No species but ours can pride itself in the possession of both mathematical and linguistic abilities. Some primates possess numerical abilities, but only we humans can talk about what we count. Are these two abilities governed by the same principles and supported by the same neural structures? Philosophers from Plato to Frege have pondered the relation between mathematical and natural languages, as these seem to share properties, having alphabets and combinatorial rules, allowing for recursion, as well as ambiguous expressions. Chomsky (1988), for instance, has speculated that “the number faculty developed as a by-product of the language faculty” (p. 169; see Changeux and Connes, 1995, for a recent similar position). In contrast to these speculations, experimental evidence, mostly from double-dissociations observed in brain-damaged patients, has suggested that neurologically, the two capacities are distinct (Dehaene and Cohen, 1997; Cohen and Dehaene, 2000; Brannon, 2005; Gelman and Butterworth, 2005).

We report an fMRI experiment that addressed the language/math question from a new angle, and helped uncover the anatomical loci of linguistic and numerical operations. Our behavioral and imaging data seem to provide a fresh perspective on this perennial debate. Guided by current models of linguistic and mathematical capacity, we separated the neural underpinnings of complex language processes from those of numerosity-related ones. Within the latter, moreover, we were successful in identifying two distinct neural systems (one for quantity estimation, the other, for comparison). This three-way distinction between neurocognitive components emerged as subjects were evaluating linguistic statements about quantity against visual scenarios. Critically, it emanated from analyses of data from a single cross-modal parametric fMRI experiment that concomitantly probed numerical cognition and language.

The current view on numerical cognition and its brain basis is rather refined (Dehaene and Cohen, 1997; Butterworth, 1999; Cantlon and Brannon, 2007), drawing on complex computational models, and on empirical results from humans and monkeys. These models parse basic arithmetical operations into distinct steps – notably estimation (the encoding of numerical size), and comparison (the calculation of numerical distance), driven by Weber’s Law, that states that the ability to perceive a minimal difference between the quantities of two stimuli is invariant across the range of possible stimulus sizes. The consequence of this law in the present context is, roughly, that the organism’s ability to compare two quantities is a function of the proportion, rather than the distance, between them (Meck and Church, 1983; Dehaene and Changeux, 1993; Verguts and Fias, 2004). Experiments on numerosity typically present a sequence of images to participants. The first image depicts a number of objects (the reference numerosity r), and is followed by other images, of which the crucial one contains the comparandum numerosity c. In most instances, r is kept constant across trials, while c is a systematically varied parameter. Typically, this sequence is not accompanied by linguistic material.

The implementation of tasks that require the estimation of r and c, and then r:c comparison, has led to the discovery of neural correlates of these two operations in the macaque brain. There, neurons in lateral intraparietal cortex (LIP) in both hemispheres that respond to a relatively broad range of numerosities were distinguished from anteriorly adjacent (VIP) and frontal neurons that code for numerical distance, namely carry out comparisons (Nieder and Miller, 2003; Piazza et al., 2004; Roitman et al., 2007). Monkeys’ numerical judgment behavior is, moreover, said to be asymmetrical, best described by Gaussian curves on a logarithmically compressed scale, whose variability is fixed across numerosities (Dehaene et al., 2003).

In humans, bilateral parietal and frontal regions modulated by numerical distance have also been identified, and shown to have similar properties: in a seminal study by Piazza et al. (2004), participants viewed a sequence of images that contained a fixed number of objects r, thereby adapting to r; the critical part then presented an image with the same objects, but whose numerosity c was parameterized. Using an fMRI adaptation paradigm (i.e., using the habituation of the fMRI signal as an indicator of repeated involvement of a region in the same kind of cognitive process), they found regions in right angular gyrus, right intraparietal sulcus (IPS), and right superior parietal lobe (SPL) that follow the relation between r and c, indicating involvement in numerical comparison: These regions present an adaptation response pattern to a numerical distance parameter (manipulated in a comparison task) that resembles monkeys’ behavioral and neural patterns.

