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Embodied number processing
Oliver Lindemanna & Martin H. Fischera
a Division of Cognitive Sciences, University of Potsdam, Potsdam, Germany
Published online: 27 Apr 2015.
To cite this article: Oliver Lindemann & Martin H. Fischer (2015) Embodied number processing, Journal of Cognitive
Psychology, 27:4, 381-387, DOI: 10.1080/20445911.2015.1032295
To link to this article: http://dx.doi.org/10.1080/20445911.2015.1032295
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EDITORIAL
Embodied number processing
The contributions to this Special Issue come from a
wide range of research groups, many of which were
brought together in September 2013 in Potsdam,
Germany, for a two-day Workshop “From Num-
bers To Knowledge—20 Years Of Spatial-Numer-
ical Associations”supported by the European
Science Foundation. As one result of this work-
shop, several researchers agreed that “embodied
number processing”is one of the most important
new emerging and unifying topics in the field of
numerical cognition.
Cognitive scientists have long held an embodied
view on cognition which assumes that symbols and
abstract concepts become meaningful only when
they refer to bodily experiences (e.g., Barsalou,
1999; Fischer & Zwaan, 2008; Glenberg, 1997;
Pulvermüller, 2005; Rueschemeyer, Lindemann,
van Elk, & Bekkering, 2009). Over the last two
decades, research on language comprehension has
provided a large amount of evidence for the idea
that conceptual knowledge is grounded through
such sensory-motor referencing. For instance, read-
ing action-related sentences that imply a particular
rotational motion (e.g., she opens the bottle) facil-
itates the execution of similar actions (e.g., rotating
a knob counter-clockwise; Zwaan & Taylor, 2006;
see also Glenberg & Kaschak, 2002). This func-
tional link between the motor system and language
comprehension has also been found at the neural
level, reflected by motor-cortical activation through
action-related language processing (e.g., Hauk,
Johnsrude, & Pulvermüller, 2004). In turn, there is
also evidence for the reverse referencing (in the
sense of a functional link) from motor actions onto
more efficient processing of action-congruent lin-
guistic descriptions (Rueschemeyer, Lindemann,
van Rooij, van Dam, & Bekkering, 2010) and
memory retrieval (Pecher, Van Dantzig, Zwaan, &
Zeelenberg, 2009; van Dam, Rueschemeyer, Bek-
kering, & Lindemann, 2013).
EMBODIED NUMERICAL COGNITION
With this increasing empirical evidence that activa-
tion of bodily representations contributes to con-
ceptual knowledge, researchers began to consider
the sensory-motor grounding of semantic proces-
sing also in the domain of mathematical cognition
(e.g., Andres, Olivier, & Badets, 2008; Fischer,
2012; Lakoff & Núñez, 2000; Lindemann, Ruesche-
meyer, & Bekkering, 2009; Moeller et al., 2012).
The present Special Issue presents the latest devel-
opments of this approach.
The role of sensory and motor codes in number
representation and mental arithmetic is important
to understand because the two codes share the
same class of information, namely knowledge
about quantities and magnitudes. Consistent with
this fact, similar parietal brain areas support mag-
nitude processing for numbers and for grasping
movements (Simon, Mangin, Cohen, Le Bihan, &
Dehaene, 2002). Behavioural priming therefore
works across the two domains. For example, large
numbers facilitate hand opening responses
(Andres, Davare, Pesenti, Olivier, & Seron, 2004)
and the grasping of objects with a power grip
(Lindemann, Abolafia, Girardi, & Bekkering,
2007) whereas small numbers prime hand closure
and precision grip actions. In turn, number proces-
sing interferes with the timing of the duration of
manual button responses (Kiesel & Vierck, 2008)
and with the use of a particular response force
(Krause, Lindemann, Toni, & Bekkering, 2014).
Number processing also impacts judgements about
© 2015 Taylor & Francis
Journal of Cognitive Psychology, 2015
Vol. 27, No. 4, 381–387, http://dx.doi.org/10.1080/20445911.2015.1032295
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motor affordances of objects, such as their grasp-
ability (Badets, Andres, Di Luca, & Pesenti, 2007).
