Evidence for Two Numerical Systems That Are Similar in Humans and Guppies

Abstract

Humans and non-human animals share an approximate non-verbal system for representing and comparing numerosities that has no upper limit and for which accuracy is dependent on the numerical ratio. Current evidence indicates that the mechanism for keeping track of individual objects can also be used for numerical purposes; if so, its accuracy will be independent of numerical ratio, but its capacity is limited to the number of items that can be tracked, about four. There is, however, growing controversy as to whether two separate number systems are present in other vertebrate species.
In this study, we compared the ability of undergraduate students and guppies to discriminate the same numerical ratios, both within and beyond the small number range. In both students and fish the performance was ratio-independent for the numbers 1-4, while it steadily increased with numerical distance when larger numbers were presented.
Our results suggest that two distinct systems underlie quantity discrimination in both humans and fish, implying that the building blocks of uniquely human mathematical abilities may be evolutionarily ancient, dating back to before the divergence of bony fish and tetrapod lineages.

Full text

Evidence for Two Numerical Systems That Are Similar
in Humans and Guppies
Christian Agrillo
1
*, Laura Piffer
1
, Angelo Bisazza
1
, Brian Butterworth
2
1Department of General Psychology, University of Padova, Padova, Italy, 2Institute of Cognitive Neuroscience, University College of London, London, United Kingdom
Abstract
Background:
Humans and non-human animals share an approximate non-verbal system for representing and comparing
numerosities that has no upper limit and for which accuracy is dependent on the numerical ratio. Current evidence indicates
that the mechanism for keeping track of individual objects can also be used for numerical purposes; if so, its accuracy will be
independent of numerical ratio, but its capacity is limited to the number of items that can be tracked, about four. There is,
however, growing controversy as to whether two separate number systems are present in other vertebrate species.
Methodology/Principal Findings:
In this study, we compared the ability of undergraduate students and guppies to
discriminate the same numerical ratios, both within and beyond the small number range. In both students and fish the
performance was ratio-independent for the numbers 1–4, while it steadily increased with numerical distance when larger
numbers were presented.
Conclusions/Significance:
Our results suggest that two distinct systems underlie quantity discrimination in both humans
and fish, implying that the building blocks of uniquely human mathematical abilities may be evolutionarily ancient, dating
back to before the divergence of bony fish and tetrapod lineages.
Citation: Agrillo C, Piffer L, Bisazza A, Butterworth B (2012) Evidence for Two Numerical Systems That Are Similar in ?Humans and Guppies. PLoS ONE 7(2):
e31923. doi:10.1371/journal.pone.0031923
Editor: Sarah Frances Brosnan, Georgia State University, United States of America
Received October 19, 2011; Accepted January 20, 2012; Published February 15, 2012
Copyright: ß2012 Agrillo et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This research was funded by research grant ‘‘Progetto di Eccellenza CARIPARO 2007’’, and ‘‘Progetto Giovani Studiosi 2010’’ from University of Padova
to Christian Agrillo. The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: christian.agrillo@unipd.it
Introduction
Numerousness, like shape, size and color, is a basic property of
our perceptual world. It has long been recognized that adults,
infants and non-human animals can instantly extract the
numerical information from a visual scene without counting the
elements [1,2,3,4]. In humans, this ability is thought to be based
on two distinct non-verbal systems that operate over different parts
of the number range [5,6,7]. One is a system for representing
approximate numerosities as analog magnitudes, and is usually
referred to as the ‘analog magnitude system’ (ANS) [8,9,10] that
has virtually no upper limit but is subject to a ratio limit in
accordance with Weber’s Law, which states that the capacity to
discriminate between two quantities becomes increasingly accurate
as the ratio between them increases. The second system is referred
to as the ‘object-file system’ and is held to depend on a mechanism
for representing and tracking small numbers of individual objects
[3,7,11]. If this object-tracking system operates by keeping track of
individual elements, it is precise but allows for the parallel
representation of a small number of objects (usually three-four
elements in adults). It is often assumed that this is the system that
supports ‘subitizing’ – the accurate reporting of the numerosity of
small sets without serial counting [6,7]. The two mechanisms
appear to differ in many respects, including speed, accuracy and
cognitive load [11,12].
