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Markus Pantsar
Aachen University of Technology
Markus Pantsar
University of Helsinki
  1. Objectivity in Mathematics, Without Mathematical Objects†.Markus Pantsar - 2021 - Philosophia Mathematica 29 (3):318-352.
    I identify two reasons for believing in the objectivity of mathematical knowledge: apparent objectivity and applications in science. Focusing on arithmetic, I analyze platonism and cognitive nativism in terms of explaining these two reasons. After establishing that both theories run into difficulties, I present an alternative epistemological account that combines the theoretical frameworks of enculturation and cumulative cultural evolution. I show that this account can explain why arithmetical knowledge appears to be objective and has scientific applications. Finally, I will argue (...)
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  2. 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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  3. Bootstrapping of integer concepts: the stronger deviant-interpretation challenge.Markus Pantsar - 2021 - Synthese 199 (3-4):5791-5814.
    Beck presents an outline of the procedure of bootstrapping of integer concepts, with the purpose of explicating the account of Carey. According to that theory, integer concepts are acquired through a process of inductive and analogous reasoning based on the object tracking system, which allows individuating objects in a parallel fashion. Discussing the bootstrapping theory, Beck dismisses what he calls the "deviant-interpretation challenge"—the possibility that the bootstrapped integer sequence does not follow a linear progression after some point—as being general to (...)
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  4. On What Ground Do Thin Objects Exist? In Search of the Cognitive Foundation of Number Concepts.Markus Pantsar - 2023 - Theoria 89 (3):298-313.
    Linnebo in 2018 argues that abstract objects like numbers are “thin” because they are only required to be referents of singular terms in abstraction principles, such as Hume's principle. As the specification of existence claims made by analytic truths (the abstraction principles), their existence does not make any substantial demands of the world; however, as Linnebo notes, there is a potential counter-argument concerning infinite regress against introducing objects this way. Against this, he argues that vicious regress is avoided in the (...)
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  5. An empirically feasible approach to the epistemology of arithmetic.Markus Pantsar - 2014 - Synthese 191 (17):4201-4229.
    Recent years have seen an explosion of empirical data concerning arithmetical cognition. In this paper that data is taken to be philosophically important and an outline for an empirically feasible epistemological theory of arithmetic is presented. The epistemological theory is based on the empirically well-supported hypothesis that our arithmetical ability is built on a protoarithmetical ability to categorize observations in terms of quantities that we have already as infants and share with many nonhuman animals. It is argued here that arithmetical (...)
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  6. Mathematical cognition and enculturation: introduction to the Synthese special issue.Markus Pantsar - 2020 - Synthese 197 (9):3647-3655.
  7. Early numerical cognition and mathematical processes.Markus Pantsar - 2018 - Theoria : An International Journal for Theory, History and Fundations of Science 33 (2):285-304.
    In this paper I study the development of arithmetical cognition with the focus on metaphorical thinking. In an approach developing on Lakoff and Núñez, I propose one particular conceptual metaphor, the Process → Object Metaphor, as a key element in understanding the development of mathematical thinking.
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  8. 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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  9. Developing Artificial Human-Like Arithmetical Intelligence (and Why).Markus Pantsar - 2023 - Minds and Machines 33 (3):379-396.
    Why would we want to develop artificial human-like arithmetical intelligence, when computers already outperform humans in arithmetical calculations? Aside from arithmetic consisting of much more than mere calculations, one suggested reason is that AI research can help us explain the development of human arithmetical cognition. Here I argue that this question needs to be studied already in the context of basic, non-symbolic, numerical cognition. Analyzing recent machine learning research on artificial neural networks, I show how AI studies could potentially shed (...)
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  10. In search of $$\aleph _{0}$$ ℵ 0 : how infinity can be created.Markus Pantsar - 2015 - Synthese 192 (8):2489-2511.
    In this paper I develop a philosophical account of actual mathematical infinity that does not demand ontologically or epistemologically problematic assumptions. The account is based on a simple metaphor in which we think of indefinitely continuing processes as defining objects. It is shown that such a metaphor is valid in terms of mathematical practice, as well as in line with empirical data on arithmetical cognition.
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  11. 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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  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. On the development of geometric cognition: Beyond nature vs. nurture.Markus Pantsar - 2022 - Philosophical Psychology 35 (4):595-616.