Similar interactions between magnitude features
have been reported for perceptual representations
in numerical tasks. The best-known example is
the frequently replicated size-congruity effect
(e.g., Tzelgov, Meyer, & Henik, 1992), which
reflects the phenomenon that the comparison of
the physical size of two digits is strongly modulated
by their numerical values. Perceptual magnitude-
related priming effects on number processing have
also been reported for luminance perception
(Cohen Kadosh, Cohen Kadosh, & Henik, 2008),
duration perception (Fabbri, Cancellieri, & Natale,
2012), and the perceived amount of tactually stimu-
lated fingers (Krause, Bekkering, & Linde-
mann, 2013).
Several different views on embodied cognition
have been proposed over recent years (cf., Mete-
yard, Cuadrado, Bahrami, & Vigliocco, 2012;
Wilson, 2002) and as a consequence there are
many theoretical inconsistencies in the numerical
cognition literature related to the term “embodi-
ment”. Even though some authors have tried to
provide terminological clarifications (e.g., Andres
& Pesenti, in press; Fischer, 2012; Moeller et al.,
2012), there is so far no consensus about necessary
and sufficient features of an embodied number
representation. Based on the finding of magnitude
priming between numbers and action, and consist-
ent with the grounding hypothesis of semantic
knowledge, we propose here that embodied
numerical cognition refers to the idea that our
representation or processing of numbers relies
obligatorily on sensory-motor activation of magni-
tudes originating from perception and action.
Importantly, this notion implies that magnitudes
for numerical processing and magnitudes for sens-
ory-motor processing share a common cognitive
metric. This grounding of the number concept in
sensory-motor magnitudes is, in our view, a neces-
sary and sufficient precondition for embodiment in
numerical cognition.
Our notion of number meaning grounded in
sensory-motor magnitude codes is not only in line
with many general cognitive theories that empha-
sise the interplay between perception, action, and
cognition (e.g., Barsalou, 1999; Gibson, 1979; Hom-
mel, Müsseler, Aschersleben, & Prinz, 2001). The
basic assumption about the association of numbers
and bodily representations is also shared by some
recent theories of number processing and arithmetic
that have not been explicitly formulated with
reference to the embodied cognition debate. For
instance, the idea of a generalised magnitude
system located in parietal cortex (Walsh, 2003,in
press) also emphasises the existence of a common
medium that codes size-related information from
different cognitive domains and from the sensory-
motor modalities. Another example for a compat-
ible approach is the neuronal recycling hypothesis
(Dehaene & Cohen, 2007), which assumes that
cultural inventions such as arithmetic utilise evolu-
tionarily older brain circuits that initially emerged
to control perception and action.
PROTHETIC AND METATHETIC
SENSORY CODES IN EMBODIED
NUMERICAL COGNITION
Given our basic definition of embodiment, which
appears to be consensual across theories, any remain-
ing theoretical discrepancies might be accounted for
by the distinction of two types of sensory-motor codes
when discussing the underlying mechanisms of
embodied numerical cognition: those stemming
from “prothetic”dimensions and those stemming
from “metathetic”dimensions. Following Stevens
(1957), a prothetic dimension is represented by an
additive process at the physiological level. That is, a
perceived increase along the prothetic dimension is
caused by additional physiological excitation. For
instance, an increase of luminance or visual size is
driven by a stronger stimulation of photoreceptor
cells on the retina. A metathetic dimension, in
contrast, is characterised by a substitutive process.
That is, a perceived change along the metathetic
dimension is caused by a substitution of one pattern
of excitation for another or by a change of the locus of
stimulation. Typical examples for metathetic con-
tinua are space or pitch perception. In other words,
variation along prothetic sensory codes refers to
quantitative differences and triggers questions such
as “How much?”, whereas variation along metathetic
sensory dimensions represent qualitative changes
which trigger “Where?”or “What?”questions.
This distinction between prothetic and meta-
thetic codes is crucial for the ongoing debate on
embodied numerical cognition, especially the
extent to which number symbols require sensory-
motor grounding. Our previous definition of
embodiment is motivated by the grounding prob-
lem of meaning constitution and assumes that
sensory-motor associations of concepts are formed
to provide a meaningful reference frame for the
representation of otherwise abstract knowledge, in
our case that of numerical size. Since only prothetic
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continua can serve as a cognitive metric to map
magnitude information without further transforma-
tions, we postulate that prothetic sensory-motor
codes are the essential features contributing to the
conceptualisation of the core meaning of numbers.
This is the first and fundamental mechanism of
embodied numerical cognition.
