Table Of ContentSPACE, TIME AND
NUMBER IN THE BRAIN
SEARCHING FOR THE FOUNDATIONS OF
MATHEMATICAL THOUGHT
SPACE, TIME
AND NUMBER
IN THE BRAIN
SEARCHING FOR THE
FOUNDATIONS OF
MATHEMATICAL THOUGHT
AN ATTENTION AND
PERFORMANCE SERIES VOLUME
Edited by
Stanislas Dehaene and Elizabeth M. Brannon
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10 11 12 13 14 15 10 9 8 7 6 5 4 3 2 1
Contents
Contributors vii 11. Neglect "Around the Clock": Dissociating
Foreword ix Number and Spatial Neglect in Right Brain
Damage 149
12. Saccades Compress Space, Time, and
I
Number 175
MENTAL MAGNITUDES
IV
AND THEIR
TRANSFORMATIONS ORIGINS OF PROTO-
MATHEMATICAL
1. Mental Magnitudes 3
2. Objects, Sets, and Ensembles 13 INTUITIONS
3. Attention Mechanisms for Counting in
Stabilized and in Dynamic Displays 23 13. Origins of Spatial, Temporal, and Numerical
Cognition: Insights from Comparative
Psychology 191
II
14. Evolutionary Foundations of the Approximate
Number System 207
NEURAL CODES FOR SPACE,
15. Origins and Development of Generalized
TIME AND NUMBER Magnitude Representation 225
4. A Manifold of Spatial Maps in the Brain 41
V
5. Temporal Neuronal Oscillations can Produce
Spatial Phase Codes 59
REPRESENTATIONAL
6. Population Clocks: Motor Timing with Neural
Dynamics 71 CHANGE AND
7. Discrete Neuroanatomical Substrates
EDUCATION
for Generating and Updating Temporal
Expectations 87
16. Foundational Numerical Capacities and the
8. The Neural Code for Number 103
Origins of Dyscalculia 249
17. Neurocognitive Start-Up Tools for Symbolic
III Number Representations 267
18. Natural Number and Natural Geometry 287
SHARED MECHANISMS FOR 19. Geometry as a Universal Mental
Construction 319
SPACE, TIME AND NUMBER?
20. How Languages Construct Time 333
21. Improving Low-Income Children's Number
9. Synesthesia: Gluing Together Time, Number,
Sense 343
and Space 123
10. How is Number Associated with Space?
Index 355
The Role of Working Memory 133
v
Contributors
Marilena Aiello Dipartimento di Psicologia, Uni- Jennifer T. Coull Laboratoire de Neurobiologie
versità degli Studi “La Sapienza”, Roma, Italy & de la Cognition, CNRS—Université de Provence,
Fondazione Santa Lucia IRCCS, Roma, Italy Marseille, France
Paola Binda Istituto di Neuroscienze del CNR, Stanislas Dehaene Collège de France, Paris,
Pisa, Italy & Italian Institute of Technology, France & INSERM, Cognitive Neuro imaging
Genova, Italy Unit, NeuroSpin center, Saclay, France
Lera Boroditsky Psychology Department, Stan- Dori Derdikman Department of Physiology,
ford University, Stanford, CA Faculty of Medicine, Technion, Haifa, Israel
Elizabeth M. Brannon Center for Cognitive Neu- Jean-Philippe van Dijck Department of Experi-
roscience, Duke University, Durham, NC mental Psychology, Ghent University, Belgium
Claudio Brozzoli ImpAct, Centre des Neuro- Fabrizio Doricchi Dipartimento di Psicologia,
sciences de Lyon, Institut National de la Santé Università degli Studi “La Sapienza”, Roma, Italy
et de la Recherche Médicale, Université Claude & Fondazione Santa Lucia IRCCS, Roma, Italy
Bernard Lyon, Bron, France
Lisa Feigenson Department of Psychological
Dean V. Buonomano Departments of Neuro - and Brain Sciences, Johns Hopkins University,
biology and Psychology, UCLA, Los Angeles, CA Baltimore, MD
Christopher Burgess UCL Institute of Cognitive Wim Fias Department of Experimental Psy-
Neuroscience and UCL Institute of Neurology, chology, Ghent University, Belgium
University College London, UK
C. R. Gallistel Department of Psychology and
Neil Burgess UCL Institute of Cognitive Neu- Rutgers Center for Cognitive Science, Rutgers
roscience and UCL Institute of Neurology, University, New Brunswick, NJ
University College London, UK
Limor Gertner Department of Psychology, and
David Burr Istituto di Neuroscienze del CNR, Zlotowski Center for Neuroscience, Ben-Gurion
Pisa, Italy & Department of Psychology, Uni- University of the Negev, Beer-Sheva, Israel
versitá Degli Studi di Firenze, Firenze, Italy
Wim Gevers Unité de Recherches en Neuro-
Brian Butterworth Institute of Cognitive Neuro- sciences Cognitives, Université Libre de
science & Department of Psychology, University Bruxelles, Belgium
College London, UK
Daniel Haun Max Planck Institute for Evoluti-
Patrick Cavanagh Laboratoire Psychologie de la onary Anthropology, Leipzig, Germany, &
Perception, Université Paris Descartes, France Max Planck Institute for Psycholinguistics,
Nijmegen, The Netherlands, & University of
Nicky Clayton Department of Experimental Psy-
Portsmouth, UK
chology, University of Cambridge, UK
Sheng He Department of Psychology, University
Roi Cohen Kadosh Department of Experimental
of Minnesota, Minneapolis, MN
Psychology and Oxford Centre for Functional
Magnetic Resonance Imaging of the Brain, Danielle Hinchey Department of Psychology,
University of Oxford, UK Harvard University, Cambridge, MA
vii
viii Contributors
Masami Ishihara Department of Health Pro- Andreas Nieder Animal Physiology Institute
motion Sciences, Tokyo Metropolitan University, of Neurobiology, University of Tübingen,
Japan Germany
Véronique Izard Laboratoire Psychologie de la
Manuela Piazza INSERM, U562, Cognitive Neu-
Perception, Université Paris Descartes, Paris,
roimaging Unit, CEA/SAC/DSV/DRM/Neu-
France & CNRS UMR 8158, Paris France &
rospin center, Gif-sur-Yvette, France, & Center for
Department of Psychology, Harvard University,
Mind/Brain Sciences and Dipar timento di Scienze