    How is knowledge of geometry developed and acquired? This central question in the philosophy of mathematics has received very different answers. Spelke and colleagues argue for a “core cognitivist”, nativist, view according to which geometric cognition is in an important way shaped by genetically determined abilities for shape recognition and orientation. Against the nativist position, Ferreirós and García-Pérez have argued for a “culturalist” account that takes geometric cognition to be fundamentally a culturally developed phenomenon. In this paper, I argue that (...)
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  14. Truth, Proof and Gödelian Arguments: A Defence of Tarskian Truth in Mathematics.Markus Pantsar - 2009 - Dissertation, University of Helsinki
    One of the most fundamental questions in the philosophy of mathematics concerns the relation between truth and formal proof. The position according to which the two concepts are the same is called deflationism, and the opposing viewpoint substantialism. In an important result of mathematical logic, Kurt Gödel proved in his first incompleteness theorem that all consistent formal systems containing arithmetic include sentences that can neither be proved nor disproved within that system. However, such undecidable Gödel sentences can be established to (...)
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  15. On Radical Enactivist Accounts of Arithmetical Cognition.Markus Pantsar - 2022 - Ergo: An Open Access Journal of Philosophy 9.
    Hutto and Myin have proposed an account of radically enactive (or embodied) cognition (REC) as an explanation of cognitive phenomena, one that does not include mental representations or mental content in basic minds. Recently, Zahidi and Myin have presented an account of arithmetical cognition that is consistent with the REC view. In this paper, I first evaluate the feasibility of that account by focusing on the evolutionarily developed proto-arithmetical abilities and whether empirical data on them support the radical enactivist view. (...)
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  16. From Maximal Intersubjectivity to Objectivity: An Argument from the Development of Arithmetical Cognition.Markus Pantsar - 2022 - Topoi 42 (1):271-281.
    One main challenge of non-platonist philosophy of mathematics is to account for the apparent objectivity of mathematical knowledge. Cole and Feferman have proposed accounts that aim to explain objectivity through the intersubjectivity of mathematical knowledge. In this paper, focusing on arithmetic, I will argue that these accounts as such cannot explain the apparent objectivity of mathematical knowledge. However, with support from recent progress in the empirical study of the development of arithmetical cognition, a stronger argument can be provided. I will (...)
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  17. The Modal Status of Contextually A Priori Arithmetical Truths.Markus Pantsar - 2016 - In Francesca Boccuni & Andrea Sereni (eds.), Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics. Cham, Switzerland: Springer International Publishing. pp. 67-79.
    In Pantsar (2014), an outline for an empirically feasible epistemological theory of arithmetic is presented. According to that theory, arithmetical knowledge is based on biological primitives but in the resulting empirical context develops an essentially a priori character. Such contextual a priori theory of arithmetical knowledge can explain two of the three characteristics that are usually associated with mathematical knowledge: that it appears to be a priori and objective. In this paper it is argued that it can also explain the (...)
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  18.  80
    Theorem proving in artificial neural networks: new frontiers in mathematical AI.Markus Pantsar - 2024 - European Journal for Philosophy of Science 14 (1):1-22.
    Computer assisted theorem proving is an increasingly important part of mathematical methodology, as well as a long-standing topic in artificial intelligence (AI) research. However, the current generation of theorem proving software have limited functioning in terms of providing new proofs. Importantly, they are not able to discriminate interesting theorems and proofs from trivial ones. In order for computers to develop further in theorem proving, there would need to be a radical change in how the software functions. Recently, machine learning results (...)
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  19. The Modal Status of Contextually A Priori Arithmetical Truths.Markus Pantsar - 2016 - In Francesca Boccuni & Andrea Sereni (eds.), Objectivity, Realism, and Proof. FilMat Studies in the Philosophy of Mathematics. Cham, Switzerland: Springer International Publishing.
    In Pantsar, an outline for an empirically feasible epistemological theory of arithmetic is presented. According to that theory, arithmetical knowledge is based on biological primitives but in the resulting empirical context develops an essentially a priori character. Such contextual a priori theory of arithmetical knowledge can explain two of the three characteristics that are usually associated with mathematical knowledge: that it appears to be a priori and objective. In this paper it is argued that it can also explain the third (...)
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  20.  16
    Why do numbers exist? A psychologist constructivist account.Markus Pantsar - forthcoming - Inquiry: An Interdisciplinary Journal of Philosophy.