The lack of a ratio effect is the main signature that allows
experimental differentiation of the object-file system from the
analog magnitude system [3]: the performance of an adult is very
similar when discriminating 3 vs. 4 or 1 vs. 4 objects, whereas we
are much more accurate in discriminating 5 from 20 objects than
15 from 20 objects. Not all the studies, however, reported a
different ratio-effect between small and large numbers [13,14,15].
For instance, in a task requiring to apply an ordinal numerical
rule, Cantlon and Brannon [16] also found evidence of a ratio-
effect in the small number range. Recent neuropsychological,
electrophysiological and brain imaging data suggest that these two
non-verbal numerical systems probably have distinct neural
substrates [17,18,19,20,21].
The analog magnitude system appears to be shared among
many vertebrates. When required to determine the larger of two
sets of elements, infants, macaques, dogs, swordtails and
mosquitofish give approximate responses and their capacity to
discriminate is strongly influenced by the numerical ratio
[4,16,22,23,24]. Some have proposed that human infants and
non-human vertebrates may also share with adults a distinct
mechanism for precisely representing quantities up to four.
However, evidence of separate systems in human infants and
animals is less clear. In some studies, infants were found to
discriminate different ratios in the small and large number range.
For instance they were found to discriminate 2 vs. 3 but not 4 vs. 6
items in a habituation task despite the identical ratio difference
[25]. Moreover, a study showed that children with Williams
syndrome report a specific impairment in large number discrim-
ination while the discrimination capacity in the small number
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range was unaffected [18]. Yet, vanMarle and Wynn [26] found
that in infants the discrimination of auditory events was ratio-
dependent also in the small number range, suggesting that infants
use a single system of analog magnitude in the auditory domain.
Regarding non-human primates, chimpanzees have been shown
to increase their reaction times to estimate the number of dots in a
large number range but not in a small number range [27], but
another study combining the data from four great apes found that
numerical ratio was the best performance predictor in the range
0–6 [28]. In support of two-system hypothesis, rhesus monkeys
successfully selected the greater group of apple slices with
comparison of 1 vs. 2, 2 vs. 3 and 3 vs. 4 [29] while they
discriminated between 4 and 8 lemons (1:2 numerical ratio) but
not between 4 and 6 (2:3). However two recent studies reported
that in rhesus monkeys accuracy was strongly affected by
numerical ratio for both small and large quantities, in agreement
with the existence of a single non-verbal mechanism over the
whole numerical range [16,30].
Several lines of evidence suggest that separate systems for small
and large numbers may exist in other vertebrates too. In a recent
work on attack/retreat decisions in free-ranging dogs [31],
Bonanni and collaborators reported that dogs spontaneously
assessed large quantities as noisy magnitudes. In contrast dogs
approached aggressively with the same probability when they
outnumbered opponents by a 1:2, 2:3 or 3:4 ratio, suggesting that
dogs discriminate two small quantities using an object-file
mechanism. Among birds, New Zealand robins can discriminate
1 vs. 2, 2 vs. 3 and 3 vs. 4 but not 4 vs. 5 and 5 vs. 6; however these
birds can also distinguish between large numbers provided that the
numerical ratio is at least 1:2 [32]. Studies on domestic chicks and
bees have demonstrated that both species can successfully
discriminate 2 vs. 3 while they fail to distinguish the same
numerical ratio when two large numbers are presented, such as 4
vs. 6 [33,34]. However, this kind of evidence for two numerical
systems may not be conclusive. In some cases there are alternative
explanations. For example, if chicks and bees can only represent
small numbers, then they will not be able to discriminate 4 from 6.
In recent years, numerical competence has been investigated in
several teleost species using either operant conditioning or
spontaneous preferences [22,23,35,36,37,38]. Two of these studies
imply the possibility that separate systems for large and small
numbers may exist even in fish. Mosquitofish have been found to
discriminate between shoals differing in numerosity when the
paired numbers were 1 vs. 2, 2 vs. 3 and 3 vs. 4 but they succeeded
with only up to a 1:2 numerical ratio (4 vs. 8 or 8 vs. 16) when they
had to discriminate between large numbers [22]. In the second
study it has been shown that one-day old guppies can discriminate
between small quantities of social companions (,4), showing an
inborn ability to elaborate small quantities, while the capacity to
discriminate large quantities (4 vs. 8) emerges later, as a
consequence of both maturation and social experience. This
developmental dissociation indirectly suggests the existence of
different systems for small and large quantities also in guppies [37].
