Mathematics and the brain

Stanislas Dehaene

www.unicog.org

Collège de France, Paris

INSERM-CEA Cognitive Neuroimaging Unit,

NeuroSpin center, CEA, Saclay, France

© Kumi Yamashita

What is mathematics?

A corpus of absolute truths, independent of the human mind :« I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our “creations,” are simply our notes of our observations » (Hardy)

A creation of the human brain :« Mathematical objects correspond to physical states of our brain » (Changeux)

A mystery : « The unreasonable effectiveness in the natural sciences. » (Wigner)« How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of physical reality? » (Einstein)

In particular, we all possess a number sense, a specific capacity to

represent the approximate cardinal of a set, and to combine these

numbers according to arithmetic operations.

We recycle this system for higher-level mathematics.

In mathematics, we formalize these intuitions using a hierarchy of

symbols and the combinations (a language of thought)

Yet these symbols always maintain a tight connection to the

underlying non-symbolic “core semantic systems”.

During its evolution, our brain was endowed with elementary

representations of space, time and number, that we share with

many animal species and lie at the foundation of mathematical

intuition.

The number sense hypothesis:

The origins of mathematicsin the evolution of the human brain

Number: a spontaneous competence in many animals

Monkeys, lions, dolphins, rats… exhibit a spontaneous competence for assessing and comparing numbers.

- Food items- Social groups

McComb, K., Packer, C., & Pusey, A. (1994). Roaring and numerical assessment in contests between groups of female lions, Panthera leo. Animal Behaviour, 47, 379-387.

…infants look longer at suchimpossible events

Core knowledge of arithmetic in infants

Babies of a few month of age discriminatenumbers and react to violations of the laws of

arithmetic and probability.

K. McCrink, K. Wynn, L. Bonatti, F. Xu, E. Spelke…

When 5 +5 does not make 10….

Number neurons in the macaque monkey(Nieder, Freedman & Miller, 2002; Nieder & Miller, 2003, 2004, 2005; Roitman, Brannon & Platt, 2007)

Macaque monkeys can decidewhether two numerosities are

same or different.

homo Sapiens

Proportion

Their intraparietal cortex contains neurons tuned to

number.

( lo g s c a le )

0

2 5

5 0

7 5

1 0 0

0

2 5

5 0

7 5

1 0 0

0

2 5

5 0

7 5

1 0 0

No

r ma

liz

ed

res

po

ns

e(

%)

0

2 5

5 0

7 5

1 0 0

0

2 5

5 0

7 5

1 0 0

1 2 3 4 5

Neuronal tuning curves

Neuron preferring 1

Neuron preferring 2

Neuron preferring 3

Neuron preferring 4

Neuron preferring 5

Number of items

1 2 84 16…

Internal logarithmic scale : log(n)

Log-Gaussian model of number sense :Approximate number is represented by a bank of neural

filters, with Gaussian tuning curves on a logarithmetic scale

Decoding numerosity from intraparietal fMRI signalsEger, Michel, Thirion, Amadon, Dehaene & Kleinschmidt, Current Biology 2009

Scan with high-res fMRI (1.5 mm voxels at 3T)

+

Memorizethe approximate number

Decode the numberfrom brain activity

0,10

0,15

0,20

0,25

0,30

0,35

0,40

0,45

0,50

0 5 10 15 20 25 30

Age (years)

Inte

rna

l w

Numerical uncertainty improves with age and education

dyscalculics

normals

Number sense gets refinedwith age and education

Piazza, Zorzi, Dehaene et al., Cognition 2010

Which number is larger ?

The Distance Effect in number comparison(first discovered by Moyer and Landauer, 1967)

52

84

31

99

larger or

smaller

than 65 ?

smaller larger 500

550

600

650

700

750

800

850

900

30 40 50 60 70 80 90 100

Dehaene, S., Dupoux, E., & Mehler, J. (1990). Journal of Experimental Psychology: Human Perception and Performance, 16, 626-641.

Response time

Target number

0

0,2

0,4

0,6

0,8

% activation

Increasing distance between the numbers

500

550

600

650

700

RT

Neural bases of the distance effect

in symbolic number comparison

99

84

59

66

Larger

or smaller

than 65 ?

smaller larger

Z=48

Pinel, P., Dehaene, S., Riviere, D., & LeBihan, D. (2001). Neuroimage, 14(5), 1013-1026.

