origins of mathematical intuitions - citeseer

THE YEAR IN COGNITIVE NEUROSCIENCE 2009

Origins of Mathematical Intuitions

The Case of Arithmetic

Stanislas Dehaenea,b,c

aINSERM, Cognitive Neuro-imaging Unit, IFR 49, Gif sur Yvette, FrancebCEA, NeuroSpin center, IFR 49, Gif sur Yvette, France

cCollege de France, Paris, France

Mathematicians frequently evoke their “intuition” when they are able to quickly andautomatically solve a problem, with little introspection into their insight. Cognitiveneuroscience research shows that mathematical intuition is a valid concept that canbe studied in the laboratory in reduced paradigms, and that relates to the availabilityof “core knowledge” associated with evolutionarily ancient and specialized cerebralsubsystems. As an illustration, I discuss the case of elementary arithmetic. Intuitions ofnumbers and their elementary transformations by addition and subtraction are presentin all human cultures. They relate to a brain system, located in the intraparietal sulcus ofboth hemispheres, which extracts numerosity of sets and, in educated adults, maps backand forth between numerical symbols and the corresponding quantities. This system isavailable to animal species and to preverbal human infants. Its neuronal organizationis increasingly being uncovered, leading to a precise mathematical theory of how weperform tasks of number comparison or number naming. The next challenge will be tounderstand how education changes our core intuitions of number.

Key words: number; neuroimaging; mathematics

What then is mathematics if it is not a unique, rigorous,

logical structure? It is a series of great intuitions carefully

sifted, and organized by the logic men are willing and able

to apply at any time.

—Morris Kline, Mathematics: The Loss of Certainty

(p. 312)

Introduction

All great mathematicians appeal to their“intuition.” One of the most famous propo-nents of this concept was the French mathe-matician Jacques Hadamard, who published asystematic inquiry into his fellow mathemati-cians’ practices (Hadamard 1945). Hadamardreported that many major mathematical dis-

Address for correspondence: Stanislas Dehaene, INSERM, CognitiveNeuro-imaging Unit, IFR 49, Gif sur Yvette, France. Voice +33 1 69 0883 90. [email protected]

coveries were preceded by long periods of un-conscious “incubation” followed by sudden in-sight. He also quoted Einstein’s well-knownintrospection: “Words and language, whetherspoken or written, do not seem to play anyrole in my thinking mechanisms. The mentalentities that serve as elements of my thoughtare certain signs or images, more or less clear,that can ‘at will’ be reproduced or combined.”Henri Poincare saw intuition as the foundationupon which the mathematical enterprise wasbased. Davis and Hersh, in their descriptionof “the mathematical experience,” went evenfurther (and probably too far) by viewing intu-ition as the single faculty that allows us to domathematics: “In the realm of ideas, of mentalobjects, those ideas whose properties are repro-ducible are called mathematical objects, andthe study of mental objects with reproducibleproperties is called mathematics. Intuition is thefaculty by which we can consider or examine

The Year in Cognitive Neuroscience 2009: Ann. N.Y. Acad. Sci. 1156: 232–259 (2009).doi: 10.1111/j.1749-6632.2009.04469.x C© 2009 New York Academy of Sciences.

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Dehaene: Origins of Mathematical Intuitions 233

these (internal mental) objects” (Davis, Hersh,& Marchisotto 1995, p. 399).

The problem with these statements is thatthey fail to define what an intuition is. Indeed,the concept, like many other theoretical con-structs of folk psychological origin (“attention,”“association,” “consciousness,” and others) re-mains suspect in present-day cognitive neuro-science because it is unclear whether it corre-sponds to a well-characterized mental entity orprocess. I would like to argue, however, thatrecent research in numerical cognition fleshesout the concept of intuition, at least within thesmall domain of elementary arithmetic. Theresults indicate that a sense of number is partof Homo sapiens’ core knowledge, present earlyon in infancy, and with a reproducible cerebralsubstrate. It permits a rapid evaluation of (1) ap-proximately how many objects are present in ascene, (2) whether this number is more or lessthan another number, and (3) how this num-ber is changed by simple operations of additionand subtraction. Its operation obeys three crite-ria that may be seen as definitional of the term“intuition”: it is fast, automatic, and inaccessi-ble to introspection. These properties, far fromimplying that this intuition is inaccessible toscientific understanding, constitute a decipher-able signature of intuition, and we shall see thata simple but precise quantitative mathematicalmodel can be proposed for some of the simplestaspects of their operation.

It will be important, in the following, toclearly distinguish symbolic and nonsymbolicaspects of elementary arithmetic. Symbolicarithmetic deals with how we understand andmanipulate numerals and number words suchas “five” or “twenty.” Nonsymbolic arithmeticis concerned with how we grasp and com-bine the approximate cardinality or “numeros-ity” of concrete sets of objects (such as visualdots, sounds, and actions). Our core knowl-edge of arithmetic is essentially nonsymbolic—the availability of number symbols varies acrosscultures and arises late in human development.Nevertheless, number symbols, once available,become strongly attached to the corresponding

nonsymbolic representations of numbers and,thereafter, a form of “second-order intuition”seems to develop, as the links between symbolsand quantities themselves become fast, auto-matic, and unconscious. In the second part ofthis chapter, we shall review current evidenceconcerning this symbol grounding problem.

Arithmetic Intuition: ACross-Cultural Universal

A recently discovered effect of numericaladaptation (Burr & Ross 2008) provides an ex-cellent introduction to nonsymbolic numericalintuition. Stare at the fixation point in Figure 1,which has 10 dots at left and 100 dots atright. Then after 30 sec of adaptation, shift toFigure 2. You should have the strong impressionthat the left display is more numerous than theright one, although both have exactly 40 dots.You may also have the erroneous impressionthat there are much less than 40 dots in bothcases (see Izard & Dehaene 2008). Both adap-tation and underestimation effects have beenfound to resist extensive manipulation of thenon-numerical parameters of the display, thusevading simple explanations in terms of size,density, or contrast.

At the moment of this writing, why adap-tation can have such a profound effect onnumerosity estimates remains largely unex-plained. Yet the fact that numerical perceptsimpose themselves upon us so immediately,automatically, and without conscious control(even if we know that the numbers are equal)points to the operation of a special and largelyautomatic processing system. As noted by Burrand colleagues, “Just as we have a direct visualsense of the reddishness of half a dozen ripecherries, so we do of their sixishness.”

Cross-cultural research suggests that thisability is present in humans without mathe-matical training. Although number words fig-ure among the most frequent lexical items andare present in many languages, it is still possi-ble to find aboriginal, Amazonian or African

234 Annals of the New York Academy of Sciences

Figure 1. Numerosity adaptation. Stare at the fixation cross for 30 sec, then see Figure 2.

Figure 2. Numerosity adaptation. After staring at Figure 1 for 30 sec, you should experi-ence the strong impression that the left display is more numerous than the right, although theyare actually identical (after Burr & Ross 2008).

cultures with a reduced lexicon for number,sometimes as limited as to include only wordsfor “one,” “two,” and “many.” In spite of thislinguistic limitation, and of their frequent lackof access to education, these people exhibit aremarkable nonverbal competence for elemen-tary arithmetic—keeping in mind that theirknowledge is approximate rather than exact.Peter Gordon (2004) observed how the Pi-raha, whose lexicon stops at two, could approx-imately match two numerosities. They couldnot perform an exact match, however (appar-ently not even with the smallest numbers, 1 and2), leading Gordon to claim that their numeri-cal cognition was “incommensurate” with ours.This terminology appears misleading, however,because on average their matching responses

were almost perfectly linearly correlated withthe true numerosity—with a standard devia-tion that also increased in direct proportion tothe inferred numerosity.

Pierre Pica, Veronique Izard, Cathy Lemer,and I studied the Mundurucu, another Ama-zonian group that has number words up to five(Pica, Lemer, Izard, & Dehaene 2004). We dis-covered that Mundurucu adults and even chil-dren possess an excellent capacity to discrim-inate two sets based on their number, decidewhich is more numerous, or even approxi-mately add or subtract two such numerosities–even with numerosities up into the 50s andmore, way beyond their naming range. Theirpsychophysical behavior was qualitatively sim-ilar to that of Western students: both were


determined by Weber’s law, which states thatthe minimal difference between two numbersthat leads to a fixed level of discriminationvaries in direct proportion to the size of thenumbers. Weber’s law has been observed witha great variety of perceptual continua such aspitch, loudness, or brightness. Its observation inthe number domain further strengthens the hy-pothesis that arithmetical intuition starts withina basic perceptual system for estimating ap-proximate number. Obviously, then, intuitionsconcerning the cardinality of sets are availableto isolated adults, even in the absence of formaleducation and a sophisticated mathematicallanguage.

