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  1. Zeno Paradox, Unexpected Hanging Paradox (Modeling of Reality & Physical Reality, A Historical-Philosophical view).Farzad Didehvar - manuscript
    . In our research about Fuzzy Time and modeling time, "Unexpected Hanging Paradox" plays a major role. Here, we compare this paradox to the Zeno Paradox and the relations of them with our standard models of continuum and Fuzzy numbers. To do this, we review the project "Fuzzy Time and Possible Impacts of It on Science" and introduce a new way in order to approach the solutions for these paradoxes. Additionally, we have a more general discussion about paradoxes, as Philosophical (...)
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  2. A Hyperintensional Two-Dimensionalist Solution to the Access Problem.David Elohim - manuscript
    I argue that the two-dimensional hyperintensions of epistemic topic-sensitive two-dimensional truthmaker semantics provide a compelling solution to the access problem. I countenance an abstraction principle for epistemic hyperintensions based on Voevodsky's Univalence Axiom and function type equivalence in Homotopy Type Theory. I apply, further, modal rationalism in modal epistemology to solve the access problem. Epistemic possibility and hyperintensionality, i.e. conceivability, can be a guide to metaphysical possibility and hyperintensionality, when (i) epistemic worlds or epistemic hyperintensional states are interpreted as being (...)
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  3. On the basic principle of number.Joosoak Kim - manuscript
    A history of the construction of number has been in line with the process of recognition about the properties of geometry. Natural number representing countability is exhibited on a straight line and the completeness of real number is also originated from the continuous property of the number line. Complex number on a plane off the number line is established and thereafter, the whole number system is completed. When the process of constructing a number with geometric features is investigated from different (...)
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  4. A hub-and-spoke model of geometric concepts.Mario Bacelar Valente - 2023 - Theoria : An International Journal for Theory, History and Fundations of Science 38 (1):25-44.
    The cognitive basis of geometry is still poorly understood, even the ‘simpler’ issue of what kind of representation of geometric objects we have. In this work, we set forward a tentative model of the neural representation of geometric objects for the case of the pure geometry of Euclid. To arrive at a coherent model, we found it necessary to consider earlier forms of geometry. We start by developing models of the neural representation of the geometric figures of ancient Greek practical (...)
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  5. The psychology of mathematics: a journey of personal mathematical empowerment for educators and curious minds.Anderson Norton - 2022 - New York, NY: Routledge.
    This book offers an innovative introduction to the psychological basis of mathematics and the nature of mathematical thinking and learning, using an approach that empowers students by fostering their own construction of mathematical structures. Through accessible and engaging writing, award-winning mathematician and educator Anderson Norton reframes mathematics as something that exists first in the minds of students, rather than something that exists first in a textbook. By exploring the psychological basis for mathematics at every level - including geometry, algebra, calculus, (...)
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  6. Naturalising Mathematics? A Wittgensteinian Perspective.Jan Stam, Martin Stokhof & Michiel Van Lambalgen - 2022 - Philosophies 7 (4):85.
    There is a noticeable gap between results of cognitive neuroscientific research into basic mathematical abilities and philosophical and empirical investigations of mathematics as a distinct intellectual activity. The paper explores the relevance of a Wittgensteinian framework for dealing with this discrepancy.
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  7. Numbers, numerosities, and new directions.Jacob Beck & Sam Clarke - 2021 - Behavioral and Brain Sciences 44:1-20.
    In our target article, we argued that the number sense represents natural and rational numbers. Here, we respond to the 26 commentaries we received, highlighting new directions for empirical and theoretical research. We discuss two background assumptions, arguments against the number sense, whether the approximate number system represents numbers or numerosities, and why the ANS represents rational numbers.
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  8. A momentum effect in temporal arithmetic.Mario Bonato, Umberto D'Ovidio, Wim Fias & Marco Zorzi - 2021 - Cognition 206 (C):104488.
    The mental representation of brief temporal durations, when assessed in standard laboratory conditions, is highly accurate. Here we show that adding or subtracting temporal durations systematically results in strong and opposite biases, namely over-estimation for addition and under-estimation for subtraction. The difference with respect to a baseline temporal reproduction task changed across durations in an operation-specific way and survived correcting for the effect due to operation sign alone, indexing a reliable signature of arithmetic processing on time representation. A second experiment (...)
