Philosophy Of Mathematics Research Articles

Status of Mathematics Teachers’ Espoused Beliefs about Nature and Teaching Learning Process of Mathematics

This study aimed to investigate the espoused beliefs held by secondary-level mathematics teachers about mathematics and its teaching and learning processes. The research employed a quantitative survey design, focusing on four districts (Chitwan, Makawanpur, Gorkha, and Parsa) as representatives of Nepal. The selection of 168 mathematics teachers from these districts utilized proportional allocation of stratified random sampling. A questionnaire, employing a Likert five-point scale, was designed to collect data on teachers' espoused beliefs. The results revealed a prevalent alignment of secondary-level mathematics teachers' espoused beliefs with the fallibilist philosophy of mathematics. Additionally, the findings indicated a strong inclination towards the use of the constructivist method in teaching mathematics. This research contributes valuable insights into the prevailing beliefs among mathematics teachers at the secondary level, providing a basis for understanding their perspectives on the nature of mathematics and preferred teaching methodologies.

Kneebone and Lakatos: at the roots of a dialectical philosophy of mathematics

Kneebone and Lakatos: at the roots of a dialectical philosophy of mathematics

“‘Here is one hand’…, ‘and here is another’”: Comments on Mark Textor's The Disappearance of Soul

ABSTRACT In this paper I pose four questions raised by Mark Textor’s The Disappearance of Soul and the Turn Against Metaphysics. These are: (1) When did Bertrand Russell abandon the subject? (2) Is acquaintance always a polyadic relation? (3) Does Idealism follow from the thesis that all relations are internal? (4) Is the development of philosophy of psychology and philosophy of mathematics linked together, that is, did the New Logic substantially influence British analytic philosophizing about the soul, or did the New Logic develop independently of theorizing about soul?

An Analysis of the Functionality of the Philosophy of Mathematics and the Philosophy of Education in Facilitating Conceptual Understanding in Mathematics Learning Activities

An Analysis of the Functionality of the Philosophy of Mathematics and the Philosophy of Education in Facilitating Conceptual Understanding in Mathematics Learning Activities

The role of Anschauung in Kant's conception of geometry

Far from being outdated, Kant’s philosophy of mathematics offers valuable insights to modern scholarship. This paper focuses on Kant’s notion of Anschauung (commonly translated as intuition) within geometry. Contrary to a widely held view, it will be shown that appealing to Kantian intuition is, in some cases, still necessary. After reviewing past and recent objections to Kant’s arguments, a new interpretation of A 716-717/B 744-745 of the first Critique (CPR) will be proposed. The role that Kant assigns to intuition in this passage, which I term 'revelatory,' remains indispensable despite modern advances in mathematics and logic. Notably, almost all English translations of the CPR, as well as the official translations in French and Greek, have disregarded this function of intuition. This paper aims to show why these interpretations are inconsistent with Kant’s philosophy of mathematics and to argue that the revelatory function of intuition withstands objections to the classical reading.

Mathematical SETIbacks: Open Texture in Mathematics as a New Challenge for Messaging Extra-Terrestrial Intelligence

ABSTRACT Beyond the obvious technical difficulties, human attempts to communicate with hypothetical Extra-Terrestrial Intelligences also present a number of philosophical puzzles. After all, an alien intelligence is likely the closest thing to a Wittgensteinian lion humanity could ever encounter. In this paper I advance a new challenge for the feasibility of communication with extra-terrestrials. The problem I raise is a practical problem that falls out of the history and philosophy of mathematics and the implementation of METI projects—specifically, the semiprime self-decryption schema of the Drake Pictures message strategy. The Drake Pictures strategy presumes that aliens share the concept ‘prime number’ with us, as understanding that concept is necessary to decrypt our message. However, if the concept ‘prime number’ exhibited open texture at any point in its history, it could have developed in a different direction than it did in our history. If that is so, an alien could have the concept ‘prime number’ and still not be capable of decrypting our messages. I argue that this new problem is more trenchant than previous arguments in both the philosophy and SETI literature, such as applications of the aforementioned Lion argument and other concerns about the possibility of long-range communication without the assumption of shared concepts.

Mathematical understanding – Common themes in philosophy and mathematics education

We present different characterizations of mathematical understanding given by mathematicians, philosophers of mathematics, and mathematics educators. One purpose is to illustrate the diversity of these characterizations. Although the descriptions of understanding may seem incompatible, the paper ends by pointing to some shared themes. They include an emphasis on qualities such as relations and unification. Additionally, we note that re-presentation, including visual representations, is thought to play a role in understanding.

