A hub-and-spoke model of geometric concepts

  1. Bacelar Valente, Mario
Aldizkaria:
Theoria: an international journal for theory, history and foundations of science

ISSN: 0495-4548

Argitalpen urtea: 2023

Alea: 38

Zenbakia: 1

Orrialdeak: 25-44

Mota: Artikulua

DOI: 10.1387/THEORIA.23729 DIALNET GOOGLE SCHOLAR lock_openDialnet editor

Beste argitalpen batzuk: Theoria: an international journal for theory, history and foundations of science

Laburpena

The cognitive basis of geometry is still poorly understood, even the ‘simpler’ issue of what kind of representation of geometric objects we have. In this work, we set forward a tentative model of the neural representation of geometric objects for the case of the pure geometry of Euclid. To arrive at a coherent model, we found it necessary to consider earlier forms of geometry. We start by developing models of the neural representation of the geometric figures of ancient Greek practical geometry. Then, we propose a related model for the earliest form of pure geometry – that of Hippocrates of Chios. Finally, we develop the model of the neural representation of the geometric objects of Euclidean geometry. The models are based on the hub-and-spoke theory. In our view, the existence of specific models opens the possibility of addressing the relationship between geometric figures and geometric objects, in a novel way, in terms of their neural representation.

Erreferentzia bibliografikoak

  • Asper, M. (2003). The two cultures of mathematics in ancient Greece. In E. Robson, & J. Stedall (Eds.). The Oxford handbook of the history of mathematics (pp. 107-132). Oxford: Oxford University Press.
  • Baddeley, A., Eysenck, M. W., & Anderson, M. C. (2020). Memory. New York: Routledge.
  • Brashear, W. (1994). Vier neue Texte zum antiken Bildungswesen. Archiv für Papyrusforschung, 40, 29-35.
  • Busard. H. L. L., & Folkerts, M. (1992). Robert of Chester’s(?) redaction of Euclid’s Elements, the so-called Adelard II version. Basel: Springer Basel.
  • Cairncross, A., & Henry, W. B. (2016). Euclid, Elements I.4 (diagram), 8-11, 14-25 (without proofs). The Oxyrhyunchus Papyri, LXXXII (Graeco-Roman memoires, No. 103), 23-38.
  • Cellucci, C. (2013). Rethinking logic: logic in relation to mathematics, evolution, and method. Dordrecht: Springer.
  • Chiou, R., & Ralph, M. A. L. (2019). Unveiling the dynamic interplay between the huband spoke-components of the brain’s semantic system and its impact on human behaviour. NeuroImage, 199, 114-126.
  • Cuomo, S. (2001). Ancient mathematics. London: Routledge.
  • Euclid (1956). The thirteen Books of the Elements (second edition, Vols. I-III). Translated with introduction and commentary by Sir Thomas L. Heath, from the critical edition of Heiberg. New York: Dover Publications.
  • Eysenck, M. W., & Keane, M. T. (2020). Cognitive psychology: a student’s handbook. London: Psychology Press.
  • Dal Magro, T., & García-Pérez, M. J. (2019). On Euclidean diagrams and geometrical knowledge. Theoria, 34, 255-276.
  • Damerow, p. (2016). The impact of notation systems: from the practical knowledge of surveyors to Babylonian geometry. In M. Schemmel (Ed.). Spatial thinking and external representation: towards a historical epistemology of space (pp. 93-119). Berlin: Edition Open Access.
  • Dillon, M. R., Duyck, M., Dehaene, S., Amalric, M., & Izard, V. (2019). Geometric Categories in Cognition. Journal of Experimental Psychology: Human Perception and Performance, 45, 1236-1247.
  • Ferreirós, J. (2016). Mathematical knowledge and the interplay of practices. Princeton: Princeton University Press.
  • Ferreirós, J., & García-Pérez, M. J. (2020). Beyond natural geometry: on the nature of proto-geometry. Philosophical Psychology, 33, 181-205.
  • Frankland, S. M., & Greene, J. D. (2020). Concepts and compositionality: in search of the brain’s language of thought. Annual Review of Psychology, 71, 273-303.
  • Friberg, J. (2005). Unexpected links between Egyptian and Babylonian mathematics. Singapore: World Scientific.
  • Friberg, J. (2007). A remarkable collection of Babylonian mathematical texts: manuscripts in the Schøyen collection cuneiform texts I. New York: Springer.
  • Giaquinto, M. (2007). Visual thinking in mathematics. New York: Oxford University Press.
  • Harari, O. (2003). The concept of Existence and the role of constructions in Euclid’s Elements. Archive for History of Exact Sciences, 57, 1-23.
  • Hoffman, P., Binney, R. J., & Ralph, M. A. L. (2015). Differing contributions of inferior prefrontal and anterior temporal cortex to concrete and abstract conceptual knowledge. Cortex, 63, 250-266.
  • Hohol, M. (2020). Foundations of geometric cognition. New York: Routledge.
  • Hohol, M., & MiÅkowski, M. (2019). Cognitive artifacts for geometric reasoning. Foundations of Science, 24,
  • 657-680. Høyrup, J. (2002). Lengths, widths, surfaces: a portrait of Old Babylonian algebra and its kin. New York: Springer.
  • Høyrup, J. (2019). Hippocrates of Chios – his Elements and his lunes: A critique of circular reasoning. AIMS mathematics, 5, 158-184.
  • Hyde, D. C. (2011). Two systems of non-symbolic numerical cognition. Frontiers in Human Neuroscience, 5, 150.
