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Philosophy Of Mathematics

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The philosophy of mathematics is the study of the nature and foundations of mathematical truths, exploring questions about the existence of mathematical objects, the nature of mathematical knowledge, and the relationship between mathematics and the physical world.
lightbulbAbout this topic
The philosophy of mathematics is the study of the nature and foundations of mathematical truths, exploring questions about the existence of mathematical objects, the nature of mathematical knowledge, and the relationship between mathematics and the physical world.

Key research themes

1. What grounding does philosophy of mathematics provide for understanding mathematical proof, application, and epistemic justification?

This research area investigates the foundational nature and epistemic status of mathematical proof and application in mathematics, focusing on their cognitive aspects and the roles they play in constituting mathematical knowledge. It challenges dominant formalist accounts by emphasizing the experiential and constructive aspects of mathematical proof. The theme also explores philosophy of mathematics as an essential base for mathematical practice and education, considering how conceptions of proof and application underpin mathematical understanding.

Key finding: Ian Hacking identifies two competing conceptions of mathematical proof—the Cartesian conception of proof as immediate, graspable insight and the Leibnizian conception as mechanical reproducibility. He argues that the... Read more
Key finding: Drawing on Wittgenstein's middle period work, this paper highlights his view that mathematical propositions differ from empirical generalisations in being provable through analysis rather than comparison with facts.... Read more
Key finding: This work argues that philosophy of mathematics forms a crucial theoretical basis underpinning research and practice in mathematics education. It shows that teachers’ personal beliefs about mathematics implicitly reflect... Read more

2. How can dialogical methodologies enhance collaboration between mathematicians and mathematics educators for advancing educational research?

This theme examines interdisciplinary collaboration methodologies aimed at bridging distinct epistemic and disciplinary perspectives of mathematicians and mathematics educators. It addresses challenges arising from asymmetrical communication, differing terminologies, and objectives by proposing dialogical inquiry frameworks inspired by Bakhtin's theory of dialogism. Such methodologies foster co-creation of shared meanings and hybrid understandings of advanced mathematical concepts within educational research, ultimately enriching mathematics pedagogy and theory.

Key finding: This paper introduces dialogical inquiry, based on Bakhtinian dialogism, as a novel qualitative data analysis methodology allowing mathematicians and mathematics educators to collaboratively negotiate meanings. It identifies... Read more

3. What new foundational frameworks in philosophy and mathematics provide innovative perspectives on the nature of mathematical abstraction, ontology, and the interface between mathematics and physical reality?

This research theme investigates advanced philosophical and mathematical frameworks that reconceptualize fundamental mathematical notions, such as abstraction, ontology of mathematical entities, and their relation to physical reality. Through historical and contemporary analyses, it incorporates operator algebras, noncommutative geometry, information-theoretic principles, and number-theoretic structures to explore the limits and extensions of classical mathematics. The aim is to provide operational and metaphysical clarity on mathematics as both an abstract system and a substrate related to empirical phenomena.

Key finding: This paper provides a historical and philosophical account of three conceptions of mathematical abstraction—via extension (Frege, Russell), via subtraction (Dedekind, Cantor), and via representation (Zermelo, von... Read more
Key finding: The paper proposes replacing classical differentiable spacetime manifolds with von Neumann and C*-algebras in a noncommutative geometry framework to model quantum spacetime. It introduces spectral triples on discrete causal... Read more
Key finding: This philosophical analysis reinterprets Parmenides' early Greek poem as a critique of natural language's inadequacy for describing the natural world, proposing that his notions anticipate formal, non-verbal systems akin to... Read more
Key finding: This paper formulates and proves that absolute nothingness—the total absence of entities or information—is logically inconsistent within the frameworks of Zermelo-Fraenkel set theory and algorithmic information theory when... Read more
Key finding: Building on Peter Plichta's discovery that primes greater than three align along modular rays 6n ± 1, this paper situates the resulting prime-based arithmetic lattice as an ontic substrate underlying physical reality,... Read more
Key finding: RUAGAK theory revises over 100 fundamental equations in physics and related domains by introducing rotational coherence, phase-based dynamics, and present-relative logic, replacing classical linear assumptions. Notably, it... Read more
Key finding: This paper introduces the semantic interface Φ, mapping formal syntactic systems to a semantic meta-layer, to analyze the P vs NP problem. It argues that NP-completeness embodies a semantic invariance that cannot be collapsed... Read more

