Imre Lakatos and Various Forms of Constructivism

伊姆雷·拉卡托斯和各种形式的建构主义

 

 Proofs and Refutations is a work situated within the philosophy of science and clearly not intended for, nor advocates a didactic position on the teaching and learning of mathematics (Pimm et al. 2008; Sriraman 2008).

《证明与反驳》是一本科学哲学方面的著作，很明显它既不是为某种教授和学习数学的方法而写的，也没有提倡这类的方法。

Pimm et al. (2008)point out that the mathematics education community has not only embraced the work but has also used it to put forth positions on the nature of mathematics (Ernest 1991) and its teaching and learning (Ernest 1994; Lampert 1990;Sriraman 2006).

Pimm等人指出数学教育团体不仅信奉这本著作，而且用它推进了数学(Ernest 1991)和它的教授与学习(Ernest 1994; Lampert 1990;Sriraman 2006).

They further state:We are concerned about the proliferating Lakatos personas that seem to exist, including a growing range of self-styled ‘reform’ or ‘progressive’ educational practices get attributed

to him. (Pimm et al. 2008, p. 469)

他们进一步指出：我们关注派生出来的Lakatos式的事物似乎存在，包括增长中的自认为是“改革”或者“进步”的教育实践也受益于他。

This is a serious concern, one that the community of mathematics educators has not addressed.

这是一个严肃的关注——一个数学教育者团体还没有论述过的关注。

Generally speaking Proofs and Refutations addresses the importance

of the role of history and the need to consider the historical development of mathematical concepts in advocating any philosophy of mathematics.

通常讲，《证明与反驳》论述了历史发展的重要性和在提倡任何一个数学哲学时需要考虑数学概念的历史发展。

In other words, the book attempts to bridge the worlds of historians and philosophers.

换句话说，这本书尝试在历史学家和哲学家之间架起一座桥梁。

As one of the early reviews of the book pointed out:

His (Lakatos’) aim is to show while the history of mathematics without the philosophy of mathematics is blind, the philosophy of mathematics without the history of mathematics is empty. (Lenoir 1981, p. 100) (italics added)

正如这本书早期评论中指出：

他的目的是在于表明：没有数学哲学的数学历史是盲目的，没有数学历史的数学哲学是空洞的。(Lenoir 1981, p. 100)

Anyone who has read Proofs and Refutations and tried to find other mathematical “cases” such as the development of the Euler-Descartes theorem for polyhedra, will know that the so called “generic” case presented by Lakatos also happens to be one of the few special instances in the history of mathematics that reveals the rich world of actually doing mathematics, the world of the working mathematician, and the world of informal mathematics characterized by conjectures, failed proofs, thought experiments, examples, and counter examples etc.

任何读了《猜想与反驳》并且试图找到类似于Euler-Descartes theorem for polyhedra（欧拉－笛卡尔的多面体定理）的数学案例的人知道:那些Lakatos（拉卡托斯）介绍的所谓“普遍”状况恰巧是数学历史上少见的、特别的状况。这些状况反映了数学面对的困难、数学家的工作，还有以猜想、失败的证明、思想实验、正例、反例等等为特征的非正式的数学。

Reuben Hersh began to popularize Proof and Refutations within the mathematics community in a paper titled, “Introducing Imre Lakatos” (Hersh 1978) and called for the community of mathematicians to take an interest in re-examining the philosophy

of mathematics.

Reuben Hersh通过一篇论文“介绍伊姆雷·拉卡托斯” 开始在数学团体中普及《证明与反驳》(Hersh 1978)。他还呼吁数学团体拾起兴趣去重新检验数学哲学。

Nearly three decades later, Hersh (2006) attributed Proofs

and Refutations as being instrumental in a revival of the philosophy of mathematics informed by scholars from numerous domains outside of mathematical philosophy,“in a much needed and welcome change from the foundationist ping-pong in the ancient style of Rudolf Carnap or Willard van Ormond Quine” (p. vii).

Hersh(2006)认为：近30年来，《猜想与反驳》在数学哲学的复兴起着重要作用，并且使得数学哲学以外众多领域的学者认识到了这点。它在改变老式的由Rudolf Carnap或者 Willard van Ormond Quine提出的“奠基人乒乓球赛”（？）这类观点时，被认为是一种迫切的需要，并且受到了欢迎。

An interest in this book among the community of philosophers grew as a result of Lakatos’ untimely death, as well as a favourable review of the book given by W.V. Quine himself in 1977 in the British Journal for the Philosophy of Science.

当Lakatos不合时宜地去逝时，哲学团体对这本书以及W.V. Quine 1977年在British Journal for the Philosophy of Science发表的推介这本书的评论，增加了兴趣。

The book can be viewed as a challenge for philosophers of mathematics, but resulted in those outside this community taking an interest and contributing to its development (Hersh 2006).

这本书被认为是对数学哲学家的一个挑战，由此导致了数学哲学家这类团体以外的人对数学哲学产生了兴趣，并且促进了它的发展。

Interestingly enough, one finds a striking analogical development in voices outside of the mathematics education community contributing to its theoretical development.

相当有趣的是，可以发现非常类似于上述的发展状况：数学教育团体以外人对数学教育的理论发展所提出的意见也促进了数学教育的发展。

In one sense the theoretical underpinnings of mathematics education has developed in parallel with new developments in the philosophy of mathematics, with occasional overlaps in these two universes.

