The Role of Notations in Mathematics

The terms of a mathematical problem become precise and concise if they are expressed in an appropriate notation, therefore notations are useful to mathematics. But are notations only useful, or also essential? According to prevailing view, they are not essential. Contrary to this view, this paper argues that notations are essential to mathematics, because they may play a crucial role in mathematical discovery. Specifically, since notations may consist of symbolic notations, diagrammatic notations, or a mix of symbolic and diagrammatic notations, notations may play a crucial role in mathematical discovery in different ways. Some examples are given to illustrate these ways.

     

Capacity and contextual constraints on product activation: Evidence from task-irrelevant fact retrieval

Three experiments tested the limiting conditions of multiplication facts retrieval in a number-matching task (LeFevre, Bisanz, & Mrkonjic, 1988). By presenting two digits as cue and by requiring participants to decide whether a subsequent numerical target had been present in the pair, we found interference when the target coincided with the product of the cue digits. This was evidence for obligatory activation of multiplication facts. Also, we showed that multiplication facts retrieval occurred even in the absence of any arithmetic context (i.e., a multiplication sign between the cue digits) and did not require processing resources (i.e., the process met the capacity criterion of automaticity; Jonides, 1981), whereas manipulation of the spatial relation between the two operands (cue digits) negatively affected retrieval. The present work appears to be unique in the context of previous similar studies on mental calculation, which invariably adopted an arithmetic task as the primary demand. We identify this difference as the reason for the failure of all previous studies in revealing independence of multiplication facts from attentional resources. Furthermore, we suggest the application of a contextual definition of automaticity to this kind of retrieval, given the fact that it might depend both on association strength and on contextual setting variables.     

2-Deoxy-d-Galactose Effects on Passive Avoidance Memorization in the Rat

2-Deoxy-d-galactose (do-gal) hinders glycoprotein fucosylation. This compound was intracerebroventricularly administered to male adult Wistar rats in order to assess whether it could exert amnesic effects on a passive avoidance response (PAR) to be learned in the light–dark box apparatus. Three experiments were performed. In the first, do-gal was administered immediately after the acquisition trials at three dosages (2, 4, and 8 μmol). It was found that only the 4-μmol dosage was followed by PAR disruption. In the second, do-gal was administered at the adequate dosage (4 μmol) either 30 min before the acquisition trial or 30 min before retrieval testing. It was found that only the preretrieval administration was followed by PAR impairment. In the third, do-gal (4 μmol) was administered in postacquisition, at increasing postacquisition delays (0.25, 1.5, 4, and 6 h). It was found that there was PAR disruption only after do-gal administration at the shortest (0.25 h) delay. The results confirm that in the rat, glycoprotein fucosylation is involved in some of the phases of memory trace processing, and they are discussed in relation to other findings in the rat and the chick.     

Learning capacities during sleep: A preliminary study in a group of schizophrenic inpatients

The relationship between sleep and learning processes is analysed in a sample of schizophrenic patients, starting from more recent hypotheses about the function of REM sleep in learning and memory processes. This is done by means of two experiments: in the first AA. evaluate the possibility to elicit a simple motor conditional reflex acquired during daytime in different sleep stages. With the second experiment daytime learning performances are evaluated with and without a reinforcement administered during REM sleep. Results for the first experiment underline a qualitative difference between REM and nREM sleep in a reflexological perspective. In nREM sleep the conditional response is better maintained than in REM sleep. The second experiment confirms the possibility to improve daytime learning performances after an additional presentation of learning material in REM. The joint study of sleep abnormalities and learning and cognitive impairment in schizophrenic patients is finally suggested.     

Selective deficit of conceptual structures in aphasia: Class versus thematic relations

Conceptual structures in aphasia were investigated by means of a nonverbal test. The understanding of class and thematic relationships was demonstrated to be selectively impaired according to varieties of aphasic deficit. Broca's aphasics showed difficulty evaluating a thematic relationship and did not differ from normal controls as far as class relationships were concerned. The opposite happened for Wernicke's aphasics whose defect seems to be selectively restricted to class relationships.     