Overall, the existence of a phylogenetically shared analog
magnitude system appears generally accepted, but authors
disagree as to whether a single analog magnitude mechanism
accounts for discrimination over the whole numerical range, or a
distinct system operates over the small number range.
In the present study, we used guppies to test one of the
predictions of the hypothesis of two separate systems; according to
this hypothesis fishes’ ability to discriminate two large numbers
should become more accurate as the difference between them
increases while, in discriminating quantities ,4, the performance
should not be affected by the numerical ratio.
Guppies were required to choose the more numerous of two
available groups of conspecifics. As reference, a group of
undergraduate students were required to estimate the larger of
two groups of dots while prevented from verbal counting. Both
species were presented with the same five numerical ratios (0.25/
0.33/0.50/0.67/0.75) both within the small number range (1–4)
and beyond it. Furthermore, we examined the possible role of
learning on performance of the fish, comparing subjects with or
without previous experience of social groups.
Materials and Methods
Ethics Statement
The Experiment involving fish complies with all laws of the
country (Italy) in which it was performed (D.M. 116192) and was
approved by ‘Ministero della Salute’ (permit number: 6726-2011).
The experiment with undergraduates was approved by the ethics
committee of the Department of General Psychology of University
of Padova and was conducted according to the Declaration of
Helsinki. Before testing, all participants gave their written consent.
Undergraduate experiment
In this experiment we adopted a procedure commonly used to
measure non-verbal numerical abilities in adults, namely a
computerized numerical judgement with sequential presentation
of the stimuli [17,36,39]. Undergraduates were required to
estimate the larger of two groups of dots while being prevented
from using verbal counting.
Participants. A total of 18 undergraduate students (three
males) between the ages of 18 and 31 (mean age: 21.33) took part
in the present study for course credits. The experiments were
carried out at the Department of General Psychology, University
of Padova. All participants had normal or corrected vision.
Stimuli and procedure. The stimuli consisted of 240 pairs of
arrays composed of different numbers of black dots. The dots
differed in size and appeared in the center of the screen on a white
background. The number of dots presented on the screen ranged
from 1 to 24. Half of the pairs were controlled for continuous
variables (cumulative surface area, density, luminance and overall
space occupied by arrays), while the other half were not. Twenty
further pairs, with identical features, were created for the initial
training phase. The stimuli were displayed on a 17-inch monitor,
using E-Prime software, in a darkened room.
After a period of dark adaptation, a short familiarization
training phase with feedback was presented. The participants
initially read the experimental instructions on the screen. A
fixation cross then appeared in the center of the screen for
1000 ms, then a group of dots was displayed in the centre of the
screen for 150 ms (Fig. 1). Following a 500 ms delay, the
participants were shown another group of dots for 150 ms. They
were required to estimate which one of the two groups was more
numerous by pressing one of two keys on the keyboard. In half of
the stimuli the larger group was presented first, in the other half
the smaller group was presented first. They were instructed to
make their responses as quickly and as accurately as possible.
Furthermore, to prevent verbal processing of the stimuli, verbal
suppression was introduced during the test by asking the
participants to continuously repeat ‘abc’. No feedback was
provided during the test.
Five numerical ratios were presented (0.25, 0.33, 0.50, 0.67,
0.75) for small (1 vs. 4, 1 vs. 3, 1 vs. 2, 2 vs. 3 and 3 vs. 4) and large
(6 vs. 24, 6 vs. 18, 6 vs. 12, 6 vs. 9, 6 vs. 8) numerical contrasts.
Reaction time is the common measure of numerical acuity in
human studies, especially when, as in our study, participants must
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perform easy discriminations [17,40,41]. In these cases, in fact,
participants commit few errors and variation in accuracy is
potentially obscured by a ceiling effect. However since accuracy
(proportion of time spent near the larger shoal) was the only
variable measured in the fish experiment, for comparison, we also
analyzed accuracy in the human experiment.
Fish experiment
Like many other social fish, single guppies placed in an
unknown environment show a strong tendency to join social
companions and, if choosing between two shoals, they exhibit a
preference for the larger one [37,42]. This spontaneous tendency
was used in this experiment to measure the ability of guppies to
discriminate between two numerosities. During their life guppies
might have different opportunity of familiarizing with large or
small shoals and this could potentially affect the experiment.
Because of this, alongside a sample of 140 adults, we tested a
sample of 200 immature fish reared in pairs and therefore with no
previous experience of social groups.