For language to be directly related to numerical cognition, aspects of language processing that pertain to quantity would be expected to have a role in numerical cognition. Past studies have either investigated linguistic and arithmetical abilities separately Dehaene et al., 2003), or focused on the relation between number words and their denotation (Dehaene and Cohen, 1997; Cohen and Dehaene, 2001; Varley et al., 2005). Investigations into the relation between the processing of sentences about numerical relations and corresponding number-containing scenarios are few and far between (McMillan et al., 2005; Troiani et al., 2009), despite the fact that we as speakers do just this naturally and frequently, as we evaluate statements we hear (or speak) against their real-life contexts. As a result, how exactly the neural substrate for numerical calculation relates to language is not entirely clear. Our project therefore involved a rather natural task, one that required participants to relate linguistic expressions to scenarios whose analysis necessitated numerical Estimation and Comparison: the scenarios contained quantities of objects, whereas the linguistic materials were sentences with quantity denoting expressions, that is, natural language quantifiers like many, few, most, or less-than-half. Successful performance required proper parsing of the visual scenarios, as well as an analysis of sentence meaning. Our contrasts, then, involved quantifiers that differed in meaning from one another.

We probed circuits for linguistic and mathematical processing in healthy participants in an fMRI experiment whose design featured two parameters, one that served as a proxy for numerical estimation, and the other, for comparison. It also featured a linguistic contrast that was orthogonal to numerosity (Polarity, the difference between positive quantifiers like more-than-half and less-than-half), which we used to detect certain aspects of the analysis of linguistic meaning which are orthogonal to numerosity. This design enabled us to measure the BOLD response related to each of these operations separately on the same data set.

A parametric proportion paradigm (PPP henceforth) was introduced: participants were asked to verify auditory sentences about proportion against visual scenarios that contained two quantities of objects, and do so under time pressure. The relevant meaning representations were created through the inclusion of a proportional (or degree) quantifier in each sentence, which was either positive, e.g., many of the circles are blue, or negative, e.g., few of the circles are blue. Each sentence was immediately followed by a visual image with an array of 50 quasi-randomly positioned circles. While the number of circles was fixed at 50 (a numerosity chosen to avoid subitizing), their radii were varied, and they were divided into two contiguous clusters of blue and yellow circles. The truth or falsity of each sentence depended on the meaning of the sentence and its match to the blue/yellow proportion (Figure 1A). To verify the sentence above, for example, participants had to first estimate c, the number of circles in the target color mentioned in the sentence (here blue), and r, the number of circles in the other color (here yellow). Next, they had to compare c to r. They then had to judge the truth-value of the sentence against the representation of the scenario (semantic truth-value judgment, SVJT). A sentence was deemed true just in case if c > r. Alternatively, as r + c = 50 in every case, a comparison of c to 25 = 50/2 would also suffice, as they sentence is true when c > 25. Either way, the successful completion of the task in a limited time window required both estimation and comparison. BOLD response as well as reaction time (RT) were measured.

The PPP design systematically varied the proportion between the colors across scenarios. Participants thus judged each sentence against eight different scenarios (created in Mathematica™), each with a different blue/yellow proportion (Figure 1B, Properties of the Images and How They Were Created in Supplementary Material). The manipulation of c (= Target Color, TarCol) modulated RT: a change in c it coupled to a change in r (because c + r = 50), and affects the numerical distance between c and r; this, in turn, affects the difficulty of the comparison component of the task – the closer c is to r (or to 25), the more difficult the task becomes, and RT is elevated relative to trials in which c is distant from r (or from 25). The PPP used sentences conveyed weak statements that are true in multiple scenarios, and so truth-value was kept constant across several values of c. Moreover, they were contrasted along the linguistic dimension of Polarity, as some contained negative quantifiers (few) whereas others were positive (many).

This design allows correlating the BOLD response separately with two parameters, and one contrast:

The experimental goal, then, was to obtain a three-way dissociation within the same fMRI data set. This goal was accomplished by identifying voxels responding to the one or the other of the uncorrelated parameters (c the monotonic estimation parameter, and RT, the non-monotonic comparison parameter), or to the semantic contrast.