Recently, Brian Butterworth and his col-leagues extended this cross-cultural researchto a group of 4- to 7-year-old indigenous Aus-tralian children (Butterworth, Reeve, Reynolds,et al. 2008). Using one-to-one matching taskssimilar to Gordon’s, but extended to includecross-modal addition and sharing conditions,they observed a high level of performance thatwas indistinguishable from that of English-speaking children. Somewhat controversially,they concluded that concepts of exact numberare already available to both groups, regardlessof language. The data, however, indicate onlyapproximate performance, subject to Weber’slaw, with a rather sharp drop for numbers abovethree, suggesting a contribution from bothestimation and subitizing (Butterworth et al.2008, Figs. 3 and 4). It is true that cross-modalmatching performance remained relativelyhigh even with numbers 5 and above, but inthe absence of more knowledge about the chil-dren’s education and the availability of finger orbody pointing symbols in this culture, it is hardto reach a firm conclusion as to whether the re-sults reflected a genuine grasp of exact numberin the absence of education, especially giventhat the matching task lends itself to a varietyof spatial cueing strategies. Uncontroversially,however, all cross-cultural studies point to thefact that approximate number is an intuitionavailable to humans regardless of language andeducation.

Intuitions of Numbers in Infancyand Early Childhood

Piaget’s influential research, summarized inThe Child’s Conception of Number (1952), initiallysuggested that young children do not have anystable, invariant representation of number, andthat knowledge of arithmetic emerges slowlyas a logical construction. Less than 30 yearslater, Rochel Gelman and Randy Gallistel’swork, summarized in The Child’s Understanding

of Number (1978), played an instrumental role inoverturning the Piagetian view. Gelman andGallistel showed that even preschoolers hadintuitions in arithmetic, since they could de-tect unexpected changes in small numerosities(the “magic” experiments) or violations in thecounting routine. From this point, it was onlya small leap to ask whether infants also have asense of number (Starkey & Cooper 1980). To-day, a large set of behavioral studies using habit-uation and violation-of-expectancy paradigmshas revealed a clear sensitivity to large num-bers in 4- to 6-month-old infants. For instance,infants discriminate when the numerosity of aset unexpectedly changes from 8 to 16 dots orvice-versa, even when non-numerical parame-ters such as density and total surface are tightlycontrolled (Xu & Spelke 2000). Infants also de-tect violations of approximate addition and sub-traction events. For instance, upon seeing fiveobjects being hidden behind a screen, then an-other five objects being added, they appear toexpect 10 objects and express a form of sur-prise through longer looking times when thescreen collapses and only five objects are re-vealed (McCrink & Wynn 2004; Wynn 1992a).

In the range of small numbers one, two,and three, the issue of numerical competencein infancy has been subject to much moredebate. Initial observations suggested numer-ical discrimination in infants and even in new-borns (Antell & Keating 1983; Bijeljac-Babic,Bertoncini, & Mehler 1991). Further studieswith tighter controls for non-numerical param-eters, however, failed to replicate these findingsand suggested that performance was driven by


low-level confounds such as the total amount ofstuff (Feigenson, Dehaene, & Spelke 2004; Mix,Levine, & Huttenlocher 1997; Simon 1999;Xu & Spelke 2000). A resolution of this de-bate now seems to be in sight. The newer re-sults indicate that infants can attend to eithernumerical or non-numerical parameters, anddo so to a variable extent depending on theexperimental design and, particularly, on thevariability of these parameters in the stimu-lus set. For instance, Feigenson (2005) demon-strated that infants attend to visual numerosity,even in the range from one to three objects, aslong as the sets comprise highly distinctive ob-jects rather than identical replicas of the sameobject. Extensive recent work by Sara Cordesand collaborators now suggests that number isindeed available to infants, even in the pres-ence of variability in other cues. Cordes andBrannon (2008) go as far as to suggest that,for infants, using number is easier than usingcontinuous extent, in the sense that a smallerproportional amount of change is required forinfants to detect changes in the former than inthe latter. A recent study using event-relatedpotentials, to be further detailed below, alsoindicates discrimination of numbers 2 versus3 in the absence of non-numerical confounds(Izard, Dehaene-Lambertz, & Dehaene 2008).

To what extent are these demonstrations ofnumerosity discrimination actually relevant tosubsequent mathematical intuitions? Althoughit has long been proposed that infants’ numeri-cal discrimination abilities provide a foundationon which children base their subsequent un-derstanding of the number domain (Dehaene1997), until recently there was little direct em-pirical support for this suggestion. Recently,however, a behavioral study in preschoolersdemonstrated that early arithmetic intuitionsare translated into higher-than-chance perfor-mance in school-relevant symbolic arithmeticproblems (Gilmore, McCarthy, & Spelke 2007).Gilmore, McCarthy, and Spelke gave 5- and 6-year-olds problems such as “Sarah has 21 can-dies, she gets 30 more, John has 34 candies—who has more?” The problems were simultane-

ously presented both orally, as spoken numer-als, and in writing as Arabic numerals. How-ever, note that the participants were preschool-ers and had therefore received no training withnumbers of that size, nor with the conceptsof addition or subtraction. Nevertheless, theyspontaneously performed much better thanchance (60%–75%), regardless of their socio-economic origins. Performance was approxi-mate and depended on the ratio of the twonumbers, again a clear signature of Weber’slaw. Importantly, variability in performancewas predictive of achievement in the school’scurriculum.

Similarly, Holloway and Ansari (2008) re-cently reported, in slightly older children agedfrom 6 to 8 years, that the variability in the dis-tance effect during number comparison is pre-dictive of mathematics achievement, but not ofreading achievement. Finally, Halberda et al.(2008) showed a tight correlation between per-formance in nonsymbolic number compari-son and achievement scores in mathematics(but not other domains) throughout the schoolcurriculum. Altogether, and although a directcausal influence has not yet been demonstrated,those results suggest that a grasp of approxi-mate numerosity and of distance relations be-tween numbers, grounded in Weber’s law, maygovern the subsequent understanding of sym-bolic arithmetic.

The emphasis on the presence of early intu-itions in young children should not be confusedwith a naıve nativist view of numerical devel-opment, according to which the foundations ofarithmetic would all be present shortly after thebirth. Clear limits on intuitions of arithmeticare evident around 2.5 to 4 years of age, whenchildren begin to acquire number words. For along period, children may know the meaningof the word one (for instance showing an abilityto provide just one object or to name a set ofnumerosity one), but not the meaning of othernumbers two, three, and above (for instance grab-bing a random number of objects when askedto provide two) (Wynn 1992). Children slowlylearn to map number words one, two, three, and


four to their corresponding quantities one byone, until suddenly they understand that eachword maps onto a different number. It maytake them at least another 6 months beforethey understand that larger number words suchas “six,” “eight,” and “ten” map onto distinctquantities, and begin to grasp their order rela-tions (Le Corre & Carey 2007). It is not untilthe age of five that they understand that a wordsuch as “eight,” if it applies to a set, continuesto apply after the objects are shuffled but ceasesto apply if the set is increased by one, doubled,or halved (Condry & Spelke 2008; Lipton &Spelke 2006; Sarnecka & Gelman 2004).

Exactly how children eventually acquirehigher-level intuitions of exact arithmetic re-mains a matter of intense theoretical debate(for specific proposals, see Gelman & Butter-worth 2005; Le Corre & Carey 2007; Nunez &Lakoff 2000; Rips, Bloomfield, & Asmuth 2008;Spelke 2003). The intensity of this discussionis inversely proportional to the quality of thedata: in truth, we know next to nothing aboutthe psychological or neural structures that un-derlie our understanding of the basic principlesof formal arithmetic such as successor function,commutativity, infinity, or induction. At the endof this review, we will briefly consider a tenta-tive proposal for how the acquisition of exactnumber concepts (e.g., exactly six) is encodedat the neural level. For present purposes, how-ever, we focus of the consensus area, which isthat preschoolers and even infants, tested withnonsymbolic displays of numerosity, exhibit ex-cellent performance even in subtle tests of com-parison, addition, and subtraction (Barth et al.2006). This early ability points to a particularcerebral substrate, to which we now turn.

Cerebral Networksof Number Sense

Calculation and the Intraparietal Area

The first imaging studies of calculation, usingsingle photon emission computerized tomogra-

phy (SPECT), positron emission tomography(PET), and functional magnetic resonanceimaging (fMRI), quickly pointed to a remark-able fact: whenever adults calculate, a re-producible bilateral parietal activation is ob-served (Appolonio et al. 1994; Dehaene et al.1996; Roland & Friberg 1985). The use ofsingle-subject fMRI demonstrated that, al-though interindividual variability is somewhatlarger than in studies of reading or face per-ception, the horizontal segment of the intra-parietal sulcus (hIPS) is always consistentlyactivated whenever adults compute simplecomparison, addition, subtraction, or multipli-cation with Arabic numerals (Chochon, Co-hen, van de Moortele, et al. 1999). Figure 3shows the location of this region relative toother landmark areas of the human parietallobe.

