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  1.  83
    Learning the Natural Numbers as a Child.Stefan Buijsman - 2019 - 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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  2.  25
    The Representations of the Approximate Number System.Stefan Buijsman - 2021 - Philosophical Psychology 34 (2):300-317.
    The Approximate Number System (ANS) is a system that allows us to distinguish between collections based on the number of items, though only if the ratio between numbers is high enough. One of the questions that has been raised is what the representations involved in this system represent. I point to two important constraints for any account: (a) it doesn’t involve numbers, and (b) it can account for the approximate nature of the ANS. Furthermore, I argue that representations of pure (...)
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  3.  24
    Acquiring Mathematical Concepts: The Viability of Hypothesis Testing.Stefan Buijsman - 2021 - Mind and Language 36 (1):48-61.
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  4.  14
    How numerals support new cognitive capacities.Stefan Buijsman - 2020 - Synthese 197 (9):3779-3796.
    Mathematical cognition has become an interesting case study for wider theories of cognition. Menary :1–20, 2015) argues that arithmetical cognition not only shows that internalist theories of cognition are wrong, but that it also shows that the Hypothesis of Extended Cognition is right. I examine this argument in more detail, to see if arithmetical cognition can support such conclusions. Specifically, I look at how the use of numerals extends our arithmetical abilities from quantity-related innate systems to systems that can deal (...)
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  5.  5
    Two Roads to the Successor Axiom.Stefan Buijsman - 2020 - Synthese 197 (3):1241-1261.
    Most accounts of our knowledge of the successor axiom claim that this is based on the procedure of adding one. While they usually don’t claim to provide an account of how children actually acquire this knowledge, one may well think that this is how they get that knowledge. I argue that when we look at children’s responses in interviews, the time when they learn the successor axiom and the intermediate learning stages they find themselves in, that there is an empirically (...)
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  6.  29
    Referring to Mathematical Objects Via Definite Descriptions.Stefan Buijsman - 2017 - Philosophia Mathematica 25 (1):128-138.
    Linsky and Zalta try to explain how we can refer to mathematical objects by saying that this happens through definite descriptions which may appeal to mathematical theories. I present two issues for their account. First, there is a problem of finding appropriate pre-conditions to reference, which are currently difficult to satisfy. Second, there is a problem of ensuring the stability of the resulting reference. Slight changes in the properties ascribed to a mathematical object can result in a shift of reference (...)
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  7. Philosophy of Mathematics for the Masses : Extending the Scope of the Philosophy of Mathematics.Stefan Buijsman - 2016 - Dissertation, Stockholm University
    One of the important discussions in the philosophy of mathematics, is that centered on Benacerraf’s Dilemma. Benacerraf’s dilemma challenges theorists to provide an epistemology and semantics for mathematics, based on their favourite ontology. This challenge is the point on which all philosophies of mathematics are judged, and clarifying how we might acquire mathematical knowledge is one of the main occupations of philosophers of mathematics. In this thesis I argue that this discussion has overlooked an important part of mathematics, namely mathematics (...)
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  8.  76
    Are the Natural Numbers Fundamentally Ordinals?Bahram Assadian & Stefan Buijsman - 2019 - Philosophy and Phenomenological Research 99 (3):564-580.
    There are two ways of thinking about the natural numbers: as ordinal numbers or as cardinal numbers. It is, moreover, well‐known that the cardinal numbers can be defined in terms of the ordinal numbers. Some philosophies of mathematics have taken this as a reason to hold the ordinal numbers as fundamental. By discussing structuralism and neo‐logicism we argue that one can empirically distinguish between accounts that endorse this fundamentality claim and those that do not. In particular, we argue that if (...)
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  9. Building blocks for a cognitive science-led epistemology of arithmetic.Stefan Buijsman - 2022 - Philosophical Studies 179 (5):1777-1794.
    In recent years philosophers have used results from cognitive science to formulate epistemologies of arithmetic :5–18, 2001). Such epistemologies have, however, been criticised, e.g. by Azzouni, for interpreting the capacities found by cognitive science in an overly numerical way. I offer an alternative framework for the way these psychological processes can be combined, forming the basis for an epistemology for arithmetic. The resulting framework avoids assigning numerical content to the Approximate Number System and Object Tracking System, two systems that have (...)
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  10.  7
    Spotting When Algorithms Are Wrong.Stefan Buijsman & Herman Veluwenkamp - forthcoming - Minds and Machines:1-22.
    Users of sociotechnical systems often have no way to independently verify whether the system output which they use to make decisions is correct; they are epistemically dependent on the system. We argue that this leads to problems when the system is wrong, namely to bad decisions and violations of the norm of practical reasoning. To prevent this from occurring we suggest the implementation of defeaters: information that a system is unreliable in a specific case or independent information that the output (...)
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  11.  13
    Building blocks for a cognitive science-led epistemology of arithmetic.Stefan Buijsman - 2021 - Philosophical Studies 179 (5):1-18.
    In recent years philosophers have used results from cognitive science to formulate epistemologies of arithmetic :5–18, 2001). Such epistemologies have, however, been criticised, e.g. by Azzouni, for interpreting the capacities found by cognitive science in an overly numerical way. I offer an alternative framework for the way these psychological processes can be combined, forming the basis for an epistemology for arithmetic. The resulting framework avoids assigning numerical content to the Approximate Number System and Object Tracking System, two systems that have (...)
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  12.  13
    How Do We Semantically Individuate Natural Numbers?†.Stefan Buijsman - forthcoming - Philosophia Mathematica.
    ABSTRACT How do non-experts single out numbers for reference? Linnebo has argued that they do so using a criterion of identity based on the ordinal properties of numerals. Neo-logicists, on the other hand, claim that cardinal properties are the basis of individuation, when they invoke Hume’s Principle. I discuss empirical data from cognitive science and linguistics to answer how non-experts individuate numbers better in practice. I use those findings to develop an alternative account that mixes ordinal and cardinal properties to (...)
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  13.  44
    Accessibility of Reformulated Mathematical Content.Stefan Buijsman - 2017 - Synthese 194 (6).
    I challenge a claim that seems to be made when nominalists offer reformulations of the content of mathematical beliefs, namely that these reformulations are accessible to everyone. By doing so, I argue that these theories cannot account for the mathematical knowledge that ordinary people have. In the first part of the paper I look at reformulations that employ the concept of proof, such as those of Mary Leng and Ottavio Bueno. I argue that ordinary people don’t have many beliefs about (...)
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  14. Second-Order Characteristics Don't Favor a Number-Representing ANS.Stefan Buijsman - 2021 - Behavioral and Brain Sciences 44.
    Clarke and Beck argue that the ANS doesn't represent non-numerical magnitudes because of its second-order character. A sensory integration mechanism can explain this character as well, provided the dumbbell studies involve interference from systems that segment by objects such as the Object Tracking System. Although currently equal hypotheses, I point to several ways the two can be distinguished.
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  15.  8
    The Role of Mathematics in Science: Hartry Field: Science Without Numbers, 2nd Edition. Oxford: Oxford University Press, 2016, 176 Pp, $74.00 HB. [REVIEW]Stefan Buijsman - 2017 - Metascience 26 (3):507-509.
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