  • Imhausen, A. (2016). Mathematics in ancient Egypt: a contextual history. Princeton: Princeton University Press.
  • Kautz, R. (2011). Chaos: the science of predictable random motion. Oxford: Oxford University Press.
  • Kemmerer, D. (2015). Cognitive neuroscience of language. New York: Psychology Press.
  • Knorr, W. R. (1986). The ancient tradition of geometric problems. Boston: Birkhäuser.
  • Kuhnke, P., Kiefer, M., & Hartwigsen, G. (2020). Task-dependent recruitment of modality-specific and multimodal regions during conceptual processing. Cerebral Cortex, 30, 3938-3959.
  • Kuhnke, P., Beaupain, M. C., Arola, J., Kiefer, M., & Hartwigsen, G. (2023). Meta-analytic evidence for a novel hierarchical model of conceptual processing. Neuroscience & Biobehavioral Reviews, 104994.
  • Lewis, M. J. T. (2004). Surveying instruments of Greece and Rome. Cambridge: Cambridge University Press.
  • Lupyan, G. (2017). The paradox of the universal triangle: Concepts, language, and prototypes. The Quarterly Journal of Experimental Psychology, 70, 389-412.
  • Mahon, B. Z., & Hickok, G. (2016). Arguments about the nature of concepts: symbols, embodiment, and beyond. Psychonomic Bulletin & Review, 23, 941-958.
  • Mandl, F. (1959). Introduction to quantum field theory. New York: Interscience Publishers.
  • Michel, C. (2021). Overcoming the modal/amodal dichotomy of concepts. Phenomenology and the Cognitive Sciences 20, 655-677.
  • Mori, L. (2007). Land and land use: the middle Euphrates valley. In G. Leick (Ed.). The Babylonian world (pp. 39-53). New York: Routledge.
  • Netz, R. (1999). The shaping of deduction in Greek mathematics: a study in cognitive history. Cambridge: Cambridge University Press.
  • Netz, R. (2004). Eudemus of Rhodes, Hippocrates of Chios and the earliest form of a Greek mathematical text. Centaurus 2004, 243-286.
  • Nguyen, J. (2020). It’s not a game: Accurate representation with toy models. The British Journal for the Philosophy of Science, 71, 1013-1041.
  • Pantsar, M. (2022). On the development of geometric cognition: beyond nature vs. nurture. Philosophical Psychology, 35, 595-616.
  • Plato (1997). Complete works. Edited with introduction and notes by J. M. Cooper. Indianapolis: Hackett Publishing Company.
  • Ralph, M. A. L. (2014). Neurocognitive insights on conceptual knowledge and its breakdown. Philosophical Transactions of the Royal Society B: Biological Sciences 369, 20120392
  • Ralph, M. A. L., Jefferies, E., Patterson, K., & Rogers, T. T. (2017). The neural and computational bases of semantic cognition. Nature Reviews Neuroscience, 18, 42-55.
  • Reutlinger, A., Hangleiter, D., & Hartmann, S. (2017). Understanding (with) toy models. The British Journal for the Philosophy of Science, 69, 1069-1099.
  • Robson, E. (2004). Words and pictures: new light on Plimpton 322. In M. Anderson, V. Katz, & R. Wilson (Eds.). Sherlock Holmes in Babylon and other tales of mathematical history (pp. 14-26). Washington: Mathematical Association of America.
  • Robson, E. (2008). Mathematics in ancient Iraq. Princeton: Princeton University Press.
  • Rodin, A. (2014). Axiomatic method and category theory. Cham. Springer.
  • Rogers, T. T., & Ralph, M. A. L. (2022). Semantic tiles or hub-and-spokes? Trends in Cognitive Sciences, 26,
  • 189-190. Rossi, C. (2007). Architecture and mathematics in ancient Egypt. Cambridge: Cambridge University Press.
  • Sablé-Meyer, M., Fagot, J., Caparos, S., van Kerkoerle, T., Amalric, M., & Dehaene, S. (2021). Sensitivity to geometric shape regularity in humans and baboons: A putative signature of human singularity. Proceedings of the National Academy of Sciences, 118, e2023123118.
  • Saito, K. (2018). Diagrams and traces of oral teaching in Euclid’s Elements: labels and references. Zentralblatt für Didaktik der Mathematik (ZDM), 50, 921-936.
  • Sokolowski, M., Hawes, Z., Peters, L., & Ansari, D. (2021). Symbols are special: an fMRI adaptation study of symbolic, nonsymbolic and non-numerical magnitude processing in the human brain. Cerebral Cortex Communications, 2, tgab048.
  • Spelke, E. (2011). Natural number and natural geometry. In S. Dehaene & E. Brannon (Eds.). Space, time and number in the brain: searching for the foundations of mathematical thought (pp. 287-317). Amsterdam: Elsevier.
  • Spelke, E., Lee, S. A., & Izard, V. (2010). Beyond core geometry: natural geometry. Cognitive Science 34, 863-884.
  • Spelke, E., & Lee, S. A. (2012). Core systems of geometry in animal minds. Philosophical Transactions of the Royal Society B: Biological Sciences, 367, 2784-2793.
  • Valente, M. B. (2020). From practical to pure geometry and back. Revista Brasileira de História da Matemática, 20, 13-33.
  • Valente, M. B. (2021). On the relationship between geometric objects and figures in Euclidean geometry. In A. Basu, G. Stapleton, S. Linker, C. Legg, E. Manalo, & p. Viana (Eds.). Diagrammatic Representation and Inference (pp. 70-78). Cham: Springer.
  • Yeomans, J. M. (1992). Statistical mechanics of phase transitions. Oxford: Oxford University Press.
  • Zhou, X., Li, M., Li, L., Zhang, Y., Cui, J., Liu, J., & Chen, C. (2018). The semantic system is involved in mathematical problem solving. NeuroImage, 166, 360-370.
Datuen iturria: Dialnet