Estamos trabajando para incorporar este artículo al repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor


During language processing, humans form complex embedded representations from sequential inputs. Here, we ask whether a “geometrical language” with recursive embedding also underlies the human ability to encode sequences of spatial locations. We introduce a novel paradigm in which subjects are exposed to a sequence of spatial locations on an octagon, and are asked to predict future locations. The sequences vary in complexity according to a well-defined language comprising elementary primitives and recursive rules. A detailed analysis of error patterns indicates that primitives of symmetry and rotation are spontaneously detected and used by adults, preschoolers, and adult members of an indigene group in the Amazon, the Munduruku, who have a restricted numerical and geometrical lexicon and limited access to schooling. Furthermore, subjects readily combine these geometrical primitives into hierarchically organized expressions. By evaluating a large set of such combinations, we obtained a first view of the language needed to account for the representation of visuospatial sequences in humans, and conclude that they encode visuospatial sequences by minimizing the complexity of the structured expressions that capture them. © 2017 Amalric et al.


Documento: Artículo
Título:The language of geometry: Fast comprehension of geometrical primitives and rules in human adults and preschoolers
Autor:Amalric, M.; Wang, L.; Pica, P.; Figueira, S.; Sigman, M.; Dehaene, S.
Filiación:Cognitive Neuroimaging Unit, CEA DSV/I2BM, INSERM, Université Paris-Sud, Université Paris-Saclay, NeuroSpin center, Gif/Yvette, France
Sorbonne Universités, UPMC Univ Paris 06, IFD, Paris, France
Institute of Neuroscience, Key Laboratory of Primate Neurobiology, CAS Center for Excellence in Brain Science and Intelligence Technology, Chinese Academy of Sciences, Shanghai, China
Instituto do Cérebro, Universidade Federal do Rio Grande do Norte, Natal, Brazil
UMR 7023 Structures Formelles du Langage CNRS, Université Paris 8, Saint-Denis, France
Department of Computer Science, FCEN, University of Buenos Aires and ICC-CONICET, Buenos Aires, Argentina
Neuroscience Laboratory, Universidad Torcuato Di Tella, Buenos Aires, Argentina
Collège de France, Paris, France
Palabras clave:adult; analytical error; comprehension; DNA structure; embedding; exposure; geometry; human; human experiment; language; rotation; algorithm; American Indian; comprehension; concept formation; cultural anthropology; education; educational model; male; mathematical phenomena; mathematics; nomenclature; physiology; preschool child; Adult; Algorithms; Child, Preschool; Comprehension; Concept Formation; Culture; Humans; Indians, South American; Language; Male; Mathematical Concepts; Mathematics; Models, Educational; Terminology as Topic
Título revista:PLoS Computational Biology
Título revista abreviado:PLoS Comput. Biol.


  • Dehaene, S., Meyniel, F., Wacongne, C., Wang, L., Pallier, C., The Neural Representation of Sequences: From Transition Probabilities to Algebraic Patterns and Linguistic Trees (2015) Neuron, 88, pp. 2-19. , 10.1016/j.neuron.2015.09.019, 6447569,.;: –
  • Yang, C., Ontogeny and phylogeny of language (2013) Proc Natl Acad Sci, 110, pp. 6324-6327. , 10.1073/pnas.1216803110, 3576720,.;: –
  • Saffran, J.R., Aslin, R.N., Newport, E.L., Statistical Learning by 8-Month-Old Infants (1996) Science, 274, pp. 1926-1928. , 943209,.;: –
  • Kabdebon, C., Pena, M., Buiatti, M., Dehaene-Lambertz, G., Electrophysiological evidence of statistical learning of long-distance dependencies in 8-month-old preterm and full-term infants (2015) Brain Lang, 148, pp. 25-36. , 10.1016/j.bandl.2015.03.005, 5865749,.;: –
