Example-driven machine-human collaboration in mathematics

在最近一项对数学家谈论的内容的研究中，我们发现，例子构成了最大的单一类别。这些可以是概念的例子，例如自然数集合是群的例子，而数字3、4和5是毕达哥拉斯三元组的例子，或者是对猜想的支持或反例，例如1和3是对两个奇数的和是奇数的猜想的反例。这项研究发现，在对话的不同点上，例子被用于不同的原因，例如，为了理解一个猜想，测试它，或扩展它。作为一个例子-在数学中使用例子，考虑以下对话，取自一个在线论坛，用来解决一个猜想：“如果点形成一个凸多边形，它很容易。”[匿名--2011年7月19日@8：08 PM]“是的。如果只有一个点不在这些点的凸包上，我们能做到吗？[托马斯·H-2011年7月19日@晚上8：09]“假设有四个点：一个等边三角形，然后是三角形中心的一个点。没有三个点是共线的。在我看来，风车不能多次使用中心点！一旦它碰到一个角点，它就会无限期地在角落里循环，永远不会回到中心点。我一定是错过了什么……‘’[Jerzy--2011年7月19日晚上8：17]“这不是真的--它会在三角形的中心和每个顶点之间交替。”[Joe-7月19日，2011@8：21 PM]在这里，我们看到人们提出简单的例子来支持一个猜想，并提出和讨论其他作为潜在问题的例子。在这个项目中，我们将以我们对例子在数学中的使用的调查为基础，并使用第三方模型生成器来设计和构建一个可以通过以下方式连接到在线数学对话的系统：1.手动和自动地诱导对话规则，以确定在对话中何时引入一个例子是合适的。发现论据模式，关于一个例子在给定的点上将扮演什么角色。手动和自动地归纳公理集作为模型生成器的输入。从模型生成器中选择有用的输出示例。将上下文信息添加到示例中，例如它是否是对上一条评论的回复，或者这个示例在这一点上感兴趣的原因。以一种对其他(人类)参与者有用的方式，将例子与上下文信息一起添加到对话中。这将把论证理论、自动推理系统以及关于人们如何做数学的人种学、认知学和哲学研究结合起来。对原型系统的评估将通过实时在数学对话中运行它，并通过各种衡量标准来观察数学家是否认为它有用，以及他们是否准备与它直接互动。此外，在项目持续时间内，我们将建立一个新的广泛的潜在研究用户网络，以确定这种跨学科工作可能采取的进一步方向。

In a recent study of what mathematicians talk about, we found that examples form the biggest single category. These may be examples of a concept, such as the set of natural numbers being an example of a group, and the numbers 3, 4, and 5 an example of a Pythagorean triple, or supporting or counterexamples to a conjecture, such as 1 and 3 being a counterexample to the conjecture that the sum of two odd integers is odd. The study found that examples are used for different reasons at different points in a conversation, for instance to understand a conjecture, to test it, or extend it.As an example of example-use in mathematics, consider the following conversation, taken from an online forum for solving a conjecture:"If the points form a convex polygon, it is easy.'' [Anonymous - July 19, 2011 @ 8:08 pm]"Yes. Can we do it if there is a single point not on the convex hull of the points?'' [Thomas H - July 19, 2011 @ 8:09 pm]"Say there are four points: an equilateral triangle, and then one point in the center of the triangle. No three points are collinear. It seems to me that the windmill can not use the center point more than once! As soon as it hits one of the corner points, it will cycle indefinitely through the corners and never return to the center point. I must be missing something here...'' [Jerzy - July 19, 2011 @ 8:17 pm]"This isn't true - it will alternate between the centre and each vertex of the triangle.'' [Joe - July 19, 2011 @ 8:21 pm]Here we see people raising simple examples in support of a conjecture, and proposing and discussing other examples as potential problems.In this project we will build on our investigations into example-use in mathematics, and employ third party model generators, to design and build a system which can interface to online mathematical conversations by:1. Manually and automatically inducing dialogue rules to determine when in a conversation it would be appropriate to introduce an example.2. Discovering argument patterns as to what role an example will play at a given point.3. Manually and automatically inducing sets of axioms as input to the model generator.4. Selecting useful output examples from the model generator.5. Adding contextual information to the examples such as whether it is a reply to a previous comment or the reason why this example is of interest at this point.6. Adding the examples alongside the contextual information into a conversation in a way which is useful to other (human) participants.This will draw together theories of argumentation, automated reasoning systems, and ethnographical, cognitive and philosophical studies of how people do mathematics. The prototype system will be evaluated by running it on mathematical conversations in real-time, and seeing, by a variety of measures, whether mathematicians regard it as useful, and whether they are prepared to interact directly with it.Furthermore, over the duration of the project, we will build a new and broad network of potential research users, in order to determine further directions which such inter-disciplinary work may take.