A perceptual account of symbolic reasoning by David Landy, Colin Allen and Carlos Zednik

In what follows, we propose an  account of symbolic reasoning according to which perception, manipulation, and perceptual imagination lie at the heart
of mathematical and logical competence. p. 1

The authors reference Landy & Goldstone.

They talk about syntactic and computational accounts, and semantic processing accounts in why systems interpret and represent mathematical relations, including mental models, conceptual metaphors etc. This shifts the focus from syntax to meaningful relations. Both tend to ignore the perceptual role of the symbols themselves, even accounts that claim some grounded origin. p. 3

They contrast this with an Andy Clark cyborg account, a more fully situated cognition version in which the symbols actively help reasoning:

the active manipulation of physical notations plays the role of “guiding” the
human biological machinery through  an abstract mathematical
problem space—one that may far exceed the space of otherwise
solvable problems.p. 3

but point out that still not enough attention is paid to the effect on perception. They elaborate on the cyborg account, calling this Perceptual Manipulations Theory.

 Rather,complexvisualandauditoryprocesses
suchasaffordancelearning,perceptualpattern-matchingand
perceptualgroupingofnotationalstructuresproducesimplified
representationsofthemathematicalproblem,simplifyingthetask
facedbytherestofthesymbolicreasoningsystem.[…]

The neuralprocessesthatPMTtakestobeinvolvedinsym-
bolic reasoningalmostneverhaveastheirprimaryfunction
the implementationofamodally representedrulesormodels.
Instead,theyincludesensorimotorsystemsforvisualgrouping
and perceptualorganization,objectrecognition,objecttrack-
ing andsymmetrydetection,amongothers.p. 4

They run through some examples of perceptual metaphors like the dy/dx one I wrote about so long ago.

¯A ∪ B = ¯A ∩ ¯B
whichbearsastrikingvisualsimilaritytoDeMorgan’slaw,
¯P ∨ Q ≡ ¯P ∧ ¯Q

p. 5

They then outline some of the evidence that supports PMT, such as evidence that the physical form of notations significantly affects their interpretation/efficacy, etc.

A corollaryoftheclaimthatsymbolicandotherformsof
mathematical andlogicalreasoningaregroundedinawidevari-
etyofsensorimotorskillsisthatsymbolicreasoningislikely
tobebothidiosyncraticandcontext-specific. p. 7

overt rule-following emerges from the fine-
tuned interactions between the perceptual and sensorimotor
systems with well-designed physical notations—symbolic rea-
soning is a form of sophisticated “symbolpushing” that happens
to adhere to the formal rules of mathematics and logic,
due to a lengthy process of cultural adaptation and pedagogical
scaffolding. p. 8

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