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Research
Cite this article: Zhou K, Bowern C. 2015
Quantifying uncertainty in the phylogenetics of
Australian numeral systems. Proc. R. Soc. B
282: 20151278.
http://dx.doi.org/10.1098/rspb.2015.1278
Received: 29 May 2015
Accepted: 20 August 2015
Subject Areas:
evolution
Keywords:
phylogenetics, linguistics, Australian languages,
evolution, cultural reconstruction
Author for correspondence:
Claire Bowern
e-mail: [email protected]
Electronic supplementary material is available
at http://dx.doi.org/10.1098/rspb.2015.1278 or
via http://rspb.royalsocietypublishing.org.
Quantifying uncertainty in the
phylogenetics of Australian numeral
systems
Kevin Zhou1 and Claire Bowern2
1
2
Department of Biomedical Engineering, Yale University, 55 Prospect St, New Haven, CT 06511, USA
Department of Linguistics, Yale University, 370 Temple St, New Haven, CT 06511, USA
KZ, 0000-0002-0351-8812; CB, 0000-0002-9512-4393
Researchers have long been interested in the evolution of culture and the
ways in which change in cultural systems can be reconstructed and tracked.
Within the realm of language, these questions are increasingly investigated
with Bayesian phylogenetic methods. However, such work in cultural phylogenetics could be improved by more explicit quantification of reconstruction
and transition probabilities. We apply such methods to numerals in the
languages of Australia. As a large phylogeny with almost universal ‘lowlimit’ systems, Australian languages are ideal for investigating numeral
change over time. We reconstruct the most likely extent of the system at the
root and use that information to explore the ways numerals evolve. We show
that these systems do not increment serially, but most commonly vary their
upper limits between 3 and 5. While there is evidence for rapid system elaboration beyond the lower limits, languages lose numerals as well as gain them.
We investigate the ways larger numerals build on smaller bases, and show
that there is a general tendency to both gain and replace 4 by combining
2 þ 2 (rather than inventing a new unanalysable word ‘four’). We develop a
series of methods for quantifying and visualizing the results.
1. Introduction
Researchers have long been interested in the evolution of culture, and the ways
in which change in cultural systems can be reconstructed and tracked. Debates
about the ways in which such systems change have focused on the relative frequency of horizontal diffusion versus vertical transmission. That is, are cultural
traits primarily inherited, or do they spread across geographical areas [1 –4]?
Within the realm of language and culture, these questions are increasingly
investigated with Bayesian phylogenetic methods. Recent research has investigated questions in kinship [5,6], basic vocabulary [7–11] and grammatical
patterns [12,13]. Such work uses Bayesian inference to design and test evolutionary
scenarios. But results are often presented without full and explicit quantification of
the uncertainties associated with models, reconstructions and trees used to formulate descent hypotheses. For example, some papers [13] use tree samples for
their phylogenetic trait analysis, yet present only a single model for transition
probabilities, without quantifying the support for that model or discussing
alternatives. Such work, important though it is, could be improved by more
explicit quantification of reconstruction and transition probabilities.
We address these issues by developing a series of methods for quantifying
and visualizing results of ancestral trait reconstruction. We test them using data
from numeral system reconstruction in Australian languages of the PamaNyungan family [14]. Numeral systems across the world show great variation
[15–17], but within Australia, the systems tend to be low limit (with an upper
bound of less than 20, and frequently less than 5). However, while there is variation in both the composition and the extent of the systems, that variation is
highly constrained. Low-limit numeral systems are not unique to Australian
languages, but are characteristic of hunter–gatherer societies throughout the
& 2015 The Author(s) Published by the Royal Society. All rights reserved.
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(a) Numeral systems
We investigate two questions about the evolution of Australia’s
numeral systems. First, we explore the phylogenetics of numeral
system limits across the Pama-Nyungan family (the most populous
family of Australian languages, covering almost 90% of the land
mass and containing approx. 70% of Australian languages).
Bowern & Zentz [14] show that upper limits vary from 3 to 20,
with the median at 4. We investigate whether limits are structured
phylogenetically. Within this domain, we test four hypotheses
about numeral evolution. One hypothesis is that languages add
numerals incrementally over time. If this is so, we would expect
the root (Proto-Pama-Nyungan) to reconstruct to 2, and for numeral
extent to increase serially approaching the tips. A second hypothesis is that upper limits are random, or the products of local
system enhancement. A third hypothesis is that languages with
low-limit numeral systems may lose numeral terms over time (an
alternative hypothesis to hypothesis 1 that limits strictly increase).