Selective semantic-lexical impairment of language comprehension in right-brain-damaged patients

Fifty right-brain-damaged patients and thirty-nine normal controls were given two tests of semantic-lexical discrimination (auditory language comprehension and reading comprehension) and a test of phoneme discrimination. The aims of the research consisted in checking: (A) if right-brain-damaged patients are significantly more impaired than normal controls on the tasks of semantic discrimination even when the influence of associated variables (such as unilateral spatial neglect or general mental impairment) is ruled out; and (B) if semantic-lexical disorders of right-brain-damaged patients are selective or if they are associated with disorders of phoneme discrimination. Results have shown that right hemispheric lesions consistently impair semantic-lexical discrimination but do not hamper phoneme discrimination. Some implications of these results are discussed.     

Newborns' preference for faces: what is crucial?

Three experiments investigated whether the presence of more elements in the upper part of a configuration (i.e., up-down asymmetry) plays a role in determining newborns' preference for facelike patterns. Newborns preferred a nonfacelike stimulus with more elements in the upper part over a nonfacelike stimulus with more elements in the lower part (Experiment 1), did not show a preference for a facelike stimulus over a nonfacelike configuration equated for the number of elements in the upper part of the configuration (Experiment 2), and preferred a nonfacelike configuration located in the upper portion of the stimulus over a facelike configuration in the lower portion of the pattern (Experiment 3). Results demonstrated that up-down asymmetry is crucial in determining newborns' face preference.     

Nonspatial attentional shifts between audition and vision

This study investigated nonspatial shifts of attention between visual and auditory modalities. The authors provide evidence that the modality of a stimulus (S1) affected the processing of a subsequent stimulus (S2) depending on whether they shared the same modality. For both vision and audition, the onset of S1 summoned attention exogenously to its modality, causing a delay in processing S2 in a different modality. That undermines the notion that auditory stimuli have a stronger and more automatic alerting effect than visual stimuli (M. I. Posner, M. J. Nissen, & R. M. Klein, 1976). The results are consistent with other recent studies showing cross-modal attentional limitation. The authors suggest that such cross-modal limitation can be produced by simply presenting S1 and S2 in different modalities and that central processing mechanisms are also, at least partially, modality dependent.     

Cross-modal re-mapping influences the Simon effect

Tagliabue, Zorzi, Umiltà, and Bassignani (2000) showed that one's practicing of a spatially incompatible task influences performance in a Simon task even when the interval between the two tasks is as long as 1 week. In the present study, three experiments were conducted to investigate whether such an effect could be found in a cross-modal paradigm, whereby stimuli in the two tasks were presented in different modalities. Subjects performed either compatible or incompatible mappings in an acoustic spatial compatibility task and, after an interval of 5 min, 24 h, or 7 days, performed a visual Simon task. Results show that the spatially incompatible mapping task affected performance in the Simon task: The Simon effect was absent for all three intervals. This pattern is similar to the results of the Tagliabue et al. study, in which both tasks were performed in the same (visual) modality. Our findings disprove possible explanations based on episodic/contextual effects and support the hypothesis of a long-lasting spatial remapping that is not modality specific.     

Derivation by prefixation in Slovenian: two aphasia case studies

The present study is concerned with two Slovenian-speaking patients who were asked to produce, in various tasks, verbs, nouns, and adjectives derived by prefixation with prepositions. Despite differences due to their specific linguistic difficulties, both patients' performance was characterized by the differential processing of prefixes and remaining components of complex words. Prepositions in prefixation were mostly preserved, and less frequently substituted, regardless of the numerous errors produced in the remaining portion of the words. These errors seem clearly determined by the morphological structure of the words and therefore appear to be authentic morphological errors. These findings contribute to the theoretical debate on mental lexical representation, speaking in favor of a morphological decomposition in processing of prefixed complex words at different processing levels.