BRAIN AND COGNITION 17, 102-115 (1991)

Acalculia: An Historical Review of Localization

HELEN J. KAHN

University of Vermont

AND

HARRY A. WHITAKER

Universite’ du Quebec d Montrkal, Canada

This article reviews the brain localization of calculation disorders (acalculia) beginning with Gall’s claim in the early 19th century for a “center” of calculation. A renewed interest in the subject arose around the time of Henschen during the first quarter of the 20th century. A summary of the cases of acalculia since Henschen leads to the conclusion that regardless of the functional modular nature of calculation ability, there is neither a localized region nor a specific hemisphere uniquely underlying the disorder. o 1991 Academic PE.S, 1~.

The possibility that the mind is organized into modules has a long and distinguished history; however, the possibility that the brain may be mod- ularly organized is a more recent idea. In his book on the modularity of the mind, Fodor (1983) introduces a thoughtfully delineated set of criteria for modular organization. He begins with the observation that Input Sys- tems are Modular. If one asked, ‘What are input systems and how many of them are there?‘, a classical answer would be: ‘There is an input system for each of the traditional sensory modes, e.g., sight, hearing, touch, taste, and smell’; however, Fodor actually proposes a model much like Franz Joseph Gall’s (1825) in which there are “highly specialized com-

Preparation of this article was supported by Fonds de la recherche en Sante du Quebec while the first author was a postdoctoral fellow at the Laboratoire Thtophile Alajouanine, Centre de Recherche, Center hospitalier Cote-des-Neiges, Montreal (Quebec). Support to the second author was by an operating grant from Fonds pour la formation de chercheurs et I’aide a la recherche (Quebec). We gratefully acknowledge the assistance provided by Dr. Brigitte Stemmer for the interpretation of German sources cited in this article. Requests for reprints and correspondence should be addressed to Helen J. Kahn at the Department of Communication Science and Disorders, University of Vermont, Burlington, VT 05405.

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0278-2626191 $3.00 Copyright 0 1991 by Academic Press. Inc. All rights of reproduction in any form reserved.

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putational mechanisms” which “generate hypotheses about the distal sources of proximal stimulation”; “ The specialization of these mechanisms consists in constraints either on the range of information they can access in the course of projecting such hypotheses or in the range of distal properties they can project such hypotheses about” (Fodor, 1983, p. 47). Fodor argues that there are nine properties of input systems which es- sentially define their modular nature:

(1) Domain Specificity

Examples of domain specific input systems are the recognition of faces, color vision, the perception of 3-dimensional figures, and a computational system that assigns grammatical descriptions to an utterance.

There are distinct psychological mechanisms that correspond to distinct stimulus domains (as in Gall’s system, according to Fodor); a modern example is categorical perception of speech versus perception of nonlin- guistic sounds. The domain may be “eccentric” (p. 52) in that the stimuli must satisfy a very specific set of criteria in order to trigger the module.

Fodor (1983) argues that there are two ways of considering mental structure as functional architecture: the first is the traditional faculty psy- chology of, for example, Plato, in which the mental faculties are indivi- duated by reference to their typical effects, e.g., they are functionally individuated. Fodor calls this approach horizontal. Typical horizontal fa- culties would be judgment, language, or memory. Using Fodor’s example,

faculty of judgment might get exercised in respect of matters aesthetic, legal, scientific, practical or moral The point is that, according to the horizontal treatment of mental structure, it is the self same faculty of judgement every time (pp. 11-12). Horizontal faculty psychology has been with us always; it seems to be the common sense theory of the mind (p. 14).

By contrast, the vertical tradition in faculty psychology has specifiable historical roots. It traces back to the work of Franz Joseph Gall . . there is a bundle of what Gall variously describes as propensities, dispositions, qualities, aptitudes and fundamental powers. . . . in the case of what Gall sometimes calls the intellectual capacities, it is useful to identify an aptitude with competence in a certain cognitive domain; in which case, the intellectual aptitudes, unlike the horizontal faculties, are distinguished by reference to their subject matter. It is of central importance to understand that, in thus insisting upon domain specificity, Gall is not simply making the conceptual point that if music is distinct from mathematics, then musical aptitude is correspondingly distinct from mathematical aptitude. Gall is also claim- ing that the psychological mechanisms which subserve the one capacity are different, de facto, from those that subserve the other. I take it that this claim is the heart of Gall’s theory (pp. 14-15).

