Monty Hall Dilemma的困难原因探讨

采用Monty Hall Dilemma的经典研究范式,探讨了大学生被试在不同提示条件下对MHD问题的解决情况,结果发现只有在条件关系提示下,MHD的作业成绩才得到了明显的改善,但是正确率也只有35.7%。这表明MHD问题困难的原因很可能是多方面的,如认为主持人的行为完全是随机事件,很难意识到参与者的最初选择与主持人打开特定盒子间的三种条件关系;多数被试作出选择的理由是非理性的,如“坚持第一选择,相信自己的直觉”;不能正确判断MHD问题的内在概率关系以及缺乏认真思考的动力等。     

论现代数学背景下的康德几何观

<正>在对数学的哲学基础研究中,20世纪通常说来有三大流派:逻辑主义、形式主义和直觉主义。但在哥德尔定理出现以后,一致性问题给逻辑主义和形式主义带来了难以克服的困难;而直觉主义又很难对数学中的"抽象实体"作出合理的解释,并且数学中有些重要定理的证明完全超出了直觉主     

关于Erdos-Moser定理的研究

本文主要研究Erdos-Moser定理。在简单介绍了反推数学的一些基础知识后,首先研究了Erdos-Moser定理的证明论强度：存在一个可计算的二元二染色函数使得任何无穷∑20集合都不是该函数的传递集,同时存在一个可计算的二元二染色函数使得每一个该函数的无穷传递集都是超免疫的。其次,我们进一步考虑了稳定性Erdos—Moser定理,证明了在二阶算术子系统RCA0下稳定性Erdos-Moser定理是不可证的并且对每一个可计算的稳定性二色二阶函数,我们构造了一个Ф’可计算的无穷传递集。     

“不可或缺性论证”与反实在论数学哲学

一、引言20世纪初的数学哲学研究侧重于探讨数学基础问题,由此产生了逻辑主义、形式主义与直觉主义等数学基础流派。进入20世纪中期以后,数学基础问题不再困扰数学家们,哲学家们也开始转向关注经典数学的本体论与认识论问题。这包括探讨抽象数学对象是否在字面意义上存在,数学定理是否为关于抽象数学对象的客观真理,我们如何能够获得、确证经典数学的知识,以及数学为何可应用等等。这些哲学问题其实是自柏拉图以来的传统哲学家们所关心的本体论问题,以及自17、18世纪经验主义与理性主义以来的传统哲学家们所关心的认识论问题,在数学领域的反…     

解分数应用题的三步曲

分数应用题是指倍数关系比较问题的分数类题,也就是分数乘、除法应用题。分数应用题是小学高年级数学教学的重点和难点,长期以来,笔者对解分数应用题的方法、规律进行了广泛地、深入地探究,成果是一一解分数应用题的三步曲。用解分数应用题的三步曲解分数应用题,行之有效,以飨读者。     

定律的逻辑概率一定为0吗?

波普尔曾论证"定律的逻辑概率为0"这一论题,以反对基于贝叶斯定理的归纳逻辑。他对该论题的证明本质上有三个:(a)独立性论证;(b)反杰弗里斯论证;(c)维度论证。本文将论证他的这三个论证均不成立。(a)论证的错误在于假定Uij的独立性;(b)论证错误地使用了古德曼类似物;(c)论证中则断定了与概率演算不一致的命题。总的来看,他的错误或许来自于对逻辑和认识论逻辑的混淆。     

哥德尔的直觉主义逻辑思想

哥德尔（Kurt Godel）是现代逻辑史上的巨匠，他在逻辑史上有两大贡献：一是他证明了罗素和怀特海在《数学原理》中提出的一阶逻辑演算的完全性定理（1930），即任何有效的一阶公式都是可证的。二是他证明了著名的不完全性定理（1931），《数学原理》的系统和集合论的ZF公理系统不足以判定能在这些系统中形式化的所有数学问题...     

