The Edge of Platonism.
The Edge of Platonism.
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柏拉图主义的边缘。
DOI：
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发表时间：
1985
期刊：
影响因子：
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通讯作者：
David Hawkins
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作者：
David Hawkins
它的早期版本大约二十年前出现；在剑桥版完成之前，作者就英年早逝了。这是一篇小文章，范围狭窄，但含义广泛。第二本是两位数学家关于数学的论文集，他们对各自学科的历史和哲学都非常了解并进行了反思。这本书细节丰富，为了说明起见，它很好地从数学的王国、门和类中汲取了样本。第三本书是由一位哲学家写的，他希望在认真关注对概念和方法论变化的理解的研究中，将对数学知识本质的讨论与其自身的历史和一般科学的历史建立更富有成效的联系。如果你遵循相同的哲学路径，你就会认出它的论证风格，这对局外人来说并不容易。我把这三本书放在一起不是为了详细回顾，而是为了一路参考。我希望走的路是就数学哲学与数学教学，特别是早期数学学习的相关性提出一些问题供讨论。我不会在哲学上开辟新的天地，尽管我会尝试打破一两块土块。菲利普·基彻想要采取一种长期以来最受蔑视的哲学立场，即约翰·斯图尔特·密尔的经验主义。尽管密尔坚信所有知识都是通过感觉经验的归纳概括得出的，但他也坚信数学的主题只是自然世界：与物理学或生物学的不同之处不是在性质上，而只是在普遍性程度上不同。对于这样的经验主义者来说，唯一的选择是形式主义，基彻对此很少关注。戴维斯和赫什给出了更多，特别是其逻辑实证主义版本。他们的批评简洁而有效，可以作为多次布道的文本。因为任何严格的形式主义数学陈述都是空洞的，它们的真值只是同义反复。例如，数学只是一种描述物理现象的语言。形式主义抛弃了除了形式化的最终产品之外的一切，因此也抛弃了工作的数学家以及所有数学猜想、论证和发现的伟大世界。与此相反的是形而上学，称为柏拉图主义。这个术语的普遍使用并没有太多关注柏拉图的著作，柏拉图从未成为自己的弟子，因此不能正确地被称为柏拉图主义者。但我们却被这些主义所困；柏拉图确实阐述了这一点，他的对话以令人难忘的方式围绕着它。但要小心。他是一位思想剧作家，有时戏剧台词并不能清醒地回答他的人物提出的问题；相反，它可能是一种酗酒，只会加深问题并阻碍简单的答案。 “勇气，”老战士说，“就是坚持自己的立场。”对话结束前，勇气已陆续被重新定义为“希望与恐惧的知识”。这似乎是一个谜题，但对话却引出了它。这是数学家的答案。这似乎与最初的猜想相去甚远，但如果你这样定义勇气，那么所有的争论就会变得更加激烈。我提到柏拉图的这一方面是因为它使柏拉图主义处于比任何帕特定义所能管理的更有趣的角度。我认为我们必须认识到一种牢固的家庭关系，例如，柏拉图式的辩证法与主导拉卡托斯对话的辩证法之间的关系。如果你研究他的证明和反驳，然后仔细看看一些对话，如《高尔吉亚》或《小希皮亚斯》或《理想国》的某些部分，那么相似之处是显而易见的。尽管主题存在各种差异，但在道德或数学方面，我们的想法相互影响的方式暗示了一种隐含的结构。通过朴素的表述、例子和证明、反例和反驳、重新表述等等，我们可以希望将这些想法视为更大、更连贯的结构的一部分，超越我们最初对它们的理解，仍然熟悉，但对于嵌入它们的新背景来说，却很奇怪。在一种情况下，勇气的美德嵌入到认知模式的背景中，在另一种情况下，多面体的元素作为向量代数中的结构重新出现。在这两种情况下，人们都怀疑翻译是否遗漏了一些东西。戴维斯和赫什讨论了拉卡托斯对发现方法的分析，并为我们提供了他们自己的案例说明。他们在乔治·波利亚（George Polya）关于启发法的丰富而多方面的工作的背景下进行了这一叙述，这是应该的。
early version of it appeared some twenty years ago; the author died young, before the Cambridge edition was completed. It is a small essay, narrow in scope and wide in the range of its implications. The second is a whole collection of essays in and about mathematics by two mathematicians who are both well informed and reflective about the history and philosophy of their discipline. The book is rich in detail, and for illustration it samples well from the kingdoms, phyla and classes of what is all, somehow, mathematics. The third book is by a philosopher who wants to bring the discussion of the nature of mathematical knowledge into a more fruitful relation to its own history and to that of the sciences generally, when studied with serious concern for the understanding of conceptual and methodological change. Its argument is in a style which you will recognize if you have followed the same philosophical paths, not easy for the outsider. I bring these three books together not for detailed review but for reference along the way. The way I wish to go is to open up, for discussion, some questions about the relevance of the philosophy of mathematics to mathematics teaching, and on early mathematics learning in particular. I shall not break fresh ground in the philosophy, though I'll try to crack a clod or two. Philip Kitcher wants to take a philosophical position which has for a long time been the most scorned, the empiricism of John Stuart Mill. Convinced as he was that all knowledge comes by way of inductive generalization from sense experience, Mill was then also committed to the belief that the subject matter of mathematics is simply the world of nature: different from physics or biology not in kind, but only in degree of generality. The only alternative for such an empiricist is that of Formalism, to which Kitcher gives little attention. Davis and Hersh give it more, particularly in its logical positivist version. Their criticism is neat and effective, and could be a text for several sermons. For any strict formalist mathematical statements are empty, their truth value is only that of tautology. Mathematics is only a language for describing phenomena of physics, for example. Formalism drops out everything except the formalized end-product, and therefore also drops out the working mathematician and all the great world of mathematical conjecture, argument and discovery. At the opposite pole is the metaphysics called Platonism. The term is commonly used without much attention to the writings of Plato, who never became his own disciple and thus cannot properly be called a Platonist. But we are stuck with the isms; Plato did formulate this one, and his dialogues hover around it, in quite unforgettable ways. But be careful. He was a dramatist of ideas, and sometimes the dramatic line is not soberly to answer the questions raised by his cast of characters; it can be, rather, a kind of insobriety which only deepens the questions and blocks the easy answers. "Courage," says the old soldier, "is standing your ground." Before the dialogue is over courage has been successively redefined as "the knowledge of hope and fear." That seems a puzzle, but the dialogue leads to it. It is a mathematician's kind of answer. It seems far from the initial conjectures, but if you define courage that way it tightens all the arguments. I mention this aspect of Plato because it puts Platonism in a more interesting light than any pat definition can manage. I think we have to recognize a strong family relation, for example, between this Platonic dialectic and that which dominates the dialogue of Lakatos. If you study his Proofs and refutations and then look carefully at some of the dialogues like Gorgias or Hippias Minor or parts of The Republic the parallel is clear. Despite all differences in subject matter there is an implicit structure, implied by the ways our ideas engage each other, in ethics or mathematics. By naive formulations, examples and proofs, counterexamples and refutations, reformulations et seq., we can hope to see these ideas as part of a larger and more coherent structure, transformed beyond our initial grasp of them, still familiar but, for the new context embedding them, strangely so. In the one case the virtue of courage gets embedded in a context of the modes of knowing, in the other the elements of polyhedra reappear as structures within a vector algebra. In both cases there is residual doubt as to whether the translation has not left something out. Davis and Hersh discuss Lakatos' analysis of the method of discovery and provide us with their own casehistory illustrations. They set this account in the context of George Polya's rich and many-sided work on heuristics, where it should be.