Subjects. The experienced subjects were adult females
because they are more gregarious than males. They were reared
in groups of 15 or more individuals. Seventy fish (14 in each
numerical contrast) were tested in small quantity discriminations,
and 70 were tested in large quantity discriminations. The
inexperienced subjects were juvenile fish tested at the onset of
their numerical abilities. A recent study showed that one-day-old
guppies could discriminate the larger shoal when the choice was
between numbers in the small number range, whereas the ability
to discriminate large quantities appeared later, at approximately
day 40 [37]. Thus, 100 one-day-old fish (20 in each numerical
contrast) were tested in small quantity discrimination tasks, and
100 40-day-old fish were tested in large quantity discrimination
tasks.
Stimuli and procedure. The experimental apparatus was
composed of three adjacent tanks (Fig. 2). The central one, the
‘subject tank’, housed the test fish (36660635 cm). At the two
ends two ‘stimulus tanks’ (36610610 cm) faced the subject tank.
The apparatus was filled with 10 cm of water. The apparatus for
Figure 1. The experimental design used in the undergraduate experiment. The participants were sequentially presented with two groups
of dots and had to estimate which group was more numerous.
doi:10.1371/journal.pone.0031923.g001
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the juvenile fish was similar to that used for the adult fish but
reduced in size. The central ‘subject tank’ was 20617.5625 cm
and the two ‘stimulus tanks’ were semi-octagonal shaped, with a
side of 6.3 cm. This apparatus was filled with 4 cm of water and
was partially modified when testing the 40-day-old fish by
enlarging the stimulus tanks (semi-octagonal sides 8.3 cm long)
and increasing the level of water (6 cm).
Two shoals containing different numbers of fish were placed
into the stimulus tanks. The subjects were individually introduced
into a transparent plastic cylinder (10 cm in diameter for the
adults, and 3.5 cm for the juveniles) in the middle of the subject
tank and allowed to acclimatize for two minutes. After this period
the subject was observed for 15 minutes. Shoal preference was
calculated as the time spent by the subject within a distance of
11 cm (4 cm when testing juveniles) from the glass facing either of
the stimulus tanks. Subjects that did not visit either stimulus sector
at least three times or spent less than 50% of the time in a choice
area were considered inactive; they were discarded and replaced
by another fish.
The same five numerical ratios of the students’ experiment were
presented to fish both for small (1 vs. 4, 1 vs. 3, 1 vs. 2, 2 vs. 3 and
3 vs. 4) and large (4 vs. 16, 4 vs. 12, 4 vs. 8, 4 vs. 6 and 6 vs. 8)
numerical contrasts.
At the end of these trials, we increased the sample size for ratio
0.67 by testing 64 additional adult females, 32 in a small quantity
discrimination (2 vs. 3) and 32 in a large quantity discrimination
(4 vs. 6).
Results
Undergraduate experiment
Small numbers. In accordance with the theory, reaction
time was not affected by numerical ratio (ANOVA F
(4,68)
= 1.474,
P= 0.220, Fig. 3). Control of continuous variables did not affect
Figure 2. The experimental apparatus used in the fish
experiment. Fish were individually placed into the middle of the
apparatus where two shoals containing different numbers of fish were
visible at the ends.
doi:10.1371/journal.pone.0031923.g002
Figure 3. The results of the undergraduate experiment. Accuracy (proportion of correct responses) and reaction time are plotted against the
numerical ratio of the contrasts for both large and small number (1–4) ranges. The performance of the participants showed ratio sensitivity for large
numbers and ratio insensitivity in the small number range. Bars represent the standard error.
doi:10.1371/journal.pone.0031923.g003
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performances (F
(1,17)
= 0.421, P= 0.525; interaction F
(4,68)
= 0.822,
P= 0.516). To test if there was a significant slope, we performed a
polynomial trend analysis [43]. No significant linear or larger
order trend was found (F
(1,17)
= 2.644, P= 0.122). A likelihood
ratio analysis (see [44] for details) also reflected the absence of ratio
effect (l= 3.67).
Comparable differences were observed in accuracy. Accuracy
was not affected by numerical ratio (F
(4,68)
= 0.572, P= 0.684),
continuous variables did not affect the performance
(F
(1,17)
= 1.285, P= 0.273) and the interaction was not significant
(F
(4,68)
= 0.383, P= 0.820). No significant trend was found
(F
(1,17)
= 0.378, P= 0.547). A likelihood ratio analysis also reflected
the absence of ratio effect (l= 1.15).