A fourth hypothesis is that languages might borrow numerals
from one another, thus leading to horizontal transfer of features
of numeral systems as well as of the words used in those systems.
Secondly, we investigate whether numeral creation and
replacement evolves phylogenetically. Languages vary in
whether their numerals are compositional or opaque. Compositional numerals are formed from two other numbers (e.g.
English ‘twenty-one’ ¼ ‘twenty’ þ ‘one’). In the Wiradjuri
language, for example, 3 is buula-nguunbaay (‘two’ þ ‘one’) and
4 is buula-buula (‘two’ þ ‘two’). There is substantial variation
across the languages, however, as to whether both 3 and 4 are
compositional or opaque, and if only one is compositional,
which numeral shows the compositional form. We investigate
this further by comparing evolutionary models where the form
of 4 is dependent on the form of 3, versus models where the
two numerals are assumed to change independently. We further
hypothesize that opaque numerals will tend to be renewed or
replaced by compositional forms.
(b) Bayesian statistics
Bayesian phylogenetic methods are relatively new to the study of
linguistic prehistory, serving as a complement to the comparative
method [19,20]. They were adapted from methods developed
for evolutionary biology upon the realization that many of the properties of evolution have direct parallels in language change [7,21].
These statistical methods, particularly Markov chain Monte Carlo
(MCMC [22]), have been successful in verifying conclusions
(c) Representation of RJMCMC reconstructions
Given that each RJMCMC run fits the data to multiple models,
there is potential for the state reconstructions to vary more
significantly than those deduced by conventional MCMC
methods, as we shall see. For example, there might be two significantly different sets of models that both explain the data
well. As such, representing the uncertainty in the reconstructions
using a simple mean and confidence interval in place of a histogram of the reconstructions would obscure the true nature of the
reconstructions. Here, we propose a new method that uses the
Shannon entropy of the reconstructions as a sorting criterion.
Briefly, the entropy of a discrete trait X with a probability mass
reconstruction p(x) is given by H(X ) ¼ 2Sxp(x)logkp(x). If
p(x) ¼ 0 for some x, then p(x)logkp(x) is defined to be 0. In most
2
Proc. R. Soc. B 282: 20151278
2. Methods
drawn from more traditional linguistic methods as well as offering
support for deeper and less obvious relationships (e.g. [10,11]). As is
well known, MCMC numerically approximates the posterior distribution of the parameters given the data, which is often intractable to
calculate analytically. It involves searching the n-dimensional space
of possible parameter values, where n is the number of parameters
that attempt to describe the data. Parameter vectors that explain the
data well, based on the likelihood and prior distributions, will be
visited frequently; those that poorly explain the data will be visited
less often. In this way, the stationary distribution of the Markov
chain informs us of the range of possible parameter values.
Here, we are interested in transition rates between the different states (i.e. numeral extent or opacity) throughout the
language tree, as well as the state reconstructions at various
nodes within the tree, which are represented as probability
mass functions, allocating different probabilities to each state in
proportion to the degree of their support. We use an extension
of MCMC known as Reversible Jump MCMC (RJMCMC; [23],
see [24] for its introduction to (and development in) evolutionary
biology), which allows the number of parameters to change. This
technique is particularly suited for evolutionary models with a
large number of possible states. RJMCMC traverses not only
the parameter space but also the model space. In this way,
models that poorly explain the data are visited less often.
In our case, the parameters are transition rates between
the different states. Given a set of k independent traits, there are
k(k 2 1) transition rates. However, calculating all of the transition
rates is not only computationally demanding but also unnecessary
because it leads to overparametrization. To reduce the number of
transition rates, we can either delete rates or restrict them to the
values of other rates, but there is no systematic method of determining these deletions and groupings without external knowledge.
RJMCMC overcomes this issue by allowing the Markov chain to
‘jump’ from one dimension to another by deleting or grouping
parameters together in addition to varying them. As with conventional MCMC, such jumps are more likely to occur if the candidate
parameter vector better fits the data. Hence, we can draw conclusions about the different transition rates based on, for example,
how often they are deleted.
