Mathematical symbols as epistemic actions

Cruz, Helen De; Smedt, Johan De

doi:10.1007/s11229-010-9837-9

Cited by 41 publications

(8 citation statements)

References 55 publications

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“…Accordingly, in terms of Burge’s social externalism, if expert mathematicians introduce a new property of the number zero, for instance, that 5 0 = 1 or 0! = 1 (De Cruz and de Smedt 2013 , p. 6), this changes the truth conditions of our beliefs about zero because the content of our representation of the number zero is determined by the experts of the relevant community. This establishes content externalism in the case of mathematical knowledge.…”

Section: External Representations With Non-derived Contentmentioning

confidence: 99%

“…Krämer ( 2003 ) calls this process ‘operative writing’: what the notation represents—that is, the content of the symbol(s)—is constituted by the symbol itself and its interaction with other symbols. This is especially persuasive in cases where new symbols are introduced and manipulated within an established practice but in ways that were not anticipated before, such as subtracting a larger number from a smaller, taking the root of a negative number, or raising to the power where the exponent is a fraction or a real number (De Cruz and de Smedt 2013 ).…”

Section: External Representations With Non-derived Contentmentioning

confidence: 99%

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Extended mathematical cognition: external representations with non-derived content

Vold

Schlimm

2019

Synthese

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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 we develop one aspect of Menary's vehicle externalist theory of cognitive integration-the process of enculturation-to respond to this longstanding objection. We offer examples of how expert mathematicians introduce new symbols to represent new mathematical possibilities that are not yet understood, and we argue that these new symbols have genuine non-derived content, that is, content that is not dependent on an act of interpretation by a cognitive agent and that does not derive from conventional associations, as many linguistic representations do.

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Section: External Representations With Non-derived Contentmentioning

confidence: 99%

Section: External Representations With Non-derived Contentmentioning

confidence: 99%

Extended mathematical cognition: external representations with non-derived content

Vold

Schlimm

2019

Synthese

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“…In relation to the literature on mathematical practice we will focus on here, the affordances and cognitive functions of various representational systems have been analyzed (e.g. Clark 1989, p.133;De Cruz and De Smedt 2013;Schlimm and Neth 2008;Zhang and Norman 1995), and historical case studies have pointed out that the choice of representational form can influence the theoretical and conceptual development of mathematics (e.g. Epple 2004;Johansen and Misfeldt 2015;Kjeldsen 2009;Steensen and Johansen 2016).…”

Section: Cognitive Support In Mathematical Practicementioning

confidence: 99%

Material representations in mathematical research practice

Johansen

Misfeldt

2018

Synthese

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Mathematicians' use of external representations constitutes an important focal point in current philosophical attempts to understand mathematical practice. In this paper, we add to this understanding by presenting and analyzing how research mathematicians use and interact with external representations. The empirical basis of the article consists of a qualitative interview study we conducted with active research mathematicians. In our analysis of the empirical material, we primarily used the empirically based frameworks provided by distributed cognition and cognitive semantics as well as the broader theory of cognitive integration as an analytical lens. We conclude that research mathematicians engage in generative feedback loops with material representations, that they use representations to facilitate the use of experiences of handling the physical world as a resource in mathematical work, and that their use of representations is socially sanctioned and enabled. These results verify the validity of the cognitive frameworks used as the basis for our analysis, but also show the need for augmentation and revision. Especially, we conclude that the social and cultural context cannot be excluded from cognitive analysis of mathematicians' use of external representations. Rather, representations are socially sanctioned and enabled in an enculturation process.

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“…Other examples of refinements of the Indian-Arabic mathematical symbol system include the introduction of negative numbers, square roots, or variables (De Cruz & De Smedt, 2013;Menary, 2015). Taken together, these developmental steps in the history of cognitive innovation gave rise to a genuinely new type of symbol system that is indispensable for mathematical practices.…”

Section: Enculturation and Opportunity Provision For New Innovative Pmentioning

confidence: 99%

Cognitive Innovation, Cumulative Cultural Evolution, and Enculturation

Fabry

2017

JOCC

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Cognitive innovation has shaped and transformed our cognitive capacities throughout history. Until recently, cognitive innovation has not received much attention by empirical and conceptual research in the cognitive sciences. This paper is a first attempt to help close this gap. It will be argued that cognitive innovation is best understood in connection with cumulative cultural evolution and enculturation. Cumulative cultural evolution plays a vital role for the inter-generational transmission of the products of cognitive innovation. Furthermore, there are at least two important functions of enculturation for cognitive innovation. First, enculturation is responsible for the ontogenetic acquisition of cognitive practices governing the interaction with innovative products. Second, successful processes of enculturation provide opportunities for subsequent innovative processes. The trans-generational trajectory of calculation from mathematical symbol systems to the first digital computers will serve as a paradigm example of the delicate interplay of cognitive innovation, cumulative cultural evolution, and enculturation.

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Mathematical symbols as epistemic actions

Cited by 41 publications

References 55 publications

Extended mathematical cognition: external representations with non-derived content

Extended mathematical cognition: external representations with non-derived content

Material representations in mathematical research practice

Cognitive Innovation, Cumulative Cultural Evolution, and Enculturation

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