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 ∧ ¯Qp. 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. 7overt 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