  • Saffran, J.R., Wilson, D.P., From Syllables to Syntax: Multilevel Statistical Learning by 12-Month-Old Infants (2003) Infancy, 4, pp. 273-284
  • Restle, F., Theory of serial pattern learning: structural trees (1970) Psychol Rev, 77, p. 481
  • Sakai, K., Kitaguchi, K., Hikosaka, O., Chunking during human visuomotor sequence learning (2003) Exp Brain Res, 152, pp. 229-242. , 10.1007/s00221-003-1548-8, 2879170,.;: –
  • Peña, M., Bonatti, L.L., Nespor, M., Mehler, J., Signal-Driven Computations in Speech Processing (2002) Science, 298, pp. 604-607. , 10.1126/science.1072901, 2202684,.;: –
  • Marcus, G.F., Vijayan, S., Rao, S.B., Vishton, P.M., Rule learning by seven-month-old infants (1999) Science, 283, pp. 77-80. , 872745,.;: –
  • Wang, L., Uhrig, L., Jarraya, B., Dehaene, S., Representation of Numerical and Sequential Patterns in Macaque and Human Brains (2015) Curr Biol, 25, pp. 1966-1974. , 10.1016/j.cub.2015.06.035, 6212883,.;: –
  • Bahlmann, J., Schubotz, R.I., Friederici, A.D., Hierarchical artificial grammar processing engages Broca’s area (2008) NeuroImage, 42, pp. 525-534. , 10.1016/j.neuroimage.2008.04.249, 8554927,.;: –
  • Monti, M.M., Parsons, L.M., Osherson, D.N., Thought Beyond Language: Neural Dissociation of Algebra and Natural Language (2012) Psychol Sci, 23, pp. 914-922. , 10.1177/0956797612437427, 2760883,.;: –
  • Amalric, M., Dehaene, S., Origins of the brain networks for advanced mathematics in expert mathematicians (2016) Proc Natl Acad Sci, , 201603205
  • Friederici, A.D., Steinhauer, K., Pfeifer, E., Brain signatures of artificial language processing: Evidence challenging the critical period hypothesis (2002) Proc Natl Acad Sci, 99, pp. 529-534. , 10.1073/pnas.012611199, 1773629,.;: –
  • Fitch, W.T., Friederici, A.D., Artificial grammar learning meets formal language theory: an overview (2012) Phil Trans R Soc B, 367, pp. 1933-1955. , 10.1098/rstb.2012.0103, 2688631,.;: –
  • Gómez, R.L., Gerken, L., Gómez, R.L., Gerken, L., Gómez, R.L., Gerken, L., Infant artificial language learning and language acquisition (2000) Trends Cogn Sci, 4, pp. 178-186. , 0782103,..;: –
  • Pothos, E.M., Theories of artificial grammar learning (2007) Psychol Bull, 133, pp. 227-244. , 10.1037/0033-2909.133.2.227, 7338598,.;: –
  • Martins, M.D., Laaha, S., Freiberger, E.M., Choi, S., Fitch, W.T., How children perceive fractals: Hierarchical self-similarity and cognitive development (2014) Cognition, 133, pp. 10-24. , 10.1016/j.cognition.2014.05.010, 4955884,.;: –
  • Martins, M.J., Fischmeister, F.P., Puig-Waldmüller, E., Oh, J., Geißler, A., Robinson, S., Fractal image perception provides novel insights into hierarchical cognition (2014) NeuroImage, 96, pp. 300-308. , 10.1016/j.neuroimage.2014.03.064, 4699014,..;: –
  • Landau, B., Gleitman, H., Spelke, E., Spatial knowledge and geometric representation in a child blind from birth (1981) Science, 213, pp. 1275-1278. , 268438,.;: –
  • Lee, S.A., Sovrano, V.A., Spelke, E.S., Navigation as a source of geometric knowledge: Young children’s use of length, angle, distance, and direction in a reorientation task (2012) Cognition, 123, pp. 144-161. , 10.1016/j.cognition.2011.12.015, 2257573,.;: –
  • Cheng, K., A purely geometric module in the rat’s spatial representation (1986) Cognition, 23, pp. 149-178. , 742991,.;: –
  • Chiandetti, C., Vallortigara, G., Is there an innate geometric module? Effects of experience with angular geometric cues on spatial re-orientation based on the shape of the environment (2007) Anim Cogn, 11, pp. 139-146. , 10.1007/s10071-007-0099-y, 7629754,.;: –
  • Spelke, E.S., Lee, S.A., Core systems of geometry in animal minds (2012) Philos Trans R Soc B Biol Sci, 367, pp. 2784-2793