Initial studies probed the exact nature of thecontribution of this region to the processingof numbers presented as Arabic numerals or asnumber words. Intraparietal activation was ob-served during a great variety of number-relatedtasks, including calculation but also larger–smaller comparison (Pinel, Dehaene, Riviere,et al. 2001) or even the mere detection of a digitamong colors and letters (Eger, Sterzer, Russ,et al. 2003). The intraparietal region seems tobe associated with an abstract, amodal repre-sentation of numbers inasmuch as it can beactivated by numbers presented in various cul-turally learned symbolic notations such as Ara-bic numerals and spelled-out or spoken num-ber words (Eger et al. 2003). Similar resultshave been consistently observed in experimentswith adults from various countries and culturesincluding France, UK, USA, Austria, Singa-pore, China, and Japan. In a direct compar-ison of Chinese and English speakers, Tanget al. (2006) observed intraparietal activationat a similar location in the IPS during calcula-tion and comparison tasks. They did howeverobserve cultural variation in other surroundingareas, particularly in left premotor cortex (moreactive in Chinese subjects) and left perisylvianareas (more active in English subjects).


Figure 3. Anatomical and functional organization of human parietal lobe areas. (A) Activations reportedin a study of six parietal functions in the same subjects (calculation, language, saccades, attention, pointing,and grasping tasks) (for details, see Simon et al., 2004; Simon et al., 2002). Calculation activates thehorizontal segment of the intraparietal sulcus (hIPS, shown in red). In all subjects, the hIPS lies posterior toan area activated by grasping movements, thought to be a plausible human homolog of monkey anteriorintraparietal area AIP (hAIP, shown in green). The hIPS is also anterior to a set of areas activated by saccadiceye movements (shown in blue, pink, and purple), one of which might be the human homolog of monkeylateral intraparietal area (hLIP). (B) Finer-grained study of the relationship between calculation, saccades,and multisensory motion in humans (Hubbard, Pinel, Jobert, et al. 2008). In macaque monkeys, tactile facialmotion, and visual flow fields activate neurons in the ventral intraparietal area (VIP), where neurons tuned tonumber are also found. The same stimuli identify a plausible human homolog of VIP (hVIP), which overlapspartially with the hIPS activation observed during symbolic calculation, but is consistently mesial to it.

Linguistic and Nonlinguistic Networksfor Arithmetic

It is likely that some of these additional, ex-traparietal activations relate to spoken or writ-ten language networks that are thought to sup-plement the core intraparietal system in orderto perform exact symbol-based calculations orto retrieve arithmetic facts from linguistic mem-ory. fMRI studies have indeed demonstratedthat the intraparietal sulcus is most activatedduring tasks that call upon quantity manip-ulations, particularly approximation of addi-

tions or subtractions, whereas another set ofareas involving the left angular gyrus and/orsurrounding perisylvian cortices shows greateractivation during operations of exact calcula-tion that depend on explicit education and of-ten rely on language-specific rote memorizing(Dehaene, Spelke, Pinel, et al. 1999; Lee 2000;Venkatraman, Siong, Chee, & Ansari 2006;Zago et al. 2008). During training with a givenset of arithmetic facts, activation progressivelyshifts from the intraparietal region to the an-gular gyrus as subjects commit these facts toverbal memory (Delazer et al. 2003; Ischebeck


et al. 2006). Altogether, therefore, those resultsmesh well with the notion of a core system ofnumber, associated with the bilateral intrapari-etal cortex and invariable across culture andeducation, and a distinct perisylvian circuit as-sociated with language- and education-specificstrategies for storing and retrieving arithmeticfacts (Dehaene & Cohen 1995). Double disso-ciations following brain lesions further supportthis basic distinction: focal intraparietal lesionscan cause drastic impairments in number sense,often affecting operations as simple as addition,subtraction, comparison or numerosity estima-tion, while lesions to perisylvian cortices or leftbasal ganglia generally impact on rote opera-tions such as the multiplication table (Cipolotti,Butterworth, & Denes 1991; Lee 2000; Lemer,Dehaene, Spelke, et al. 2003).

Instrumental in characterizing the role ofthe intraparietal sulcus in core number sensehave been neuroimaging studies that relied onnonsymbolic presentations of number as setsof dots or as series of tones (Castelli, Glaser, &Butterworth 2006; Piazza, Izard, Pinel, et al.2004; Piazza, Mechelli, Price, et al. 2006;Piazza, Pinel, Le Bihan, et al. 2007) Attendingto the numerosity of such stimuli suffices toinduce a strong bilateral activation of the IPS(Castelli et al. 2006; Piazza et al. 2006). Evenpassively looking at a set of dots suffices toencode its numerosity and adapt to it, so thatthe intraparietal sulcus later shows a reboundfMRI response when the number is changedby a sufficient amount (Piazza et al. 2004).This fMRI adaptation method has also beenused to demonstrate a convergence of symbolicand nonsymbolic presentations of numberstoward a common representation of quantityin the IPS (Piazza et al. 2007).

Early Parietal Response to Numberin Infancy

When in development does the parietal cor-tex first begin to respond to number? fMRIand event-related potentials (ERPs) have shownthat number-related parietal activations, par-

ticularly in the right hemisphere, are alreadypresent in 4-year-old children as they attendto the numerosity of sets (Cantlon, Brannon,Carter, et al. 2006; Temple & Posner 1998).Thus, the parietal mechanism of numerosityextraction seems to be already functional priorto arithmetic education in humans. As notedabove, behavioral studies of numerical discrim-ination imply the presence of a functional nu-merosity processing system at an even ear-lier age, during infancy. Indeed, Berger et al.(2006) measured ERPs in 7-month-old infantsas they viewed short movies depicting correctand incorrect nonsymbolic arithmetic opera-tions (1 + 1 = 2 vs. 1 + 1 = 1). The de-tection of these arithmetic violations was ac-companied by a clear negativity recorded overanterior electrodes. This reaction was similarto the classical error-related negativity, thoughtto arise from the anterior cingulate cortex,which indexes error detection and correction.Presumably thus, this experiment reflected nu-merical competence only indirectly, without di-rectly pinpointing to the cortical origins of thiscompetence.

To visualize more directly the brain’s re-sponses to number in infants, we recordedevent-related potentials from 3-month-old in-fants, while they were presented with a con-tinuous stream of sets of objects (Izard et al.2008). Within a given run, most sets had thesame numerosity and were made of the sameobjects (e.g., repeatedly presented various im-ages of three ducks). However, occasionally atest image would appear that could differ fromthe habituation images in either number, ob-ject identity, or both. Across different groupsof subjects, the numbers involved ranged from2 versus 3 to 4 versus 8 or 4 versus 12. Inall cases, event-related potentials revealed thatthe infants’ brain detected the two types ofchanges (number vs. object), yet with differentunderlying circuitry. Source reconstruction, us-ing an accurate model of the infant’s corticalfolds, suggested that the right parietal cortexresponded to numerical novelty, while the leftoccipitotemporal cortex responded to object


novelty. This ventral/dorsal double dissocia-tion is similar to what has been observed inadults and 4-year-olds (Cantlon et al. 2006;Piazza et al. 2004): fMRI adaptation showsthat the fusiform gyrus reacts to changes inobject identity but not in number, and whilethe parietal regions react to changes in num-ber but not in object identity. Thus, the resultssuggest that the well-known ventral–dorsal dis-sociation between object identity (“what”), ver-sus object location, size, and motor affordance(“where” and “how”) (Goodale & Milner 1992)may already be in place at 4 months of age,and that number belongs to the parametersthat are quickly extracted by the dorsal parietalpathway even in infants.

Specificity of the Intraparietal Region

An important and only partially resolved is-sue concerns whether any part of the intra-parietal is uniquely specialized for number. Inthe present state of knowledge, the answer is anuanced one. The intraparietal sulcus clearlycontains a specialized subsystem for numberin the sense that its activation during calcula-tion cannot be reduced to simpler sensorimotorfunctions such as attention or response plan-ning. However, there does not seem to be a sin-gle, isolated piece of cortex that responds solelyto number—parameters of object size and lo-cation also seem to be coded by intermingledneuronal circuits distributed within the samegeneral area of IPS.

As far as the first point is concerned, theparietal activation putatively associated withcore “number sense” occupies a fixed locationrelative to other parietal areas involved insensory, motor, and attentional functions (seeFig. 3). Our group used fMRI to study thecerebral organization of six different functionspreviously associated with parietal systems:finger pointing, manual grasping, visualattention orienting, eye movement, writtenword processing, and calculation (Simon et al.2004; Simon, Mangin, Cohen, et al. 2002). Allsubjects showed a reproducible geometrical

layout of activations associated with these func-tions. Most notably, activation uniquely evokedby calculation was observed in the depth ofthe intraparietal sulcus and was surroundedby a systematic front-to-back arrangement ofactivation associated with grasping, pointing,attention, eye movement, and language-relatedactivations. The systematicity of this organi-zation was confirmed by subsequent researchusing more selective experiments specificallydesigned to isolate grasping, saccade, andattention-related functions (for review, seeCulham, Cavina-Pratesi, & Singhal 2006).