Large numbers. In the large quantity range the reaction
time increased with decreasing numerical ratio (F
(4,68)
= 31.889,
P,0.001), while the continuous variable factor was not significant
(F
(1,17)
= 2.880, P= 0.108; interaction F
(4,68)
= 2.469, P= 0.053,
Fig. 3). There was a significant trend (linear trend: F
(1,17)
= 57.302,
P,0.001; quadratic trend: F
(1,17)
= 8.250, P= 0.011).
Similarly, accuracy decreased with decreasing numerical ratio
(F
(4,68)
= 8.564, P,0.001), control of the continuous variables did
not affect performance (F
(1,17)
= 0.205, P= 0.657) and no interac-
tion was found (F
(4,68)
= 0.359, P= 0.837). There was a significant
trend (linear trend: F
(1,17)
= 24.348, P,0.001; quadratic trend:
F
(1,17)
= 8.303, P= 0.010).
On the whole, accuracy did not differ between the small
number range and the large number range (t
(17)
= 1.877,
P= 0.078). However, for the ratio of 0.75 the participants were
significantly more accurate in the small number range than the
large number range (t
(17)
= 2.197, P= 0.042).
Fish experiment
Data were analysed separately for small number and large
number range with a two (experience: juveniles/adults) by five
(numerical ratio: 0.25/0.33/0.50/0.67/0.75) between-subject
ANOVA.
Small numbers. The proportion of time spent near the
larger shoal was not influenced by either numerical ratio
(F
(4,169)
= 0.047, P= 0.996) or experience (F
(1,169)
= 0.030, P=
0.864), and the interaction was not significant (F
(4,169)
= 0.242,
P= 0.914, Fig. 4). No significant trend was found (F
(4, 169)
= 0.045,
P= 0.876). Likelihood ratio analysis also reflected the absence of
ratio effect (l= 3.68).
Large numbers. In the large number range, the proportion
of time spent near the larger shoal was influenced by numerical
ratio (F
(4,169)
= 3.190, P= 0.015) but not by experience
(F
(1,169)
= 0.300, P= 0.585; interaction: F
(4,169)
= 0.341, P= 0.850,
Fig. 4). There was a significant linear trend (F
(4, 169)
= 3.312,
P,0.001).
Figure 4. The results of the fish experiment. The proportion of time spent near the larger shoal is plotted against the numerical ratio of the
contrasts for both large and small number (1–4) ranges. The performance of the fish (both experienced and inexperienced fish) adhered to the same
patterns as for humans in the two numerical ranges, with ratio sensitivity only being shown in the large number range. Bars represent the standard
error.
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Within the small number range, preference for the larger group
was statistically significant for all ratios (0.25: t
(33)
= 3.265,
P= 0.003; 0.33: t
(33)
= 3.260, P= 0.003; 0.50: t
(33)
= 3.537,
P,0.001; 0.67: t
(33)
= 4.003, P,0.001; 0.75: t
(33)
= 3.230,
P= 0.003); in the large number range, the fish spent significantly
more time near the larger shoal for ratios 0.25 (t
(33)
= 6.922,
P,0.001), 0.33 (t
(33)
= 5.788, P,0.001) and 0.50 (t
(33)
= 4.466,
P,0.001), but not for ratios 0.67 (t
(33)
= 1.481, P= 0.148) or 0.75
(t
(33)
= 1.159, P= 0.255).
No single ratio significantly differed between the two ranges
(two sample t-test, df = 66 all P.0.169). Yet all comparisons had a
very low statistical power (0.052#x#0.279) and therefore lack of
significance could be due to type II error. Sample size calculation
indicates that, for each comparison of our experiment, a minimum
of 90 subjects was needed in order to reach a 80% power to detect
a 20% difference between treatment groups with a two-sided test
and alpha = 0.05 [45].
To test if lack of significance of between-range comparisons was
due to inadequate sample size we increased the sample size for
one numerical ratio (0.67), by testing 64 additional adult fish, 32
in a small number discrimination (2 vs. 3) and 32 in a large
number discrimination (4 vs. 6). The difference between the 2 vs.
3 and the 4 vs. 6 contrasts became statistically significant
(independent t-test t
(130)
= 2.688, P= 0.008). As observed with
humans, as discrimination became more difficult, the fish tended
to be more accurate in the small number range; they could in fact
discriminate between two numbers with a 0.67 or 0.75 ratio in the
small number range, whereas they required a 2:1 ratio when
tested with large numbers.
