Mathematics and Spiritual Interpretation: A Bridge to Genuine Interdisciplinarity

This article is a spiritual interpretation of Leonhard Euler's famous equation linking the most important entities in mathematics: e (the base of natural logarithms), π (the ratio of the diameter to the circumference of a circle), i (√‐1),1 , and . The equation itself (e^πi+1 = 0> ) can be understood in terms of a traditional mathematical proof, but that does not give one a sense of what it might mean. While one might intuit, given the significance of the elements of the equation, that there is a deeper meaning, one is not in a position to get at that meaning within the discipline of mathematics itself. It is only by going outside of mathematics and adopting the perspective of theology that any kind of understanding of the equation might be gained, the significant implication here being that the whole mathematical field might be a vast treasure house of insights into the mind of God. In this regard, the article is a response to the monograph by George Lakoff and Rafael Núñez, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being (2000), which attempts to approach mathematics in general and the Euler equation in particular in terms of some basic principles of cognitive psychology. It is my position that while there may be an external basis for understanding mathematics, the results are somewhat disappointing and fail to reveal the full measure of meaning buried within that equation.     

Who can escape the natural number bias in rational number tasks? A study involving students and experts

Many learners have difficulties with rational number tasks because they persistently rely on their natural number knowledge, which is not always applicable. Studies show that such a natural number bias can mislead not only children but also educated adults. It is still unclear whether and under what conditions mathematical expertise enables people to be completely unaffected by such a bias on tasks in which people with less expertise are clearly biased. We compared the performance of eighth‐grade students and expert mathematicians on the same set of algebraic expression problems that addressed the effect of arithmetic operations (multiplication and division). Using accuracy and response time measures, we found clear evidence for a natural number bias in students but no traces of a bias in experts. The data suggested that whereas students based their answers on their intuitions about natural numbers, expert mathematicians relied on their skilled intuitions about algebraic expressions. We conclude that it is possible for experts to be unaffected by the natural number bias on rational number tasks when they use strategies that do not involve natural numbers.     

The Fibonacci Life‐Chart Method (FLCM) as a foundation for Carl Jung's theory of synchronicity

Since the scientific method requires events to be subject to controlled examination it would seem that synchronicities are not scientifically investigable. Jung speculated that because these incredible events are like the random sparks of a firefly they cannot be pinned down. However, doubting Jung's doubts, the author provides a possible method of elucidating these seemingly random and elusive events. The author draws on a new method, designated the Fibonacci Life‐Chart Method (FLCM), which categorizes phase transitions and Phi fractal scaling in human development based on the occurrence of Fibonacci numbers in biological cell division and self‐organizing systems. The FLCM offers an orientation towards psychological experience that may have relevance to Jung's theory of synchronicity in which connections are deemed to be intrinsically meaningful rather than demonstrable consequences of cause and effect. In such a model synchronistic events can be seen to be, as the self‐organizing system enlarges, manifestations of self‐organized critical moments and Phi fractal scaling. Recommendations for future studies include testing the results of the FLCM using case reports of synchronistic and spiritual experiences.     

关于数字卦与六十四卦符号体系之形成问题

将见于商周器物上的数字卦与传逝文献的有关记载结合起来分析,商周时期应已存在用两个符号记写的六十四卦体系;周初陶拍所见易卦与传本《周易》相同的非覆即反的排列方法,也表明当时的筮书应是用两个确定的符号记写的;六十四卦应如文献所记是由八卦重合而成的,而这种八卦形成的前提同样是将其记写符号确定为两个;构成八卦的阴阳爻应是按阴阳观念将偶数记成“一一”,将奇数记戚“—”的产物。     

王弼"得意忘象"解《易》方法辨析

对王弼解《易》的传统理解,学者多认为是"扫象"说。结合王弼《周易略例》和《周易注》进行分析,可以发现,这种看法明显偏颇。诚然,在解《易》的方法上,王弼主张"得意忘象",但他在解《易》过程中意在强调"象"的工具性和"意"的目的性。王弼此种解《易》路数并未"尽黜象数"。实际上,王弼对汉代以来之象数既有所扫,又有所保留。与其说王弼解《易》是"尽黜象数",不如说是"扫象阐理"。王弼"得意忘象"这一解《易》方法,开启了中国传统哲学对经典的解读思路,不但开义理解《易》之先河,也发宋明义理易学之先声。本文通过分析王弼解《易》的这一方法论内容与特点,进一步揭示其在中国哲学史与易学史上的理论意义。     