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  9. How to Frame Understanding in Mathematics: A Case Study Using Extremal Proofs.Merlin Carl, Marcos Cramer, Bernhard Fisseni, Deniz Sarikaya & Bernhard Schröder - 2021 - Axiomathes 31 (5):649-676.
    The frame concept from linguistics, cognitive science and artificial intelligence is a theoretical tool to model how explicitly given information is combined with expectations deriving from background knowledge. In this paper, we show how the frame concept can be fruitfully applied to analyze the notion of mathematical understanding. Our analysis additionally integrates insights from the hermeneutic tradition of philosophy as well as Schmid’s ideal genetic model of narrative constitution. We illustrate the practical applicability of our theoretical analysis through a case (...)
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  10. Arnošt Kolman’s Critique of Mathematical Fetishism.Jakub Mácha & Jan Zouhar - 2020 - In Radek Schuster (ed.), The Vienna Circle in Czechoslovakia. Springer. pp. 135-150.
    Arnošt Kolman (1892–1979) was a Czech mathematician, philosopher and Communist official. In this paper, we would like to look at Kolman’s arguments against logical positivism which revolve around the notion of the fetishization of mathematics. Kolman derives his notion of fetishism from Marx’s conception of commodity fetishism. Kolman is aiming to show the fact that an entity (system, structure, logical construction) acquires besides its real existence another formal existence. Fetishism means the fantastic detachment of the physical characteristics of real things (...)
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  11. Mathematical cognition and enculturation: introduction to the Synthese special issue.Markus Pantsar - 2020 - Synthese 197 (9):3647-3655.
  12. Descriptive Complexity, Computational Tractability, and the Logical and Cognitive Foundations of Mathematics.Markus Pantsar - 2020 - Minds and Machines 31 (1):75-98.
    In computational complexity theory, decision problems are divided into complexity classes based on the amount of computational resources it takes for algorithms to solve them. In theoretical computer science, it is commonly accepted that only functions for solving problems in the complexity class P, solvable by a deterministic Turing machine in polynomial time, are considered to be tractable. In cognitive science and philosophy, this tractability result has been used to argue that only functions in P can feasibly work as computational (...)
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  13. Skolem’s “paradox” as logic of ground: The mutual foundation of both proper and improper interpretations.Vasil Penchev - 2020 - Epistemology eJournal (Elsevier: SSRN) 13 (19):1-16.
    A principle, according to which any scientific theory can be mathematized, is investigated. That theory is presupposed to be a consistent text, which can be exhaustedly represented by a certain mathematical structure constructively. In thus used, the term “theory” includes all hypotheses as yet unconfirmed as already rejected. The investigation of the sketch of a possible proof of the principle demonstrates that it should be accepted rather a metamathematical axiom about the relation of mathematics and reality. Its investigation needs philosophical (...)
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  14. 《我是一个奇怪的循环》的回顾由道格拉斯·霍夫施塔特 (2007)(Review of I Am a Strange Loop by Douglas Hofstadter (2007)) (审查修订 2019).Michael Richard Starks - 2020 - In 欢迎来到地球上的地狱 婴儿,气候变化,比特币,卡特尔,中国,民主,多样性,养成基因,平等,黑客,人权,伊斯兰教,自由主义,繁荣,网络,混乱。饥饿,疾病,暴力,人工智能,战争. Las Vegas, NV USA: Reality Press. pp. 105-120.
    霍夫施塔特牧师从原教旨主义自然主义教会的最新讲道。像他更出名(或臭名昭著的无情的哲学错误)的工作戈德尔,埃舍尔,巴赫,它有一个肤浅的合理性,但如果人们明白,这是猖獗的科学主义,混合真正的科学问题与哲学 问题(即,只有真正的问题是我们应该玩什么语言游戏),然后几乎所有的兴趣消失。我提供了一个基于进化心理学和维特根斯坦工作的分析框架(自从我最近的著作中更新)。 那些希望从现代两个系统的观点来看为人类行为建立一个全面的最新框架的人,可以查阅我的书《路德维希的哲学、心理学、心神 (Mind) 和语言的逻辑结构》维特根斯坦和约翰·西尔的《第二部》(2019年)。那些对我更多的作品感兴趣的人可能会看到《会说话的猴子——一个末日星球上的哲学、心理学、科学、宗教和政治——文章和评论2006-201 9年第3次(2019年)和自杀乌托邦幻想21篇世纪4日 (2019).
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  15. Extended mathematical cognition: external representations with non-derived content.Karina Vold & Dirk Schlimm - 2020 - Synthese 197 (9):3757-3777.