To model numerical system extent, we use BayesTraits [25] to
infer ancestral states at different nodes. To investigate correlated
evolution for numerical opacity, we examine two classes of
models, one that assumes independent evolution and one that
assumes dependent evolution, which we then compare via the
Bayes factor (BF) [26]. Fitting the data to both models and comparing their posterior likelihoods via the BF indicates whether
coevolution is favoured. We adopt the RJMCMC approach, but
we modify the coevolution model of opacity of ‘3’ and ‘4’ by
introducing two new paired states (electronic supplementary
material figures S1 and S4). These are required because the opacity of ‘4’ is ternary—opaque, compositional, or not attested. We
retain the opacity of ‘3’ as a binary term because no languages
have a term for ‘4’ but not for ‘3’.
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world, which presumably have a lesser functional need for a
more general system [15]. Variation in Australian numeral
systems stems not only from the systems’ upper extents but
also from whether larger numbers are composed of smaller
numbers (e.g. 3 ¼ 2 þ 1, 4 ¼ 2 þ 2, etc.; compare English
‘twenty-five’) or show unanalysable, opaque etymology
(such as English ‘three’ and ‘four’) [15,16]. While previous
work [16,17] recognizes some variation in the extent of lowlimit numeral systems, there has not been any investigation
into how low-limit systems change over time, particularly
with respect to their upper limits. Do such systems gradually
increment, for example? And can such systems lose numerals,
and revise their upper limits downwards? Finally, Australian
languages are also claimed to show extensive horizontal transfer of features [18], but Bayesian phylogenetic trait analyses
allow us to investigate the level of phylogenetic signal in a
dataset. All these factors make Australian numeral systems
an ideal test case for phylogenetic investigation.
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To investigate the transition rates that figure importantly in the
analyses, we rank the rates. Ranking rates by their speed across
the MCMC chain allows us to quantify which rates are relatively
fast: for each state in the RJMCMC chain, we ranked the rates
from fastest to slowest and tabulated the proportion of the
chain each rate was fastest. We also plot (for comparison) the
proportion of runs for which each rate is deleted. Doing so
exploits the joint Bayesian relationship among all the parameters,
while merely comparing the mean rate, as some authors have
previously done [24], ignores the joint relationship.
3. Results
(a) Extent
Figure 1 displays the entropy-sorted reconstructions at
various ancestral nodes throughout the tree. The reconstructions vary in certainty at any given node and across all
nodes. The closer to zero the reconstructions cluster on the
x-axis, the lower the entropy and the greater the confidence
in the reconstruction. For example, the Mayi and Ngayarta
subgroup node reconstructions have entropies approaching
0, where an extent of four and three (respectively) are confidently reconstructed. On the other hand, the Central New
South Wales node is not as confidently reconstructed, as illustrated by the relatively high entropy levels, even though
reconstructions of an extent of 4 predominate.
Marrngu provides an illustration of conflicting models.
Lower entropy reconstructions favour a limit of three, but the
peak density for reconstructions indicates greater overall
support for limit-five, although with less certainty. Likewise,
Paman ancestral state reconstructions suggest two solutions;
either limit-three or limit-five. Since reconstructions of limitthree are allocated a higher posterior probability across the
entropy range, they are to be preferred. The root (ProtoPama-Nyungan) reconstruction appears to be limit-four.
Certainty of the reconstructions is affected by the degree of posterior support for the particular node in the tree. As Pagel et al.
3
Proc. R. Soc. B 282: 20151278
(d) Analysis of rates
[25] put it, ‘the probability that the node exists puts an upper
limit on how certain one can ever be about the ancestral state
of that node’. Most of the nodes under study here have probabilities in the sample at or close to 1; however, it should be
borne in mind that trait reconstruction confidence can never
exceed the probability of the node it is assigned to.
Low-limit numeral systems do not only increase their limits
over time, and they do not simply increase limits serially.
Languages can also lose numerals, supporting hypothesis 3,
and contrary to strong versions of hypotheses 1 and 2. In the
Central subgroup of Pama-Nyungan, for example, the most
likely reconstruction is limit-four, but Karnic, one of the
daughter subgroups, has substantial support for limit-three.