从某种意义上来说，数学教育的理论支柱和数学哲学的发展是平行发展的，偶尔这两个领域还会存在重叠。

Lakatos is an important bridge between these two universes.

拉卡托斯是两个领域之间一个重要的桥梁。

Proofs and Refutations was intended for philosophers of mathematics to be cognizant of the historical development of ideas.

《猜想与反驳》的目的是让数学哲学家审视数学思想的历史发展。

Yet, its popularization by Reuben Hersh (and Philip Davis) gradually led to the development of the so called “maverick” traditions in the philosophy of mathematics, culminating in the release of Reuben Hersh’s (2006) book 18 Unconventional Essays on the Nature of Mathematics—a delightful collection of essays written by mathematicians, philosophers,sociologists, an anthropologist, a cognitive scientist and a computer scientist.

然而，在 Reuben Hersh (and Philip Davis)推动下，它的普及逐渐导致了数学哲学里“非常规”的惯例。这种惯例在Reuben Hersh (2006)的书《非传统数学18篇》出版时达到了顶峰。这本书收录了由数学家、哲学家、社会学家、人数学家、认知方面科学家写的文章。

 These essays are scattered “across time” in the fact that Hersh collected various essays written over the last 60 years that support the “maverick” viewpoint.

Hersh收集了散布在近60年的时间内的各种文章，用于支持“非常规”这种观点。

His book questions what constitutes a philosophy of mathematics and re-examines foundational questions without getting into Kantian, Quinean or Wittgensteinian linguistic quagmires.

他的书探索了什么构成了数学的哲学，以及在重新审视了各种基础性问题的同时没有落入康德式的、奎因式的或者维特根斯坦式的语言泥沼之中。

In a similar vein the work of Paul Ernest can be viewed as an attempt to develop a maverick philosophy, namely a social constructivist philosophy of mathematics (education).

Paul Ernest的著作具有相同的脉落，也可以被当作一种尝试。这种尝试在于建立一种“非常规”哲学——社会建构主义的数学（教育）哲学。

We have put the word education in parentheses because Ernest

does not make any explicit argument for an associated pedagogy as argued by Steffe(1992).

我们把教育放入括号里，是因为，正如Steffe(1992)所说的那样，Ernest并没有表达明确的与教育学相关的观点。

Does Lakatos’ work have any direct significance for mathematics education?

Lakatos的著作对于数学教育是否有直接的意义呢？

Can Lakatos’ Proofs and Refutations be directly implicated for the teaching and learning of mathematics?

拉卡托斯的《猜想与反驳》有没有直接涉及数学的教授与学习呢？

We would argue that it cannot be directly implicated.

我们认为《猜想与反驳》并没有直接涉及到这个方面。

However Proofs and Refutations may very well serve as a basis for a philosophy of mathematics,such as a social constructivist philosophy of mathematics, which in turn can be used a basis to develop a theory of learning such as constructivism.

然而《猜想与反驳》可以很快地作为数学哲学的基础，正如建构主义哲学用之于数学；相应地《猜想与反驳》也可以作为基础来发展学习理论，正如建构主义用之于学习理论。

This is a position that Steffe (1992) advocated, which has gone unheeded.

这正是Steffe (1992)所提倡的观点，这个观点还未得到重视。

Les Steffe in his review of Ernest’s (1991) The Philosophy of Mathematics Education wrote:Constructivism is sufficient because the principles of the brand of constructivism that is currently called “radical” (von Glasersfeld 1989) should be simply accepted as the principles of what I believe should go by the name Constructivism.

Les Steffe在他对Ernest的书《数学教育哲学》的回顾中写到：（译注：英文大写的、唯一的、特定的）建构主义已经充分了，因为标名为建构主义的原则目前被称为“激进的” (von Glasersfeld 1989)，但是我认为这些原则应该只被称为（译注：英文大写的、唯一的、特定的）建构主义。

It seems to me that the radical constructivism of von Glasersfeld and the social constructivism of Ernest are categorically two different levels of the same theory.

我认为von Glasersfeld的激进建构主义和Ernest的社会建构主义确切地说是相同理论的不同层次。

Constructivism (radical), as an epistemology, forms the hard core of social constructivism, which is a model in what Lakatos (1970) calls its protective belt.

大写的建构主义（激进的）作为认识论，形成了社会建构主义稳固的核心，而社会建构主义作为一个模型在拉卡托斯的说法应该被称为一种保护带，即建构主义的保护带。

Likewise, psychological constructivism is but a model in the protective belt of the hard-core principles of Constructivism.

同样地，心理学的建构主义也是一个模型，处于大写的建构主义的核心原则的保护带里。

These models continually modify the hard-core principles, and that is how a progressive research program that has interaction as a principle in its hard core should make progress.

这些模型持续地改进核心原则，这也是一些进步的研究和核心原则相互影响后为什么会取得进步的原因。

 It is a lot easier to integrate models in the protective belt of a research program that has been established to serve certain purposes than it is to integrate epistemological hard cores. (Steffe 1992, p. 184)

融入处于保护带中的模型——保护带是为了某种目的而建立起来的，要比融入认识论的核心容易得多(Steffe 1992, p. 184)。