Our claim does not exclude that also metathetic
sensory-motor codes can support the acquisition of
embodied numerical knowledge. In fact, several
studies on spatial-numerical associations (for review
see Hubbard, Piazza, Pinel, & Dehaene, 2005)and
on finger-based number representations (Andres &
Pesenti, in press) provide evidence for stable associa-
tions between numbers and metathetic sensory-
motor codes. This secondary type of embodiment
can best be understood in the context of associative
learning theories (cf. Fischer, 2012). There it reflects
the fact that any sensory and motor correlates that
we perceive consistently during the acquisition of
numerical concepts become associated with number
meaning. In other words, any bodily correlate from
metathetic or not-magnitude-related dimensions that
supports conceptual processing might become an
integral part of an embodied number representation
(Lindemann & Fischer, in press; Moeller et al. 2012).
For instance, when we experience that ordered
sequences are constantly arranged from the left to
the right side of space then we develop over time a
particular spatial-numerical long-term memory asso-
ciation (Göbel, Shaki, & Fischer, 2011).
Such culturally mediated associations are quite
flexible and do not reflect the length of their
acquisition phase. Instead, they are situation-
specific mappings that exert a powerful influence
on performance. A relatively weaker constraint
on performance is imposed by the bodily nature
of our knowledge acquisitions. We experience
sensory features of objects, such as their weight
and size, as prothetic sensory-motor features that
converge in our brains (Walsh, 2003,in press). As
a consequence, all number meaning should be
seen as a result of the physical constraints of the
world within which cognitive capacities are uni-
versally adapted: for instance, objects are distinct
from each other and cannot occupy the same
place at the same time. This is a prerequisite for
their countability and for our perception of
numerosities in the first place. In order to account
for the wide range of influences of the culture
and the context on numerical cognition, it seems
helpful to distinguish levels of embodiment of
number knowledge with characteristically different
performance signatures (e.g., situated, embodied,
grounded; Fischer, 2012).
THE PRESENT CONTRIBUTIONS
The present Special Issue brings together a wide
range of approaches to study embodied numerosity
and highlights the current state of this active field of
research. The opening contribution by Rugani,
Vallortigara, and Regolin (2015) reviews evolution-
ary origins of spatial asymmetries in magnitude
processing. As one example, their work on domestic
chicks utilises a food localisation task that begins
with a training phase in which an elongated food
tray is always oriented along the chick’s mid-sagittal
plane and contains food in a proximal opening, thus
requiring a short distance to find the reward. In the
test phase the food tray is turned 90° so that it
extends perpendicular to the chick’s approach but it
is unclear whether the “proximal”opening is now
on the left or right side. The fact that a majority of
chicks spontaneously explore the left side first is
taken as evidence for a brain lateralisation for
ordered sequences. Very recent research from the
same group confirms this notion and provides even
evidence for a left-to-right preference in chicks
when associating numerosities with space (Rugani,
Vallortigara, Priftis, & Regolin, 2015). This appar-
ent similarity between chicks and humans is docu-
mented by the finding that three-day-old chicks
prefer to look for food behind an object on their left
side when it shows a relatively small numerosity and
behind an object on the right side when it shows a
relatively large numerosity. The results of Rugani
and colleagues call into question the proposed
linguistic mediation or cultural emergence of spa-
tially directional associations between number
and space (Göbel et al., 2011) and place a stronger
emphasis on inherited compared to learned map-
pings. This interpretation is in line with embodied
cognition approaches emphasising a phylogenetic
shaping of the nervous system in accordance
with invariant physical constraints of the world
(Fischer, 2012).
The current Special Issue contains two contri-
butes that study the ontogenetic development of
embodied number representation. Stapel, Hunnius,
Bekkering, and Lindemann (2015) investigate the
development of the approximate numbers system
(Dehaene, 1997) by measuring children’s associa-
tions of number words with visual non-symbolic
numerosities. It has recently been proposed that the
“sense”for numbers does not behave like a classical
EDITORIAL 383
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physiological sensory system because the shape of
the psychophysical function of the perceived intens-
ity or numerosity seems to depend on age and
acquired knowledge about the number system
(Siegler & Opfer, 2003). However, in the context
of an embodied view, Stapel et al. argue against the
idea of developmental changes in the approximate
number system and hypothesise that the often
reported shift from logarithmic to linear number
representation might be the result of the employed
number-to-position method, which requires chil-
dren to indicate numbers by pointing on a spatial,
that is, metathetic continuum (Siegler & Opfer,
2003). Stapel et al. therefore developed a numer-
osity production task that does not require a
transformation of prothetic number magnitudes to
metathetic spatial codes. Interestingly, their test of
5- and 8-year-old children with this new method
reveals a very linear association of symbolic and
non-symbolic numerosities already in the younger
children. Their findings therefore show no evidence
for developmental changes in the underlying ana-
logue magnitude and rather suggest that perform-
ance differences between age groups are driven by
differences in verbal knowledge and the range of
number words that are appropriately conceptua-
lised by the child.