Number sense and the horizontal segment of the intraparietal sulcus (HIPS)

z = 44 x = 39x = - 48

Left hemisphere Right hemisphere

50 %

22 %

z = 49

HIPS

Axial slice

Dehaene, S., Piazza, M., Pinel, P., & Cohen, L. (2003). Cognitive Neuropsychology

• All numerical tasks activate this region(e.g. addition, subtraction, comparison, approximation, digit detection…)

• This region fulfils two criteria for a semantic-level representation:-It responds to number in various formats (Arabic digits, written or spoken words), more than to other categories of objects (e.g. letters, colors, animals…)-Its activation varies according to a semantic metric (numerical distance, number size)

Studies of the Mundurucu:Arithmetic and geometry in the absence

of formal mathematical education

• Pica, Lemer, Izard, & Dehaene, Science, 2004

• Dehaene, Izard, Pica & Spelke, Science, 2006

• Dehaene, Izard, Spelke & Pica, Science, 2008

• Izard, Pica, Spelke & Dehaene, PNAS, 2011

Brésil

Transamazonian

highway

Rio Tapajos

Rio JuruenaRio Cururu

Rio Teles Pires

Missão Velha

Jacareacanga

Rio das tropas

KatõSai Cinza

40

km

Success in approximate addition and comparison

Pica, Lemer, Izard, & Dehaene, Science, 2004

All Munduruku

French subjects

Distance between numbers(Ratio of n1+n2 and n3)

Percent success

1 2 3 4

0.5

0.6

0.7

0.8

0.9

1

Failure in

exact subtraction of

small quantities

Pica, Lemer, Izard, & Dehaene, Science, 2004

Magnitude of n1

Percent success

2 4 6 8

0.2

0.4

0.6

0.8

1

All Munduruku

French subjects

Brain networks for high-level mathematics in professional mathematicians

Amalric & Dehaene, PNAS 2016

Subjects = Professional mathematicians (n=15) Comparison with professors of humanities of matched academic standing, but without

mathematical training (n=15).

Main task = perform a fast intuitive judgment on spoken statements (classify them as true, false, or meaningless)

+ Calculation localizer : « please compute seven minus three » vs hearing control sentences.

+ Visual localizer : one-back task with various categories of stimuli:

Sentence presentation

Reflectionperiod

Motor response

Resting period

1 s mean = 4.6 ± 0.9 s 4 s 2 s 7 s

Alertingsound

Alertingsound

Brain areas for mathematical expertise in mathematicians :ventrolateral temporal, intraparietal, and dorsal prefrontal cortices

Meaningfulmath > non-math in

mathematicians

Contrast Math > non-Mathrestricted to meaningful stimuli, during reflection period

Even

t-re

late

d a

vera

ge

L IPS[-52 -43 56]

L fusiform[-52 -56 -15]

R fusiform[55 -52 -18]

R IPS[55 -35 56]

time (s)

statement statement

statement statement

L BA 44d[-46 6 31]

Interaction: Meaningful math > meaningful non- math in Mathematicians > Controls

The math-network activation is found only in mathematicians

z = -16 z = 51

Meaningfulmath > Non-Math in mathematicians

L intraparietal

0 5 10 15 20-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 5 10 15 20-2

-1.5

-1

-0.5

0

0.5

1

1.5

Mathematicians Controls

L inferior temporal

0 5 10 15 20

-1

-0.5

0

0.5

1

0 5 10 15 20

-1

-0.5

0

0.5

1

Mathematicians Controls

Left pSTS/AG [-53 -67 27] Right pSTS/AG [58 -65 28]

Mathematicians Controls Mathematicians Controls

% bold

Left MTG [-62 -12 -20] Right MTG [64 -7 -21]

Mathematicians Controls Mathematicians Controls

Meaningful math > non-mathin mathematicians

Meaningful non-math > mathin both groups

AnalysisAlgebraTopologyGeometryNon-math

Activation to meaningful sentences in:

General semantic knowledge activates areas completely different from those involved in mathematical thinking

Controversy over the language of mathematics

Galileo: « This book [the universe] is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it. »How does mathematical language relate to natural language?

According to Noam Chomsky, “the origin of the mathematical capacity lies in an abstraction from linguistic operations”.

According to Albert Einstein (and many other physicists and mathematicians), « words and language, whether written or spoken, do not seem to play any part in my thought processes. The psychological entities that serve as building blocks for my thought are certain signs or images, more or less clear, that I can reproduce and recombine at will.»