    In this paper, I study the kind of questions we can ask about the existence of numbers. In addition to asking whether numbers exist, and how, I argue that there is also a third relevant question: why numbers exist. In platonist and nominalist accounts this question may not make sense, but in the psychologist account I develop, it is as well-placed as the other two questions. In fact, there are two such why-questions: the causal why-question asks what causes numbers to (...)
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  21. The Great Gibberish - Mathematics in Western Popular Culture.Markus Pantsar - 2016 - In Brendan Larvor (ed.), Mathematical Cultures: The London Meetings 2012-2014. Springer International Publishing. pp. 409-437.
    In this paper, I study how mathematicians are presented in western popular culture. I identify five stereotypes that I test on the best-known modern movies and television shows containing a significant amount of mathematics or important mathematician characters: (1) Mathematics is highly valued as an intellectual pursuit. (2) Little attention is given to the mathematical content. (3) Mathematical practice is portrayed in an unrealistic way. (4) Mathematicians are asocial and unable to enjoy normal life. (5) Higher mathematics is ...
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  22.  21
    Book Review of “Numbers and the Making of Us: Counting and the Course of Human Cultures” by Caleb Everett.Paula Quinon & Markus Pantsar - 2018 - Journal of Numerical Cognition 4 (2).
  23. Assessing the “Empirical Philosophy of Mathematics”.Markus Pantsar - 2015 - Discipline Filosofiche:111-130.
    Abstract In the new millennium there have been important empirical developments in the philosophy of mathematics. One of these is the so-called “Empirical Philosophy of Mathematics”(EPM) of Buldt, Löwe, Müller and Müller-Hill, which aims to complement the methodology of the philosophy of mathematics with empirical work. Among other things, this includes surveys of mathematicians, which EPM believes to give philosophically important results. In this paper I take a critical look at the sociological part of EPM as a case study of (...)
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  24. Mitä Gödelin epätäydellisyysteoreemoista voidaan päätellä filosofiassa?Markus Pantsar - 2011 - Ajatus 68.
    Tutkin tässä artikkelissa Kurt Gödelin epätäydellisyysteoreemojen tulkintoja filosofiassa. Aihepiiri kattaa valtavan määrän eri tulkintoja tekoälystä fysiikkaan ja runouteen asti. Osoitan, että kriittisesti tarkasteltuna kaikki radikaalit epätäydellisyysteoreemojen sovellukset ovat virheellisiä.
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    Twitter and the aphoristic (re)turn in thought, knowledge and education.Steve Fuller, David Gorman, Val Dusek, Markus Pantsar, Babette Babich, Thomas Basbøll & Sharon Rider - 2023 - Educational Philosophy and Theory 55 (13):1436-1449.
    David GormanNorthern Illinois UniversityThe official topic of Steve Fuller’s editorial is aphorisms, but I think that it is early days in his thinking about this interesting genre. He mentions them...
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  26. Frege, Dedekind, and the Modern Epistemology of Arithmetic.Markus Pantsar - 2016 - Acta Analytica 31 (3):297-318.
    In early analytic philosophy, one of the most central questions concerned the status of arithmetical objects. Frege argued against the popular conception that we arrive at natural numbers with a psychological process of abstraction. Instead, he wanted to show that arithmetical truths can be derived from the truths of logic, thus eliminating all psychological components. Meanwhile, Dedekind and Peano developed axiomatic systems of arithmetic. The differences between the logicist and axiomatic approaches turned out to be philosophical as well as mathematical. (...)
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    Assessing the “Empirical Philosophy of Mathematics”.Markus Pantsar - 2015 - Discipline filosofiche. 25 (1):111-130.
    In the new millennium there have been important empirical developments in the philosophy of mathematics. One of these is the so-called “Empirical Philosophy of Mathematics” of Buldt, Löwe, Müller and Müller-Hill, which aims to complement the methodology of the philosophy of mathematics with empirical work. Among other things, this includes surveys of mathematicians, which EPM believes to give philosophically important results. In this paper I take a critical look at the sociological part of EPM as a case study of sociological (...)
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    Naturalizing Logico-Mathematical Knowledge: Approaches from Philosophy, Psychology and Cognitive Science.Markus Pantsar - 2019 - Philosophical Quarterly 69 (275):432-435.
    Naturalizing Logico-Mathematical Knowledge: Approaches from Philosophy, Psychology and Cognitive Science. Edited by Bangu Sorin.
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