(2) Mandatory Operation

The operation of the input system is mandatory both in the sense that when the right stimulus is present, the module must recognize it and in the sense that other control processes cannot prevent the module from

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functioning. Fodor observes that you cannot help hearing a sentence when you hear an utterance in a language that you know.

(3) Limited Access

There is only limited central access to the mental representations that input systems compute. One does not have equal access to all the different levels of processing involved in the module, the representations that con- stitute the final consequences of input processing are fully and freely available to the cognitive processes that eventually result in overt behavior.

(4) Fast Operation

Input systems are fast, which also implies that they are efficient.

(5) Information Encapsulation

Input systems are informationally encapsulated, that is, they do not use information from other modules or from other sources, as, for example, in feedback. Fodor notes that this proposition is subject to further in- vestigation.

(6) Shallow Output

Input analyzers have “shallow” outputs, that is, they are “phenome- nologically” salient; an example is that a module in the visual system would access color and shape, but not access, directly, information about molecular structure (Fodor, 1983, pp. 93-96).

(7) Fixed Neural Architecture

Input systems are associated with fixed neural architecture that is, in the sense of “privileged paths of informational access” which facilitate fast and direct access (Fodor, 1983, p. 98). Although Fodor himself does not strongly argue that this implies a consistent geographic or morpholog- ical localization in the brain, it has certainly been so construed by the most classical neurologists and neuropsychologists.

This may be contrasted with the position of the so-called connectionists who take the position that the neural architecture underlying all such faculties is generalized and distributed.

(8) Specific Breakdown Patterns

Input systems exhibit characteristic and specific breakdown patterns; the syndromes of the agnosias and the aphasias are classic evidence for this characteristic. Intuitively, acalculia provides additional support, but as we shall see, not without controversy.

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(9) Fixed Ontogeny

The ontogeny of input systems exhibits a characteristic pace and se- quencing. Fodor suggests that aspects of language acquisition and early visual capacities of infants are consistent with a modular sequencing. The neural mechanisms subserving input analysis develop according to specific, endogenously determined patterns under the impact of environmental releasers.

Following Fodor (1983) it is currently in vogue to point out that Franz Joseph Gall’s early 19th century theory of functional brain modules pre- saged contemporary neurofunctional theories. However, theories that the cortical substance of the brain is divisible into structurally and functionally independent units antedate Gall. We find relatively modular-sounding statements about brain-behavior correlations in the 17th century work of Thomas Willis (1664/1965; cited in Clarke & O’Malley, 1968) and in the 18th century work of Georg Prochaska (1784/1851) and Johann C. Lavater (1789-1810). Both Prochaska and Lavater were acknowledged by Gall. Nonetheless, as Fodor (1983), Young (1970) Ackerknecht (1958), Fancher (1990), Harrington (1987), Boring (1957), and many others have pointed out, Gall’s neuropsychological model was the first well-developed theory of brain organization which, although unusual in respect to some of the brain modules proposed, was nonetheless impeccably modern in other respects, e.g., its reliance on clinical-pathological correlation (Whi- taker & Grou, 1991). Among the 27 independent mental faculties given their own brain site in Gall’s system was “the source of numbers, math- ematics” which to Gall was discernible on the skull lateral and posterior to each orbit, thus approximately an inferior frontal lobe location. The Organ of Language, which Gall had originally divided into two separate faculties which he called “speech signs” and “word signs,” was located directly behind the orbit and thus more anterior and more inferior to the Organ of Calculation, although still close to it.