  • Dillon, M.R., Huang, Y., Spelke, E.S., Core foundations of abstract geometry (2013) Proc Natl Acad Sci, 110, pp. 14191-14195. , 10.1073/pnas.1312640110, 3940342,.;: –
  • Dillon, M.R., Spelke, E.S., Core geometry in perspective (2015) Dev Sci, 18, pp. 894-908. , 10.1111/desc.12266, 5441089,.;: –
  • Dehaene, S., Izard, V., Pica, P., Spelke, E., Core Knowledge of Geometry in an Amazonian Indigene Group (2006) Science, 311, pp. 381-384. , 10.1126/science.1121739, 6424341,.;: –
  • Izard, V., Pica, P., Dehaene, S., Hinchey, D., Spelke, E., (2011) Space, Time and Number in the Brain, pp. 319-332. ,, pp. –. Elsevie
  • Gilmore, C.K., McCarthy, S.E., Spelke, E.S., Non-symbolic arithmetic abilities and mathematics achievement in the first year of formal schooling (2010) Cognition, 115, pp. 394-406. , 10.1016/j.cognition.2010.02.002, 0347435,.;: –
  • Halberda, J., Mazzocco, M.M.M., Feigenson, L., Individual differences in non-verbal number acuity correlate with maths achievement (2008) Nature, 455, pp. 665-668. , 10.1038/nature07246, 8776888,.;: –
  • Dehaene, S., Izard, V., Spelke, E., Pica, P., Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures (2008) Science, 320, pp. 1217-1220. , 10.1126/science.1156540, 8511690,.;: –
  • Lakoff, G., Núñez, R.E., (2000) Where mathematics comes from: how the embodied mind brings mathematics into being, , New York Basic Book
  • Spelke, E., Lee, S.A., Izard, V., Beyond core knowledge: Natural geometry (2010) Cogn Sci, 34, pp. 863-884. , 10.1111/j.1551-6709.2010.01110.x, 0625445,.;: –
  • Fodor, J.A., (1975) The Language of Thought, , Harvard University Pres
  • Fodor, J.A., (1983) The Modularity of Mind: An Essay on Faculty Psychology, , MIT Pres
  • Romano, S., Sigman, M., Figueira, S., LT2C2s: A language of thought with Turing-computable Kolmogorov complexity (2013) Pap Phys, 5
  • Goodman, N.D., Tenenbaum, J.B., Gerstenberg, T., (2015) The conceptual mind: New directions in the study of concepts, ,
  • Piantadosi, S.T., Tenenbaum, J.B., Goodman, N.D., The Logical Primitives of Thought: Empirical Foundations for Compositional Cognitive Models (2016) Psychol Rev
  • Piantadosi, S.T., Tenenbaum, J.B., Goodman, N.D., Bootstrapping in a language of thought: A formal model of numerical concept learning (2012) Cognition, 123, pp. 199-217. , 10.1016/j.cognition.2011.11.005, 2284806,.;: –
  • Yildirim, I., Jacobs, R.A., Learning multisensory representations for auditory-visual transfer of sequence category knowledge: a probabilistic language of thought approach (2015) Psychon Bull Rev, 22, pp. 673-686. , 10.3758/s13423-014-0734-y, 5338656,.;: –
  •, olomon CJ, Papert S. A case study of a young child doing Turtle Graphics in LOGO. Proceedings of the June 7–10, 1976, national computer conference and exposition. ACM; 1976. pp. 1049–1056; Leyton, M., A process-grammar for shape (1988) Artif Intell, 34, pp. 213-247
  • Leyton, M., (2001) A generative theory of shape, , Berlin; New York Springe
  • Li, M., Vitanyi, P., (2013) An Introduction to Kolmogorov Complexity and Its Applications, , Springer Science & Business Medi
  • Cilibrasi, R., Vitányi, P.M., Clustering by compression (2005) IEEE Trans Inf Theory, 51, pp. 1523-1545
  • Grunwald, P., (2004) ArXiv Prepr Math0406077, ,
  • Bradmetz, J., Mathy, F., Response times seen as decompression times in Boolean concept use (2006) Psychol Res, 72, pp. 211-234. , 10.1007/s00426-006-0098-7, 7093950,.;: –
  • Mathy, F., Feldman, J., What’s magic about magic numbers? Chunking and data compression in short-term memory (2012) Cognition, 122, pp. 346-362. , 10.1016/j.cognition.2011.11.003, 2176752,.;: –
  • Mathy, F., Fartoukh, M., Gauvrit, N., Guida, A., Developmental Abilities to Form Chunks in Immediate Memory and Its Non-Relationship to Span Development (2016) Front Psychol, 7
  • Feldman, J., The simplicity principle in human concept learning (2003) Curr Dir Psychol Sci, 12, pp. 227-232