Several subgroups, including Paman and Western PamaNyungan, show support for a reconstruction limit of three, in
comparison to the root, with limit-four.
Transition rates provide further information about which
changes in extent are particularly common. A transition rate
from numeral extent x to y is denoted by qxy. Thus, q34 is the
transition rate from state 3 to state 4 (or, in real-world terms,
the transition from a language having three numerals to
having four). Rates that are frequently deleted have less
importance in the model. We rank rates by speed and plot
two pieces of information: the proportion of runs in which
the rate is the fastest, and the proportion in which the rate
is deleted (figure 2). Three rates stand out as particularly
fast: q34, q35 and q43. This implies that in our data, limited
elaboration of the numeral system is common (from extent
3 to extent 4 or 5), as is some simplification (from extent 5
to 4). These results also imply that strict stepwise elaboration
of low-limit systems is not necessary, given the rate of extent
elaboration from 3 to 5.
The next fastest rate is q49. This perhaps implies that there
is a threshold for numerical complexity (that is, if elaboration
beyond 4 or 5 is found, significant rather than marginal elaboration is found). This accords with research on acquisition of
numerals [27,28], which shows that children encode the
numerals involved in subitizing (1–3) earlier. They map
small numerosities to specific lexical items and only later generalize to higher numerals [27]. Rates q37, q47 and q57 are
overall slow and are most frequently deleted. Stepwise rates
(e.g. q34, q45) are, on the whole, faster than rates where
limits increase or decrease non-incrementally (e.g. q35, q24),
implying that while hypothesis 1 is false, stepwise incrementation is still more common than non-stepwise elaboration.
These data are consistent with a situation in which numeral
extent rates fluctuate stochastically, but where variation in
numeral extent is mostly (though not entirely) concentrated
around accretion and deletion of numerals between 3 and
5. Hypothesis 3, that languages can lose numerals, even in
low-limit systems, is thus well-supported in our dataset.
The final hypothesis concerns the borrowing of numeral
systems. The data suggest elaboration of numeral systems
in the languages of the South subgroup (Kulin, Waka-Kabi,
Central NSW and Lower Murray). While there is no direct
evidence from the forms of the numerals that borrowing
has taken place, it is striking that the only subgroup outside
the Southern clade with numeral extent above 5 is ThuraYura, a subgroup that is directly adjacent to the Lower
Murray languages. Wati, the subgroup with the next highest
extent reconstruction, borders Thura-Yura to the west. This
implies that horizontal transmission may well play a role in
the elaboration of numeral systems, supporting hypothesis 4.
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applications, k ¼ 2 and the units are in bits; however, for convenience we assume k to be the number of possible values X can take
so that the entropy is bounded between 0 and 1. Entropy has
many interpretations depending on the application, but here it
is useful to think of it as how certain the reconstruction is. For
example, an entropy of 0 indicates that one particular value of
X has a probability of 1, meaning that the reconstruction is completely certain. On the other hand, a maximum entropy of 1
indicates that the reconstruction is uniform across all the
states—that is, it is completely uncertain.
By sorting reconstructions based on their entropies, we achieve
a better sense of the diversity of the reconstructions returned by
RJMCMC. Furthermore, as we will see, this representation is
more compact than using histograms and provides more information. Generating one-dimensional histograms is an inherently
marginalizing procedure—in other words, it discards any joint
relationships. Our procedure displays the joint reconstruction
and simultaneously visualizes all the reconstructions given by
the Markov chain. Also of interest is the distribution of the entropies themselves, the shape of which conveys the overall degree
of certainty the procedure was able to confer. Hence, we weighted
each sorted entropy by how frequently each reconstruction of that
entropy appeared. Doing so accentuates the reconstructions and
the entropies that appear most often. Estimates of the entropy
density were given by spline interpolation of histograms.
