Holistic representation of negative numbers is formed when needed for the task

Past research suggested that negative numbers are represented in terms of their components—the polarity marker and the number (e.g., Fischer & Rottmann, 2005 Fischer, M. and Rottmann, J. 2005. Do negative numbers have a place on the mental number line?. Psychology Science, 47(1): 22–32.  [Google Scholar]; Ganor-Stern & Tzelgov, 2008 Ganor-Stern, D. and Tzelgov, J. 2008. Negative numbers are generated in the mind. Experimental Psychology, 55(3): 157–163.  [Google Scholar]). The present study shows that a holistic representation is formed when needed for the task requirement. Specifically, performing the numerical comparison task on positive and negative numbers presented sequentially required participants to hold both the polarity and the number magnitude in memory. Such a condition resulted in a holistic representation of negative numbers, as indicated by the distance and semantic congruity effects. This holistic representation was added to the initial components representation, thus producing a hybrid holistic-components representation.     

Components representation of negative numbers: Evidence from auditory stimuli detection and number classification tasks

Past research suggested that negative numbers could be represented in terms of their components in the visual modality. The present study examined the processing of negative numbers in the auditory modality and whether it is affected by context. Experiment 1 employed a stimuli detection task where only negative numbers were presented binaurally. Experiment 2 employed the same task, but both positive and negative numbers were mixed as cues. A reverse attentional spatial–numerical association of response codes (SNARC) effect for negative numbers was obtained in these two experiments. Experiment 3 employed a number classification task where only negative numbers were presented binaurally. Experiment 4 employed the same task, but both positive and negative numbers were mixed. A reverse SNARC effect for negative numbers was obtained in these two experiments. These findings suggest that negative numbers in the auditory modality are generated from the set of positive numbers, thus supporting a components representation.     

Essai de représentation par des nombres réels d'une analyse infinite des notions individuelles dans une infinité de mondes possibles

The aim of this study is to try to make use of real numbers for representing an infinite analysis of individual notions in an infinity of possible worlds.As an introduction to the subject, the author shows, firstly, the possibility of representing Boole's lattice of universal notions by an associate Boole's lattice of rational numbers.But, in opposition to the universal notions, definable by a finite number of predicates, an individual notion, cannot (as Leibniz pointed out) admits this sort of definition, because each state of an individual subject is characterized by the values (present or absent, applicable or inapplicable) taken by an infinite number of predicates, each of whom may appear or disappear in the next state.The notion of degree of identification of an individual notion is then introduced and arithmetized by a rational number.As an individual notion can be defined by a convergent succession of degrees of identification, the real characteristic number of such an individual notion can be defined by the corresponding convergent succession of rational numbers, satisfying Cauchy's conditions for the convergence of successions.

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Essai de représentation par des nombres réels d'une analyse infinite des notions individuelles dans une infinité de mondes possibles

     

序成六虚上下,数生太极乾坤--今本《周易》卦序排列数学规律三探

今本<周易>序卦是一件完美的数学作品.序卦的分布规律体现于一系列的数列之中,但是,假如这些排列规律彼此间缺乏关联性,显得孤立、分散,便难以真正体现序卦排列的数学规律性.经过更深入的研究,笔者发现这些排列规律并非彼此分散、互不相干的,而是互有关联,整合成一个完美的统一体.序卦排列数学规律其鲜明的特点有四:其一是连续性,其二是周期性,其三是对称性,其四是关联性.连续性、周期性体现其简易性,对称性、关联性体现其统一性.简易性、统一性体现其完美性.     

Infants’ auditory enumeration: Evidence for analog magnitudes in the small number range

Vigorous debate surrounds the issue of whether infants use different representational mechanisms to discriminate small and large numbers. We report evidence for ratio-dependent performance in infants’ discrimination of small numbers of auditory events, suggesting that infants can use analog magnitudes to represent small values, at least in the auditory domain. Seven-month-old infants in the present study reliably discriminated two from four tones (a 1:2 ratio) in Experiment 1, when melodic and continuous temporal properties of the sequences were controlled, but failed to discriminate two from three tones (a 2:3 ratio) under the same conditions in Experiment 2. A third experiment ruled out the possibility that infants in Experiment 1 were responding to greater melodic variety in the four-tone sequences. The discrimination function obtained here is the same as that found for infants’ discrimination of large numbers of visual and auditory items at a similar age, as well as for that obtained for similar-aged infants’ duration discriminations, and thus adds to a growing body of evidence suggesting that human infants may share with adults and nonhuman animals a mechanism for representing quantities as “noisy” mental magnitudes.     

THERE'S MORE TO TIME THAN TICKING AWAY

Time is an element that each of us experiences in the core of our being. Yet it also is one of the great mysteries in our conceptual grasp of reality. The notion of time has therefore been reflected upon and explored by thinkers and scientists since ancient times. In this essay I relate the multiple ways in which Antje's Jackelén's scholarly and stimulating work Time and Eternity analyzes the historical, philosophical, theological, and scientific perspectives on the notion of time lived and its relation to the conceptual endless time that we call eternity, and offer some of my own contextual reflections on the topic.     

数字表征的表象赋义效应:来自数字线估计任务的证据

已有研究中数字线估计任务几乎都使用纯数字。本研究以二到六年级儿童为被试,采用纯数字任务和赋义数字任务来探索赋义表象对数字表征形式的影响。结果表明,对0~1000的数字赋义后,对数模型的解释力上升,而线性模型的解释力下降;表象大小对于赋义数字的估计影响显著,大表象赋义提高了对数模型解释力而降低了数字估计的准确性,小表象的影响比较微弱。     

Taurek, Numbers and Probabilities

In his paper, “Should the Numbers Count?" John Taurek imagines that we are in a position such that we can either save a group of five people, or we can save one individual, David. We cannot save David and the five. This is because they each require a life-saving drug. However, David needs all of the drug if he is to survive, while the other five need only a fifth each. Typically, people have argued as if there was a choice to be made: either numbers matter, in which case we should save the greater number, or numbers don't matter, but rather there is moral value in giving each person an equal chance of survival, and therefore we should toss a coin. My claim is that we do not have to make a choice in this way. Rather, numbers do matter, but it doesn't follow that we should always save the greater number. And likewise, there is moral value in giving each person an equal chance of survival, but it doesn't follow that we should always toss a coin. In addition, I argue that a similar approach can be applied to situations in which we can save one person or another, but the chances of success are different.     

五行、九宫与八卦--胡渭《易图明辨》"五行、九宫"说述评

胡渭对后人以"五行生成数"和"九宫数"为"河图"、"洛书"的来龙去脉进行了剖析,辨明了五行、九宫与八卦的关系,指出,春秋战国时期的阴阳五行之说,是一套独立的系统,它的五行生成之数说与<周易>中的天地之数说还是相矛盾的,也没和八卦发生任何关系."九宫数"原本与九数图的"河图"(或称"洛书")也毫无关系,也是后人硬把它们扯到一起的.胡渭的考证虽有待进一步的深入,但他认为五行数、九宫数与八卦、古河图原不相涉的说法,还是无可辩驳的.