Reviewer #3 (Public review):
Summary:
The authors report converging evidence from behavioral studies as well as several brain-imaging techniques that geometric figures, notably quadrilaterals, are processed differently in visual (lower activation) and spatial (greater) areas of the human brain than representative figures. Comparison of mathematical models to fit activity for geometric figures shows the best fit for abstract geometric features like parallelism and symmetry. The brain areas active for geometric figures are also active in processing mathematical concepts even in blind mathematicians, linking geometric shapes to abstract math concepts. The effects are stronger in adults than in 6-year-old Western children. Similar phenomena do not appear in great apes, suggesting that this is uniquely human and developmental.
Strengths:
Multiple converging techniques of brain imaging and testing of mathematical models showing special status of perception of abstract forms. Careful reasoning at every step of research and presentation of research, anticipating and addressing possible reservations. Connecting these findings to other findings, brain, behavior, and historical/anthropological to suggest broad and important fundamental connections between abstract visual-spatial forms and mathematical reasoning.
Weaknesses:
I have reservations of the authors' use of "symbolic." They seem to interpret "symbolic" as relying on "discrete, exact, rule-based features." Words are generally considered to symbolic (that is their major function), yet words do not meet those criteria. Depictions of objects can be regarded as symbolic because they represent real objects, they are not the same as the object (as Magritte observed). If so then perhaps depictions of quadrilaterals are also symbolic but then they do not differ from depictions of objects on that quality. Relatedly, calling abstract or generalized representations of forms a distinct "language of thought" doesn't seem supportable by the current findings. Minimally, a language has elements that are combined more or less according to rules. The authors present evidence for geometric forms as elements but nowhere is there evidence for combining them into meaningful strings.
Further thoughts
Incidentally, there have been many attempts at constructing visual languages from visual elements combined by rules, that is, mapping meaning to depictions. Many written languages like Egyptian hieroglyphics or Mayan or Chinese, began that way; there are current attempts using emoji. Apparently, mapping sound to discrete letters, alphabets, is more efficient and was invented once but spread. That said, for restricted domains like maps, circuit diagrams, networks, chemical interactions, mathematics, and more, visual "languages" work quite well.
The findings are striking and as such invite speculation about their meaning and limitations. The images of real objects seem to be interpreted as representations of 3D objects as they activate the same visual areas as real objects. By contrast, the images of 2D geometric forms are not interpreted as representations of real objects but rather seemingly as 2D abstractions. It would be instructive to investigate stimuli that are on a continuum from representational to geometric, e. g., real objects that have simple geometric forms like table tops or boxes under various projections or balls or buildings that are rectangular or triangular. Objects differ from geometric forms in many ways: 3D rather than 2D, more complicated shapes; internal features as well as outlines. The geometric figures used are flat, 2-D, but much geometry is 3-D (e. g. cubes) with similar abstract features. The feature space of geometry is more than parallelism and symmetry; angles are important for example. Listing and testing features would be fascinating.
Can we say that mathematical thinking began with the regularities of shapes or with counting, or both? External representations of counting go far back into prehistory; tallies are frequent and wide-spread. Infants are sensitive to number across domains as are other primates (and perhaps other species). Finding overlapping brain areas for geometric forms and number is intriguing but doesn't show how they are related.
Categories are established in part by contrast categories; are quadrilaterals and triangles and circles different categories? As for quadrilaterals, the authors say some are "completely irregular." Not really; they are still quadrilaterals, if atypical. See Eleanor Rosch's insightful work on (visual) categories. One wonders about distinguishing squashed quadrilaterals from squashed triangles.
What in human experience but not the experience of close primates would drive the abstraction of these geometric properties? It's easy to make a case for elaborate brain processes for recognizing and distinguishing things in the world, shared by many species, but the case for brain areas sensitive to abstracting geometric figures is harder. The fact that these areas are active in blind mathematicians and that they are parietal areas suggest that what is important is spatial far more than visual. Could these geometric figures and their abstract properties be connected in some way to behavior, perhaps with fabrication, construction or use of objects? Or with other interactions with complex objects and environments where symmetry and parallelism (and angles and curvature--and weight and size) would be important? Manual dexterity and fabrication also distinguish humans from great apes (quantitatively not qualitatively) and action drives both visual and spatial representations of objects and spaces in the brain. I certainly wouldn't expect the authors to add research to this already packed paper, but raising some of the conceptual issues would contribute to the significance of the paper.