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.

     

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.     

Response strategies and the Simon effect

The study investigated whether the Simon effect, and its facilitation and interference components, shows up in reaction time (RT) or in movement time (MT), depending on the response strategy. Experiment 1 replicated a study by Hietanen and Rämä. Subjects had to press one of two lateralised keys in response to one of two stimuli. The stimuli were presented in the center (neutral condition) or to the left or right side (corresponding or non-corresponding conditions). To press the response key, a reaching movement was necessary, and both RT and MT were recorded. One group of subjects showed an RT facilitation effect and an MT interference effect. Another group of subjects showed both MT facilitation and MT interference effects. It was hypothesized that the two groups used different response strategies. In Exps. 2 and 3, the subjects were explicitly instructed to use the two strategies that were hypothesized for Exp. 1. The results showed that whether facilitation and interference manifest themselves in RT or MT depends on the response strategy adopted by the subjects.     

Mental representation of prepositional compounds: evidence from Italian agrammatic patients

The processing of Prepositional compounds (typical Neo-latin noun--noun modifications where a head noun is modified by a prepositional phrase, e.g., mulino a vento, windmill) was preliminarily studied with a group of six agrammatic aphasic patients, and, in more detail, with a further agrammatic patient (MB). Omission was the most frequent error type in naming, whereas in the other tasks (repetition, reading, writing, and completion) errors were mostly substitutions of the target preposition. This happened even with fully lexicalized compound forms, i.e., those forms where the linking preposition is syntactically and semantically opaque. These findings are interpreted in terms of a dual-route theory of lexical access to morphologically complex words.