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Mathematical Intuition and the Cognitive Roots of Mathematical Concepts

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Abstract

The foundation of Mathematics is both a logico-formal issue and an epistemological one. By the first, we mean the explicitation and analysis of formal proof principles, which, largely a posteriori, ground proof on general deduction rules and schemata. By the second, we mean the investigation of the constitutive genesis of concepts and structures, the aim of this paper. This “genealogy of concepts”, so dear to Riemann, Poincaré and Enriques among others, is necessary both in order to enrich the foundational analysis with an often disregarded aspect (the cognitive and historical constitution of mathematical structures) and because of the provable incompleteness of proof principles also in the analysis of deduction. For the purposes of our investigation, we will hint here to a philosophical frame as well as to some recent experimental studies on numerical cognition that support our claim on the cognitive origin and the constitutive role of mathematical intuition.

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Notes

  1. For more on this epistemological perspective, see Bailly and Longo (2006) and related papers by Longo, since Longo (1998).

References

  • Bailly F, Longo G (2004) Space, time and cognition. In: Peruzzi (ed) From the standpoint of mathematics and natural science. Invited paper, mind and causality, Benjamins, Amsterdam, pp 149–199

  • Bailly F, Longo G (2006) Mathématiques et sciences de la nature. La singularité physique du vivant. Hermann, Paris (Introduction in English and in French downloadable; ongogin translation in English)

    Google Scholar 

  • Bailly F, Longo G (2009) Geometric schemes for biological time. Invited lecture,. In: Boniface J, Miquel PA (eds) Conference on “Episodic memory and time in neurophysiology”, Strasbourg, October 2007 (version française dans “Logique du vivant”, n. 13, Noesis-Vrin)

  • Banks WP, Coleman MJ (1981) Two subjective scales of number. Percept Psychophys 29:95–105

    Google Scholar 

  • Banks WP, Hill DK (1974) The apparent magnitude of number scaled by random production. J Exp Psychol 102:353–376

    Article  Google Scholar 

  • Berthoz A (1997) Le sens du mouvement, Odile Jacob, Paris (English transl. The brain sens of movement, Harvard University Press, 2000)

  • Bottazzini U, Tazzioli R (1995) Naturphilosophie and its role in Riemann's Mathematics. Revue d’Histoire des Mathématiques 1:3–386

  • Burr D, Ross J (2008) A visual sense of number. Curr Biol 18:425–428

    Article  Google Scholar 

  • Cauty A (1988) Sémantique de la mise en signes du nombre : une vision ordinale. Amerindia, 13

  • Connes A (1994) Non-commutative geometry. Academic Press, New York

    Google Scholar 

  • Dehaene S (1997) La Bosse des Maths. Editions Odile Jacob, Paris

    Google Scholar 

  • Dehaene S, Bossini S, Giraux P (1993) The mental representation of parity and numerical magnitude. J Exp Psychol Gen 122:371–396

    Article  Google Scholar 

  • Dehaene S, Cohen L (2007) Cultural recycling of cortical maps. Neuron 56:384–398

    Article  Google Scholar 

  • Dubucs J (1999) Beth, Kant et l’intuition mathématique. Philos Sci 3(4):93–134

    Google Scholar 

  • Enriques F (1935) Philosophie scientifique et empirisme logique. In: Actes du Congrès international de philosophie scientifique, Hermann, Paris

  • Faracovi O, Speranza F (1998) Federigo Enriques. Filosofia e storia del pensiero scientifico, Belforte, Livorno

  • Fischbein E (1987) Intuition in science and mathematics, an educational approach. Springer, Berlin

    Google Scholar 

  • Galton F (1880) Visualised numerals. Nature 252–256

  • Gelman R, Gallistel CR (1978) The child’s understanding of number. Harvard University Press, Cambridge

    Google Scholar 

  • Gould S (1989) Wonderful life. WW. Norton, New York

    Google Scholar 

  • Harrington L et al (eds) (1985) H. Friedman’s research on the foundations of mathematics. Elsevier, North Holland

    Google Scholar 

  • Herrenschmidt C (2007) Les trois écritures. Gallimard, France

    Google Scholar 

  • Hilbert D (1899) Grundlagen der Geometrie, Leipzig (trad. fran., Dunod, 1971)

  • Hubbard EM, Piazza M, Pinel P, Dehaene S (2005) Interactions between number and space in parietal cortex. Nat Rev Neurosci 6(6):435–448

    Article  Google Scholar 

  • Hunting RP (2003) Part-whole knowledge in preschool children. J Math Behav 22:217–235

    Article  Google Scholar 

  • Husserl E (1933) The origin of geometry (trad. fran. PUF, 1962)

  • Izard V, Dehaene S (2008) Calibrating the mental number line. Cognition 106:1221–1247

    Article  Google Scholar 

  • Jacob SN, Nieder A (2009) Notation-independent representation of fractions in the human parietal cortex. J Neurosci 29(14):4652–4657

    Article  Google Scholar 

  • Kant E (1944/2004) Critique de la raison pure. Traduction française, 7èmé édition Quadrige, P.U.F., Paris

  • Kline M (1980/1989) Mathématiques: la fin de la certitude. traduction française, Christian Bourgois éditeur

  • Lakoff G, Nunez R (2000) Where mathematics comes from: how the embodied mathematics creates mathematics. Basic Books, New York

    Google Scholar 

  • Longo G (1998) The mathematical continuum, from intuition to logic. In: Petitot J et al (eds) Naturalizing phenomenology. Stanford University Press, Stanford

