Alexander Calandra 2009-10-19 Catherine Angels on a Pin Alexander Calandra 2009-10-19 Catherine

Introduction to the Text This article was written at a time when the whole of the United States was thrown into a panic over the launching of the first satellite (Sputnik) by the Russians, because this seemed to prove that the former Soviet Union had left the US far behind in science and technology. Many Americans believed that there must be something seriously wrong with their educational system. The author, in writing this article, seemed to think that part of the nation’s problem in education was the traditional teaching and testing methods, which emphasized book knowledge rather than students’ originality and creativity.

Introduction to the Text Education is also a great challenge in China. Problems are many. One of them is perhaps the fact that our education is too closely geared to tests and examinations. In fact, often our effort seems to serve the purpose of enabling students to handle them properly. Tests and examinations are necessary and useful of course. But their purposes are limited, and so is their usefulness. If they are overused, or if we take the means for the ends, they will lead to serious consequences. This text can give us some food for thought.

Alexander Calder Alexander Calandra is now Professor of Emeritus of Physical Sciences at Washington University in St. Louis, Missouri. The present text is adapted from “Angels on the Head of a Pin: A Modern Parable” which first appeared in Saturday Review. Dec. 21, 1968 and has, since then, become a classic (or an often quoted) case on the problems of American education.

Angles on a pin The title of the text “Angles on a Pin” comes from the much-talked about question: “How many angles can dance on the head of a pin?” which is used to ridicule those people who asked meaningless questions about the Bible in the Middle Ages. It is also used ironically to describe the kind of questions that philosophers ponder.

Angles on a pin Why is the article entitled “Angels on a Pin”? I.Medieval scholastics were fond of debating such meaningless questions as “How many angels can dance on the point of a pin?” “Did Adam have a navel?” and “Do angels defecate?” The emerging sciences replaced such “scholarly” debates with experimentation and appeals to observable fact. II. Callandra seems to be suggesting that “exploring the deep inner logic of a subject in a pedantic (学究的；迂腐的) way is similar to the empty arguments of scholasticism.” He compares this to the “new math”, so much in the news in the 60s, which attempted to replace rote memorization (死记硬背) of math with a deeper understanding of the logic and principles of mathematics, and he seems to be deriding (嘲笑) that effort, too.

There was considerable speculation in the Middle Ages about the nature ofangels -- what form they had, whether they had substance, etc.  Some of thediscussion were taken up by University students and faculty as debating exercises.  One particularly famous exercise was the question, "How many angels can dance on the head of a pin?"  The answer is either one (if angels have material substance) or an infinite number (if they do not).     This is often cited as an example of the useless and pedantic degeneracy of medieval Catholic theology...

About the last paragraph The last paragraph of the story quoted above seems jarringly inconsistent with the remainder of the story. In particular, the mentions of "the new mathematics", "scholasticism", and "the Sputnik-panicked classrooms of America" are irrelevant to both the subject and the lesson of the story. During an exchange of e-mails with Prof. Richard Hake during August 2000 about the origin of the barometer story, I noticed that this discordant last paragraph does not appear in a 1964 version of Calandra's story, and also does not appear in the version in Calandra's 1961 book The Teaching of Elementary Science and Mathematics, which suggests to me that the last paragraph was added by an editor at Saturday Review in 1968. A copy of the 1964 version is posted at the end of a document at Harvard University. I regard this legendary story as important for what it says about teaching and creativity, and I only care a little bit about the origin of this legend. The last paragraph of the story quoted above seems jarringly inconsistent with the remainder of the story. In particular, the mentions of "the new mathematics", "scholasticism", and "the Sputnik-panicked classrooms of America" are irrelevant to both the subject and the lesson of the story. During an exchange of e-mails with Prof. Richard Hake during August 2000 about the origin of the barometer story, I noticed that this discordant last paragraph does not appear in a 1964 version of Calandra's story, and also does not appear in the version in Calandra's 1961 book The Teaching of Elementary Science and Mathematics, which suggests to me that the last paragraph was added by an editor at Saturday Review in 1968. A copy of the 1964 version is posted at the end of a document at Harvard University. I regard this legendary story as important for what it says about teaching and creativity, and I only care a little bit about the origin of this legend.