Magnitude, Matter, and Kant’s Principle of Mechanism

Abstract For Kant, inquiry into nature requires seeking to explain all material wholes merely mechanically, in terms of their parts. There is no consensus on how he justifies this Principle of Mechanism. I argue that Kant seeks to derive this claim about part and wholes neither from his laws of mechanics, nor from the mere discursivity of our understanding (two standard options in the literature), but instead from a priori principles laid out in the first Critique, which govern parts, wholes, and magnitudes. These principles are also fundamental to Kant’s account of mathematics. Therefore, Kant’s Principle of Mechanism and his philosophy of mathematics have common foundations.

Philosophy of music in the context of mathematics

In the philosophy of mathematics, two opposing trends have historically developed: apriorism (Pythagoras) and empiricism (Aristotle) and both of them have had their impact on the development of music. Philosophers of Ancient Greece and the East noted the important role of music; it was used in religious rites, which served as the basis for social and state relations. The article presents a joint panorama of the development (by historical eras) of the philosophy of mathematics and music and notes their close relationship. But if the Philosophy of Mathematics has long been included in the section of sciences «Philosophical Questions of Science», the Philosophy of Music has not yet acquired such a full-fledged status, although philosophers, specialists in aesthetics, art criticism, composers and musicians have dedicated their works to it. This can be attributed to the fact that since the time of Pythagoras, science has not advanced much in explaining the phenomenon of music. The article attempts to explain this paradox and offer as a hypothesis possible options for studying this topic «Mathematics and Music». The development of brain sciences, high technologies and cognitive sciences can contribute to the success in the development of the Philosophy of Music. The philosophy of music explores ontological, epistemological and axiological problems, the essence, historical and social existence of music, as well as the features of its interaction with art, literature, mathematics, natural and humanitarian sciences. Applied issues of the philosophy of music are the formation of ethical and cognitive guidelines for life and ideological ideals in humans. Music finds its practical embodiment in medicine, psychology, education and various means of communication. The latter is becoming the most important direction of modern philosophy and science. At present, mathematics contributes to the development of new integral directions in music associated with light, color and many other characteristics.

Mathematics and society reunited: The social aspects of Brouwer's intuitionism

Brouwer's philosophy of mathematics is usually regarded as an intra-subjective, even solipsistic approach, an approach that also underlies his mathematical intuitionism, as he strived to create a mathematics that develops out of something inner and a-linguistic. Thus, points of connection between Brouwer's mathematical views and his views about and the social world seem improbable and are rarely mentioned in the literature. The current paper aims to challenge and change that. The paper employs a socially oriented prism to examine Brouwer's views on the construction, use, and practice of mathematics. It focuses on Brouwer's views on language, his social interactions, and the importance of group context as they appear in the significs dialogues. It does so by exploring the establishment and dissolution of the significs movement, focusing on Gerrit Mannoury's influence and relationship with Brouwer and analyzing several fragments from the significs dialogues while emphasizing the role Brouwer ascribed to groups in forming and sharing new ideas. The paper concludes by raising two questions that challenge common historical and philosophical readings of intuitionism.

Are Mathematical Objects ‘sui generis Fictions’? Some Remarks on Aquinas’s Philosophy of Mathematics

AbstractThis contribution proposes an interpretation of Thomas Aquinas’s philosophy of mathematics. It is argued that Aquinas’s philosophy of mathematics is a coherent view whose main features enable us to understand it as a moderate realism according to which mathematical objects have an esse intentionale. This esse intentionale involves both mathematicians’ intellectual activity and natural things being knowable mathematically. It is shown that, in Aquinas’s view, mathematics’ constructive part does not conflict with mathematical realism. It is also held that mathematics’ imaginative reasoning is coherent with Aquinas’s doctrine of formal abstraction and his realistism. It focuses on some of Aquinas’s texts, which it places within their textual and doctrinal context and interprets them in the light of some historical elements.

A Taxonomy for Set-Theoretic Potentialism

Abstract Set-theoretic potentialism is one of the most lively trends in the philosophy of mathematics. Modal accounts of sets have been developed in two different ways. The first, initiated by Charles Parsons, focuses on sets as objects. The second, dating back to Hilary Putnam and Geoffrey Hellman, investigates set-theoretic structures. The paper identifies two strands of open issues, technical and conceptual, to clarify these two different, yet often conflated, views and categorize the potentialist approaches that have emerged in the contemporary debate. The final outcome is a taxonomy that should help researchers navigate the rich landscape of modal set theories.