Through the good offices of Jean Baptiste Bouillaud, colleague and disciple of Gall, the idea of a frontal localization for language was actively, if not persuasively, argued following Gail’s death in 1828. On several separate occasions between 1830 and 1860, at formal meetings of the orthodox scientific and medical societies in France, Bouillaud debated his peers on the matter of cortical localization of function. On each occasion, Bouillaud’s point of reference was the work of Gall and the primary example was the frontal localization of language, all prior to the more celebrated publications of Pierre Paul Broca commencing in 1861.

In Great Britain, and to a lesser extent in the U.S., modularity was epitomized in the discredited work of the phrenologists who, by the end of the 19th century, had developed very elaborately detailed models of skull-brain-behavior relations, quite striking in contrast to those of Fer-

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tier, Bastian, Liepmann, Lichtheim, Charcot, Grasset, and Dejerine. Al- though the empirical basis of the phrenological models had strayed from its original sources in normal and pathological behavior (Whitaker & Grou, 1991), the phrenologists never lost sight of the Organ of Calculation. It would be interesting, although aside from our goals in this paper, to speculate on why neither the phrenologists nor the orthodox neurologists and psychologists of the 19th century chose to analyze calculation into its component parts; assuredly they must have observed acalculic patients with, for example, preserved addition and impaired multiplication. Pro- posals for a comprehensive cognitive model of calculation are quite recent (McCloskey, Caramazza, & Basili, 1985); since its focus has been the description of normal cognitive functioning based on data from brain- damaged subjects and not on lesion localization per se, we will not review that work in detail here. However, it is interesting to note that in the articles where site of lesion has been mentioned, mathematical dysfunction has arisen from frontal, temporal, parietal, and temporoparietal lesions in the left hemisphere as well as in the right hemisphere (McCloskey, Aliminosa, & Sokol, 1991; McCloskey, Caramazza, & Basili, 1985; McCloskey, Sokol, & Goodman, 1986; Sokol & McCloskey, 1988).

Salomon E. Henschen appears to have been the first individual to use the term “acalculia”, the inability to perform the operations concerned with arithmetic. It was thought to occur as the primary symptom of a focal lesion; often it is associated with other disorders such as aphasia or agnosia. Prior to Henschen’s work, Lewandowsky and Stadelmann (1908) reported one of the first published cases of an acquired calculation dis- order; there were a few other contributions to the localization of acalculia. Peritz (1918) argued for a “calculation center” in the left angular gyrus; Sittig (1917) considered the retrorolandic area in the left hemisphere to be involved in calculation. Considering the question of the brain repre- sentation of arithmetic function, Henschen (1919) offered a neuropsy- chological model, replete with suggested localization, for acalculia; he believed that modern usage of numbers could be traced to generations of education and continuous training which “reformed” existing brain cells. Functional centers were formed by the differentiation of those nerve cells involved in habitual behavior. Henschen also appears to have ac- cepted the Lamarckian view that these functionally “reformed cells” were inherited through the generations.

The grand debate over geographic functional “centers” in the brain has been a continuous feature of neuropsychology since Gall first proposed them. Focussing just on the question of acalculia, however, the major opposition to a center for calculation came from Luria (1966) who argued against “centers” of mathematical function (and other global “centers” as well) by observing that there was more than one type of disturbance of number concepts and mathematical operations. Luria noted that these

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disturbances could be associated with brain damage in many areas of the brain, e.g., inferoparietal and parietal-occipital lesions. The available facts today seem to support Luria’s view. In this regard, it is interesting to note that Henschen (1919) believed that aphasia localization theories had led to erroneous assumptions for language as well as calculation. He believed that aphasiologists of the late 19th and early 20th century were too busy building constructs without first investigating whether such con- structs corresponded to anatomically distinct localizations. Instead, Henschen advocated the following, “bottom- up” method for such re- search. He urged his colleagues to begin their research with patients who had the simplest and smallest anatomical lesions and then to investigate the clinical consequences of these lesions. After fully characterizing the behavioral consequences of these small lesions, inquiry then would pro- ceed to more extensive lesions. He maintained that psychic functions such as language or calculation were composed of numerous components and that the question of localization applied not just to the function, but rather to each component of that function, antedating current modular ap- proaches to cognitive architecture (Shallice, 1988).