    Vehicle externalism maintains that the vehicles of our mental representations can be located outside of the head, that is, they need not be instantiated by neurons located inside the brain of the cogniser. But some disagree, insisting that ‘non-derived’, or ‘original’, content is the mark of the cognitive and that only biologically instantiated representational vehicles can have non-derived content, while the contents of all extra-neural representational vehicles are derived and thus lie outside the scope of the cognitive. In this paper (...)
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  16. Introduction.Andrew Aberdein & Matthew Inglis - 2019 - In Andrew Aberdein & Matthew Inglis (eds.), Advances in Experimental Philosophy of Logic and Mathematics. London: Bloomsbury Academic. pp. 1-13.
    There has been little overt discussion of the experimental philosophy of logic or mathematics. So it may be tempting to assume that application of the methods of experimental philosophy to these areas is impractical or unavailing. This assumption is undercut by three trends in recent research: a renewed interest in historical antecedents of experimental philosophy in philosophical logic; a “practice turn” in the philosophies of mathematics and logic; and philosophical interest in a substantial body of work in adjacent disciplines, such (...)
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  17. A fresh look at research strategies in computational cognitive science: The case of enculturated mathematical problem solving.Regina E. Fabry & Markus Pantsar - 2019 - Synthese 198 (4):3221-3263.
    Marr’s seminal distinction between computational, algorithmic, and implementational levels of analysis has inspired research in cognitive science for more than 30 years. According to a widely-used paradigm, the modelling of cognitive processes should mainly operate on the computational level and be targeted at the idealised competence, rather than the actual performance of cognisers in a specific domain. In this paper, we explore how this paradigm can be adopted and revised to understand mathematical problem solving. The computational-level approach applies methods from (...)
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  18. Foundations of geometric cognition.Mateusz Hohol - 2019 - London-New York: Routledge.
    The cognitive foundations of geometry have puzzled academics for a long time, and even today are mostly unknown to many scholars, including mathematical cognition researchers. -/- Foundations of Geometric Cognition shows that basic geometric skills are deeply hardwired in the visuospatial cognitive capacities of our brains, namely spatial navigation and object recognition. These capacities, shared with non-human animals and appearing in early stages of the human ontogeny, cannot, however, fully explain a uniquely human form of geometric cognition. In the book, (...)
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  19. Functional explanation in mathematics.Matthew Inglis & Juan Pablo Mejía Ramos - 2019 - Synthese 198 (26):6369-6392.
    Mathematical explanations are poorly understood. Although mathematicians seem to regularly suggest that some proofs are explanatory whereas others are not, none of the philosophical accounts of what such claims mean has become widely accepted. In this paper we explore Wilkenfeld’s suggestion that explanations are those sorts of things that generate understanding. By considering a basic model of human cognitive architecture, we suggest that existing accounts of mathematical explanation are all derivable consequences of Wilkenfeld’s ‘functional explanation’ proposal. We therefore argue that (...)
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  20. The Enculturated Move From Proto-Arithmetic to Arithmetic.Markus Pantsar - 2019 - Frontiers in Psychology 10.
    The basic human ability to treat quantitative information can be divided into two parts. With proto-arithmetical ability, based on the core cognitive abilities for subitizing and estimation, numerosities can be treated in a limited and/or approximate manner. With arithmetical ability, numerosities are processed (counted, operated on) systematically in a discrete, linear, and unbounded manner. In this paper, I study the theory of enculturation as presented by Menary (2015) as a possible explanation of how we make the move from the proto-arithmetical (...)
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  21. Cognitive and Computational Complexity: Considerations from Mathematical Problem Solving.Markus Pantsar - 2019 - Erkenntnis 86 (4):961-997.
    Following Marr’s famous three-level distinction between explanations in cognitive science, it is often accepted that focus on modeling cognitive tasks should be on the computational level rather than the algorithmic level. When it comes to mathematical problem solving, this approach suggests that the complexity of the task of solving a problem can be characterized by the computational complexity of that problem. In this paper, I argue that human cognizers use heuristic and didactic tools and thus engage in cognitive processes that (...)
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  22. How to Frame a Mathematician.Bernhard Schröder, Martin Schmitt, Deniz Sarikaya & Bernhard Fisseni - 2019 - In Stefania Centrone, Deborah Kant & Deniz Sarikaya (eds.), Reflections on the Foundations of Mathematics: Univalent Foundations, Set Theory and General Thoughts. Springer Verlag.