Wittgenstein on mathematical facts

AbstractThe status of mathematical facts has long been taken to be unclear in Wittgenstein's philosophy of mathematics, and often, it seems that he wants to eliminate mathematical facts in favour of facts about our beliefs or behaviour. In this paper, I argue that by reading Wittgenstein as a radical conventionalist, we can give a reading of the relevant passages according to which Wittgenstein doesn't deny that there are mathematical facts, but rather denies that one needs a metaphysical account of what mathematical facts are and how they relate to the world that goes beyond the minimal claims of radical conventionalism—that empirical facts about how we would find natural to project our training into new cases and the constitution of concepts by our agreement are enough. At the end of the paper, I discuss the implications of this reading on how to understand a rule's determination of its own application.

The Algorithmicity of Mathematical Cognition

This article purports to establish the philosophical inappropriateness of using established theorems in mathematical logic, such as Gödel's (1931) first incompleteness theorem, in order to conclude that human minds have a non-algorithmic nature. First, I will argue that the ongoing debate in the philosophy of mathematics concerning absolute provability is fully independent of the question whether our brains are biologically instantiated computers or not. Second, through a combination of evolutionary considerations and the phenomenon of vagueness, I will demonstrate the fragility of the ill-defined notion of an idealized mathematical mind, which plays a key role in the anticomputationalist arguments. Third, I will make the case that all these arguments adopt a psychologically implausible conception of the nature of the high-level algorithms behind our mathematical abilities. Lastly, I look at the recent advances in artificial intelligence in order to illustrate the philosophical sleight of hand on which the anticomputationalist argument operates.

Vittorio Hösle on the Systematicity and Historicity of Modern Philosophy: An Attempt of a Paradigmatic Analysis of Kant's Philosophy of History

Since the current volume of “NaUKMA Research Papers in Philosophy and Religious Studies” presents the first Ukrainian translation of Vittorio Hösle’s essay “The Place of Kant’s Philosophy of History in the History of the Philosophy of History,” this article provides a comprehensive analysis of the work of its author, a contemporary German and American philosopher who is a prominent figure in the “objective idealism of intersubjectivity.” The focus is on the methodology effectively employed by Hösle in his research, which combines historical and philosophical analysis of the works of ancient and modern philosophers with a paradigmatic approach to their thinking. A paradigmatic analysis enables us to examine the topic under study from the perspective of modern philosophy. Hösle’s research is primarily characterized by conceptual and systematic interests, even when delving into the works of eminent philosophers of the past, such as Plato’s philosophy of mathematics. Hösle effectively employs this methodology in his study of Kant’s philosophy of history, revealing how the moral and normative orientation of history, as outlined by Kant, is complemented, even complicated in a certain way, by the realistic anthropology of the author of the “Critique of Pure Reason.” The article also analyzes several other works by Hösle. The analysis demonstrates how, through paradigmatic and systematic methodology, Hösle identified four phases of the development of ancient philosophy, each characterized by distinct thematic and epistemological aspects. This methodology necessitates researchers to depart from the linear-progressive approach of the history of philosophy to focus on the cyclical movement of philosophical thought. Attention is drawn to the problem of the relationship between the “Science of Logic” and real philosophy (philosophy of nature and the philosophy of spirit), as well as the presence of intersubjectivity in Hegel’s philosophy. Various Hösle’s works demonstrate his ability to analyze the classical texts and cinematic masterpieces, as well as the controversial trends of the modern world and the challenges faced by humanity in the 21st century, including Russia’s full-scale war against Ukraine. These challenges require new approaches in international politics and law.

Innate responses to numerousness reveal neural activation in different brain regions in newly-hatched visually naïve chicks

Whether non-symbolic encoding of quantity is predisposed at birth with dedicated hard-wired neural circuits is debated. Here we presented newly-hatched visually naive chicks with stimuli (flashing dots) of either identical or different numerousness (with a ratio 1:3) with their continuous physical appearance (size, contour length, density, convex hull) randomly changing. Chicks spontaneously tell apart the stimuli on the basis of the number of elements. Upon presentation of either fixed or changing numerousness chicks showed different expression of early gene c-fos in the visual Wulst, the hippocampal formation, the intermediate medial mesopallium, and the caudal part of the nidopallium caudolaterale. The results support the hypothesis that the ability to discriminate quantities does not require any specific instructive experience and involves a neural network with several populations of number-selective neurons. Evidence for innateness of non-symbolic numerical cognition have implications for both neurobiology and philosophy of mathematics.