Neuronal recycling model: During its evolution, our brain was endowed with non-verbal representations of space, time andnumber, that we share with many animal species.In mathematics, we formalize these intuitions using a hierarchy ofsymbols, yet these symbols remain attached to the underlying non-verbal “core semantic systems”.

Language and math areas are distinct

1

2

3

4

56

7

AnalysisAlgebraTopologyGeometryNon-math

0 5 10 15 20

-1

0

1

2

3

statement

Temporal pole0 5 10 15 20

-3-2-101234567

statement

Anterior temporal

Language areas are only transiently activated during sentence presentation

Meaningful math > non-mathin mathematicians

Meaningful non-math > mathin both groups

Math “recycles” the cortical networks for number recognition and calculation.

z = 52z = -14

Math > Non-math reflection

Numbers > Other pictures

Calculation > Sentence processing

Intersection

Parietal areas for number sense

(Dehaene et al 2003)

The visual number form area

(Shum, Hermes, Parvizi)

y = -53

z = -17

Checkers Faces Bodies Tools Houses Formulas Numbers Words

z = -17

y = -53

Mathematicians Controls

MathematiciansControls

Increased activation to mathematical expressionsin the left inferior temporal gyrus (-53, -64, -17)

Chk Fac Bod Too Hou For Num Wor0

2

4

Response at [-52.5, -64, -17]

Gmath

Gnonmath

z = -17

Expansion of responses to numbers and formulas in mathematicians

Motivation circuits are activated during the sentence listening period

Sentence presentation

Reflectionperiod

Motor response

Resting period

1 s mean = 4.6 ± 0.9 s 4 s 2 s 7 s

Alertingsound

Alertingsound

Activation of the head of the caudate nucleus, involved in motivation and execution attention, only for the preferred domain : math in mathematicans, non-math in controls

Can mathematical networks developin the absence of any visual experience?

Saunderson, Lucasian professor of mathematics

Is visual experience essential to the development of mathematical or numerical concepts ?

Remarkably, there are actually many examples of blind mathematicians in the history of mathematics:- Leonhard Euler was blind during the two last decades of his life.- Nicholas Saunderson became blind in his first year and yet became the Lucasian professor of Mathematics at Cambridge.

Do they acquire mathematics in a completely different manner?

We studied 3 blind mathematicians, one of whom has demonstrated a remarkable theorem in contact geometry.

Amalric, Denghien and Dehaene, Developmental Cognitive Neuroscience, 2017

Sighted group

Blind group

y = -54

y = -58

Experiment 2. Activations to simpler mathematical statements

Blind subject A: became blind between 3 and 10, teaches

number theory and geometry at a major French university

Recycling of occipital cortex in blind mathematicians

Experiment 1

Math > non-math in Blind > Sighted

Experiment 2

y = -84 y = -90y = -94y = -99

y = -84 y = -90y = -94y = -90

Conclusions:Blind mathematicians use similar brain areas as sighted mathematicians – and these activations again spare language areas.They also activate occipital areas, suggesting the recycling of this region for higher-level cognitive functions.The math network can develop independently of visual experience.

Blind subject B: became blind at 11; top-level mathematician,

demonstrated a major theorem in contact geometry

Blind subject C: anophtalmic; research engineer in a French computer science lab

Conclusions: On the origins of human mathematical abilities

Mathematics is a language, but that language does not appeal to classical language areas of the brain.

Mathematics builds upon ancient, non-linguistic foundations :core knowledge of number, but also of continuous quantities, space, time…shared with many other animal species.

Such core knowledge can emerge in the absence of any visual experience, and developmental evidence suggests that the corresponding brain circuits are available from birth.

Hypothesis: Humans are special in their ability to1. Discretize representations using symbols

(e.g. moving from approximate to exact representations of number)2. Combine these representations productively in a language of thought

Education: Mathematical games are helpful when they are grounded in foundational representations (number, space) and help cement the links with the appropriate symbols.

Arithmetic board games teach math in kindergarten

TheNumberRace.comteaches number sense and counting

www.TheNumberRace.com

TheNumberCatcher.comteaches base 10 and 2-digit numbers

www.TheNumberCatcher.com

Research shows that playing simple board games improves number comprehensionand helps at-risk children progress in arithmetic.