    Google Scholar 

  • Longo G (2002) Reflections on Incompleteness, or on the proofs of some formally unprovable propositions and prototype proofs in type theory. In: Callaghan et al (eds) Types for proofs and programs, Durham, (GB), Dec. 2000; Lecture notes in computer science, vol 2277, Springer, pp 160–180

  • Longo G (2005) The cognitive foundations of mathematics: human gestures in proofs and mathematical incompleteness of formalisms. In: Okada M et al (eds) Images and reasoning. Keio University Press, Tokyo, pp 105–134

    Google Scholar 

  • Longo G (2007) Laplace, Turing and the imitation game impossible geometry: randomness, determinism and programs in Turing’s test. In: Epstein R, Roberts G, Beber G (eds) The Turing test sourcebook. Kluwer, Dordrecht, The Netherlands

    Google Scholar 

  • Longo G (2009) From exact sciences to life phenomena: following Schrödinger and Turing on programs, life and causality. Inf Comput 207(5):543–670

    Article  Google Scholar 

  • Longo (2010) Theorems as constructive visions. Invited lecture, ICMI 19 conference on proof and proving, Taipei, Taiwan, May 10–15 (to appear)

  • Opfer JE, Siegler RS (2007) Representational change and children’s numerical estimation. Cogn Psychol 55:169–195

    Article  Google Scholar 

  • Paris J, Harrington L (1978) A mathematical incompleteness in Peano Arithmetic. In: Barwise J (ed) Handbook of mathematical logic. North Holland

  • Paul T (2007) Discret-continuous and classical-quantum. Math Struct Comp Sci 17:177–183

    Article  Google Scholar 

  • Petitot J (2008) Neurogéométrie de la vision. Modèles mathématiques et physiques des architectures fonctionnelles, Les Editions de l'Ecole Polytechnique, Distribution Ellipses, Paris

  • Piaget J (1967a) Logique et connaissance scientifique. Encyclopédie de la Pléiade, Paris

  • Piaget J (1967b) Biologie et connaissance. Éditions de la Pléïade, Paris

  • Pica P, Lemer C, Izard V, Dehaene S (2004) Exact and approximate arithmetic in an Amazonian indigene group. Science 306:499–503

    Article  Google Scholar 

  • Poincaré H (1905/1997) La Valeur de la Science. Champs Flammarion, Paris

  • Poincaré H (1908/1993) L’invention mathématique. conference published in: Bulletin de l’Institut Général Psychologique. 3:175–187, reprinted following: Essai sur la psychologie de l’invention dans le domaine mathématique (by Hadamard J), éditions Jacques Gabay

  • Shepard RN, Kilpatrick DW, Cunningham JP (1975) The internal representation of numbers. Cogn Psychol 7:82–138

    Article  Google Scholar 

  • Smith L (2002) Reasoning by mathematical induction in children’s arithmetic. Chapters 1 and 2. Elsevier Pergamon, Oxford

    Google Scholar 

  • Starkey P, Spelke ES, Gelman R (1983) Detection of intermodal numerical correspondences by human infants. Science 222:179–181

    Article  Google Scholar 

  • Starkey P, Spelke ES, Gelman R (1990) Numerical abstraction by human infants. Cognition 36:97–127

    Article  Google Scholar 

  • Teissier B (2005) Protomathematics, perception and the meaning of mathematical objects. In: P. Grialou, G. Longo et M. Okada (Dir.), Images and Reasoning, Keio University Press, Tokyo, pp 135–146

  • Tieszen R (2005) Phenomenology, logic, and the philosophy of mathematics. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Viarouge A, Hubbard EM, Dehaene S, Sackur J (2010) Number line compression and the illusory perception of random numbers. Exp Psychol (in press)

  • Visetti Y-M, Rosenthal V (2003) Köhler. Belles Lettres, Paris, France

    Google Scholar 

  • Walsh V (2003) A theory of magnitude: common cortical metrics of time, space and quantity. Trends Cogn Sci 7(11):483–488

    Article  Google Scholar 

  • Wigner E (1960) The unreasonable effectiveness of mathematics in the natural sciences. Comm Pure Appl Math XIII:1–14

    Article  Google Scholar 

  • Wynn K (1992) Addition and subtraction by human infants. Nature 358:749–750

    Article  Google Scholar 

  • Xuan B, Zhang D, He S, Chen X (2007) Larger stimuli are judged to last longer. J Vis 7(10):1–5

    Article  Google Scholar 

  • Zebian S (2005) Linkages between number concepts, spatial thinking, and directionality of writing: the SNARC Effect and the REVERSE SNARC effect in English and Arabic monoliterates, biliterates, and illiterate Arabic speakers. J Cogn Cult 5(1–2):165–190

    Article  Google Scholar 

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Authors and Affiliations

  1. CNRS et Ecole Normale Superieure et CREA, Ecole Polytechnique, Paris, France

    Giuseppe Longo

  2. Psychology and Human Development Dpt., Peabody College, Vanderbilt University, Nashville, TN, USA

    Arnaud Viarouge

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  1. Giuseppe Longo
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  2. Arnaud Viarouge
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Correspondence to Giuseppe Longo.

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Longo, G., Viarouge, A. Mathematical Intuition and the Cognitive Roots of Mathematical Concepts. Topoi 29, 15–27 (2010). https://doi.org/10.1007/s11245-009-9063-6

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