Patro, Nuerk, and Cress (2015) investigate
counting habits in preschool children and provide
an interesting new example of the involvement of
the body of preliterate children in the association
between numbers and space. The authors demon-
strate that typically observed cultural counting
preferences, such as left-to-right counting of pre-
schoolers from Western cultures, are modulated by
contextual influences and body-related metathetic
references. First, it matters which hand is used for
counting, since there was a preference to start
counting ipsilateral from the used hand. Second,
these hand-based effects on spatial counting strat-
egies are enlarged for objects within reaching
distance. However, it needs to be emphasised that
these body-related biases on counting seem to
disappear with increasing age and with the onset
of formal schooling. This shows that bodily and
cultural characteristics of number representations
might become more or less pronounced, depending
on the particular age of the participant. Taken
together, the two studies of Stapel et al. (2015) and
Patro et al. (2015) are in line with the notion that
sensory and motor experiences are particularly
important in the early phases of numeracy devel-
opment and during the acquisition of number
knowledge.
Another example for the importance of embodi-
ment in children’s number knowledge is the spon-
taneous use of fingers when acquiring the meaning
of number words and learning to count. As men-
tioned above, several studies have shown that
finger-based number representation has an influ-
ence on number processing in adults. Wasner,
Möller, Fischer, and Nuerk (2015) follow up on
this line of research and provide a systematic
classification of the different numerical principles,
such as ordinality, cardinality, and 1-to-1-corres-
pondence, which are all involved in finger-based
magnitude representation. Their analyses of adults’
use of fingers in different situations, as well as their
study of finger-to-number mapping with twisted
arms, reveal that counting gestures and “montring”
postures (a term derived from the French “mon-
trer”—to show) are not identical for most numbers.
Moreover, there is no relation between finger-to-
number mappings and the assignment of fingers for
counting. The authors emphasise the flexible
recruitment of number–finger associations, depend-
ing on the context and the meaning to be conveyed.
The absence of a unitary embodied relation
between fingers and numbers demonstrates the
independence of hand-based and numerical repre-
sentations and questions the role of finger associa-
tions in the semantic representation of numbers.
This limitation does not necessarily argue against
embodied accounts of number processing. Instead,
Wasner et al.’s finding is in line with our distinction
between prothetic and metathetic sensory-motor
codes and seems to suggest that metathetic codes
are merely transient sensory-motor associations
that are not crucial for the grounding of the core
number meaning (see also Krause et al., 2013 for
the same conclusion).
The contribution of metathetic sensory and
motor codes to number processing has mostly
been studied with spatial behaviour. A more thor-
ough assessment of the embodied nature of number
concepts is to remove such spatial features from the
task requirements, as happens during the recording
of brain activity in passive settings. Schuller, Hoff-
mann, Goffaux, and Schiltz (2015) ask healthy
adults to discriminate the colour of a lateralised
target object after numbers are presented as unin-
formative attentional cues at fixation that require no
response (cf. Fischer, Castel, Dodd, & Pratt, 2003).
Even though the authors do not find an influence of
the small and large numbers on target detection
times, the analyses of event-related potentials
clearly indicate that numbers affect attentional
processing. The fact that attentional modulation
384 EDITORIAL
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through number magnitude occurs in the absence of
behavioural effects is important because there have
been occasional failures to replicate the attentional
cueing effect with centrally presented digits. Schul-
ler et al. interpret their results in the light of similar
findings as evidence for the robustness of the
attentional cueing effect across situations, thus
tapping into more profound and perhaps embodied
aspects of cognition.