    Frames are a concept in knowledge representation that explains how the receiver, using background information, completes the information conveyed by the sender. This concept is used in different disciplines, most notably in cognitive linguistics and artificial intelligence. This paper argues that frames can serve as the basis for describing mathematical proofs. The usefulness of the concept is illustrated by giving a partial formalisation of proof frames, specifically focusing on induction proofs, and relevant parts of the mathematical theory within which the (...)
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  23. ¿“Natural” y “Euclidiana”? Reflexiones sobre la geometría práctica y sus raíces cognitivas.Jose Ferreiros & Manuel Garcia - 2018 - Theoria : An International Journal for Theory, History and Fundations of Science 33 (2):325-344.
    We discuss critically some recent theses about geometric cognition, namely claims of universality made by Dehaene et al., and the idea of a “natural geometry” employed by Spelke et al. We offer arguments for the need to distinguish visuo-spatial cognition from basic geometric knowledge, furthermore we claim that the latter cannot be identified with Euclidean geometry. The main aim of the paper is to advance toward a characterization of basic, practical geometry – which in our view requires a combination of (...)
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  24. Kierkegaard on Variation and Thought Experiment.Eleanor Helms - 2018 - Kierkegaard Studies Yearbook 23 (1):33-54.
  25. Learning the Natural Numbers as a Child.Stefan Buijsman - 2017 - Noûs 53 (1):3-22.
    How do we get out knowledge of the natural numbers? Various philosophical accounts exist, but there has been comparatively little attention to psychological data on how the learning process actually takes place. I work through the psychological literature on number acquisition with the aim of characterising the acquisition stages in formal terms. In doing so, I argue that we need a combination of current neologicist accounts and accounts such as that of Parsons. In particular, I argue that we learn the (...)
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  26. Intentional explanation as a cognitive function of applied mathematics.V. P. Kazaryan - 2017 - Liberal Arts in Russia 6 (1):18-32.
    Modern applied mathematics is focused on global problems of civilization. Its ultimate aim is to provide human socio-cultural activity with tool and project. That is why applied mathematics nowadays usually gives scientific explanation typical to sociological knowledge - an intentional explanation. In the article, a question is discussed about the abilities of mathematics to explain. This question was put by J. Brown in the article published in the journal ‘Epistemology and Philosophy of Science‘. The philosophy of mathematics, as well as (...)
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  27. Brain Projective Reality: Novel Clothes for the Emperor.Arturo Tozzi, James F. Peters, Andrew A. Fingelkurts, Alexander A. Fingelkurts & Pedro C. Marijuán - 2017 - Physics of Life Reviews 21:46-55.
    First of all, we would like to gratefully thank all commentators for the attention and effort they have put into reading and responding to our review paper [this issue] and for useful observations that suggest novel applications for our framework. We understand and accept that some of our claims might appear controversial and raise skepticism, because the overall neural framework we have proposed is difficult to frame in established categories, given its strong multidisciplinary character. To make an example, Elsevier is (...)
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  28. Improving Algebraic Thinking Skill, Beliefs And Attitude For Mathematics Throught Learning Cycle Based On Beliefs.Widodo Winarso & Toheri - 2017 - Munich University Library.
    In the recent years, problem-solving become a central topic that discussed by educators or researchers in mathematics education. it’s not only as the ability or as a method of teaching. but also, it is a little in reviewing about the components of the support to succeed in problem-solving, such as student's belief and attitude towards mathematics, algebraic thinking skills, resources and teaching materials. In this paper, examines the algebraic thinking skills as a foundation for problem-solving, and learning cycle as a (...)
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  29. Envisioning Transformations – The Practice of Topology.Silvia De Toffoli & Valeria Giardino - 2016 - In Brendan Larvor (ed.), Mathematical Cultures: The London Meetings 2012-2014. Springer International Publishing. pp. 25-50.
    The objective of this article is twofold. First, a methodological issue is addressed. It is pointed out that even if philosophers of mathematics have been recently more and more concerned with the practice of mathematics, there is still a need for a sharp definition of what the targets of a philosophy of mathematical practice should be. Three possible objects of inquiry are put forward: (1) the collective dimension of the practice of mathematics; (2) the cognitives capacities requested to the practitioners; (...)