Nonetheless, Henschen accepted the classical neurological model of language as an anatomical-functional system which contains two receptive and two motor foci. He used the term psychischer Vet-band or psychic union/formation, believing that lesions in one point of the system would be reflected to other points of the system, resulting in several foci which were affected at the same time. The foci and their associational connec- tions worked closely together but each focus was somewhat autonomous. If any single focus was disturbed through insult to the brain, language capability would be deficient in some form. Consistent with his view of language and brain relations, Henschen believed that calculation incor- porated a receptive and a motor form. He distinguished between numerals (number in its lexical form) Zuhlen and figures (number in its numerical form) Ziffern. Both forms received their appropriate information through acoustic and optical pathways. Henschen concluded that with existing evidence it was possible to characterize the psychic processes involved in talking and in calculating as functionally dissimilar in character (one cannot refrain from reminding that Gall made the same claim). To support this model, Henschen cited 122 cases of word blindness in which 71 of these subjects could still read figures, believing that there must be separate centers for word reading and figure reading. Moreover, he reported 51 subjects who demonstrated both word and figure blindness, seemingly supporting his contention that these two centers were in close proximity in the brain. The theory that symptom cooccurrence implies geographic proximity goes back at least to the phrenological case studies of Alexander Hood in 1824 (Whitaker & Grou, 1991).

In order to properly localize acalculia, Henschen argued, the motor

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centers must be differentiated from the sensory centers. The motor center, for instance, would be identified by the fact that a lesion in the 3rd convolution of the left frontal lobe would affect the continuous counting of numbers or the ability to write figures. The sensory center, on the other hand, would be characterized by figure blindness as a consequence of a form of visual agnosia (Seefenblindheit) caused by a lesion in the left angular gyrus extending to the intraparietal fissure. Henschen found figure agraphia (inability to write figures) to be more difficult to localize since only a few clinical cases were known at that time. In a later paper, Henschen (1925) advanced the position that extensive lesions of the left hemisphere which resulted in aphasia, figure blindness, figure agraphia, and acalculia only infrequently produced number deafness. Finally, Henschen (1926) implicated right hemisphere involvement in acalculia, although he placed no calculation centers in the nondominant hemisphere. In some cases, Henschen thought that the right hemisphere participated in calculation in a compensatory manner but only when the lesion in the left hemisphere was very large.

In 1908, Lewandowsky and Stadelmann presented a case study in which a left occipital hematoma resulted in difficulty with addition and sub- traction as well as for regrouping two- and three-place numbers. They concluded from this evidence that the left occipital lobe contained the center for calculation. However, they cautioned that even with similar lesions, one does not see the same acalculic deficit in each individual, because different individuals interpret the incoming stimuli in his/her own unique way.

Hans Berger (1926) reviewed 18 cases from his patients’ files who suf- fered from some type of calculation disorder; his review led him to identify primary acalculia (which develops independently from other impairments) and secondary acalculia (which develops in conjunction with disturbances of memory, attention, language, etc.). In that paper, Berger discussed three cases in depth, noting the site of the lesion in each case. The first case was a patient who was unable to divide; biopsy revealed a left hemisphere glioma reaching into posterior temporal and occipital lobes, with both the angular gyrus and supramarginal gyrus intact. Berger pointed out that this finding was in contrast to Henschen (1919, 1926), who had thought the angular gyrus to be always implicated in calculation. The second case involved a patient who could not subtract or divide, although the appearance of symptoms was intermittent. This patient had a glioma in the left paracentral lobe, invading neighboring funicular gyrus, corpus callosum, and left occipital lobe. Berger’s third case was a 32-year-old woman who had sustained a total loss of division and multiplication. She presented with a left thalamic glial sarcoma. Large areas of the left hemi- sphere were involved, with the lesion extending throughout most of the temporal lobe and occipital lobe. In addition to her calculation deficit,

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this patient (No. 3) also was aphasic, amnestic, and had great difficulty following written instructions. Berger thought it was important to note that in all three cases, there was not a total loss of calculation. Like Henschen, Berger rejected the concept of a single calculation center cor- responding to a specific brain site; he argued that there were components working together which orchestrate the different mathematical functions.