The present design differs from previous numerosity experiments, and experiments that used linguistic stimuli to study numerical cognition (Cohen and Dehaene, 2001; McMillan et al., 2005; Troiani et al., 2009) by the following major aspects: first, we used two (c, RT) instead of a single parameter; second, a variable comparandum c is typically compared to a fixed reference numerosity r, and so r + c is not constant. As we used scenarios that were suited to express proportion, r and c were related, such that r + c = 50. Our paradigm and the standard one, however, are inter-translatable, and the relationship between them is formally well-defined (see The Mapping Between the Current PPP and Standard Numerosity Experiments in Supplementary Material for a mathematical translation between the two paradigms). Finally, the language materials included six different proportion or degree quantifiers – words like many, most, and few – that (1) denote relations between cardinalities of sets (as opposed to specific numbers), (2) that presuppose both estimation and comparison, and (3) that are weak, in that their truth-value is preserved across multiple scenarios. This last property makes them suitable to parametric designs, because it allows a sentence that contains such a quantifier to keep its truth-value constant across different numerosities (Table 1). We used several quantifiers, distinguished along the Polarity dimension (positive vs. negative), which enabled us to correlate the BOLD signal with a contrast that is purely linguistic, and does not bear on numerosity, even if both sides of this contrast come into contact with numerosity-related scenarios (see Materials and Methods, and The Meaning of the Proportional Quantifiers Used in Supplementary Material for further discussion).

All procedures were approved by the ethics committee of the Medical School at RWTH Aachen University.

The separation between semantic Polarity, numerical Estimation, and numerical Comparison was achieved via decisions that participants made on visual scenarios in which numerosity (and subsequently proportion) was parameterized. These scenarios were presented in contrasting linguistic contexts. Though embedded in a tightly controlled design, the PPP was implemented in a rather naturalistic verification task: we verify sentences daily, in communicative acts that require us to answer a yes/no question. These can range from the most mundane topics (Are you wearing a clean shirt?) to complex ones that contain quantifiers (Is every man in the room wearing a tie?). We also engage in verification when we evaluate statements, just like in the PPP task (Most of the paintings in this gallery are by Lucian Freud). The choice of proportional quantifiers, whose calculation requires both Estimation and Comparison, helped in keeping the probing method constant while systematically varying both the numerosities and task difficulty so as to allow for the bi-parametric analysis through the PPP.

There are two novel aspects here: First, these data were obtained for the first time from the same set of subjects in a single study, instead of merging together disparate studies of different scopes and quality. Second, and most importantly, it is the spatial relation of the semantic Polarity regions to the arithmetical parts, exposed through the parametric analyses of the PPP’s Proportion and RT parameters that served as proxies for Estimation and Comparison, respectively. Not only were these two neurocognitive components successfully separated in humans for the first time on the same data set (and in keeping with earlier findings – Piazza et al., 2004; Cohen Kadosh et al., 2005 – and sketches – Dehaene et al., 2003), but also, their spatial relation to activations during semantic analysis was demonstrated: Bilateral parietal regions were shown to be entrusted with numerical Estimation (via the c parameter), and adjacent regions on the opposite side of the IPS that support numerical Comparison (via the RT parameter, uncorrelated with the previous one). A new picture of a processing sequence along the “dorsal stream” seems to be emerging. The original notion of the dorsal stream was that of “where” pathway in visual perception, relevant for localizing objects in space. The concept of a functional processing pathway connecting occipito-temporal brain regions to the frontal cortex via the parietal lobe was supported by an anatomical connectivity study (Caspers et al., 2011). In this study, analogies between white matter tracts in humans and different kinds of monkeys were established. Interestingly, though, the existence of such anatomical connectivity does not imply information about the role of the dorsal stream in cognitive processing. A recent hypothesis (Friederici et al., 2012) relates the maturation of the dorsal stream to linguistic, and in particular syntactic, development from childhood to adulthood. Parallel to that notion, research in the field of numerosity processing (Dehaene et al., 2003) suggested a functional parcelation within the parietal aspect of the dorsal stream: (1) The angular gyrus is regarded as a region for manipulation of verbal number representations. (2) The horizontal aspect of the IPS is thought to house the mental number line. (3) The superior parietal cortex, finally, might relate to attentional processing when orientation along the mental number line is concerned.