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  30. Decimal fraction representations are not distinct from natural number representations - evidence from a combined eye-tracking and computational modeling approach.Elise Klein Stefan Huber, Hans-Christoph Nuerk Klaus Willmes & Korbinian Moeller - 2016 - In Philippe Chassy & Wolfgang Grodd (eds.), Abstract mathematical cognition. [Lausanne, Switzerland]: Frontiers Media SA.
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  31. Mathematical Cognition: A Case of Enculturation.Richard Menary - 2015 - Open Mind.
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  32. Intuição e Conceito: A Transformação do Pensamento Matemático de Kant a Bolzano.Humberto de Assis Clímaco - 2014 - Dissertation, Universidade Federal de Goiás, Brazil
  33. An Inquiry into the Practice of Proving in Low-Dimensional Topology.Silvia De Toffoli & Valeria Giardino - 2014 - In Giorgio Venturi, Marco Panza & Gabriele Lolli (eds.), From Logic to Practice: Italian Studies in the Philosophy of Mathematics. Cham: Springer International Publishing. pp. 315-336.
    The aim of this article is to investigate specific aspects connected with visualization in the practice of a mathematical subfield: low-dimensional topology. Through a case study, it will be established that visualization can play an epistemic role. The background assumption is that the consideration of the actual practice of mathematics is relevant to address epistemological issues. It will be shown that in low-dimensional topology, justifications can be based on sequences of pictures. Three theses will be defended. First, the representations used (...)
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  34. The Mathematical Facts Of Games Of Chance Between Exposure, Teaching, And Contribution To Cognitive Therapies: Principles Of An Optimal Mathematical Intervention For Responsible Gambling.Catalin Barboianu - 2013 - Romanian Journal of Experimental Applied Psychology 4 (3):25-40.
    On the question of whether gambling behavior can be changed as result of teaching gamblers the mathematics of gambling, past studies have yielded contradictory results, and a clear conclusion has not yet been drawn. In this paper, I bring some criticisms to the empirical studies that tended to answer no to this hypothesis, regarding the sampling and laboratory testing, and I argue that an optimal mathematical scholastic intervention with the objective of preventing problem gambling is possible, by providing the principles (...)
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  35. Ahilej i dvosmislenosti u pojmu beskonačnosti - Meršićev pristup [Achilles and the Ambiguities in the Concept of the Infinite - Meršić's Approach].Srećko Kovač - 2009 - Prilozi Za Istrazivanje Hrvatske Filozofske Baštine 35 (1-2):83-97.
    Mate Meršić (Merchich, 1850-1928) sees the origin of Zeno’s paradox ‘Achilles’ in the ambiguities of the concept of the infinity. According to him (and to the tradition started by Gregory St. Vincent), those ambiguities are resolved by the concept of convergent geometric series. In this connection, Meršić proposes a general ontological theory with the priority of the finite over the infinite, and, proceeding from Newton’s concept of fluxion, he develops a modal interpretation of differential calculus.
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  36. The role of mathematical symbols in the development of number conceptualization: The case of the Minus sign.Joëlle Vlassis - 2008 - Philosophical Psychology 21 (4):555 – 570.
    In mathematics education, students' difficulties with negative numbers are well known. To explain these difficulties, researchers traditionally refer to obstacles raised by the concept of NEGATIVE NUMBERS itself throughout its historical evolution. In order to improve our understanding, I propose to take into consideration another point of view, based on Vygotsky's principles, which define a strong relationship between signs such as language or symbols and cognitive development. I show how it is of great interest to consider students' difficulties with negatives (...)
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  37. Crunchy Methods in Practical Mathematics.Michael Wood - 2001 - Philosophy of Mathematics Education Journal 14.
    This paper focuses on the distinction between methods which are mathematically "clever", and those which are simply crude, typically repetitive and computer intensive, approaches for "crunching" out answers to problems. Examples of the latter include simulated probability distributions and resampling methods in statistics, and iterative methods for solving equations or optimisation problems. Most of these methods require software support, but this is easily provided by a PC. The paper argues that the crunchier methods often have substantial advantages from the perspectives (...)
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  38. Natur und mathematisches Erkennen: Vorlesungen, gehalten 1919-1920 in Gottingen by David Hilbert; David E. Rowe. [REVIEW]Skuli Sigurdsson - 1993 - Isis 84:600-602.
  39. Les fondements psycho-linguistiques des mathématiques.Gerrit Mannoury - 1934 - Bussum,: Pays-Bas, F. G. Kroonder.