# Mathematical Logic

Processing mathematical formulae is a cognitive task that most of us are required to perform every day. The more abstract the formulae, the more difficult their processing seems to be.

To date, it has never been clarified why more abstract realizations of a common underlying structure are perceived as being more difficult to process than more concrete ones, even though the formulae are structurally isomorphic. If it is not the structure, the difference must lie in the semantic domain over which the formulae are interpreted.

This hypothesis is tested in the present study. Previous research has already investigated the neural basis of number processing quite extensively: either as single items or in simple arithmetic and algebraic calculations [1] , [2] , [3] , [4]. The intraparietal sulcus IPS was found to systematically activate for all these number-related tasks.

Therefore, this region was taken as a key region for the representation of numerical quantity for recent reviews see [5] or [6]. Neuroimaging research on the processing of syntax in mathematics is sparse. Activation of some frontal regions, i. These brain regions in the prefrontal cortex, although crucial for processing mathematical formulae, are not specific to the domain of mathematics.

Rather, the prefrontal cortex has been allocated to aspects of cognitive control [8] , [9] , [10] , [11] and working memory [12] , [13] , [14]. These two aspects, cognitive control and working memory, are also relevant for the processing of structured sequences [15] , [16] , [17] , and for the judgment of relations in the language and non-language domains [18] , [19] , respectively.

Here, we focused on the neural basis of processing the semantic content of mathematical formulae, i. Therefore, we designed a functional magnetic resonance imaging fMRI experiment, using syntactically well-formed hierarchical formulae written in a standard first-order language. The formulae themselves were interpreted in either the domain of abstract algebra or arithmetic, i. Participants were required to read these formulae and to decide whether they were true or false statements. In contrast to our previous study on syntax, where we used formulae in which the first occurrence of a grammatical error rendered the whole expression incorrect, and no further processing was required to achieve a well-formed judgment, in this study the calculation of the Boolean truth value needed the evaluation of the entire phrase.

By keeping the structures identical but varying the domain or the truth value, we expected to disentangle the specific roles of various prefrontal areas with respect to cognitive control and working memory and, further, to shed light on the neural substrate of processing difficulty.

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We expected to find a bilateral fronto-parietal network when comparing the different algebraic and arithmetic conditions, consisting of regions previously identified to be involved in the processing of the syntax of hierarchical expressions, and domain specificity manifesting itself as local differences in activation strength or volume extension. Moreover, we predicted a differential time course of activations for brain regions involved in calculating the truth value of the formulae depending on the semantic domain.

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Therefore, two separate analyses were conducted: one for activations 4 s post stimulus and another for activations 8 s post stimulus presentation. The percentage of correct answers given was very high. The pair-wise differences for the percentage of correct answers given across the four stimuli sub-types were not significant at calculated using a paired -test. In all tables and figures, coordinates , are reported in three-dimensional Talairach space [20].

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Capital refers to the statistical -value, which is also used to give the numerical values of local activation extrema. Finally, volumes of activation clusters are denoted in cubic millimeters mm. To isolate the entire network of regions involved in the active tasks, we compared the blood-oxygen-level-dependent BOLD activity for each of the four formulae conditions and the baseline trials no task.

Significant brain activation was observed for all formulae compared to the baseline, in an extended fronto-parietal network, which included the parietal, dorsal occipital, inferior temporal, premotor and prefrontal cortices, as well as the insula bilaterally. Further, we found subcortical structures such as the basal ganglia and the thalamus to be involved see Figure 1. These subcortical structures are known to be functionally and structurally linked to the prefrontal cortices [21] , [22] , [23] , [24].

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The figure shows the boundaries of activation clusters separately for each mathematical condition compared to baseline, corresponding to uncorrected and mapped onto a reference brain single subject. The colors represent: true algebraic light blue , false algebraic white , true arithmetic yellow and false arithmetic red conditions. A Top row from left to right: coronal , sagittal and axial section. B Bottom row from left to right: coronal , sagittal and axial section. Having defined the base network, we then probed how it was specifically activated with respect to the domain algebra vs.

To allow a direct comparison between semantic domains which is independent of possible different truth values, we compared the true algebraic and the true arithmetic conditions. We assumed that the BOLD signals in the early period would mainly reflect transient encoding activations, whereas the long period would be dominated by control, arithmetic-logical and maintenance operations. For the early period, i. In the parietal cortex, an extended region was activated, which involved the ventral parts of the precuneus PCU and the adjacent portion of the dorsal posterior cingulate area BA 31 , but did not extend into the superior parietal lobule SPL.

In addition, the right middle temporal gyrus BA 39 , the left fusiform gyrus BA 37 , and the bilateral parahippocampal gyri PhG were more active for algebra than for arithmetic. Further, we found the basal ganglia including the caudate nucleus and the globus pallidus and the thalamus to be significantly more involved see Table 1 , Figure 2A.