Head (1926) made brief reference to calculation disorders in his book Aphasia and Kindred Disorders of Speech. He considered each variety of aphasia to be associated with a distinct form of arithmetic disorder, anal- ogous to certain recent claims that the prosodic and aphasic disorders parallel each other anatomically and functionally. Head cited Pick (1925) in agreeing that the failure to manipulate numbers was likely to be cor- related with temporal-parietal lesions and would occur in association with defective perceptions and comprehension of shape (Gestalt Auffassung). A few years later, Krapf (1937) noted that if acalculia could be divided into more than one genetically and structurally different type, then a satisfactory localization might be found. With a primary emphasis on the sensory dimensions of acalculia, Krapf argued for a lesion localization in the occipital lobe but cautioned other investigators to also consider the dynamic or motoric aspect of acalculia, since calculation showed char- acteristics of action (Handlungscharakter). Parietal acalculia should show the same motor characteristics and be differentiated on the same basis as the now-familiar apraxias, thus, he proposed the terminology ideokinetsche and konstruktive acalculia. Krapf believed that the konstruktive form was the purest form of acalculia and proposed a lesion localization in the angular gyrus.

Goldstein’s (1948) study of language disorders argued that acaculia typically followed lesions in the parietal-occipital region and occasionally the frontal lobes. He declared that most published cases of acalculia had described lesions in the occipital lobe, citing Lewandowsky and Stadel- mann (1908), Henschen (1919), and Peritz (1918). In these cases, Gold- stein believed the calculation disorders arose from a visual deficit, even though dissociations for cases such as visual alexia in which a patient can read numbers but not letters would seem to suggest otherwise. Such dissociations did not occur, according to Goldstein, because there were different “centers” for letters than for numbers, but rather because num- bers are more concrete and carry intrinsic meaning of their own whereas letters do not. In the case of visual alexia, for instance, there may also be a disturbance of visualization affecting a patient who normally relies on such cues for the multiplication table when solving these types of problems. Unlike Henschen (1919) and later Hecaen and colleagues (1961), Goldstein believed that no proof existed showing that the minor hemisphere (typically, the right) participates in calculation.

In 1961, two articles were published on acalculia, one by Hecaen,

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Angelergues, and Houillier, and the other by Cohn. Both articles noted that the majority of acalculia cases exhibited bilateral lesions primarily in parietal-occipital regions. HCcaen, et al. (1961) argued that in all cases, the parietal lobes constituted the central zone for calculation and that both sides of the brain were needed to integrate calculation processes. However, in the cases cited, lesions in the left hemisphere were generally parietal while right hemisphere lesions were generally temporal or occip- ital; a small number had lesions in the parietal lobe. In support of their contention, HCcaen et al. (1961) cited Potzl (1952), who had emphasized the importance of parietal integration. Luria (1966) also noted that the parietooccipital region, in addition to its role in linguistic operations, is equally critical for arithmetic. He believed that a disturbance in mathe- matical operations could be used as a diagnostic tool, suggesting com- promised parietooccipital areas. In support of the parietal hypothesis, Luria perceived the historical roots of number operations as a “geometry” in which the development of spatial concepts led to proficiency in math- ematics. He related this idea to Piaget’s assumption that the development of cognition in the child requires action and the manipulation of external field space establishes reference points for math. Such reference points would be compromised in the event of a parietooccipital lesion. In such cases, one might find only the multiplication tables preserved because they represent a well-established speech-motor stereotype.