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4
1
Yolngu
Kartu
0.55
1
1
Kanyara-Mantharta
1
1
Ngayarta
1
Wati
1
1
0.95
Ngumpin-Yapa
1
1
Marrngu
1
1
Warluwarric
1
Arandic
0.87
1
Thura-Yura
1
Waka-Kabi
1
1
1
Central NSW
1
Southern
1
Lower Murray
1
1
Kulin
0.94
0.97
0.77
1
1
Northern
0.59
Paman
0.74
Maric
1
Central
1
Kalkatungic
Mayi 1
Karnic
1
0.98
0.98
Figure 1. Entropy plots for the Pama-Nyungan root and major subgroups (that is, selected internal tree nodes), plotted on a consensus tree derived from Bowern &
Atkinson [10]. Major subgroups of the family are labelled. Internal node labels give the posterior probabilities for the nodes. Colours for the reconstructions are dark
blue, 3; light blue, 4, green, 5; yellow, 6; red, 7; brown, 8þ. Upper limit on gamma-distributed rate prior ¼ 2.
(b) Opacity
We next consider the internal structure of numeral systems.
As described above, languages vary in the extent to which
higher numerals are built from smaller numerals; that is,
whether they are compositional, or opaque. One obvious
question is whether the compositionality of higher numerals
depends on that of lower numerals; for example, if the word
for ‘4’ is based on 2 þ 2, does that imply that the word for ‘3’
is also compositional (based on 2 þ 1)? Compositionality
changes over time, through a process known in linguistics
as grammaticalization [29]. As shown in electronic supplementary material, figure S1, all possible combinations of
opaque and compositional numerals exist in our dataset,
though at unequal frequencies.
To test for correlated evolution of compositionality, we ran
two models. In the first, numerals were assumed to evolve
independently. In the second, they were assumed to evolve
dependently. The BF was then compared to evaluate the
extent of support for one model compared with the other.
A BF range of 7.8–10.4, verified from three independent Markov chains, indicates that the dependent model is
superior to the independent model, offering strong support
for correlated evolution between the opacity of ‘3’ and ‘4’.
Electronic supplementary material, figures S9 and S10 give
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0.86
Western
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2
3
4
5
6
7
8+
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deleted
maximum
0.30
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0.25
proportion
5
0.20
0.15
0.10
0.05
q3
4
q3
5
q4
3
q4
9
q5
3
q4
5
q3
2
q2
4
q9
4
q9
5
q6
2
q5
4
q2
3
q2
5
q6
4
q5
9
q2
6
q4
2
q9
3
q6
5
q6
3
q2
9
q5
2
q7
5
q4
6
q7
9
q2
7
q7
4
q7
2
q6
9
q7
6
q3
6
q9
7
q9
2
q3
9
q6
7
q5
6
q9
6
q7
3
q5
7
q4
7
q3
7
rates
Figure 2. Rank frequencies of transition rates for numeral extent, showing (in dashed red) the proportion of samples in which that rate is the fastest and (in solid
blue) the proportion of samples in which the rate is deleted. Rates are given in the form qxy, which denotes the estimate rate of evolution from trait value x to trait
value y. (Online version in colour.)
the entropy-sorted reconstructions at ancestral nodes using
both models.
If we compare the reconstructions produced by the
dependent and independent models, we note that some reconstructions are similar (e.g. Paman, Kalkatungic) while others
are different (e.g. Root, Kulin and Mayi). The South node is
interesting in that although state 1 (compositional 3 and 4) has
the greatest a posteriori probability under both models, the
dependent model sometimes gives state 5 (i.e. compositional 3
and absent 4) as a strong alternative. Furthermore, the entropy
ranges differ; in general, it appears that the dependent model
produces reconstructions with a broader range of entropies,
often extending closer to 0. It also tends to produce results without a clear reconstruction across entropies. Furthermore, if we
compare the reconstructions with those considering extent
data (without regard for numeral etymology, from figure 1
above), we see that there is overall consistency between the
two sets of reconstructions, and subgroups reconstructed with
extent 3 tend to be reconstructed with absent 4. Within Western
Pama-Nyungan, for example, both Warluwaric and Yolngu
reconstruct with high probability to extent-3 numeral systems.
For Warluwaric, compositional 3/absent 4 is the most likely
node value, while for Yolngu opaque 3/absent 4 is most likely.
To get a sense of the mechanism of correlated evolution,
we can compare the various transition rates of the dependent
model. Figure 3 summarizes the reconstructed transition rates
in the dependent model, showing particular dependencies.
Arrows are weighted by frequency of the rate occurring in
the sample. Therefore, the thickest arrows (e.g. between 0,0
and 0,1 where opaque 4 becomes compositional) show the
transitions that are most relevant for the model.