罗素的逻辑主义及其在数理逻辑史上的地位

20世纪初,在逻辑和数学中发现了许多悖论,包括罗素本人所发现的悖论(后被称为罗素悖论)。这些悖论动摇了数学的基础,史称第三次数学危机。为了解决这一次数学危机,罗素提出了逻辑主义的纲领,并得到一些著名的逻辑学家的支持,成为数理逻辑中的三大学派之一。本文旨在对罗素的逻辑主义作出全面的科学的评述。一、数学概念和数学定理的推导罗素的逻辑主义包含两个部分:(1)数学概念可以通过显定义从逻辑概念推导出来;(2)数学定理可以通过纯逻辑推演(即一阶逻辑演算)由逻辑公理推导出来。罗素所使用的逻辑概念有:命题联结词(否定,析取,合取,蕴涵)…     

四面体定理、四元论和四元医学模式(上)

1623年,当代科学的创始人伽利略在《关于两种科学的对话》中说,"自然这部书是用数学语言写成的,字母就是三角形、圆和其他几何图形.没有这些手段,人就不可能理解任何东西."[1]四面体定理是笔者在近年发现的一个几何定理[2],将对人类的世界观产生较大的影响,实现认识论和方法论的辩证统一.该定理有望在生命科学领域得到深入应用,本文试对此进行论证和阐述.     

“文化数学”理念在高中数学教学中渗透的实践研究

著名数学教育家张奠宙先生说：“数学奥赛得了名次,高考成了状元,其实他们的数学懂得并不多。”学生数学考试成绩很高,基础很好,但是并不真正懂得数学,数学素养水平不高。从前些年的“高分低能”,到如今的“状元”不懂多少数学。我们的数学教育教学到底出了什么问题？我认为高中数学教学过于应试化。教学只关注数学形式,忽视数学实质,重视数学结论,轻视数学过程,对概念、公式、定理和性质的教学只注重本身的理解,把课堂教学过程简单地看成“知识传递”的过程,缺乏以学生为主体的课堂学习文化。本文以“文化数学”理念（即用文化等数学元素包装的数学）在高中课堂教学中渗透的实践研究为抓手,谈谈一些具体做法,供大家参考。     

Baumann on the Monty Hall problem and single-case probabilities

Peter Baumann uses the Monty Hall game to demonstrate that probabilities cannot be meaningfully applied to individual games. Baumann draws from this first conclusion a second: in a single game, it is not necessarily rational to switch from the door that I have initially chosen to the door that Monty Hall did not open. After challenging Baumann’s particular arguments for these conclusions, I argue that there is a deeper problem with his position: it rests on the false assumption that what justifies the switching strategy is its leading me to win a greater percentage of the time. In fact, what justifies the switching strategy is not any statistical result over the long run but rather the “causal structure” intrinsic to each individual game itself. Finally, I argue that an argument by Hilary Putnam will not help to save Baumann’s second conclusion above. See Moser and Mulder (1994, pp. 115–116, 118).     

Single-case probabilities and the case of Monty Hall: Levy’s view

In Baumann (American Philosophical Quarterly 42: 71–79, 2005) I argued that reflections on a variation of the Monty Hall problem throws a very general skeptical light on the idea of single-case probabilities. Levy (Synthese, forthcoming, 2007) puts forward some interesting objections which I answer here.     

Probability,rational single-case decisions and the Monty Hall Problem

The application of probabilistic arguments to rational decisions in a single case is a contentious philosophical issue which arises in various contexts. Some authors (e.g. Horgan, Philos Pap 24:209–222, 1995; Levy, Synthese 158:139–151, 2007) affirm the normative force of probabilistic arguments in single cases while others (Baumann, Am Philos Q 42:71–79, 2005; Synthese 162:265–273, 2008) deny it. I demonstrate that both sides do not give convincing arguments for their case and propose a new account of the relationship between probabilistic reasoning and rational decisions. In particular, I elaborate a flaw in Baumann’s reductio of rational single-case decisions in a modified Monty Hall Problem.     