  • Feldman, J., Minimization of Boolean complexity in human concept learning (2000) Nature, 407, pp. 630-633. , 10.1038/35036586, 1034211,.;: –
  • Hochberg, J., McAlister, E., A quantitative approach, to figural “goodness” (1953) J Exp Psychol, 46, pp. 361-364. , 3109140,.;: –
  • Izard, V., Pica, P., Spelke, E.S., Dehaene, S., Flexible intuitions of Euclidean geometry in an Amazonian indigene group (2011) Proc Natl Acad Sci, 108, pp. 9782-9787. , 10.1073/pnas.1016686108, 1606377,.;: –
  • Pica, P., Exact and Approximate Arithmetic in an Amazonian Indigene Group (2004) Science, 306, pp. 499-503. , 10.1126/science.1102085, 5486303,.;: –
  • Giaquinto, M., (2005) Visualization, explanation and reasoning styles in mathematics, pp. 31-55. ,, pp. –. Springe
  • Pizlo, Z., Sawada, T., Li, Y., Kropatsch, W.G., Steinman, R.M., New approach to the perception of 3D shape based on veridicality, complexity, symmetry and volume (2010) Vision Res, 50, pp. 1-11. , 10.1016/j.visres.2009.09.024, 9800910,.;: –
  • Westphal-Fitch, G., Huber, L., Gomez, J.C., Fitch, W.T., Production and perception rules underlying visual patterns: effects of symmetry and hierarchy (2012) Philos Trans R Soc B Biol Sci, 367, pp. 2007-2022
  • Machilsen, B., Pauwels, M., Wagemans, J., The role of vertical mirror symmetry in visualshape detection (2009) J Vis, 9, p. 11
  • Lee, S.A., Spelke, E.S., Children’s use of geometry for reorientation (2008) Dev Sci, 11, pp. 743-749. , 10.1111/j.1467-7687.2008.00724.x, 8801130,.;: –
  •, he Internet Classics Archive | Meno by Plato [Internet]. [cited 7 Apr 2016]; Goldin, A.P., Pezzatti, L., Battro, A.M., Sigman, M., From ancient Greece to modern education: Universality and lack of generalization of the Socratic dialogue (2011) Mind Brain Educ, 5, pp. 180-185
  • Vallortigara, G., Sovrano, V.A., Chiandetti, C., Doing Socrates experiment right: controlled rearing studies of geometrical knowledge in animals (2009) Curr Opin Neurobiol, 19, pp. 20-26. , 10.1016/j.conb.2009.02.002, 9299120,.;: –
  • Lourenco, S.F., Huttenlocher, J., The Representation of Geometric Cues in Infancy (2008) Infancy, 13, pp. 103-127
  • Kotovsky, K., Simon, H.A., Empirical tests of a theory of human acquisition of concepts for sequential patterns (1973) Cognit Psychol, 4, pp. 399-424
  • Restle, F., Serial patterns: The role of phrasing (1972) J Exp Psychol, 92, p. 385
  • Restle, F., Serial pattern learning: Higher order transitions (1973) J Exp Psychol, 99, p. 61
  • Restle, F., Burnside, B.L., Tracking of serial patterns (1972) J Exp Psychol, 95, p. 299. , 071911,.;:
  • Shepard, R.N., Hovland, C.I., Jenkins, H.M., Learning and memorization of classifications (1961) Psychol Monogr Gen Appl, 75, pp. 1-42


---------- APA ----------
Amalric, M., Wang, L., Pica, P., Figueira, S., Sigman, M. & Dehaene, S. (2017) . The language of geometry: Fast comprehension of geometrical primitives and rules in human adults and preschoolers. PLoS Computational Biology, 13(1).
---------- CHICAGO ----------
Amalric, M., Wang, L., Pica, P., Figueira, S., Sigman, M., Dehaene, S. "The language of geometry: Fast comprehension of geometrical primitives and rules in human adults and preschoolers" . PLoS Computational Biology 13, no. 1 (2017).
---------- MLA ----------
Amalric, M., Wang, L., Pica, P., Figueira, S., Sigman, M., Dehaene, S. "The language of geometry: Fast comprehension of geometrical primitives and rules in human adults and preschoolers" . PLoS Computational Biology, vol. 13, no. 1, 2017.
---------- VANCOUVER ----------
Amalric, M., Wang, L., Pica, P., Figueira, S., Sigman, M., Dehaene, S. The language of geometry: Fast comprehension of geometrical primitives and rules in human adults and preschoolers. PLoS Comput. Biol. 2017;13(1).