针尖上能站多少个天使是一个典型的中世纪神哲学的问题。 中世纪哲学以托马斯阿奎那哲学为代表，都是意图用希腊哲学尤其是亚里士多德哲学的概念来讨论神学命题。他们笃信“恩典成全自然”，也就是说神既然造了世界，让人类有理性，那么用理性来理解信仰就是完全有可能的。 亚里士多德认为，事物都是由形式和质料组成的。形式是一类事物的原则，是使一类事物成为一类事物的东西；而质料在与形式结合之前什么也不是，因为与形式结合才有了区分，才能形成个体事物。 举例来说，人是理性动物，理性就是人的形式，而身体在这里可以说是人的质料（注意这不同于唯物主义精神和物质的二分）。 亚里士多德还认为质料是潜在的，而形式是现实的，所以最高的存在上帝因为是最具现实性因而是纯粹形式的。 天使也是纯粹形式的，他与上帝的区别在于天使是被造物。 而之所以讨论针尖上站几个天使，正是因为天使是纯形式，没有质料，也就是说没有占有空间，所以即使针尖上能站百亿个天使也丝毫不足为奇。 ps，这是纯哲学讨论，中世纪哲学的意义这些年也越来越受重视，过去我们简单的说中世纪是西方的黑暗史这种说法似乎早就过时了。 不错，在中世纪得历史当中有和我们的社会历史非常隔膜的东西，象宗教迫害等等。我很奇怪一直有人以宗教迫害来否定宗教，我觉得这就象我们以焚书坑儒来否定儒学一样。一种文明总有他的两面性，西方文明有，东方文明又何尝无？

    中世纪的时候，对于天使的本性有很多争论：它们是什么形态，存不存在物质形态，等等。一些争论被用在训练辩论的本领上。一个非常著名的问题是：多少天使能够在一个针尖上跳舞？答案是一个（如果天使有物质形态）或者无数个（天使没有物质形态）。     这经常被作为一个对中世纪神学无用和迂腐的例子......     所以，“针尖上的天使”在实际使用的时候，指一些非常琐碎，没有意义但是却可以辩论的问题，“先有鸡还是先有蛋”之类的。但是，抛开它的意义，这是一个非常有趣的比喻。它本身就是一个谜题，还可以暗示困境，轻盈，神秘，矛盾......很多很多

经院哲学（Scholasticism）通常又被称为繁琐哲学，它因讲究繁琐的概念辨析和逻辑论证而得名。在这方面，比较典型的例子有安瑟尔谟关于上帝存在的本体论证明、托马斯·阿奎那关于上帝存在的宇宙论和目的论证明，以及其他哲学家对于一些极其无聊的问题——例如天堂里的玫瑰花有没有刺、亚当和夏娃有没有肚脐眼等——的繁琐论证。这些逻辑论证就其内容来说是毫无意义的，因为上帝、天堂以及亚当、夏娃等等都只是信仰的对象，关于他（它）们的任何具体特性（包括存在本身）的描述都只能是想象的结果，既不能在感性的范围内加以直观，也不能在理性的范围进行证明。对于这些信仰对象进行理性的逻辑论证，就如同用磅秤来称精神一样荒谬。但是从形式上来看，经院哲学的这种做法却培养了一种理性的精神，即通过逻辑的论证而不是单凭狂热的信仰来确定真理。经院哲学导致了理性神学的产生（托马斯·阿奎那是理性神学的最著名的代表），理性神学虽然仍旧承认信仰的至高无上性和不可动摇性，但是它所倡导的那种注重逻辑证明的方法却无疑为近代理性主义的勃兴埋下了最初的种子。

How many angels can dance on the head of a pin How many angels can dance on the head of a pin? From Wikipedia, the free encyclopedia The question how many angels can dance on the head of a pin? has been used many times as a trite dismissal of medieval angelology in particular, of scholasticism in general, and of particular figures such as Duns Scotus and Thomas Aquinas.[1] Another variety of the question is How Many Angels Can Sit On The Head Of A Pin? In modern usage, this question serves as a metaphor for wasting time debating topics of no practical value.[

It is still a matter of discussion whether this precise topic has a historical foundation, in actual writing or disputation from the European Middle Ages. One theory is that it is an early modern fabrication[4], used to discredit scholastic philosophy at a time when it still played a significant role in university education. James Franklin has raised the scholarly issue, and mentions that there is a seventeenth century reference in William Chillingworth's Religion of Protestants.[5], where he accuses unnamed scholastics of debating " Whether a Million of Angels may not fit upon a needles point?" This is earlier than a reference in the 1678 The True Intellectual System Of The Universe by Ralph Cudworth. H.S. Lang, author of Aristotle's Physics and its Medieval Varieties (1992), says (p. 284): "The question of how many angels can dance on the point of a needle, or the head of a pin, is often attributed to 'late medieval writers' ... In point of fact, the question has never been found in this form".

The modern version in English (usually a needle, rather than a pin) dates back at least to Richard Baxter. In his 1667 tract "The Reasons of the Christian Religion," Baxter reviews opinions on the materiality of angels from ancient times, concluding "And Schibler with others, maketh the difference of extension to be this, that Angels can contract their whole substance into one part of space, and therefore have not partes extra partes. Whereupon it is that the Schoolmen have questioned how many Angels may fit upon the point of a Needle?“ Other possibilities are that it is a surviving parody or self-parody, or debating training topic. But identifies a close parallel in a fourteenth century mystical text.