The coupling between numbers and spatial
sensory processes has also been investigated by
Ranzini et al. (2015). These authors, however, are
interested in the reverse relation and examine
whether a manipulation of visuospatial attention
has an effect on number processing. Healthy adult
participants try to fixate on horizontally moving
patterns of vertical stripes. This optokinetic stimu-
lation technique is known to induce eye position
drifts in the direction of movement, followed by
intermittent re-positioning of the gaze. While being
exposed to optokinetic stimulation, participants
classify auditorily presented numbers by either their
parity or magnitude. While the expectation was to
see a clear modulation of the strength of spatial-
numerical associations due to joint overt and covert
attention shifts, this is only the case for explicit
magnitude processing. Combined with previous
work discussed in their article, the results of Ranzini
et al. add to the growing body evidence of motor
contributions to conceptual processing and enable
the weighting of different motor components, such
as attention shifts and movement execution.
The study of Shaki, Sery, and Fischer (2015)
aims to extend embodied signatures from the mere
coding of number meaning to the cognitive process
of merging and manipulating magnitude informa-
tion. The role of sensory and motor representations
in the realm of mental arithmetic is so far not well
understood and we believe that it constitutes an
important generalisation of the empirical base in
support of embodied number processing (see
Fischer & Shaki, 2014; Wiemers, Bekkering, &
Lindemann, 2014). In their present study, the
authors ask adults to produce lines that match in
length either the magnitudes of single digits or the
outcomes of addition problems. Lines representing
a given number are longer in the addition com-
pared to the single digit condition, perhaps reflect-
ing our intuition that “adding makes more”.
Interestingly, for identical operands the partici-
pants produce longer lines when the first operand
is larger, thus violating a basic law of arithmetic.
Another violation is found when comparing unit
increments across the range of operands, where the
same unit is expressed as an ever-increasing line
length when the other operand becomes larger. In
addition to illustrating the situatedness of number
meanings, this study is perhaps the start of a larger
research programme on heuristics and biases in
mental arithmetic.
Complementing the previous reports, the last
two studies in this Special Issue focus on motor
movements as an indicator of embodied number
knowledge. Ganor-Stern and Goldman (2015)
make use of an increasingly popular approach to
document spatial biases in hand movements
through the recording of mouse cursor trajectories
(for methodological concerns pertaining to this
approach, see Fischer & Hartmann, 2014). The
advantage of trajectory analysis over traditional
reaction time analysis is the additional evidence
for the emergence of decisions over time. In the
present study, Ganor-Stern and Goldman use the
mouse-tracking approach to test a model that
proposes a rapid processing strategy for numbers
that constitute the smallest or largest number in a
range. Participants move a mouse cursor from the
lower end of the screen to the larger of two numbers
that are displayed at both sides at the top of the
screen. In support of this model, the authors find
that participants move faster in time and more
directly in space towards the targets when the
number pairs include such end values, i.e., the
number 1 or 9. Interestingly, this end effect is larger
for 1 (which was never a target when present) than
for 9 (which was always a target when present).
The contribution by Bloechle, Huber, and
Moeller (2015) also investigates number interfer-
ence effects in the execution of motor actions and
extends the idea of embodied representations to
the processing of multi-digit numbers and the
place-value structure of the Arabic number system.
The authors present two two-digit numbers and
require participants to point to the numerically
large one. The analysis of pointing end-locations
reveals that participants tend to point to the decade
digit of the number and that this pointing bias is
modulated by unit-decade compatibility (Nuerk,
Weger, & Willmes, 2001), that is, by structural
relations within and between the multi-digit num-
bers. Taken together, the findings of Blöchle et al.
document an influence of place-value processing
on the execution of motor actions and provide an
important addition to our understanding of the
metathetic embodied mechanisms that affect the
conceptual representation of symbolic numbers.
Together, the contributions to this Special
Issue document an increasing enthusiasm among
EDITORIAL 385
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cognitive scientists for an embodied understanding of
numerical cognition. This is no small change of
perspective because numbers were long thought to
be the knowledge domain par excellence for abstract,
amodal symbol manipulation that likens the human
mind to a computer. The authors of the present
contributions hope that others will follow their lead
and that this work eventually provides guidance for
meaningful and effective recommendations in the
teaching and rehabilitation of numerical cognition.
This Special Issue would not have been possible
without the help of our anonymous reviewers and
the editorial assistance from Taylor & Francis, as
well as the guidance by the editor-in-chief and the
support of the editorial board members. We thank
all of them.
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Potsdam, Potsdam, Germany
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