Tic-Tac-Toe Theory-An Escape From The Combinatorial Chaos

摘要：Tac-Tac-toe理论--从组合混乱中脱身数学是非常成功的推广(微积分、现代代数、代数几何等)；另一方面，它对经济或国际象棋等非传统复杂系统的论述令人惊讶地少之又少。复杂系统的根本问题是什么？简短的答案是：巨大的可能性空间。据估计，国际象棋中的总棋数超过了十分之一，这比可观察到的宇宙中的粒子数量还要多。离散数学中的大多数问题都会导致相同规模的案例研究。统计力学面临着同样的问题：一摩尔气体包含大约10到23个粒子；潜在的动力学非常复杂。统计力学的基本思想是处理先验概率；通常在相空间中等概率(这是违反直觉的：概率如何进入牛顿力学？)研究人员发展了ATIC-Tac-Toe理论，这是统计力学方法的惊人模拟：它可以确定无限多个自然游戏数的精确值(同样的悖论：概率如何进入国际象棋？)。这本关于这一主题的研究者的书(700页)将于2007年在剑桥大学出版社出版。理解复杂系统是当代数学(以及理论计算机科学和经济学等)的主要挑战。游戏是完美的自然模型。此外，分析游戏可能是让学生品尝数学发现的最好方式；这是最自然的数学实验。将博弈论的这一新分支(研究者的Tic-Tac-toe理论)整合到数学教育中是一个非常有前途的方面。最后，理解复杂的游戏可能会让我们对人类智力是如何工作的有独特的洞察力。

Abstract: Tic-Tac-Toe Theory--an escape from the combinatorial chaosMathematics is spectacularly successful doing generalizations (Calculus, Modern Algebra, Algebraic Geometry, etc.); on the other hand, it could say surprisingly little about nontraditional complex systems like the economy or Chess. What is the fundamental problem with complex systems? The short answer is: the immense space of possibilities. The estimated total number of moves in Chess is more than ten-to-the-hundred, which is more than the number of particles in the observable universe. The majority of problems in Discrete Mathematics lead to cases studies of the same size. Statistical Mechanics faces the same problem: a mole of gas contains aboutten-to-23 particles; the underlying dynamic is incredibly complicated. The basic idea of Statistical Mechanics is to work with a priory probabilities; usually equiprobability in the phase space (it is counterintuitive: how does probability enter Newtonian mechanics?). The investigator develops aTic-Tac-Toe Theory which is a striking analog of the Statistical Mechanics approach: it can determine the exact value of infinitely many natural Game Numbers (the same paradox: how does probability enter Chess?). The investigator's book (700 pages) about the subject will come out in2007 at Cambridge University Press.Understanding complex systems is a major challenge for contemporary Mathematics (and Theoretical Computer Science, and Economy, etc.). Games are the perfect natural models. Also analyzing games is perhaps the best way to give the students a taste of mathematical discovery; it is the most natural mathematical experimentation. To integrate this new branch of game theory (the investigator's Tic-Tac-Toe Theory) into math education is an extremely promising aspect. Finally, understanding complex games may give unique insight into how human intelligence works.