In search of $$\aleph _{0}$$ ℵ 0 : how infinity can be created

Pantsar, Markus

doi:10.1007/s11229-015-0775-4

Cited by 20 publications

(16 citation statements)

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“…Carey 2009;Izard et al 2008;Sarnecka and Carey 2008;Carey et al 2017;Beck 2017;Cheung and Le Corre 2018). Yet others (Spelke 2011a;Pantsar 2014Pantsar , 2015vanMarle et al 2018) argue that both core cognitive roles play a crucial role in the process.…”

mentioning

confidence: 99%

Mathematical cognition and enculturation: introduction to the Synthese special issue

Pantsar

2019

Synthese

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confidence: 99%

Mathematical cognition and enculturation: introduction to the Synthese special issue

Pantsar

2019

Synthese

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“…The focus of Lakoff and Núñez's work is on mathematical cognition. Both their work and its subsequent development (Núñez, 2005;Pantsar, 2015Pantsar, , 2018 provide broad support for the idea that mathematical thinking is in many ways embodied. However, Lakoff and Núñez also develop an account of the ontology or metaphysics of mathematics.…”

Section: Lakoff and Núñez On The Philosophy Of Mathematicsmentioning

confidence: 93%

“…The Basic Metaphor of Infinity has been further developed as a "double-scope conceptual blend" by Núñez (2005). Critical discussions of the Basic Metaphor of Infinity and several alternative accounts are reviewed in Pantsar (2015), who develops his own "process object" metaphor for infinity, whereby we map directly from our understanding of unending processes (e.g., the process of calculating successive numbers in the Fibonacci sequence) to an understanding of objects ("the" Fibonacci sequence as a mathematical object).…”

Section: Mathematical Knowledge As a System Of Conceptual Metaphorsmentioning

confidence: 99%

“…For example, consider the evidence that our fingers make it relatively easy to grasp the abstract truths of basic arithmetic, while transfinite cardinals are more difficult to grasp as they have no immediate correspondence with the physical world. Thus, we are prone to make errors like assuming that infinity is less than infinity plus one (Pantsar, 2015). So embodied mathematics shows us when it is easier or harder to grasp abstract objects, and which cognitive structures and processes facilitate or hinder the development and comprehension of new mathematical concepts.…”

Section: The Metaphysical Neutrality Of Embodied Mathematics Researchmentioning

confidence: 99%

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Metaphor and the Philosophical Implications of Embodied Mathematics

Winter

Yoshimi

2020

Front. Psychol.

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Embodied approaches to cognition see abstract thought and language as grounded in interactions between mind, body, and world. A particularly important challenge for embodied approaches to cognition is mathematics, perhaps the most abstract domain of human knowledge. Conceptual metaphor theory, a branch of cognitive linguistics, describes how abstract mathematical concepts are grounded in concrete physical representations. In this paper, we consider the implications of this research for the metaphysics and epistemology of mathematics. In the case of metaphysics, we argue that embodied mathematics is neutral in the sense of being compatible with all existing accounts of what mathematical entities really are. However, embodied mathematics may be able to revive an older position known as psychologism and overcome the difficulties it faces. In the case of epistemology, we argue that the evidence collected in the embodied mathematics literature is inconclusive: It does not show that abstract mathematical thinking is constituted by metaphor; it may simply show that abstract thinking is facilitated by metaphor. Our arguments suggest that closer interaction between the philosophy and cognitive science of mathematics could yield a more precise, empirically informed account of what mathematics is and how we come to have knowledge of it.

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“…I will also not provide a more general account of our knowledge of actual infinity, which has been the aim of e.g. Lakoff and Nuñez (2000) and Pantsar (2015). I only intend to provide an account of how we can acquire number concepts that match the successor axiom.…”

Section: Introductionmentioning

confidence: 99%

Two roads to the successor axiom

Buijsman

2018

Synthese

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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 viable alternative. I argue that they could also learn it on the basis of a method that has to do with the structure of the numeral system. Specifically, that they (1) use the syntactic structure of the numeral system and (2) attend to the leftmost digits, the one with the highest place-value. Children can learn that this is a reliable method of forming larger numbers by combining two elements. First, a grasp of the syntactic structure of the numeral system. That way they know that the leftmost digit receives the highest value. Second, an interpretation of numerals as designating cardinal values, so that they also realise that increasing or adding digits on the lefthand side of a numeral produces a larger number. There are thus two, currently equally well-supported, ways in which children might learn that there are infinitely many natural numbers.

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In search of $$\aleph _{0}$$ ℵ 0 : how infinity can be created

Cited by 20 publications

References 43 publications

Mathematical cognition and enculturation: introduction to the Synthese special issue

Mathematical cognition and enculturation: introduction to the Synthese special issue

Metaphor and the Philosophical Implications of Embodied Mathematics

Two roads to the successor axiom

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