As concerns the second point, several studieshave now contrasted intraparietal activationsduring judgments of number versus other con-tinuous dimensions such as physical size, loca-tion, angle, and luminance (Cohen Kadosh &Henik 2006; Cohen Kadosh et al. 2005; Fias,Lammertyn, Reynvoet, et al. 2003; Kaufmannet al. 2005; Pinel, Piazza, Le Bihan, et al. 2004;Zago et al. 2008). The results indicate that IPSactivations do not cluster neatly into distinctregions specific for a given quantitative param-eter. Rather, the activations show considerableoverlap. This overlap is particularly strong fornumber and location (Zago et al. 2008) and fornumber and size (Kaufmann et al. 2005; Pinelet al. 2004), although a partial subspecializa-tion for number processing is occasionally re-ported (Cohen Kadosh et al. 2005). There isalso considerable overlap between the activa-tions induced number and letter comparisons(Fias, Lammertyn, Caessens, et al. 2007).

Two interpretations of these overlapping ac-tivations have been proposed. Several authorshave proposed that the intraparietal sulcus ac-tivation during arithmetic reflects a generalfunction that is not specific to numbers (Fiaset al. 2007; Shuman & Kanwisher 2004; Walsh2003). Vincent Walsh, for instance, proposesthat the parietal lobe contributes to a “gener-alized magnitude system,” which encompassesrepresentations of space, time, and number. Al-ternatively, and while not denying that mag-nitude manipulation and transformation maybe an overarching function of parietal-lobe


areas, my colleagues and I have proposed thatthe overlapping activation need not reflect alack of neural specialization. Rather, analogquantities such as number, location, size, lu-minance, or time may well be coded by neu-ronal assemblies that are specialized, yet inter-mixed within the same voxels (Pinel et al. 2004).Under this interpretation, it may be wrongto generalize to parietal cortex the model ofextreme specialization and discrete “modules”with sharp boundaries that emerged from stud-ies of the fusiform face responses (Kanwisher,McDermott, & Chun 1997; Tsao, Freiwald,Tootell, et al. 2006). Rather, neuronal popu-lations coding for number would be highly dis-tributed in the intraparietal sulcus and wouldbe intertwined and overlapping with represen-tations of other quantitative parameters. Al-though the specialization debate is still farfrom being settled, as we shall now see, recentmonkey neurophysiology nicely supports theconcept of overlapping but specialized neu-ral populations (Tudusciuc & Nieder 2007, seebelow).

Neural Codes for Numberin the Macaque Monkey

Intraparietal Neurons Tunedto Numerosity

Human neuroimaging studies predicted thatif a precursor of human numerical abilitiesexisted in monkeys it might lie in the depthof the intraparietal sulcus, anterior to regionsinvolved in memorized eye movements (plau-sibly relating to monkey lateral intraparietalarea LIP) and posterior to regions involved ingrasping objects (plausibly relating to monkeyanterior intraparietal area AIP) (Simon et al.2002, see Fig. 3). Indeed, in the same year,two independent groups of electrophysiologistsworking in the awake macaque monkey iden-tified number-coding neurons within and nearthe intraparietal sulcus (Nieder, Freedman, &Miller 2002; Nieder & Miller 2004; Sawamura,Shima, & Tanji 2002). Although similar neu-rons were also found in the prefrontal cor-

tex, PFC neurons responded with a longer la-tency and showed greater delay-related activity,suggesting that the parietal neurons constitutea primary numerosity code, which prefrontalneurons held on-line during the delayed match-to-sample task.

The monkey intraparietal neural code fornumerosity may be the evolutionary precursoronto which the human invention of arithmeticencroached. First, the analogy in cerebral lo-cation is striking. Numerosity-tuned neuronsare mostly found in the depth of the intra-parietal sulcus and often show visual flow-field responses, compatible with a location inventral intraparietal area VIP (Tudusciuc &Nieder 2007). Likewise, human fMRI stud-ies have located a plausible homolog of areaVIP (Bremmer et al. 2001; Sereno & Huang2006) at a location close to and overlappingwith that of number-related responses (seeHubbard, Piazza, Pinel, et al. 2005). Recentlyour group further attempted to clarify the ho-mologies between numerical processing regionsand macaque areas LIP and VIP (Hubbardet al., submitted). To explore this question, weused physiologically inspired localizer tasks ofsaccadic eye movement, visual flow fields, andtactile face stimulation to identify plausible hu-man homologs of LIP and VIP (see Fig. 3). Wethen examined the topographical relations be-tween these activations and calculation-relatedactivations. We also measured voxel-by-voxelcorrelations between these spots of activation.The results indicate that the hIPS activationduring symbolic calculation overlaps partiallywith the human homolog of area VIP (thoughit is generally more lateral), suggesting a par-tial homology between human and macaqueparietal regions (see Fig. 3).

The profile of response of these neurons alsofits with the code for numerosity inferred fromhuman studies. The neurons studied by Niederand Miller are tuned to a particular numeros-ity (Nieder & Merten 2007; Nieder & Miller2003, 2004). For instance it is possible to findsome neurons that show a peak of firing fornumerosity 1, another set for numerosity 2,


3, . . . all the way to numbers in the 30s. Fur-thermore, the tuning curves of these neuronsshow Gaussian variability on a log scale. Forinstance, neurons that show a firing peak whenfour objects are present also respond to threeor five objects, and much less to one or to 10objects. Collectively, these neurons form a dis-tributed representation subject to Weber’s law:each number is not coded in an exact manner,but only approximately and with an impreci-sion that increases linearly with number. Thelog-Gaussian neural code that was identified byNieder and Miller in the macaque monkey isidentical to the representation thought to un-derlie numerical judgments in human subjects,as derived either from behavioral research, asmentioned earlier, or from fMRI adaptationstudies (Piazza et al. 2004, 2007).

Recent studies from Andreas Nieder’s labo-ratory have further clarified the extent to whichthis code is abstract and specific to the numberdomain. Initially studies used only visual setsof simultaneously presented objects, but recentresearch has shown that, in some neurons atleast, the code is abstract enough to respondto both sequential and simultaneous presen-tations of number (Nieder, Diester, & Tudus-ciuc 2006). Furthermore, at this single-neuronlevel, the code appears to be specific for num-ber. Number neurons maintain their numeri-cal selectivity in the face of considerable vari-ation in the size, location, and nature of theobjects in the set (Nieder et al. 2002). Con-versely, distinct populations of neurons code forline length (Tudusciuc & Nieder 2007). Thesetwo types of neurons are intermingled withinthe same areas of the IPS, and can some-times be recorded under the same electrodetip. Again, this distributed overlapping organi-zation of multiple neural codes for quantitativedimensions is very similar to that inferred fromhuman fMRI (Pinel et al. 2004).

Multiple Codes for Number

Recently, Roitman et al. (2007) uncoveredanother type of neural code for numerosity in

area LIP. Neurons in LIP behave differentlyfrom the IPS neurons discovered by Niederand Miller in several respects. First, LIP neu-rons are not tuned to number. Rather, theirfiring rate varies monotonically with numeros-ity, either increasing or decreasing sharply withthe logarithm of the number of objects in theneuron’s receptive field. Second, these neuronshave limited receptive fields and thus respondsolely to the local numerosity within a certainretinotopic area, not to the total numerosityacross the whole visual field.

Why would two quite distinct codes—monotonic versus tuned cells—coexist in thesame individual? One possibility is that themonotonic cells are needed in order to computethe tuned-cell representation, so that mono-tonic and tuned codes would constitute twodistinct stages in the computation of an in-variant numerical representation. Jean-PierreChangeux and I have presented a theoreti-cal model of how numerosity can be extractedfrom visual displays (Dehaene & Changeux1993), later elaborated by others (Verguts &Fias 2004; Verguts, Fias, & Stevens 2005). Ourmodel illustrates how approximate numeros-ity can be extracted from a retinotopic mapthrough three successive stages: (1) retinotopiccoding of object locations regardless of objectidentity and size, thus yielding a normalized,constant amount of activation for each ob-ject; (2) summing of the activation on the ob-ject location map, thus yielding a representa-tion of approximate numerosity by accumula-tion neurons; (3) thresholding of the activationin accumulation neurons by neurons with in-creasingly higher thresholds, yielding a bankof numerosity detector neurons each tuned toa specific numerosity. With only small mod-ifications, the accumulation neurons may beidentified with the LIP monotonic cells, andthe numerosity detector cells with the VIPtuned cells. Anatomically, indeed, LIP neu-rons project directly to VIP. Furthermore, LIPnumber neurons have receptive fields, whereasVIP number neurons seem to respond to thenumerosity of the whole display, consistent


with their receiving inputs from many LIPneurons.

Thus, the Dehaene–Changeux model hasplausibility, but considerable neurophysiologi-cal work will be needed to verify its key hy-potheses about the role of LIP–VIP circuitry ininvariant numerosity extraction. An importantlimit of the present data, which should be bornein mind, is that at present the two types of nu-merical codes (monotonic and tuned cells) havebeen found by different labs, in different areas,in different monkeys trained to perform differ-ent tasks. Thus, it is not known whether thesetwo codes co-exist within the same animals.