The data provide evidence for the ways in which languages
both lose and gain numerals. The three fastest rates involve the
creation of a compositional 4 either through replacing an earlier
opaque 4 or incrementing a smaller system. Note, however, that
an opaque 4 may be also be both gained or lost. This behaviour
is in contrast to 3 which, in our data, becomes opaque only if 4 is
also opaque. This is presumably in large part because compositional 3 is usually ‘2 þ 1’; this leads to a pressure for 4 to be
compositional in base 2 as well (i.e. ‘2 þ 2’; see further discussion of this by Bowern and Zentz [14]). These findings make
sense in a model of language change where new elements in
(1)
0,0
(2)
0,1
0,–
(5)
1,–
(6)
1,0
1,1
(3)
(4)
Figure 3. Summary of the transition rates with support for a model of
dependent trait evolution for numeral opacity. This information summarizes
graphically the same information in electronic supplementary material, figure
S14, which gives relative speeds of all rates. Arrows are weighted by the transition speed, with fastest rates the heaviest. 0 denotes a compositional form,
1 an opaque one. Thus 0,0 is the state in which both numerals 3 and 4 are
compositional; 0,1 is a state with compositional 3 but opaque 4 and so forth.
Solid line denotes a missing state (these are languages with upper limit of
three numerals, and hence no form for 4).
a numeral system are created and renewed analogically to
other numerals. An alternative would be where speakers of
languages recruit opaque terms from other domains [15]. This
is the usual source of words for ‘5’, which frequently originate
in words for ‘hand’ in Pama-Nyungan languages (infrequently,
words for 5 based on ‘3 þ 2’ are found). This is not the dominant mode of evolution of terms for 4 in Pama-Nyungan
languages, however. These data also imply a cycle of grammaticalization where numerals (particularly 4) tend to be
innovated compositionally, grammaticalized or fossilized so
that their origin becomes opaque to their speakers, then lost
or replaced by a new compositional form.
Compositional 3 (e.g. 2 þ 1) and opaque 4 seems to be an
unstable state, with fast transition rates leading to loss of 4,
replacement of opaque 4 with compositional 4, or replacement
of compositional 3 with opaque 3. Finally, 4 may be gained or
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4. Conclusion
Data accessibility. See electronic supplementary materials for coded datasets and references to original sources.
Authors’ contributions. C.B. and K.Z. developed the research; K.Z. wrote
the code for the analysis; C.B. sourced data; K.Z. and C.B. analysed
the data; C.B. and K.Z. wrote the paper.
Competing interests. We declare we have no competing interests.
Funding. This work was funded by NSF grants BCS-0844550 and
BCS-1423711.
Acknowledgements. Thanks to members of the Pama-Nyungan/Historical Language Lab at Yale, and to Mark Pagel.
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6
Proc. R. Soc. B 282: 20151278
In conclusion, entropy sorting provides a convenient way to
visualize the relative likelihoods of reconstructions of multistate
data at different nodes in the phylogeny. Both entropy plots and
rate comparisons allow a way to evaluate the outcomes of Bayesian phylogenetic analysis. We use this approach to investigate
the phylogenetics of small numeral systems in Australia. We
reconstruct the most likely extent of the system at the root and
use that information to explore the ways in which numeral systems evolve. We show that the mode of evolution among lowlimit numeral systems in Australia is variation between limits
of 3, 4 and 5, and that languages may both lose and gain
numerals across time. We provide evidence for a rapid elaboration of numerals above 5, and that there is some evidence
that higher numeral extents have spread areally. Below extent
5, however, languages fluctuate in their upper limits across
time. These findings argue against a model of numeral evolution
which begins with gradual incrementation or elaboration. We
investigate the etymologies of numerals and show that numeral
4 can be lost or gained independently of the structure of 3,
numeral 4 is usually innovated compositionally.
rspb.royalsocietypublishing.org
lost, whether 3 is opaque or compositional. This implies that the
upper limit of a numeral system is, at least to some extent, independent of numeral etymology, as one would expect if these are
true numerals. If the numeral words in these languages are ad
hoc compounds rather than true numerals, we would expect
there to be a clearer dependency between the presence of a compositional 3 and a compositional 4.