Our study relates to the hypothesis by Dehaene et al. (2003) in that it provides an empirical basis for a bi- (or even tri-)partition of the parietal aspect of the dorsal stream, which was obtained with the same set of participants in the same experimental session. Regions for numerical Estimation and Comparison were found, which anatomically correspond to the angular and SPL regions referred to by Dehaene et al. (2003) and which are indeed separated by the IPS. Moreover, while the present numerosity results are consistent with similar ones from the macaque (Nieder and Miller, 2003; Roitman et al., 2007; Piazza and Izard, 2009; Caspers et al., 2011), our data, obtained from humans instead of animals, for the first time establish a clear connection to linguistic analysis of quantity, which builds upon and extends previous findings: Whereas a fronto-parietal network emerges for Estimation and Comparison, linguistic–semantic analysis is most prominent in the left inferior frontal cortex. Taken together, our data characterize processing along the dorsal stream from inferior parietal to inferior frontal regions when numerical processing is linked to linguistic–semantic processing. With respect to the hypothesis by Friederici et al. (2012), the present study may suggest that the dorsal stream is indeed involved in linguistic processing. However, over and above its supposed role for the emergence of syntactic abilities, our data suggest its additional importance for the transformation from visual percepts of numerosities into cognitive and linguistic formats. Interestingly, this account parallels the syntax hypothesis in so far as it stresses the difference between monkey and man with respect to linguistic abilities.

We thus found a potential human/monkey homology for successive steps of numerosity assessment, but moreover, established a direct connection to language. At the same time, we provided evidence for the neural modularity of language and arithmetic, revealed through a task that matches linguistic representations with numerical ones. Such matching has been used before to distinguish numbers from numerosities (Cohen and Dehaene, 2001), and in the context of previous work on quantification (McMillan et al., 2005; Troiani et al., 2009). Our contribution, supporting the modularity of language and arithmetic, stems from the fact that the task we used recruits not only lexical or sentential, but also complex compositional, resources, thereby better approximating the much discussed Chomsky’s “mental organ for language.” Whereas parietal regions were involved only in arithmetical but not semantic analysis, frontal regions, by contrast, appear to be involved in both. This pattern suggests that the left inferior frontal cortex contributes to the evaluation of the numerical representations provided in the parietal regions. This is consistent with the fact that in the macaque brain, number-selective neurons (Estimation) in inferior frontal cortex respond later than number-sensitive (Comparison) parietal neurons (Piazza and Izard, 2009). This temporal delay suggests an order, whereby frontal modules evaluate the output from parietal modules. In humans, this frontal contribution relates to semantic structure and may thus additionally be linked to linguistic processing.

Interestingly, functional division between Estimation and Comparison similar to that in parietal cortex was also found in both frontal lobes. Likewise, the semantic Polarity analysis was prominent in the left frontal lobe in area 45. Evidence for frontal involvement in numerical cognition has been available (Piazza et al., 2007), but the present results are surprising, as they seem to suggest commonalities in the functional organization between the inferior frontal cortex and the peri-IPS regions. McMillan et al. (2005), who studied the relation between natural language quantifiers and numerosity, have argued for a functional distinction, such that frontal regions house working memory, and parietal ones are entrusted with knowledge of numbers. The present study demonstrated the involvement of both frontal and parietal cortex areas likewise in response to Estimation and Comparison, even thought these processes may require working memory resources to different extents. Thus, the present data provide no evidence in support of the hypothesis by McMillan et al. (2005). Moreover, their distinction that was made with respect to the underlying anatomy, however, is possibly not fine-grained enough. Given the significant architectonic difference between areas 44 and 45 (involved in the Comparison network) on the one hand, and Brodmann’s area 47 (involved in the Estimation network) on the other hand (Amunts et al., 2010), it seems difficult to maintain that all frontal clusters uniformly support the same function, be it working memory or other, during numeric cognition. Certainly, there is a wealth of evidence for structural and functional connectivity between anterior IPS areas and ventral–posterior inferior frontal cortex (Uddin et al., 2010; Caspers et al., 2011; Mars et al., 2011). But whether these really involve working memory loops (Vigneau et al., 2011), or rather a house a sequential evaluation algorithm for the evaluation of previously estimated numerosities, as proposed here on the basis of a well-controlled experiment, will be the objectives of future research of the underlying structural and functional connectivity. The significance of this work, then, is in exposing a rich array of regions that provide a rare window into the intriguingly complex neural system that handles the flow of information between neural substrates for linguistic and numerical cognition.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The Supplementary Material for this article can be found online at http://www.frontiersin.org/Evolutionary_Neuroscience/10.3389/fnevo.2012.00004/abstract