It is interesting to note, however, that theLIP numerosity code uncovered in monkeysby Roitman et al. (2007) has all the propertiesneeded to account for the recently discoveredhuman numerosity adaptation effect discussedin the introduction (Burr & Ross 2008). Hu-man numerosity adaptation is retinotopic, andit extends to across a large range of numbers:adaptation to 30 dots changes the perceptionof 400 dots, which would be impossible if num-ber was encoded by cells tuned to these spe-cific quantities, but makes sense if the adaptedrepresentation is a monotonic code. Thus, itis likely that humans also possess a monotonicnumerosity code.

Roggeman et al. (2006) also presented prim-ing evidence in support of the co-existence oftwo number codes in humans, compatible withthe notion of monotonic versus tuned cells.They asked subjects to name, as quickly as pos-sible, Arabic numerals or numerosities of sets ofdots. In both cases, numbers ranged from 1 to5, and each target was preceded by a brief butvisible prime (83 ms), also from the same range,masked by a random-line pattern (49 ms). ForArabic numerals, priming depending symmet-rically on the distance between the numbers,as previously observed with purely subliminalstimuli (Naccache & Dehaene 2001; Reynvoet,Brysbaert, & Fias 2002). For numerosities pre-sented as sets of dots, however, priming showedan asymmetrical step function: nonsymbolicprimes did not just facilitate the nearby num-

bers but in fact caused priming for all smallernumbers. This finding seems consistent with amonotonic code, with the additional specificproperty that increasing numbers of neuronsare recruited as the number of items increases(Roggeman et al. 2006 term this a summationcode). Very interestingly, these results suggestthat the dominant code, which drives behav-ioral priming in humans, differs for symbolicversus nonsymbolic numbers: the tuned-cellcode would be used for symbolic numerals,and some version of the monotonic cell codefor numerosities presented as sets of objects.It remains to be seen, however, whether thesepriming results are not just due to counting pro-cesses deployed for small numbers, and howthey can be reconciled with the fMRI evidencefor tuned-cell priming with larger numerositiesof sets of dots (Piazza et al. 2004).

Subitizing and Estimation

It is likely, indeed, that small numbers receivespecial treatment, implying that there must beyet other codes for number that underlie ourstrong arithmetic intuitions. For more than acentury now, the small numbers 1, 2, and 3have been thought to involve a partially distinctrepresentational subsystem (Bourdon 1908;Feigenson et al. 2004; Jevons 1871). In humanadults, the phenomenon of subitizing refers tothe fact that these small numerosities 1, 2, and3, when presented as sets of dots, can be veryquickly and accurately identified and named,without the need for a deployment of serialattention in order to count (Piazza, Giacomini,Le Bihan, et al. 2003). Until recently, it waspossible to maintain that this performance wasperhaps due simply to the lower end of the es-timation range—with small numbers, Weber’slaw would ensure an excellent discriminationof 1 from 2 and 2 from 3, thus resulting in anapparent subitizing effect. This interpretationwas strengthened by Nieder and Miller’sobservation that, in the monkey IPS, allnumerosities in the range 1–30 are seamlesslycoded by cells with log-Gaussian tuning, and


that this coding scheme, which does not exhibitany discontinuity for small numbers, sufficesto explain the monkey’s behavior in numericalsame–different tasks (Nieder & Miller 2003).

My laboratory, however, recently tested andrejected this interpretation of the humansubitizing effect in a recent behavioral study(Revkin, Piazza, Izard, et al. 2008). We used amasked forced-choice paradigm in which par-ticipants named the numerosity of sets with ei-ther 1–8 or 10–80 items, matched for discrim-ination difficulty. The first case corresponds tothe classical subitizing task. The second case isanalogous in all respects, except that the num-bers are all multiplied by 10. If Weber’s lawheld, then participants should behave identi-cally in the ranges 1–8 and 10–80, that is, theyshould have no difficulty in “subitizing” 10, 20,or 30 dots (once they know that only decadenumbers are presented). The results, however,showed a clear violation of Weber’s law, witha much higher precision over numerosities 1–4in comparison to 10–40. These results argueagainst the single estimation system hypothesisand support the notion of a dedicated mecha-nism for apprehending small numerosities—aconclusion similar to that reached by devel-opmental studies of infant numerical abilities(Feigenson et al. 2004). Although we currentlyhave very little idea of how this system is or-ganized at the neural level, it seems clear thata very quick and automatic grasp of the nu-merosities 1, 2, and 3 is part of the humanintuition of numbers.

The Sense of Ordinality

A final aspect of number representation thatseems to rely on a partially dissociable systemis the sense of ordinality—the knowledge ofwhich item in a fixed series comes first, sec-ond, third, and so on. Elegant experiments byTurconi and Seron have demonstrated that theordinality and cardinality meanings of num-bers, although often associated, are not synony-mous to human subjects (Turconi, Campbell,& Seron 2006; Turconi, Jemel, Rossion, et al.

2004; Turconi & Seron 2002). Surprisingly,judging whether 2 is smaller than 5 is a dif-ferent task than judging whether 2 comes be-fore 5. The function that relates response timesto numerical distance is distinct: it shows aclassical distance for cardinality judgements(faster RTs with larger distances), but a par-tially reversed effect with ordinal judgements(fast RTs to consecutive pairs such as 4 and 5)(Turconi et al. 2006). Event-related potentialsare also subtly different (Turconi et al. 2004),and knowledge of ordinality and cardinality,although often associated in brain-lesioned pa-tients (Cipolotti et al. 1991), can be dissociatedin Gerstmann’s syndrome following a left pari-etal lesion (Turconi & Seron 2002).

Few studies to date have attempted to iden-tify the cerebral bases of ordinal knowledge, butthe available evidence suggests that ordinal andcardinal knowledge may involve partially dif-ferent neural circuits within overlapping pari-etofrontal areas. Number comparison, whichinvolves the cardinal meaning of numbers,and letter comparison, which involves ordinalknowledge, activate very similar parietofrontalnetworks in humans (Fias et al. 2007). Further-more, during ordinal comparisons, the left in-traparietal area shows a distance effect (smalleractivation for more distant rank orders) whichis very similar to that observed during num-ber comparison (Marshuetz, Reuter-Lorenz,Smith, et al. 2006).

While the same overall regions are involved,recordings of single neurons in the IPS ofmacaque monkeys suggest that the fine-grainedcircuits are partially different (Nieder et al.2006). When monkeys received either paral-lel or serial presentations of sets of dots, specificneurons tracked rank order during serial pre-sentation, but very few of these neurons showeda joint sensitivity to the numerosity of simulta-neously presented sets of dots. However, acti-vation ultimately converged toward the sameneural population for serial versus parallel pre-sentations at the end of the sequence, when thefinal number was known and had to be heldin working memory. Many other studies have


recorded from neurons sensitive to rank orderin a variety of prefrontal, cingulate, frontal eyefield, or caudate areas (for review, see Nieder2005). For instance, Jean-Paul Joseph and hiscollaborators have repeatedly reported neuronsthat fire either to the first, the second, or thethird action in a motor sequence, regardless ofthe particular action being performed (Procyk,Tanaka, & Joseph 2000). It remains unknownwhether these same neurons would also be in-volved in cardinal number representation. Allin all it seems that one must add ordinal repre-sentations to the list of number-relevant neuralcodes that are available to humans and otherprimates.

From Number Neuronsto Arithmetic Intuitions

The Nieder–Miller studies suggest that car-dinal number is encoded by a bank of neuronsin area VIP, each tuned to a particular number.Furthermore, these studies are precise enoughto quantitatively characterize, in a mathemat-ical manner, the nature of the code. The fir-ing rate of a numerosity-sensitive neuron thatresponds preferentially to numerosity p, in re-sponse to a range of stimulus numerosities n,traces a bell-shaped curve which is Gaussianon a log scale and has a maximal firing peakat the location p. All neurons seem to have asimilar width of tuning (once plotted on a logscale)–there seems to be a single neural Weberfraction that defines the degree of coarsenesswith which neurons encode numerosity.

This code, which I term log-Gaussian coding,is remarkably similar to that postulated in theDehaene and Changeux (1993) neural networkmodel of numerosity processing. fMRI suggeststhat a very similar code is available in humans,in a plausibly homologous area (Piazza et al.2004). I recently developed a detailed mathe-matical theory of how humans take decisions innumber-related tasks, based on this underlyinglog-Gaussian code (Dehaene 2007). Assumingthat this is the main neuronal code underlying

our sense of number, we can reconstruct, ina mathematical manner, most if not all of theproperties of our intuitions of number duringsimple numerical cognition tasks.