The color bar indicates -values uncorrected and applies to both A and B. For the long period, i. In the posterior part of the brain, we found the ventral left and right inferior temporal gyrus ITG; BA 37 , the dorsal left and right inferior parietal lobuli, the precuneus, and the right superior parietal lobule SPL; BA 7 to be more activated for algebra than for arithmetic see Table 2 and Figure 2B.

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The separate analyses for the two time windows suggested that the dorsolateral prefrontal cortex BA 9 , the medial and middle frontal cortex BA 8, BA 46 , and the inferior frontal gyrus BA 47 only come into play during a later processing stage; a stage when the actual truth value is calculated or re-evaluated.

In a next step we focused on the time course of activation in 10 regions of interest ROI. These regions had previously been discussed to be involved in processing formulae BA 10, BA 6, BA 47, BA 7 , reasoning BA 47 , and cognitive control DLPFC , in reading fusiform gyrus and in mediating information thalamus in the literature [4] , [7] , [8] , [25] , [26] , [27].

The average time course of the hemodynamic response for these ROIs for all 15 participants, and for all five conditions was evaluated. In such an analysis, differences in time to peak can be interpreted as differences in processing speed, whereas differences in amplitude indicate the amount of involvement of the particular region. Within each region, guided by the specifications we took from the literature, e. All cubes measured mm and were made out of the 26 adjacent voxels around each center.

The color bar indicates -values uncorrected. The curves represent percent signal changes associated with the true algebraic blue , false algebraic red , true arithmetic yellow , false arithmetic green and baseline wine red conditions, averaged over a volume within each anatomical region and subjects. The x-axis shows time measured from stimulus onset in seconds and the -axis shows the normalized percent signal change. The percentage change of the BOLD signal for every time series of each individual subject was evaluated for a period of 9 s after stimulus onset, and interpolated with an interval of ms.

## Subject:Mathematical logic

Then the individual time series were averaged over all participants volume-wise. Subsequently, all time series were normalized, such that each one initiated at zero at the onset. The time course of activation across the 8 seconds informs on the involvement of the 10 different brain regions for the true and false algebraic and arithmetic conditions and the baseline.

While some of the regions do not show differences between the conditions neither in the early nor in the late period BA 10 others indicate a shift of involvement from the early to the late period. These shifts are discussed in more detail below. This fMRI study investigated differences in the processing of the truth value of structurally identical and syntactically well-formed hierarchical mathematical formulae of two different types: abstract algebra and integer numbers.

Therefore, the resulting activation differences in processing these formulae should be domain-specific and not structure-specific. Further, we focused on determining the base network active tasks vs. The human brain is organized into different and partially competing cortical networks [4] , [8] , [25] , [28] , [29].

This intrinsic organization has to be taken into account in order to interpret the fMRI activations recorded in the present study. Resting-state functional connectivity analysis rs-fcMRI has proven to be particularly efficient in identifying large-scale polysynaptic cortical circuits, some of which have been defined in earlier work. According to Fox et al. Data are displayed on the lateral, medial, and dorsal surfaces of the left and right hemispheres. Figure adapted from Vincent et al. The DAS, consisting of portions of the intraparietal and superior frontal cortices, is involved in goal-directed top-down selection, whereas the HCMS, a ventral fronto-parietal network, subserves the bottom-up detection of salient or unexpected stimuli.

Further, the HCMS has a role as an alerting system for the DAS in cases where the perceived signals are outside the current focus of attention [28]. An extension of the attention and perception framework [28] to episodic memory was proposed by Cabeza et al. According to Cabeza and colleagues, the superior parietal cortex SPL , as part of the DAS, should subserve top-down memory retrieval, search and verification, and the inferior parietal cortex IPL , as part of the HCMS, should be engaged in high-confidence recollection.

The interaction between attention and memory, as a possible fundamental mechanism underlying working memory WM , is increasingly being investigated. Bledowski et al. The combined results of a number of studies [28] , [29] , [31] , [32] lead to the conclusion that WM emerges from the interaction of neuroanatomically dissociable components that are part of superordinate structures such as the DAS and the HCMS. The third network found, the frontoparietal control system FPCS [29] , is neuroanatomically located between the DAS and the HCMS, and it includes the lateral prefrontal cortex, the anterior cingulate cortex and the inferior parietal lobule.

These regions have previously been linked with cognitive control and decision-making [4] , [8] , [9] , [11] , [33] , [34]. The functional role of the FPCS is assumed to integrate information from the anti-correlated dorsal-attention and hippocampal-cortical memory systems [29] , [30].