A woman with a posterior left hemisphere lesion who was unable to identify her own fingers or the fingers of others was first described in 1924 by Josef Gerstmann. A few years later, he saw another patient with finger agnosia who was also unable to write (Gerstmann, 1927). In 1930, Gerstmann described a patient with both of the above deficits as well as impaired calculation and right-left disorientation (Gerstmann, 1930). Ger- stmann’s hypothesis was that due to a “body schema” deficit arising from a lesion in the angular gyrus in the left or the language-dominant hem- isphere, the four behavioral symptoms cooccurred as a distinct syndrome. Although earning Gerstmann his eponym in medical history, Gerstmann’s Syndrome has been the focus of considerable dispute. Some have simply denied the validity of the syndrome altogether by claiming that even when all the behavioral components of the syndrome were present, other deficits were evident as well. For example, Heimburger, DeMeyer, and Reitan (1964) examined a large number of patients, finding that more than 20% of the sample exhibited only one of the deficits of the syndrome. Only 5% of the patients had all four deficits, but they were aphasic as well. In the latter group, lesions were typically larger than just the left angular gyrus, often involving the temporal lobe. Similarly, Poeck and Orgass (1966) found that symptoms of acalculia were more often associated with constructional apraxia than any of the other components of Gerstmann’s Syndrome, while finger agnosia associated with oral apraxia. Just two

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patients exhibited three of the four deficits; none had a lesion in the dominant hemisphere. Critchley (1967) argued that the presence of other deficits such as constructional apraxia was, in fact, due to a deficit in spatial organization, not body schema. Benton (1961, 1977) pointed out that there was not enough empirical evidence supporting the existence of a full syndrome, since many of the components associated more frequently with aphasia than with the other components of Gerstmann’s Syndrome. The controversy surrounding the Gerstmann’s Syndrome as an autono- mous neurological disorder is still an unresolved issue, with some studies supporting it (Assal & Jacot-Descombes, 1984; Benson & Denckla, 1969) and others rejecting it (Benton, 1977).

There are a few cases reported in the literature in which acalculia was the only intellectual function impaired. For instance, following the clas- sifications of acalculia proposed by HCcaen et al. (1961), Benson and Weir (1972) reported a case of acquired anarithmetia (impaired calculation). This case had a focal lesion in the left parietal lobe which contradicted, among others, Hecaen’s model insisting that anarithmetia would only be found in cases of diffuse cortical damage. Thus the site of lesion was not considered unusual, only the type of acalculia. The patient’s spontaneous speech was fluent and there were no difficulties in comprehension or repetition. He was able to read single letters, words, and numbers, but did show errors in writing. Benson and Weir’s patient thus presented with an acquired acalculia not belonging to either the aphasic or the visuospatial varieties.

In 1977, Collignon, Leclerc, and Mahy examined calculation in 26 pa- tients with both left and right hemisphere lesions and concluded that calculation can be localized not only to either hemisphere but also to different brain regions. In the majority of cases reported, acalculia was associated with other disorders-constructional apraxia, aphasia, alexia, and agraphia-following unilateral lesions, predominantly in the parietal lobe but also in all other brain regions. Grafman, Passafiume, Faglioni, and Boller (1982) also tested both right and left hemisphere brain-damaged patients. They administered a range of calculation tasks including reading and writing numbers, magnitude comparison of numbers, and simple and complex (requiring regrouping) addition, subtraction, multiplication, and division problems. The results showed that most patients could complete the number tasks without error except for multiplication and division. The patients with left hemisphere brain damage performed worse than those with right hemisphere disease. Grafman et al. (1982) also classified patients according to lesion site within the hemisphere and found that overall, patients with left posterior lesions were least accurate on the calculation tasks followed by the right posterior lesion group. The most accurate groups were those patients with left and right anterior lesions. Dahmen, Hartje, Bussing, and Sturm (1982) found that left hemisphere