Numerosity Discriminationand Comparison

The first and simplest set of tasks consistsin deciding whether a certain number, pre-sented as a set of dots, is equal to, largerthan or smaller than a fixed numerical refer-ence. The assumption that numerosity is rep-resented by a random, Gaussian distribution ofactivation over an internal continuum (in thiscase the log of the input number) brings nu-merosity judgement within the realm of classi-cal psychophysics and signal detection theory.It readily predicts that discrimination of twonumerosities, as measured by d-prime, shouldimprove linearly with distance on the logarith-mic internal continuum (i.e., with the logarithmof the ratio of the two numbers), with the slopeof that effect reflecting the Weber fraction (i.e.,the degree of precision of the numerosity repre-sentation). Van Oeffelen and Vos (1982) testedthis model against human numerosity discrim-ination data, and found it to be quite accurate.Manuela Piazza and I also verified this modelin human adults during larger/smaller andsame/different numerosity judgements (Piazzaet al. 2004). Interestingly, the precision of dis-crimination varies during human development.Ratio-dependent performance has been ob-served during numerosity discrimination in hu-man infants as early as 6 months of age (Lipton& Spelke 2003), but the Weber fraction ap-pears to decrease with age: 6-month-old ba-bies discriminate numerosities in a 2:1 ratio(Weber fraction of 1.0), but fail to discriminatenumerosities in a 3:2 ratio (Weber fraction 0.5),while 9-month-old babies can (Lipton & Spelke2003). Adults can discriminate numbers thatare within 10%–15% of each other (Halberda& Feigenson 2008; Piazza et al. 2004; Pica et al.2004). A recent study indicates that between 3and 6 years of age, the Weber fraction decreases


smoothly with age, yet without achieving adultlevels (Halberda & Feigenson 2008).

Recently, with Manuela Piazza and MarcoZorzi, we titrated the evolution of the Weberfraction using an identical numerosity compar-ison task in normally developing kindergart-ners, 10-year-olds, and young adults as well asdyslcalculic children (Piazza et al., submitted).The log-Gaussian model provided an excellentfit to performance curves at all ages. Note thatfor kindergartners, the task was very difficultbecause the numbers to be compared were al-ways within 25% of each other, and hence per-formance never exceeded 80% correct, evenat the largest numerical distances. In this task,indeed, as in many psychophysical tasks, partic-ipants often have an impression of respondingusing only their “intuition,” but performancecurves demonstrate that this intuition is not atall random and can be captured in great detailby our knowledge of numerosity coding in theparietal lobe. Furthermore, the Weber fractiondecreased exponentially from about 0.40 at age3 to 0.25 at age 10 and 0.15 in adults (resultsquantitatively similar to Halberda & Feigen-son 2008). Most importantly, it was higher indyscalculic children aged 8–12 years with se-vere deficits in arithmetic (mean Weber frac-tion = 0.34, comparable to kindergartners).Indeed, the Weber fraction predicted the sever-ity of these children’s impairment in sym-bolic arithmetic tasks, but not in other do-mains such as word reading. Similar results byHalberda and Feigenson (2008) confirm thatthe psychophysics of numerosity discriminationcan be a sensitive quantitative estimator of nu-merical understanding across development.

Response Times in Number Comparison

Signal detection theory is an idealized de-scription and does not take into account thetime taken to reach a decision. To simulta-neously model decision times and error rates,more sophisticated models of accumulationof evidence have been proposed (for reviews,see Gold & Shadlen 2002; Link 1992; Smith

& Ratcliff 2004; Usher & McClelland 2001).These models assume that, in order to reach adecision, the brain needs to sequentially accu-mulate evidence toward each of the availableresponse alternatives. Due to the inescapablepresence of sources of noise in the nervous sys-tem (either in the input, in its internal repre-sentation by noisy neurons, or in the decisionmechanism itself), the accumulated evidencewill vary in time as well as from trial to trial,forming a “random walk.” Most current mod-els assume that a firm decision is reached, andthe motor response is launched, only once theaccumulated evidence in favor of one of thealternatives has reached a critical threshold.

Assuming a log-Gaussian internal codingof number, it is possible to derive mathe-matical equations relating response time, er-ror rate, and numerical distance in simplenumerical tasks of larger/smaller comparisonand same/different judgements (for details andequations, see Dehaene 2007). As already notedby Link (1975, 1990, 1992), the accumulationof evidence or diffusion-to-bound model pro-vides a compact account of many aspects of thenumber comparison data. For error rates, themodel predicts a logistic function of distanceon the internal continuum, which is virtuallyindistinguishable from the error function (inte-gral of a Gaussian) predicted by signal detectiontheory. For response times, however, the modelcorrectly predicts a dependency on numericaldistance that is less peaked than the error curveand is well approximated by an inverse func-tion. The model also predicts a specific linearrelation between response times and a trans-formed function of error rates.

For the sake of illustration, Figure 4 plotsdata from an experiment by Cantlon and Bran-non (2006), in which the very same behav-ioral task of numerosity comparison was ap-plied to macaque monkeys and adult humans.Two monkeys and 11 human subjects were pre-sented with arrays of dots ranging from 2 to 30dots, and had to select the smaller array or, inother blocks, the larger array. Figure 4 showsthe averaged data and fits of the accumulation


Figure 4. Mathematical modeling of responsetimes and error rates during a numerosity compari-son task (data from Cantlon & Brannon 2006). Forboth monkeys and humans, panels illustrate the de-pendency of RTs (top left) and errors (top right)on distance between the numbers, measured bythe log of their ratio (i.e., the distance betweentheir logarithmic internal representations). The bottom

of evidence model. The accumulation of ev-idence model has only three free parameters(nondecision time, slope of accumulation as afunction of the difference in logs of the twonumbers, and decision boundary). Althoughthe quantitative fit is not perfect, this modelclearly suffices to capture many qualitative fea-tures of performance, including the shape ofRTs, error rates, and their interrelations.

It should also be noted that, with only oneadditional hypothesis, the accumulation of evi-dence model can also account for how perfor-mance changes when the number-comparisontask is performed concurrently with anothertask in a dual-task or “psychological refractoryperiod” setting. The only new hypothesis is thatthe decision stage creates a serial bottleneckin processing, while predecision (sensory) andpostdecision (motor) stages can operate in par-allel for the two tasks (for details, see Sigman &Dehaene 2005, 2006).

Numerosity Labeling and Naming

A more complex task consists in asking sub-jects to label numerosities using a set of sym-bols, either Arabic numerals or number words.For instance, one may ask human subjects toname sets of dots ranging from 10 to 100 withround numbers such as the decade names “ten”to “ninety” (Izard 2005). As exemplified inFigures 1 and 2, this is not necessarily an easytask, but once again, subjects are ready to ven-ture an “intuitive” response, which turns out tobe tightly correlated with the actual numeros-ity. Even chimpanzees can be trained to labelnumerosities using the Arabic digits 1 through

←−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−left panel shows the relation between mean RT andtransformed error rates predicted by the accumula-tion of evidence model (for details, see Dehaene2007; Link 1992). The bottom right panel showshow the amount of evidence accumulated at eachtime step, inferred from a fit of the accumulation ofevidence model, varies linearly with log ratio, aswas expected according the log-Gaussian codingscheme.


9 (Matsuzawa 1985; Tomonaga & Matsuzawa2002).

The theory for such numerosity labelingtasks, developed by Veronique Izard (2005;Izard & Dehaene 2008), assumes that sub-jects simply map segments of their internallog-Gaussian continuum onto the requested re-sponse labels. Humans, of course, can deploymore complex counting strategies, but thesecan be discouraged by fast stimulus presenta-tion times and avoidance of very small num-bers. In animals, and perhaps also in humancultures with few number words and no overtcounting system, the assumption that subjectsdo not count serially, but merely apply symboliclabels to their mental representations of ap-proximate numerosity, seems realistic (Dehaene& Mehler 1992; Gordon 2004; Pica et al. 2004;Tomonaga & Matsuzawa 2002). The theorytherefore assumes that, for each target nu-merosity n, subjects generate an internal dis-tribution of activation on the log-Gaussian in-ternal continuum and respond with the verballabel r whose position on the number line Log(r)

falls closest to Log(n). This strategy implies thatthe number line continuum is divided into dis-tinct response domains according to a set ofresponse criteria forming a response grid.

One difficulty is that subjects’ responses areoften poorly calibrated and underestimate thetrue numerosity. For instance, as you may judgefor yourself in Figure 1, it is quite commonto respond “fifty” to a set of 100 or 200 dots(Krueger 1982; Minturn & Reese 1951). Whilethe origins of this illusion remain unknown,Veronique Izard showed that it could be cap-tured by a simple assumption: subjects use anaffine rescaling of their response grid, referringto location a Log(r)+b instead of Log(r) for re-sponse r. The above model provided a remark-ably good fit to human subjects’ numerositynaming data in the range 10–100 (Izard & De-haene 2008). All subjects were initially miscal-ibrated and severely underestimated numeros-ity. In those cases, the model predicts a powerlaw relation between the presented numerosityand the subject’s mean response, which is ex-

actly what was observed. A single example of anumerosity–name pairing was sufficient to re-calibrate the responses to a quasilinear relation,a process that was well captured by a change inthe parameters a and b.