Cambridge, MA
della Cognizione e della Formazione University of
Sophie Jacquin-Courtois ImpAct, Centre des Trento, Italy
Neurosciences de Lyon, Institut National de la
Pierre Pica “Formal Structure of Language”,
Santé et de la Recherche Médicale, Université
CNRS and Université Paris 8, Paris, France
Claude Bernard Lyon, Bron, France, & Mouvement
et Handicap, Hôpital Henry Gabrielle, Hospices Geetha B. Ramani Department of Human Devel-
Civils de Lyon, St Genis Laval, France
opment, University of Maryland, College Park,
Fiona Jordan Max Planck Institute for MD
Psycholinguistics, Nijmegen, The Netherlands
John Ross Department of Psychology Uni-
R. Laje Departments of Neurobiology and
versity of Western Australia, Nedlands, Perth,
Psychology, UCLA, Los Angeles, CA, & Depart-
Western Australia
ment of Science and Technology, University of
Quilmes, Argentina Yves Rossetti ImpAct, Centre des Neurosciences
de Lyon, Institut National de la Santé et de la
Matthew R. Longo Department of Psychological
Recherche Médicale, Université Claude Bernard
Sciences, Birbeck, University of London, UK
Lyon, Bron, France, & Mouvement et Handicap,
Stella F. Lourenco Department of Psychology,
Hôpital Henry Gabrielle, Hospices Civils de
Emory University, Atlanta, GA
Lyon, St Genis Laval, France
Dustin J. Merritt Center for Cognitive Neuro-
science, Duke University, Duke University, Nicolas W. Schuck Department of Psychology,
Durham, NC Humboldt Universität zu Berlin, Germany
Concetta Morrone Department of Physiological Robert S. Siegler Department of Psychology,
Sciences, University of Pisa & Scientific Carnegie Mellon University, Pittsburgh, PA
Institute Stella Maris, Pisa, Italy
Elizabeth S. Spelke Department of Psychology,
Edvard I. Moser Kavli Institute for Systems
Harvard University, Cambridge, MA
Neuroscience and the Centre for the Biology of
Memory, Norwegian University of Science and Giorgio Vallortigara Centre for Mind/Brain
Technology, Trondheim, Norway Sciences, University of Trento, Rovereto, Italy
Foreword
by Stanislas Dehaene and Elizabeth M. Brannon
The knowledge of first principles, as space, evolutionary processes and neural mecha-
time, motion, number, is as sure as any of those nisms which may universally give rise to
which we get from reasoning. And reason must
Kantian intuitions.
trust these intuitions of the heart, and must base
But a second issue motivates our present
on them every argument.
Blaise Pascal, Pensées focus on the representations of space, time
(translated by W. F. Trotter) and number: they all raise deep computational
issues for cognitive neuroscience. In all three
What do the representations of space, domains, the nervous system must encode
time and number have in common that jus- and compute with quantities. Behavioral evi-
tifies our dedicating an entire book to them? dence suggests that these computations can
In his Critique of Pure Reason, Immanuel Kant be remarkably accurate, even in miniature
famously argued that they provide “a priori organisms such as desert ants or in imma-
intuitions” that precede and structure how ture systems such as the infant brain. Animal
we experience our environment. Indeed, spatial navigation implies the mental stor-
these concepts are so basic to our understand- age of spatial coordinates and their updating
ing of the external world that we find it hard through path integration [1]. Temporal deci-
to imagine how any animal species could sions imply that memorized representations
survive without possessing mechanisms for of time are subjected to operations analogous
spatial navigation, temporal orienting (e.g., to addition, subtraction and comparison [3].
time-stamped memories), and elementary In the number domain, human infants as well
numerical computations (e.g., choosing the as other animal species readily anticipate the
food patch with the largest expected return) outcome of analogs of arithmetic operations
[1]. In the course of their evolution, humans performed with concrete sets of objects [4].
and many other animal species have internal- Does this mean that a common set of coding
ized basic codes and operations isomorphic to and computation mechanisms underlies quan-
the physical and arithmetic laws that govern tity manipulations in all three domains? Do
the interaction of objects in the external world these systems share similar brain circuitry? An
[2]. Indeed, there is now considerable evi- exciting research program consists of mapping
dence that space, time and number are part of the range of the possible implementations of
the essential toolkit that adult humans share quantitative operations in the nervous system,
with infants and with many other nonhu- and testing whether evolution has arrived at
man animals. One of the main purposes of the the same computational solutions in distinct
present book is therefore to review this work organisms or for distinct domains.
in detail. From grid cells to number neurons, The 24th Attention & Performance meet-
the richness and variety of the mechanisms ing on “Space, Time, and Number: Cerebral
used by animals and humans, including Foundations of Mathematical Intuitions”, held
infants, to represent the dimensions of space, from July 6 to 10, 2010, in Vaux de Cernay near
time and number is bewildering and suggests Paris, was organized with this goal in mind: to
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