Philosophical Investigations into AI Alignment: A Wittgensteinian Framework

We argue that the later Wittgenstein’s philosophy of language and mathematics, substantially focused on rule-following, is relevant to understand and improve on the Artificial Intelligence (AI) alignment problem: his discussions on the categories that influence alignment between humans can inform about the categories that should be controlled to improve on the alignment problem when creating large data sets to be used by supervised and unsupervised learning algorithms, as well as when introducing hard coded guardrails for AI models. We cast these considerations in a model of human–human and human–machine alignment and sketch basic alignment strategies based on these categories and further reflections on rule-following like the notion of meaning as use. To sustain the validity of these considerations, we also show that successful techniques employed by AI safety researchers to better align new AI systems with our human goals are congruent with the stipulations that we derive from the later Wittgenstein’s philosophy. However, their application may benefit from the added specificities and stipulations of our framework: it extends on the current efforts and provides further, specific AI alignment techniques. Thus, we argue that the categories of the model and the core alignment strategies presented in this work can inform further AI alignment techniques.

А.Ф. Лосев о числе, математике и музыке

The article is devoted to the analysis and development of A.F. Losev’s philosophy of number. A review of existing research on this issue is provided. Definitions of arrhythmology, eidetic and hyletic numbers are given according to the early works of the philosopher of the 1920s. The roots of the term “hyletics” in E. Husserl and the change in its meaning in Losev are noted. The subject of the philosophy of number as a separate discipline is considered. The difference between number and numbering, between hyletic and real number is revealed. Examples are given of the irreducibility of numbers and the non-commutativity of operations with numbers, which can only be interpreted through a dialectical approach to mathematics. The early Losev’s philosophy of mathematics is considered in the light of Gödel's theorem formulated later. An introductory sketch of possible non-classical mathematics (a paradigm similar to quantum physics) based on correlation calculus is given. The difference between functional and correlation dependence is considered. The reductionism that is characteristic of modern mathematics is criticized. In this regard, a new view of the ancient cosmology of the Pythagoreans and Platonists and the resulting conclusions about the nature of space and time are interpreted. The final part of the study characterizes the connection between A.F. Losev’s philosophy of music with the understanding of time in the general cosmological worldview.

Human Research Methodology

Scientific research methods are essential. Each science has its subject matter and different methodologies that produce different cognitive results. Science is human life, however, science becomes a metaphysical theory, it is governed by the methodology of philosophy in history or mathematics. Philosophy and mathematics are theoretical sciences, making the related sciences into models, ideals, and pure reason. The inability of science becomes human misfortune, science becomes strange to people. True science starts from human life and becomes a methodology for researching related scientific disciplines. When studying the outside world, people uphold the principles of objectivity, comprehensiveness, development, specific history, and practice. When studying life, the object is perceived that human beings are living in the process of creating abilities and needs that make life rich and diverse. Humans are both the subject and the object of scientific research. Interdisciplinary sciences have in common that they take human life as the object of research.

«Тонкие» объекты Эйстейна Линнебо: проблема референции

Introduction. The introduction formulates the problem of abstract objects for the philosophy of mathematics, epistemology, and metaphysics. Special attention is paid to the problem of reference. Theoretical analysis. The first section examines Linnebo’s proposed concept of abstract objects as “thin” i.e., not imposing substantial demands on the world. The second section is devoted to the criteria of identity as a special tool for introducing abstract objects into discourse. Linnebo interprets abstraction principles as a particular kind of identity criteria. They allow fixing reference and ensuring its success. The third section presents metasemantic arguments used by Linnebo to justify the preference for a realistic interpretation of the semantics of singular terms. The fourth section criticizes Linnebo’s proposed mechanism of referential access to abstract “thin” objects. In our view, the metasemantic constraints proposed by Linnebo are insufficient to determine the uniqueness of the referent. An additional principle is necessary. Lewis’s “referential magnetism” can be considered as such, but it threatens the concept of “thin” objects. Conclusion. In conclusion, the strengths of Linnebo’s project are acknowledged. His interpretation better reflects how we actually use language. The importance of Linnebo’s theses and ideas for contemporary philosophy is noted.