The numerosity labeling model supposes theexistence of a rapid association process that canquickly convert a quantity to the correspond-ing verbal or Arabic symbol, or vice versa. In-deed, there is now considerable evidence thatsuch a conversion process exists and can beextremely rapid and automatic (Piazza et al.2007). Perhaps most relevant to the concept ofarithmetical intuition, there is evidence that ac-cess to the quantity representation from num-ber symbols is sufficiently efficient to occurwithout consciousness. A flashed digit or num-ber word, masked by letter or symbol stringsthat make it subjectively invisible, can never-theless prime a subsequent visible digit dur-ing a number comparison task (Dehaene et al.1998). In this paradigm, the bilateral IPS showsfMRI repetition suppression when the samenumber is presented twice, even when the in-put notation differs (e.g., “ONE” followed by1) (Naccache & Dehaene 2001). Behavioralstudies show that priming depends monoton-ically on the numerical distance between theprime and the target, being larger for the pairONE→1 than for TWO→1, THREE→1, orFOUR→1 (Reynvoet et al. 2002). Altogether,these findings suggest that the parietal quan-tity representation can be contacted noncon-sciously by the mere sight of a digit or numberword—there is, thus, a form of “second-order”intuition for number symbols, which is createdby associating these arbitrary labels to mean-ingful quantities.

Changes in Arithmetic IntuitionsInduced by the Acquisition of

Number Symbols

Does the acquisition of expertise with num-ber symbols solely consist in the laying downof a mapping with preexisting representationsof quantity? A recent neural network of the


numerosity labeling process suggests otherwiseand implies that symbol acquisition may lead todeeper changes in the quantity system (Verguts& Fias 2004). Verguts and Fias used nonsuper-vised learning in a network exposed either tonumerosity information alone or to numeros-ity paired with an approximate symbol. Whennonsymbolic information alone was presented,the network developed numerosity detectorssimilar to Nieder and Miller’s neurons: theyexhibited tuning curves that had a Gaussianshape with fixed width when plotted on a log-arithmic numerosity axis. After pairing of thenonsymbolic numerosity inputs with symbolicinformation, the numerosity detector units be-came tuned to symbols as well, but with twokeys differences. First, the tuning curves weremuch sharper when symbolic inputs were pro-vided: the simulated neurons essentially had adiscrete peak of firing at their preferred value,with only a small distance effect for other num-bers. Second, the tuning curves now exhibited afixed width for all the numbers tested (1 through5). Thus, the network developed a new type ofrepresentation, linear with fixed variability.

Verguts and Fias’s proposal has the poten-tial to explain several aspects of human intu-ition for number symbols. Once the mappingis learned, there would be instantaneous trans-fer across nonsymbolic and symbolic formats ofinputs—explaining for instance that preschool-ers perform better than chance with additionsand subtractions of large numbers presented insymbolic form (Gilmore et al. 2007). Becausethe neurons’ tuning curve is narrower for sym-bolic than for nonsymbolic inputs, calculationwould be much more precise with the formerthan with the latter. For instance, errors in num-ber comparison would be much more frequentwith nonsymbolic than with symbolic stimuli,and performance should depend on the ratioof the numbers in the former case and on thelinear distance in the latter—two predictionsthat are upheld in actual data (for details, seeDehaene 2007).

Manuela Piazza, Philippe Pinel, and I re-cently used fMRI adaptation to test directly

the hypothesis of a common parietal code forsymbolic numerals and nonsymbolic numerosi-ties (Piazza et al. 2007). For a whole fMRI run,subjects attended to the repeated presentationof an approximate quantity presented either asa set of dots (e.g., 17, 18, or 19 dots) or as anArabic numeral (the numerals 17, 18, or 19).As expected, within intraparietal regions iso-lated using an independent subtraction task, thefMRI signal adapted over the course of about40 sec. We then introduced sparse deviants thatcould be close or far from the adaptation value(e.g., 20 or 50) and that appeared either in thesame or in a different notation. The intrapari-etal cortex signal showed a distance-dependentrecovery from adaptation which, crucially, wasobserved even when the numerical notationchanged from dots to Arabic and vice versa,indicating that there must be populations ofnumerosity detector neurons coding for numer-ical quantity independently of the symbolic ornonsymbolic format of input.

Interestingly however, in left parietal cortex,the effect was asymmetrical. When adaptationwas to dots and the deviants were Arabic nu-merals, there was recovery of adaptation tofar but not to close quantities. However, whenadaptation was to Arabic numerals and the de-viants were dots, there was recovery of adapta-tion to both close and far quantities (e.g., adap-tation to 17, 18, 19, recovery to both 20 and50). This finding suggests that the quantitiesevoked by Arabic numerals may be more pre-cise than those evoked by nonsymbolic sets ofdots. Hence, in the left hemisphere at least, theneuronal populations adapted by Arabic stim-uli may be narrower than those evoked by dotspresentations, as proposed in Verguts and Fias’s(2004) model. The right-hemispheric intrapari-etal region, perhaps because it is not in directinterconnection with left-hemispheric languageareas, seems to exhibit a much lesser influenceof exposition to number symbols, if any.

The idea that acquisition of number sym-bols improves the representation of numbers inthe left parietal lobe receives support from sev-eral sources. First, developmental fMRI studies


point to left parietal cortex as a crucial sitewhose activity changes during arithmetic de-velopment, compatible with the hypothesis thatthis region serves as a hub where abstract quan-tity information meets with left-hemisphericcodes for Arabic and verbal number symbols(Dehaene & Cohen 1995). Ansari and Dhital(2006) observed a greater effect of the distancebetween numbers during a comparison task inthis region in adults compared to 10-year-olds,suggesting an increasing involvement of this re-gion with increasing age. By imaging childrenof different ages during an identical arithmetictask, Rivera et al. (2005) observed increasingactivation with age in the left parietal andleft lateral occipitotemporal cortex, compatiblewith the hypothesis that these areas respectivelycome to encode numerical quantities and num-ber symbols with increasingly refined precision.

Language and arithmetic are jointly lateral-ized to the left hemisphere in the majority ofright-handed adults. The above developmentalmodel suggests that this “colateralization” re-flects a causal interaction during developmen-tal, with language lateralization preceding andcausing a progressively increasing lateraliza-tion of numerical representations in the parietallobe. Recently, Philippe Pinel and I put this the-ory to a test using a large database of fMRI dataduring spoken language processing and calcu-lation (Pinel et al. 2009). We designed a “colat-eralization analysis” over 209 healthy subjects,investigating whether variations in the degreeof left-hemispheric asymmetry that character-ize the brain organization for language are mir-rored in the asymmetry of areas involved innumber processing. As predicted, we observedthat the degree of asymmetry in the activationof the posterior superior temporal sulcus dur-ing a sentence-reading task correlates stronglyand selectively with the degree of asymmetry ofcalculation-induced activation in the intrapari-etal sulcus. This finding is compatible with thehypothesis that during childhood education,hemispheric asymmetries for language partiallyalter the organization of the brain networks forarithmetic.

Very recently, the issue of how cultural sym-bols modify the organization of the neural net-works for number has become addressable inthe monkey. Diester and Nieder (2007) pre-sented the first study of the cerebral mecha-nisms of number symbol acquisition in non-human primates. They trained two monkeyson a symbolic match-to-sample task that re-quired matching the shape of an Arabic nu-meral, ranging from 1 to 4, to the correspond-ing numerosity of a set of dots. In dorsolateralprefrontal cortex, Diester and Nieder foundmany neurons that coded for number indepen-dently of the symbolic or nonsymbolic notationused to convey it. Each neuron had a similartuning curve for number, regardless of the for-mat of presentation. Surprisingly however, inintraparietal cortex the vast majority of neuronswere specialized either for Arabic numerals orfor numerosities, but not both. Thus, only dor-solateral prefrontal cortex seemed capable ofencoding the arbitrary relation between sym-bols and their numerical meaning. In humans,considerable training may ultimately lead to atransfer to specialized posterior brain systems.In the Rivera et al. (2005) fMRI study of arith-metic development, indeed, a massive decreasewas observed in prefrontal activity as a func-tion of age, suggesting that the automatizationof mental arithmetic is accompanied by a pro-gressive transfer from anterior generic to pos-terior specialized circuits.

Linking Numbers to Space

A final aspect of human numerical intuitionthat seems to be heavily influenced by educa-tion and cultural background is the linking ofnumbers with space. This mapping plays anessential role in mathematics, from measure-ment and geometry to the study of irrationalnumbers, Cartesian coordinates, the real num-ber line, and the complex plane. In most hu-man adults, the mere presentation of an Arabicnumeral automatically elicits a spatial bias inboth motor responding and attention orienting


(Dehaene, Bossini, & Giraux 1993; Fischer,Castel, Dodd, et al. 2003; Hubbard et al. 2005;Zorzi, Priftis, & Umilta 2002): even when per-forming a task as simple as deciding whethera digit is odd or even, or whether it is largeror smaller than 5, small numbers are automati-cally mapped to the left side of space, and largenumbers to the right side Spatial-NumericalAssociation of Response Codes, or (SNARCeffect).

We recently reported a similar systematic in-terference pattern during mental arithmetic,which we termed “operational momentum,”and which suggests that a spatial code anda sense of motion are automatically activatedduring mental arithmetic (Knops, Viarouge,& Dehaene 2009; McCrink, Dehaene, &Dehaene-Lambertz 2007). When subjects wereshown two successive sets of dots and askedto approximate their sum or their difference,their estimates always overshot the correct out-comes, as if they moved “too far” toward largenumbers during addition, and too far towardsmall numbers during subtraction. A small butsimilar effect was found with symbolic addi-tions and subtractions of two-digit Arabic nu-merals. Furthermore, when picking one out ofseveral plausible results displayed on a com-puter screen, participants were spatially biasedtoward selecting choices that appeared on thetop right side of screen during addition, and onthe top left during subtraction.

