage of female students enrolling in M.Sc. Mathematics programs, with female enrolment surpassing that of male students during this period; the percentage of female students enrolling in the M.Phil. program in Mathematics has been rapidly increasing, the enrolment rate of female students has surpassed that of male students; more male students enrolled in the Mathematics Ph.D. program compared to females, but the percentage of female enrolment steadily rose over the years. In the academic session of 2021-2022, the percentage of female enrolment surpassed male enrolment, with 51% of students being female (Ankur Nandi et al 2024).

We explore Derrida's theory of writing in connection with issues in the philosophy of logic and mathematics. In the first part of this essay, we show how Derrida's notion of ‘context’, understood as a differential system of marks that enable the inscription, identification and differentiation of meaning, accounts for the construction and verification of truth-valued propositions. Propositional truth-valued meaning always refers to some normative given context specifying its appearing, construction, functioning, and verification rules. Intentional consciousness does not produce these contextual rules; quite the opposite, consciousness must first follow them to constitute propositional truth-valued meaning as an intentional object. In the second part of this essay, we apply the notion of context to an analysis of the intuitionist theory of evident judgements and the intuitionist attempt to rehabilitate the ‘knowing subject’ as verifying power. Our analysis of the intuitionist theory of propositional meaning will confirm the central tenet of Derrida's theory of writing. The ‘originary act’ of the ‘knowing subject’ is secondary to the contextual construction and verification of propositional truth-valued meaning.

Abstract How the semantic significance of numerical discourse gets determined is a metasemantic issue par excellence. At the sub-sentential level, the issue is riddled with difficulties on account of the contested metaphysical status of the subject matter of numerical discourse, i.e., numbers and numerical properties and relations. Setting those difficulties aside, I focus instead on the sentential level, specifically, on obvious affinities between whole numerical and non-numerical sentences and how their significance is determined. From such a perspective, Frege’s 1884 construction of number, while famously mathematically untenable, fares better metasemantically than extant alternatives in the philosophy of mathematics.

The current disruptive era demands many changes in various areas of life. Likewise, the technological transformation has significant unavoidable consequences. This is where it is necessary to consider a strong foundation for students from the elementary level. This article examines the application and reflection of philosophy on the content of elementary school mathematics in Indonesia. The background of this study is the need for an era where there is a need for constructive mathematics learning from an early age towards learning the Industrial Revolution 4.0. The purpose of this study is to identify (1) the nature of symptoms or objects of application and reflection of philosophy in elementary mathematics (ontological reasons), (2) how to obtain or manage symptoms or objects (epistemological reasons), (3) the benefits of symptoms or objects (axiological reasons), and two-way understanding of phenomena and objects (hermeneutics). Data is obtained based on relevant sources such as journal articles, books, and previous research results supported by qualitative analysis. The results of this study indicate the need to reveal the application and reflection of philosophy in low-grade mathematics content through the ontology of mathematics, epistemology, axiology, and hermeneutics.

Abstract The International Humanities Council has established a new international research network Diversity of Mathematical Research Cultures & Practices (DMRCP) at the Universität Hamburg. In a tripartite contribution, we outline and discuss the specific philosophical approach that DMRCP seeks to promote for which we use the term ‘empirical philosophy of mathematics’: the contribution is therefore programmatic and methodological, rather than a contribution to a specific philosophical research question. This article forms Part III of the triptych.

The article is devoted to a cognitive analysis of the work of the Russian poet of the romantic direction, translator, prose writer and philosopher who was interested in mathematics, Dmitry Vladimirovich Venevitinov. Today, critically thinking people are thinking about translating philosophical reasoning about the sociocultural unity of mathematics and literature into a practical plane. A feature of Venevitinov's poetic, prose and journalistic heritage is a deep philosophical content in the direct meaning of the word and although he had no direct followers, his spiritual influence is subtly felt.

This article explores the nature of the philosophy of Islamic education and its integration with the philosophy of mathematics in the learning process. The philosophy of Islamic education is rooted in the Qur’an and As-Sunnah and is further reinforced by the thoughts of Muslim philosophers. It encompasses three main aspects: ontology (the nature of existence), epistemology (the sources and methods of acquiring knowledge), and axiology (values and purposes). Islamic education aims not only to shape individual character but also to build a holistic educational system based on Islamic values. In this context, education is not merely a process of knowledge transfer but a means of fostering moral and spiritual development in learners. On the other hand, the philosophy of mathematics from the perspective of Islamic education emphasizes that mathematics is not solely an exact science of arithmetic, but also a discipline closely related to ethical and moral values in Islam. Mathematics is viewed as part of the divine order of Allah’s creation, reflecting His greatness. Therefore, the integration of Islamic educational philosophy and mathematical philosophy in learning offers a deeper approach to mathematical concepts. This approach not only enhances students’ cognitive understanding of the material but also instills spiritual awareness and values of faith and piety. By combining scientific and religious aspects, the learning process becomes more systematic, meaningful, and aligned with the comprehensive goals of Islamic education.

This study aims to explore the views of the philosophy of mathematics in the context of developing project assignments in mathematics education. Through a qualitative approach based on a literature review, this study analyzes various theories and perspectives related to the philosophy of mathematics, including the pragmatic and constructivist approaches. The main focus of this study is how the philosophy of mathematics influences project-based learning and the development of project assignments based on the inquiry method. This study uses sources from various academic databases, such as Google Scholar, ResearchGate, and JSTOR, with a range of articles from 2000 to 2023. The results of the analysis show that the application of the philosophy of mathematics, especially constructivism and the pragmatic approach, plays an important role in encouraging collaboration, problem-solving skills, and the development of artifacts in project assignments. These findings provide insight that project assignments can be an effective means of developing mathematical understanding that is applicable and relevant to students' real-life contexts Kata Kunci: philosophy of mathematics, project assignments, project-based learning, mathematics education, constructivism