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10Rigor and formalizationSynthese 203 (3): 1-18. 2024.This paper critically examines and evaluates Yacin Hamami’s reconstruction of the standard view of mathematical rigor. We will argue that the reconstruction offered by Hamami is premised on a strong and controversial epistemological thesis and a strong and controversial thesis in the philosophy of mind. Secondly, we will argue that Hamami’s reconstruction of the standard view robs it of its original philosophical rationale, i.e. making sense of the notion of rigor in mathematical practice.
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30What are we doing when we perceive numbers?Behavioral and Brain Sciences 44. 2021.Clarke and Beck rightly contend that the number sense allows us to directly perceive number. However, they unnecessarily assume a representationalist approach and incur a heavy theoretical cost by invoking “modes of presentation.” We suggest that the relevant evidence is better explained by adopting a radical enactivist approach that avoids characterizing the approximate number system as a system for representing number.
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35Radicalizing numerical cognitionSynthese 198 (Suppl 1): 529-545. 2020.In recent decades, non-representational approaches to mental phenomena and cognition have been gaining traction in cognitive science and philosophy of mind. In these alternative approach, mental representations either lose their central status or, in its most radical form, are banned completely. While there is growing agreement that non-representational accounts may succeed in explaining some cognitive capacities, there is widespread skepticism about the possibility of giving non-representationa…Read more
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13How to Leave Descartes BehindEpistemology and Philosophy of Science 55 (3): 91-105. 2018.Both mainstream cognitive science and analytic philosophy of mind remain wedded to the Cartesian picture of the mind as an isolated, self-sufficient, and constitutively individual phenomenon. However, recently approaches to the mind (e.g. extended mind thesis, enactivism) that depart from the standard view have emerged. Aunifying thread that runs through these approaches can be summed up in the slogan: “to understand mental phenomena one cannot do away with the environment”. Differences between …Read more
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13Not All Problems Are Equal: Is Varela's Concept of a Problem Transferable to Mathematics Education Research? (review)Constructivist Foundations 13 (1): 175-177. 2017.I examine to what extend Varela’s remarks on problem-solving can be applied to mathematical problem-solving. I argue that despite similarities between Varela’s epistemological model and recent advances in mathematics education research on problem-solving, trying to fit ideas and concepts from the latter domain in the Varelian mold runs the risk of misconstruing fundamental aspects of mathematical problem-solving.
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1016What Is Left of the Active Externalism Debate?European Journal of Philosophy 25 (4): 1614-1639. 2017.Since the publication of Clark and Chalmers' Extended Mind paper, the central claims of that paper, viz. the thesis that cognitive processes and cognitive or mental states extend beyond the brain and body, have been vigorously debated within philosophy of mind and philosophy of cognitive science. Both defenders and detractors of these claims have since marshalled an impressive battery of arguments for and against “active externalism.” However, despite the amount of philosophical energy expended,…Read more
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26Radical Enactivism and Ecological Psychology: Friends or Foes?Constructivist Foundations 11 (2): 320-322. 2016.Open peer commentary on the article “Perception-Action Mutuality Obviates Mental Construction” by Martin Flament Fultot, Lin Nie & Claudia Carello. Upshot: We examine whether there are any irreducible contradictions between ecological psychology and radical enactivism. We concentrate on two points of contention between the two approaches: the relevance of neural structures in understanding perception and the use of semantically loaded concepts in theorizing about perception.
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Uitgebreid, complementair, of omvattend? Het waar en het hoe van het mentaleAlgemeen Nederlands Tijdschrift voor Wijsbegeerte 104 (3). 2012.
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Strikt finitisme en de wiskundige praktijkAlgemeen Nederlands Tijdschrift voor Wijsbegeerte 102 (3): 202-205. 2010.
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9Hilbert's Tenth Problem for Rings of Rational FunctionsNotre Dame Journal of Formal Logic 43 (3): 181-192. 2002.We show that if R is a nonconstant regular (semi-)local subring of a rational function field over an algebraically closed field of characteristic zero, Hilbert's Tenth Problem for this ring R has a negative answer; that is, there is no algorithm to decide whether an arbitrary Diophantine equation over R has solutions over R or not. This result can be seen as evidence for the fact that the corresponding problem for the full rational field is also unsolvable
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21Elimination theory for addition and the Frobenius map in polynomial ringsJournal of Symbolic Logic 69 (4): 1006-1026. 2004.We develop an elimination theory for addition and the Frobenius map over rings of polynomials. As a consequence we show that if F is a countable, recursive and perfect field of positive characteristic p, with decidable theory, then the structure of addition, the Frobenius map x→ xp and the property ‘x∈ F', over the ring of polynomials F[T], has a decidable theory.
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55The extent of memory. From extended to extensive mindIn Danièle Moyal-Sharrock, Volker Munz & Annalisa Coliva (eds.), Mind, Language and Action: Proceedings of the 36th International Wittgenstein Symposium, De Gruyter. pp. 391-408. 2015.
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19Non-representationalist cognitive science and realismPhenomenology and the Cognitive Sciences 13 (3): 461-475. 2014.Embodied and extended cognition is a relatively new paradigm within cognitive science that challenges the basic tenet of classical cognitive science, viz. cognition consists in building and manipulating internal representations. Some of the pioneers of embodied cognitive science have claimed that this new way of conceptualizing cognition puts pressure on epistemological and ontological realism. In this paper I will argue that such anti-realist conclusions do not follow from the basic assumptions…Read more
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