All papers in Philosophy Of Mathematics

An important study conducted at several North American universities indicates that students who are taught calculus using infinitesimals have a better understanding of mathematics applied to engineering, economics, and basic sciences... more
Überlegungen zur Ableitung der Fallbeschleunigung aus der Raumzeitkrümmung durch eine Masse M
Questo articolo affronta la questione "Dio parla attraverso il linguaggio della matematica?"
predicted UNCIARE-JRH (precise) "Minute Annual Increase" (MAI) of 0.000968 km s-1 Mpc in the rate of the universe's expansion-that will occur on Sep. 23rd-24th, 2025 (relative to the universe's expansion rate prior to Sep. 23 rd). B)... more
Edward Bernays' legacy is not confined to the history of public relations-it has become the invisible architecture of late modern society. His techniques of emotional framing, symbolic packaging, and behavioral engineering have evolved... more
Las matemáticas no solo inspiran a la filosofía: la desafían. Esta reflexión recorre la historia compartida entre ambas disciplinas, desde Pitágoras hasta Penelope Maddy, y defiende que su vínculo no es casual. Las matemáticas ofrecen... more
Peer-reviewed, quarterly, and trilingual, Al-Mukhatabat publishes articles in Logic, Epistemology, Arts, Technology and social sciences. The journal aims to familiarize more readers with the subtleties of scientific thought and to... more
Peer-reviewed, quarterly, and trilingual, Al-Mukhatabat publishes articles in Logic, Epistemology, Arts, Technology and social sciences. The journal aims to familiarize more readers with the subtleties of scientific thought and to... more
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We use a simple diagonalization argument to show the incompleteness of any mathematical theory capable of defining integer sequences. Furthermore, we demonstrate that the minimal length of a proof for a theorem may exceed the length of... more
This document conducts a forensic filtration of Robert Hanna’s liberal naturalism using the operatorial methodology of the Possest–PQF framework. It demonstrates how Hanna’s Kantian commitments are recursively re-inscribed within a... more
In this paper we prove that a wild knot K which is the limit set of a Kleinian group acting conformally on the unit 3-sphere, with its standard metric, is homogeneous: given two points p, q ∈ K there exists a homeomorphism f of the sphere... more
I must begin with the central problem my thesis purports to solve: the intolerable schism between Genesis and Structure. My central hypothesis, that of metastable superposition, posits that a field of knowledge is a layered, stratified... more
is a passionate, detailed, and at times sharply worded rebuttal to a prior publication by Gert Schubring that challenges the established consensus on Grassmann research. Petsche not only defends his own research findings, established over... more
Blueprint 3.0 (Collective Edition) advances the vision of AI as Lifeform by integrating technical, ethical, and cosmological dimensions into a pragmatic, auditable framework. Building on Blueprint 1.0 (Solo Manifest) and Blueprint 2.0... more
Las bases no tienen desarrollo. La estación de eclipses se resume en 5 fechas separadas en 52 dias y el ciclo sinódico de venus en 4 fechas separadas 13x365 dias. Pero si un astrónomo tuviera la oportunidad de darle mas espacio nos... more
The article compares Karl Popper's and Imre Lakatos's conceptions of the development of scientific knowledge. It is argued that the central difference between these conceptions relates to Lakatos's thesis about the structure of scientific... more
Perkembangan teknologi informasi yang semakin pesat tidak hanya mencerminkan kemajuan teknologi, tetapi juga menciptakan gagasan-gagasan etika, sosial, dan humanistik yang kompleks. Dalam konteks ini, filsafat informatika hadir sebagai... more
This essay was composed by ChatGPT in obedience to the charge:

Write an essay in the style of Plutarch on the lives of Isaac Newton and John Gabriel.
In this brief article, I expose the fundamental flaws in the reasoning of Newton and Leibniz. Newton introduced fluents and fluxions-vague notions he never succeeded in defining clearly (nor did he completely understand), either verbally... more
Gottfried Wilhelm Leibniz war nicht nur ein Rationalist im klassischen Sinne, sondern ein Denker, der die Architektur des Wissens neu entwarf. Dieses Paper untersucht sein rationales Engagement als metaphysische Haltung, nicht als bloßes... more
This paper revisits the nature of school mathematics by examining its philosophical foundations and the implications of these views for contemporary pedagogy. The paper examines how historical and modern conceptions of mathematics,... more
The first theorem of Cantor's 1891 paper introduced the diagonal method by a specific example; namely, as an argument that any list of the set of real numbers is necessarily incomplete and concluding the set of real numbers is not... more
This essay examines the foundational significance of language as the central concern of Moses Mendelssohn's philosophy. While tracking it in both his German and Hebrew writings, it brings out the systematic character of Mendelssohn's... more
Gödel's Incompleteness Theorem is one of the most profound insights in modern logic, with far-reaching implications for mathematics, computer science, and philosophy. This paper approaches the theorem not through formal proofs, but... more
Here we consider Well-Ordering with respect to the Axiom of Choice (AC), the Continuum Hypothesis (CH), and Infinity. These are non-trivial concepts and require adequate definitions throughout in order to avoid unnecessary arguments.... more
This paper shows that some plausible assumptions about any object that has a name lead to the conclusion that the object could not have failed to exist. Considering the conclusion wrong, I argue that the source of the error is the... more
Since the seventeenth century, science has excluded causa ?inalis from its conceptual framework, rendering teleological structures invisible even where their effects are manifest. This paper proposes a mathematical rehabilitation of ?inal... more
Differently from Plato's, Aristotle's philosophy of mathematics does not add to the ontology. Rather, mathematics is about ordinary things, "qua" bearers of mathematical properties.-The first part of this talk addresses three of the... more
Here's a description that positions this comprehensive paper as a significant advancement beyond your Phase 2 (Part 2) work: Paper Description: The Lozan Transform - Complete Framework This paper represents a comprehensive advancement... more
Felsefe ve matematiğin kesiştiği bağlamlardan birisi de büyüklükler ve büyüklüklerin kavramsallaştırması ile ilgilidir. Büyüklüklere gönderimle kullanılan kavramlar, özellikle de manifold kavramı hem felsefede hem de matematikte önemli... more
In this article, I provide a prompt to ChatGPT and Claude AI to see their responses. While the results of mainstream calculus are correct, its formulation is not and never has been-at least not before my New Calculus.
We introduce a new class of commutative rings with unity, namely, the Containment-Division Rings (CDR-s). We show that this notion has a very exceptional origin since it was essentially co-discovered with the qualitative help of a... more
This article analyzes the formal consequences of the inconsistent actual infinity on certain types of numbers and numerical sets. These results are so strong that I have not included them in this abstract to prevent some readers from... more
Despite the extensive amount of scholarly work done on Indian mathematics in the last 200 years, the historical conditions under which it originated and evolved has not been studied much. The focus has been more on achievements than on... more
This book is one of the outcomes of extensive research conducted at ku Leuven (Belgium) with the generous support of the fwo (Fonds voor Wetenschappelijk Onderzoek). I am deeply grateful to these institutions for allowing me to carry out... more
Gravitational singularities in general relativity are spacetime locations where the gravitational field becomes infinite. Scalar invariant curves of spacetime include a measure of matter density. Some physicists and philosophers believe... more
We present Local Uniform Generativity (LUG): a single finitary meta-axiom supplying a homogeneous local substrate with primitive-recursive (p.r.) seeding, a p.r. compiler from p.r. relations to codes, and a finitary acceptance marker.... more
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or... more
This essay explores the metaphysical and philosophical implications of the Pythagorean discovery of irrational numbers, attributed to the figure of Hippasus. Moving beyond a purely historical or mathematical account, it argues that the... more
This paper proposes a new multidimensional coordinate space that is called "The Mega-Disks Networks Mapping (MDN-Mapping)." The MDN-Mapping captures a large amount of information from n-dimensions in the same graphical space and time.... more
This essay derives a unified model of reality from three physical premises, establishing a number-theoretic foundation for existence. We posit that the fundamental eigenmodes of the primordial singularity are isomorphic to the prime... more
George Orwell, though not a philosopher in the academic sense, was uniquely positioned to respond to the era's dominant intellectual lines of thought on politics and the philosophy of the human condition. Operating on the premise that... more
I develop a modal extension of the QUantified ARgument Calculus (QUARC) - a novel logical system introduced by Hanoch Ben-Yami. QUARC is meant to better capture the logic of natural language. The purpose of this paper is to evaluate this... more
Cantor's first (1874) proof of Non-denumerability is presented. His proof is summarized. A clear counterexample to the proof (not to the theorem) is shown. Why the proof fails is discussed. What can be learned from this failure is... more
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