Iconicity in mathematical notation: Commutativity and symmetry

Wege, Theresa Elise; Batchelor, Sophie; Inglis, Matthew; Mistry, Honali; Schlimm, Dirk

doi:10.5964/jnc.v6i3.314

Cited by 7 publications

(4 citation statements)

References 20 publications

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“…Intriguingly, the relationship between visual symmetry and abstract thought may extend to other forms of transformation invariance beyond the concepts tokened by predicates in natural language. A recent paper explored the relationship between the visual symmetry of mathematical symbols and the related mathematical concept of commutativity (i.e., where the order of operations is interchangeable, as for addition and multiplication; Wege et al, 2020). The authors found that participants have a strong tendency to associate vertically symmetrical symbols (e.g., ▲) with commutativity, as compared to when these same symbols are rotated 90 degrees (e.g., ▶), which is known to diminish the salience of visual symmetry (Wagemans, 1997).…”

Section: Discussionmentioning

confidence: 99%

Where word and world meet: Language and vision share an abstract representation of symmetry.

Hafri¹,

Gleitman²,

Landau³

et al. 2023

Journal of Experimental Psychology: General

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Symmetry is ubiquitous in nature, in logic and mathematics, and in perception, language, and thought. Although humans are exquisitely sensitive to visual symmetry (e.g., of a butterfly), symmetry in natural language goes beyond visuospatial properties: many words point to abstract concepts with symmetrical content (e.g., equal, marry). For example, if Mark marries Bill, then Bill marries Mark. In both cases (vision and language), symmetry may be formally characterized as invariance under transformation. Is this a coincidence, or is there some deeper psychological resemblance? Here we asked whether representations of symmetry correspond across language and vision. To do so, we developed a novel cross-modal matching paradigm. On each trial, participants observed a visual stimulus (either symmetrical or nonsymmetrical) and had to choose between a symmetrical and nonsymmetrical English predicate unrelated to the stimulus (e.g., "negotiate" vs. "propose"). In a first study with visual events (symmetrical collision or asymmetrical launch), participants reliably chose the predicate matching the event's symmetry. A second study showed that this "language-vision correspondence" generalized to objects and was weakened when the stimuli's binary nature was made less apparent (i.e., for one object, rather than two inward-facing objects). A final study showed the same effect when nonsigners guessed English translations of signs from American Sign Language, which expresses many symmetrical concepts spatially. Taken together, our findings support the existence of an abstract representation of symmetry which humans access via both perceptual and linguistic means. More broadly, this work sheds light on the rich, structured nature of the language-cognition interface.

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Section: Discussionmentioning

confidence: 99%

Where word and world meet: Language and vision share an abstract representation of symmetry.

Hafri¹,

Gleitman²,

Landau³

et al. 2023

Journal of Experimental Psychology: General

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show abstract

“…23 It is described as "the reversed initial letter of the word contains [contiene]" in [Peano 1889b, 6], 24 whereas the symbol or included, he notes that "this figurative notation is very convenient and fairly widespread" [Peano 1916[Peano -1917. That the use of iconic mathematical symbols has indeed cognitive advantages has recently been shown in [Wege, Batchelor et al 2020]. 23.…”

Section: Iconicity and Mnemonicsmentioning

confidence: 95%

Peano on Symbolization, Design Principles for Notations, and the Dot Notation

Schlimm¹

2021

philosophiascientiae

Self Cite

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Peano a été l'une des forces motrices dans le développement du formalisme mathématique actuel. Dans cet article, nous étudions son approche particulière de la conception notationnelle et présentons quelques caractéristiques originales de ses notations. Pour motiver l'approche de Peano, nous présentons d'abord sa vision de la logique comme méthode d'analyse et son désir d'un symbolisme rigoureux et concis pour représenter les idées mathématiques. Sur la base à la fois de sa pratique et de ses réflexions explicites sur les notations, nous discutons des principes qui ont guidé Peano dans l'introduction de nouveaux symboles, le choix des caractères et la mise en forme des formules. Enfin, nous examinons de plus près, d'un point de vue systématique et historique, l'une des innovations les plus marquantes de Peano, à savoir l'usage de points pour regrouper des sous-formules.

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“…Intriguingly, the relationship between visual symmetry and abstract thought may extend to other forms of transformation invariance beyond the concepts tokened by predicates in natural language. A recent paper explored the relationship between the visual symmetry of mathematical symbols and the related mathematical concept of commutativity (i.e., where order of operations is interchangeable, as for addition and multiplication) (Wege et al, 2020). The authors found that participants have a strong tendency to associate vertically symmetrical symbols (e.g., ▲) with commutativity, as compared to when these same symbols were rotated 90 degrees (e.g., ▶), which is known to diminish the salience of visual symmetry (Wagemans, 1997).…”

Section: Beyond Language and Vision: Symmetry In Mathematical Reasoningmentioning

confidence: 99%

Where word and world meet: Language and vision share an abstract representation of symmetry

Hafri¹,

Gleitman²,

Landau³

et al. 2022

Preprint

View full text Add to dashboard Cite

Symmetry is ubiquitous in nature, in logic and mathematics, and in perception, language, and thought. Although humans are exquisitely sensitive to visual symmetry (e.g., of a butterfly), symmetry in natural language goes beyond visuospatial properties: many words point to abstract concepts with symmetrical content (e.g., equal, marry). For example, if Mark marries Bill, then Bill marries Mark. In both cases (vision and language), symmetry may be formally characterized as invariance under transformation. Is this a coincidence, or is there some deeper psychological resemblance? Here we asked whether representations of symmetry correspond across language and vision. To do so, we developed a novel cross-modal matching paradigm. On each trial, participants observed a visual stimulus (either symmetrical or non-symmetrical) and had to choose between a symmetrical and non-symmetrical English predicate unrelated to the stimulus (e.g., "negotiate" vs. "propose"). In a first study with visual events (symmetrical collision or asymmetrical launch), participants reliably chose the predicate matching the event's symmetry. A second study showed that this "language-vision correspondence" generalized to objects and was weakened when the stimuli's binary nature was made less apparent (i.e., of one object, rather than two inward-facing objects). A final study showed the same effect when nonsigners guessed English translations of signs from American Sign Language, which expresses many symmetrical concepts spatially. Taken together, our findings support the existence of an abstract representation of symmetry which humans access via both perceptual and linguistic means. More broadly, this work sheds light on the rich, structured nature of the language-cognition interface.

show abstract

Iconicity in mathematical notation: Commutativity and symmetry

Cited by 7 publications

References 20 publications

Where word and world meet: Language and vision share an abstract representation of symmetry.

Where word and world meet: Language and vision share an abstract representation of symmetry.

Peano on Symbolization, Design Principles for Notations, and the Dot Notation

Where word and world meet: Language and vision share an abstract representation of symmetry

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