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lesioned patients (including those with both Broca’s and Wernicke’s aphasia) performed significantly worse on arithmetic tasks than a right hemisphere retrorolandic lesion group. In addition, the Wernicke’s aphasia patients whose lesions were predominantly left temporoparietal, made more errors, in general, on the tasks than the Broca’s aphasia group whose lesions were in the left frontal lobe. Jackson and Warrington (1986) also tested right and left hemisphere brain-damaged patients, although they did not specify the unambiguous sites of the lesions. The two brain- damaged groups were given an oral test of addition and subtraction prob- lems (Graded Difficulty Arithmetic Test) as well as the Wechsler Adult Intelligence Scale (WAZS) Arithmetic subtest and the WAZS Digit Span subtest. The left hemisphere brain-damaged group produced significantly more errors than the right hemisphere brain-damaged group on the Graded Dificulty Arithmetic Test. Interestingly, a large proportion of the left hemisphere group showed normal performance on the WAZS Digit Span subtest while there was no difference between the two groups on the WAZS Arithmetic subtest.

In recent years it has been demonstrated that localization for number processing and calculation is not confined to cortical structures. Ojemann (1974) interrupted backward counting and simple calculations (serial sub- traction) of a patient undergoing a therapeutic stereotaxic thalamotomy operation, when he stimulated both the right and the left ventrolateral thalamus. During left thalamic stimulation, the rate of counting was ac- celerated, accompanied by a corresponding increase in errors in calcu- lation. Conversely, with right thalamic stimulation, counting slowed but calculation was still marked by significant errors. Ojemann suggested that these findings demonsrate that either the thalamus is directly involved in number operations or pathways ascending to the cerebral cortex to, from, or through the thalamus are involved in the calculation system. Whitaker, Habinger, and Ivers (1985) reported a patient with a left, lenticular cau- date lesion who was unable to orally multiply. Likewise, Corbett, McCusker, and Davidson (1986) reported a 60-year-old woman with a left subcortical infarct, implicating the caudate nucleus, putamen, the anterior limb of the internal capsule, and periventricular white matter. The patient could not subtract, multiply, or divide without error.

There are a few cases reported in which calculation abilities are disas- sociated from other intellectual functions. Anderson, Damasio, and Da- masio (1990) reported a case of a woman who developed both alexia and agraphia but showed no disturbances in language and number operations. Alexia with agraphia is relatively rare and generally occurs with nonfluent aphasia due to frontal lesions. Following surgery in the left dorsolateral frontal lobe (Brodmann’s field 6), the patient was unable to read or write letters, words, or sentences. However, all numbers were easily written and read and there were no errors in calculation. This would seem to

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imply that the neurological substrate for the reading and writing of num- bers is separate from that for letters and words. However, Henderson (1987) described a series of patients who all had left inferior occipital infarcts and showed severe alexia for both numbers and letters. This implies that both numbers and letters use some of the same pathways for the transfer of visual numeric information. Warrington (1982) reported a physician who became acalculic after a left posterior parietal lesion due to a hematoma. The patient’s knowledge of digits was intact as was his concept of magnitude and quantity. However, he was unable to correctly perform simple addition, subtraction, and multiplication problems.

Calculation is an interesting “faculty,” seemingly a prime candidate for the designation “modular” (Fodor, 1983; Shallice, 1988). Certain com- ponents, such as the multiplication table (up to 12 x 12, at least) appear to have an independent automatized representation, regardless of how one views their internal structure. There are clearly sets of rules for addition, subtraction, etc., which, although bearing some resemblance to linguistic rules, are most certainly independent of language. Yet, from the studies and papers just reviewed, it is evident that disorders of cal- culation-acalculia-may follow lesions in anterior and posterior brain, left or right hemisphere, and both cortical and subcortical structures. Although these facts do not disprove that the brain basis for the faculty of calculation is modular, they at least offer a formidable obstacle to the typical (common sense) conception or interpretation of modularity as having a discrete or independent brain substrate. We do not believe, however, that these facts render network or connectionist models any more viable. In fact, as Fodor (1983) may have meant to imply, the morphological or geographical argument is not likely to be the crucial piece of evidence in support of or in evidence against, modularity.

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