归纳推理的多样性效应及其机制探索

归纳推理结论范围大于前提范围的这一特点决定了归纳结论的不确定性,寻求前提的多样化是增强结论可靠度的重要途径。已有研究通过寻找证据、力度判断、特征扩展与归宿选择等研究方法发现前提类别覆盖范围、前提概率原则可以较好地解释多样性的内在机制。事实上,多样性效应的实质是大数原则在归纳推理中的运用,其产生的心理过程包括差异识别、大数信息抽取与大数信息运用三个步骤     

《周易》对《林兰香》叙事艺术的影响

《林兰香》是一部优秀的世情小说,它的叙事艺术有独特的风格,它模仿《易经》,采用了以“数”谋篇布局的叙事方法．《林兰香》八卷六十四回,六十四是《易经》卦数之总．《林兰香》叙事以八回作为一个单元,每八回有一个相对完整的故事或主题．这个故事或主题又往往孕育着与之不合谐乃至对立的因素．《林兰香》“倚数”结撰,其谋篇布局之法,体现了作者对人生命运的思考,在艺术上则形成了“循环往复”之美．     

信息提供者数量对职务信息完整性的影响

本研究探讨3种不同实验情境下,信息提供者数量对职务分析信息完整性及击中完整性的影响。方差分析表明:a)4名任职者可提供足够完整的职务信息;b)职务分析员数量与信息完整性的关系在情境间不稳定;c)职务分析实践中,采用4名任职者提供信息最合理,如可排除动机因素影响,采用3名任职者提供信息最合理;d)采用4名分析员凭经验提供职务信息最合理;如可阅读职务说明书,采用3名分析员最合理。     

In touch with numbers: Embodied and situated effects in number magnitude comparison

The idea of embodied numerosity denotes that seemingly abstract number concepts (e.g., magnitude) are rooted in bodily experiences and situated action. In the present study we evaluated whether there is an embodied representation of the place–value structure of the Arabic number system and if so whether this representation is influenced by situated aspects. In a two-digit number magnitude comparison task participants had to directly touch the larger of two numbers. Importantly, pointing responses were systematically biased toward the decade digit of the target number. Additionally, this leftward bias towards the tens digit was reduced in unit–decade incompatible number pairs. Thereby, we demonstrated an influence of place–value processing on manual pointing movement. Our results therefore corroborate the notion of an embodied representation of the place–value structure of Arabic numbers which is modulated by situated aspects.     

Children's Understanding of the Natural Numbers’ Structure

When young children attempt to locate numbers along a number line, they show logarithmic (or other compressive) placement. For example, the distance between “5” and “10” is larger than the distance between “75” and “80.” This has often been explained by assuming that children have a logarithmically scaled mental representation of number (e.g., Berteletti, Lucangeli, Piazza, Dehaene, & Zorzi, 2010 ; Siegler & Opfer, 2003 ). However, several investigators have questioned this argument (e.g., Barth & Paladino, 2011 ; Cantlon, Cordes, Libertus, & Brannon, 2009 ; Cohen & Blanc‐Goldhammer, 2011 ). We show here that children prefer linear number lines over logarithmic lines when they do not have to deal with the meanings of individual numerals (i.e., number symbols, such as “5” or “80”). In Experiments 1 and 2, when 5‐ and 6‐year‐olds choose between number lines in a forced‐choice task, they prefer linear to logarithmic and exponential displays. However, this preference does not persist when Experiment 3 presents the same lines without reference to numbers, and children simply choose which line they like best. In Experiments 4 and 5, children position beads on a number line to indicate how the integers 1–100 are arranged. The bead placement of 4‐ and 5‐year‐olds is better fit by a linear than by a logarithmic model. We argue that previous results from the number‐line task may depend on strategies specific to the task.