Discussion
Previous studies have reported remarkable similarities in the
performance of non-verbal numerical tasks among humans, apes
and monkeys, suggesting the existence of the same basic numerical
systems among primates [3,29,46,47,48]. Here we provide
evidence of a similar correspondence in numerical abilities
between humans and teleost fishes.
When tested in the same numerical tasks, the students and
guppies showed almost identical performance patterns. In both
species, the ability to discriminate between large numbers (.4) was
approximate and strongly dependent on the ratio between the
numerosities. In contrast, in both fish and students, discrimination
in the small number range was not dependent on ratio and
discriminating 3 from 4 was as easy as discriminating 1 from 4.
Likelihood ratio analyses indicate that the lack of ratio effect is
3.68 times more likely than the alternative hypothesis in fish and
3.67 times in students. As a consequence, the discrimination of
larger numerical ratios, 0.67 and 0.75, is easy in the small number
range for both species, but becomes relatively more difficult
(among students) or even impossible (among fish) when confronted
with large numbers.
It is possible that, in fish, the different discriminative ability in
the two numerical ranges is the consequence of a different
encounter rate with large or small social groups and thus a
different familiarity with large and small numbers. However, the
observation that juveniles raised in pairs, and therefore with no
previous experience in comparing social groups, showed the same
pattern as experienced adults rules this possibility out and indicates
that numerical systems are probably innate in fish. The
observation that some numerical abilities are exhibited by guppies
at birth is certainly remarkable and is line with evidence that both
chicks and newborns display some rudimentary numerical skills
[49,50]. This reinforces the proposition by some authors of a core
knowledge of number, a system of innate numerical representation
shared among the different non-human species [3,51].
Since numerosity normally co-varies with other physical
attributes such as the total area occupied by objects, one may
argue that in the fish experiment subjects were using these cues
instead of numerical information to solve the task. This possibility
was not checked in the present study. However in two previous
studies we showed that guppies and mosquitofish can discriminate
between two schools of fish using the numerosity information only,
both within the small number range or outside it [37,52] and that
mosquitofish can discriminate between sets of geometric figures in
both numerical ranges even after all continuous variables were
controlled for. In particular a recent study using a training
procedure has shown no difference in the learning rate between
fish trained to use numerical information only and fish trained to
use continuous quantities only, suggesting that the number per se
is not more cognitively demanding than continuous quantities
[53]. On the other hand students showed a very similar
performance whether continuous variables were controlled or
not, thus making the comparisons fully legitimate.
One can argue that the guppy and student experiments differ in
many respects. In particular students were tested in a sequential
presentation, whereas fish saw the simultaneous presentation of the
numerosity pairs. However in this respect the two experiments
may differ less than may appear at first glance. In our test situation
the fish could never see the two stimuli binocularly simultaneously.
It was possible for a subject to observe both shoals only when
swimming perpendicularly to the main axis, which occurred very
rarely during a test. Moreover, in this position, each stimulus was
seen with a different eye; in this situation, as a consequence of the
lateralization of social recognition [54,55] and reduced inter-
hemispheric transfer of information [56,57], fish cannot be
expected to guess the larger group.
The difference in ratio-dependence suggests the existence in
fish, as in humans, of two distinct non-verbal mechanisms of
numerical representation, one for numbers 1–4 and one for large
quantities. Yet the hypothesis that a precise object-file mechanism
does underlie small number discrimination also predicts higher
performance in the small number range, a finding not evident in
our study. However, the lack of a statistical difference between
ranges observed in fish experiment may be due to the limit of the
procedure adopted. Previous studies using this procedure have
shown that accuracy in selecting the larger shoal never exceed
70% even with very easy numerical ratios (guppies [37,58],
mosquitofish [22], topminnows [59], angelfish [60]). Due to large
measurement variance combined with small sample size in each
numerical contrast, these statistical tests suffered from low power
and hence lack of significance may be attributable to a great
probability of making a type II error. Sample size calculation
indicates that in our experiment a threefold sample size was
needed to obtain an adequate statistical power when two
treatments had to be compared. As confirmation, a statistical
difference emerged between the large and small number range
after we increased the sample size in the 0.67 ratio. Regarding this
latter evidence, previous work has reported that macaques,
mosquitofish, chicks and bees could distinguish two from three
items but failed with the same ratio in the large number range
[22,29,33,34,61], even though none of these studies could provide
a statistical test to support a difference in performance between the
two ranges. Our result highlights the possibility that, as in other
research fields [62,63,64], many studies are underpowered to
detect statistical differences among subgroups.