These momentum and numerical–spatial in-terference effects may originate from the factthat the intraparietal region, which is activeduring number processing and calculation, isremarkably close and often overlapping withareas engaged in the coding of spatial dimen-sions such as size, location, and gaze direc-tion (Hubbard et al. 2005; Pinel et al. 2004;Simon et al. 2002). In particular, the putativehuman homolog of area LIP is also active dur-ing some number processing tasks (Dehaene,Piazza, Pinel, et al. 2003), fueling the specula-tion that the VIP–LIP circuitry is partially recy-cled for mental arithmetic in humans (Hubbardet al. 2005). We recently used fMRI to demon-

strate a numerical interference effect in the pu-tative human homolog of area LIP (Hubbard,Pinel, Jobert, et al. 2008): during the classi-cal SNARC task (parity judgment), large num-bers evoked slightly more activation in left LIP,consistent with a rightward shift of attentionin space, while small numbers evoked slightlymore activation in right LIP.

Although there is considerable evidence thatnumber–space mappings are an integral partof numerical intuition, the exact shape of thismapping seems to be heavily influenced by cul-ture and education. First, the direction of thenumber–space association—small numbers tothe left, large numbers to the right—varies withthe cultural environment, particularly the di-rection of reading, as it tends to be reduced,canceled, or even reversed in right-to-left read-ers (Dehaene et al. 1993; Shaki & Fischer2008; Zebian 2005). Indeed, multiple map-pings may co-exist in the same individual: Chi-nese readers preferentially associate Arabic nu-merals with the horizontal axis, and Chinesenumber words with the vertical axis (Hung,Hung, Tzeng, et al. 2008). Second, recent ex-periments have documented a remarkable de-velopmental shift in children’s conception ofhow numbers map onto space (Booth & Siegler2006; Siegler & Booth 2004; Siegler & Opfer2003). When asked to point toward the cor-rect location for a spoken number word on aline segment labeled with 1 at left and 100 atright, even kindergarteners understand the taskand behave nonrandomly, systematically plac-ing smaller numbers at left and larger num-bers at right. However, they do not distributethe numbers evenly in a linear manner. Rather,they devote more space to small numbers, thusimposing a compressed and seemingly logarith-mic mapping. For instance they will place num-ber 10 near the middle of the interval 1 through100. Beran et al. (2008) recently reported sim-ilar logarithmic responding with a bisectiontask in monkeys and 4- to- 5-year-old children.A shift from logarithmic to linear mappingoccurs later in development, between first andfourth grade, depending on experience and the


range of numbers tested (Booth & Siegler 2006;Siegler & Booth 2004; Siegler & Opfer 2003).My colleagues and I recently showed that thisshift depends on culture and education: it doesnot occur spontaneously in the Mundurucu,even in adult subjects and even in the range ofnumbers 1 to 10 (Dehaene, Izard, Spelke, et al.2008). The Mundurucu do have strong intu-itions of number–space mappings, since theycan systematically map numbers to a segmentbearing the numerosities 1 and 10 in a mono-tonic manner. However, their responses arelogarithmically spaced. Even bilingual adultswho could count in Portuguese, although theymapped Portuguese words onto the line seg-ment in a linear manner, still mapped sets ofdots and Mundurucu number words using alogarithmic scale.

This compressive, logarithmic response canbe easily explained by the log-Gaussian model.One must merely assume that subjects reporta rating of the psychological distance of eachnumber to the endpoints of the segment. Onthe log scale, distance covaries with numericalratio. For instance, 3 is equally distant from 1and from 9 because these numbers are all ina 3:1 ratio. Thus, it is natural or “intuitive”for the Mundurucu and for young children toplace 3 in the middle of 1 and 9. The pro-gressive replacement of this intuitive similar-ity scale by a linear scale is compatible withVerguts and Fias’s (2004) model, in which expo-sure to number symbols sharpens neural tun-ing until all neurons have a fixed width irre-spective of the size of the numbers involved,so that Weber’s law no longer applies to sym-bolic numerals when they are represented atthis conceptual level. As discussed above, itis tempting to speculate that this sharpeningand linearization effect only occurs in the leftbut not in the right parietal cortex. Such aco-existence of logarithmic and linear repre-sentations of numbers, even within the brainsof educated adults, would explain why evenAmerican adults continue to judge that 5 ismore similar to 9 than to 1, with a similaritymetric that is captured by a logarithmic scale

(Shepard, Kilpatrick, & Cunningham 1975). Itwould also account for why adults are unable torate whether a sequence of random numbers isspread equally across the number continuum—they always rate as “most random” a sequencethat oversamples small numbers, as if they weresampling from a compressive internal con-tinuum (Banks & Coleman 1981; Viarouge,Hubbard, Dehaene, et al. 2008).

The present emphasis on number–space in-teractions should not be taken to deny thatother types of automatic associations also con-tribute to our arithmetic intuition. Consider-able behavioral evidence indicates a tight andautomatic link between number and the lo-cation and size of hand movements (Andres,Davare, Pesenti, Olivier, et al. 2004; Andres,Ostry, Nicol, et al. 2008; Lindemann, Abolafia,Girardi, et al. 2007; Moretto & di Pellegrino2008; Song & Nakayama 2008). Fingers andnumbers are of course highly associated bycounting practices in the course of arithmeticdevelopment, and this is reflected in auto-matic number–finger associations in humanadults (Andres, Seron, & Olivier 2007; Di Luca,Grana, Semenza, et al. 2006; Sato, Catta-neo, Rizzolatti, et al. 2007), although number–space interactions appear to dominate overnumber–finger associations when the two arein conflict (Brozzoli et al. 2008). Although theneurophysiological basis of number–hand in-teractions is currently unknown, it is tempt-ing to speculate that it arises from interac-tions between numerical representations inthe hIPS and representations of hand shapeduring grasping in the nearby area AIP (seeFig. 3). Indeed, activation is occasionally re-ported at the location of the putative humanhomolog of area AIP during number process-ing (e.g., Fias et al. 2007; Hubbard et al. 2005).Through its central location in the parietallobe and its high interconnectivity with thesurrounding regions, the hIPS certainly hasmany opportunities for learning associationsof number with other spatial and movement-related domains in the course of arithmeticdevelopment.


Figure 5. Number–space mapping task used in the Mundurucu (after Dehaene et al.2008). Each target number was presented as a set of dots, a stream of tones, or a Mundurucuor Portuguese number word (note that we used very rarely uttered Mundurucu expressions for7 and 9). Participants were asked to point to the corresponding location on the number line.The sample data shown at bottom illustrate that the Mundurucu participants understood thetask and mapped numbers systematically, yet with a logarithmic scale, which was not seen incontrol American participants.


Conclusion

Our arithmetic intuition consists of a com-plex web of knowledge, ranging from the quickassessment of the approximate numerosity of30 dots to the realization that numerosity mustchange when elements are added to a set, orto the immediate realization that 8 is largerthan 3 and that 13 + 34 = 97 is false. Ourintuition is not even necessarily consistent, aswhen we judge that 5 is more similar to 9than to 1, yet falls in the middle of the 1–9 interval. In the past 20 years, research hasconsistently demonstrated that the core of ourintuition—a logarithmic sense of approximatenumerosity—finds its origin in an internal rep-resentation of numbers that dates back to ourinfancy, that we share with other animal species,and that relates to a bilateral parietal cir-cuitry. Much less clear is how this intuitiongets transformed and refined in the course ofeducation. Experience with number symbols,counting, and measurement probably all con-tribute to the “linearization” of our numericalintuition—the shift from an informal logarith-mic representation of approximate numerosityto a formal, symbolic, linear representation ofnumber and arithmetic. In this chapter, I spec-ulated that this transformation relates to theprogressive transformation of number codes,particularly in the left parietal lobe, “recycling”this region until it can be accessed automati-cally through the arbitrary cultural symbols ofArabic numerals (Dehaene & Cohen 2007). Ialso proposed that a dormant logarithmic codemay remain present in right intraparietal cor-tex. Future research should concentrate on theempirical testing of these two hypotheses, butalso on the important issue of how arithmeticintuition can be fostered by classroom practicein normal as well as dyscalculic children.

Acknowledgments

This theoretical synthesis benefited from use-ful discussion with many colleagues includingElizabeth Spelke, Manuela Piazza, Pierre Pica,

Veronique Izard, Ed Hubbard, Andre Knops,Laurent Cohen, Ghislaine Dehaene-Lambertz,and two anonymous referees. I gratefullyacknowledge support from Institut Nationalde la Sante et de la Recherche Medicale(INSERM), Commissariat a l’Energie Atom-ique (CEA), and a McDonnell foundation cen-tennial fellowship.

Conflicts of Interest

The author declares no conflicts of interest.

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