The psychology of the Monty Hall problem: discovering psychological mechanisms for solving a tenacious brain teaser

The Monty Hall problem (or three-door problem) is a famous example of a "cognitive illusion," often used to demonstrate people's resistance and deficiency in dealing with uncertainty. The authors formulated the problem using manipulations in 4 cognitive aspects, namely, natural frequencies, mental models, perspective change, and the less-is-more effect. These manipulations combined led to a significant increase in the proportion of correct answers given by novice participants, largely because of the synergy of frequency-based formulation and perspective change (Experiments 1, 2). In a raining study (Experiment 3) frequency formulation and mental models, but not Bayes's rule training, showed significant positive transfer in solving related problems.     

A Logical Analysis of Monty Hall and Sleeping Beauty

Hintikka and Sandu’s independence-friendly (IF) logic is a conservative extension of first-order logic that allows one to consider semantic games with imperfect information. In the present article, we first show how several variants of the Monty Hall problem can be modeled as semantic games for IF sentences. In the process, we extend IF logic to include semantic games with chance moves and dub this extension stochastic IF logic. Finally, we use stochastic IF logic to analyze the Sleeping Beauty problem, leading to the conclusion that the thirders are correct while identifying the main error in the halfers’ argument.     

A randomised Monty Hall experiment: The positive effect of conditional frequency feedback

The Monty Hall dilemma (MHD) is a notorious probability problem with a counterintuitive solution. There is a strong tendency to stay with the initial choice, despite the fact that switching doubles the probability of winning. The current randomised experiment investigates whether feedback in a series of trials improves behavioural performance on the MHD and increases the level of understanding of the problem. Feedback was either conditional or non-conditional, and was given either in frequency format or in percentage format. Results show that people learn to switch most when receiving conditional feedback in frequency format. However, problem understanding does not improve as a consequence of receiving feedback. Our study confirms the dissociation between behavioural performance on the MHD, on one hand, and actual understanding of the MHD, on the other. We discuss how this dissociation can be understood.     

Focusing failures in competitive environments: explaining decision errors in the Monty Hall game,the Acquiring a Company problem,and multiparty ultimatums

This paper offers a unifying conceptual explanation for failures in competitive decision making across three seemingly unrelated tasks: the Monty Hall game (Nalebuff, 1987), the Acquiring a Company problem (Samuelson & Bazerman, 1985), and multiparty ultimatums (Messick, Moore, & Bazerman, 1997). We argue that the failures observed in these three tasks have a common root. Specifically, due to a limited focus of attention, competitive decision makers fail properly to consider all of the information needed to solve the problem correctly. Using protocol analyses, we show that competitive decision makers tend to focus on their own goals, often to the exclusion of the decisions of the other parties, the rules of the game, and the interaction among the parties in light of these rules. In addition, we show that the failure to consider these effects explains common decision failures across all three games. Finally, we suggest that this systematic focusing error in competitive contexts can serve to explain and improve our understanding of many additional, seemingly disparate, competitive decision‐making failures. Copyright © 2003 John Wiley & Sons, Ltd.     

Working memory capacity and a notorious brain teaser: the case of the Monty Hall Dilemma

The Monty Hall Dilemma (MHD) is an intriguing example of the discrepancy between people's intuitions and normative reasoning. This study examines whether the notorious difficulty of the MHD is associated with limitations in working memory resources. Experiment 1 and 2 examined the link between MHD reasoning and working memory capacity. Experiment 3 tested the role of working memory experimentally by burdening the executive resources with a secondary task. Results showed that participants who solved the MHD correctly had a significantly higher working memory capacity than erroneous responders. Correct responding also decreased under secondary task load. Findings indicate that working memory capacity plays a key role in overcoming salient intuitions and selecting the correct switching response during MHD reasoning.