The results reported here differ from those found in a very
recent study on angelfish which discriminated 2 vs. 3 but not 3 vs.
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4 fish [60]. Since both guppies and mosquitofish appear to
discriminate 3 from 4 fish [22,37] there might be taxonomic or
ecological differences in the numerical abilities among fish species.
Other explanations are however possible. Since poeciliids do not
form tight shoals, during the experiment, all stimulus fish of one
group are usually simultaneously visible to the subject. Angelfish,
by contrast, are characterized by a wide lateral body surface and
tend to form coherent and synchronized schools; thus, as the
number of individuals increases some individuals could often block
the sight of other shoalmates, making it difficult for a subject to
accurately estimate group numerosity.
Some recent studies found that accuracy was affected by
numerical ratio for both small and large numbers leading some
researchers to question the existence of two separate numerical
systems [16,65]. It is unclear why some studies find that
performance in the range 1–4 is independent of ratio and some
do not. A possible explanation for this literature inconsistency,
recently proposed by some authors (e.g. [26,66,67]), is that small
quantities may be represented by both analog magnitude and
object-file mechanisms, and that recruitment of one or the other
system may depend on context and previous experience. In line
with this hypothesis, a recent study found that, unlike controls, a
sample of participants with an expertise that requires years of
training in estimation of magnitudes showed the typical signature
of the analog magnitude system, ratio effect, in the small number
range too [Agrillo and Piffer, unpublished].
However that may be, the lack of a ratio effect in the 1–4
numerical range does not necessarily entail the existence of a
separate system. As pointed out by Gallistel and Gelman [9], the
difference in ratio effect between large and small numbers could
occur because there is so little error in the analog magnitude
representations of 1, 2, 3, and 4 that they are highly
distinguishable from one another, and thus coarse behavioral
measures of the underlying processes (as they often are with both
human and non-human experiments), fail to evidence a ratio
dependence when in fact such a relationship may exist.
Our results do not help to clarify this issue, being compatible
with both hypotheses. However, even assuming the existence of a
single system of analog magnitude with a different ratio sensitivity
in the range 1–4 and beyond it, the similarity in human and guppy
experiments in the steepness of the slope in both ranges is again
strongly suggestive of similar systems of numerical representation.
Other studies have provided evidence to support strong
similarities between teleosts and primates. Swordtails, mosquito-
fish, angelfish and climbing perch appear to adhere to Weber’s
Law when discriminating between two large quantities
[22,23,38,68] and mosquitofish trained to discriminate between
large sets of geometric figures were found to be equally efficient in
discriminating 4 vs. 8 items or 100 vs. 200 items, exactly like the
college students tested with the same stimuli [36]. Naturally, as
with most comparative data, it is always possible that a strong
similarity in cognitive abilities is the product of convergent
evolution and that similar performance reflect very dissimilar
underlying mechanisms.
It might seem surprising to discover similar numerical abilities
in humans and in guppies, especially when considering that the
brain size of a guppy is less than a thousandth of that of primates.
However, it is clear from recent literature that the cognitive
abilities of fish have been greatly underestimated and that teleosts
are capable of complex behaviors such as individual recognition,
transmission of cultural traditions, cooperation, copying behavior
and deception, which have traditionally been associated with the
evolution of large cortical areas in mammals and birds [69,70]. On
the other hand, it is also possible that a cognitive function such as
numerical discrimination, which is apparently complex, may
actually be based on relatively simple neural circuits, as suggested
by a recent neural network study [71].
On a more general note, the evolution of numerical abilities in
animal species is still a largely unexplored field and future research
is needed to understand the origin of the quantitative systems
shown by vertebrates. If numerical abilities have evolved many
times independently in different taxa it would be challenging to
understand which selective constraints have shaped them in a
converging fashion. On the other hand, the results of this
comparative study admits the possibility of common mechanisms
between primates and basal vertebrates, suggesting that the
evolutionary emergence of numerical abilities may be very
ancient, possibly dating back to before the teleost-tetrapod
divergence.
Author Contributions
Conceived and designed the experiments: CA AB. Performed the
experiments: CA LP. Analyzed the data: LP BB. Wrote the paper: CA
AB BB.
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