实验是创立数学思想甚至数学新领域的传统方法之一。最著名的一个例子就是 Fermat 猜想，它是 Fermat 尝试为著名的 Fermat 方程寻找整数解时进行的推测。这个猜想导致了一个完整知识领域的创建，但是直到数百年之后才被证明。 本书基于俄罗斯数学大师 V. I. Arnold 先生的演讲，介绍了数学研究的几个新方向。所有这些方向都是基于作者进行的数值实验，它们引出了目前仍未解决（既未被证明也未被证伪）的新猜想。猜想的内容从几何和拓扑（平面曲线和光滑函数的统计数据）到组合函数（组合复杂性和随机排列）再到代数和数论（连分数和 Galois 群）。对于每个主题，作者都描述了问题并展示了导致他做出特定猜想的数值结果。在大多数情况下，作者都指出了读者怎样才能接近这些猜想（至少通过进行更多的数值实验）。 本书以 Arnold 先生的独特风格写成，适合广泛的数学人群阅读，从有兴趣独立探索数学的不寻常领域的高中生，到本科生和研究生，直至想获得一种新颖、有些非传统的数学研究观点的研究人员。

前辅文
Introduction
Lecture 1. The Statistics of Topology and Algebra
  §1. Hilbert's Sixteenth Problem
  §2. The Statistics of Smooth Functions
  §3. Statistics and the Topology of Periodic Functions and Trigonometric Polynomials
  §4. Algebraic Geometry of Trigonometric Polynomials
  Editor's notes
Lecture 2. Combinatorial Complexity and Randomness
  §1. Binary Sequences
  §2. Graph of the Operation of Taking Differences
  §3. Logarithmic Functions and Their Complexity
  §4. Complexity and Randomness of Tables of Galois Fields
  Editor's notes
Lecture 3. Random Permutations and Young Diagrams of Their Cycles
  §1. Statistics of Young Diagrams of Permutations of Small Numbers of Objects
  §2. Experimentation with Random Permutations of Larger Numbers of Elements
  §3. Random Permutations of p2 Elements Generated by Galois Fields
  §4. Statistics of Cycles of Fibonacci Automorphisms
  Editor's notes
Lecture 4. The Geometry of Frobenius Numbers for Additive Semigroups
  §1. Sylvester's Theorem and the Frobenius Numbers
  §2. Trees Blocked by Others in a Forest
  §3. The Geometry of Numbers
  §4. Upper Bound Estimate of the Frobenius Number
  §5. Average Values of the Frobenius Numbers
  §6. Proof of Sylvester's Theorem
  §7. The Geometry of Continued Fractions of Frobenius Numbers
  §8. The Distribution of Points of an Additive Semigroup on the Segment Preceding the Frobenius Number
  Editor's notes
Bibliography