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math-thinking-lenses

探索数学问题时的多视角思维透镜。用公理化、对偶、对称、谱、变分、极端、归纳、计数等视角切换思路，帮助突破单一思路的僵局。当探索问题思路、形成猜想或评估证明路径时使用。

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# 数学思维透镜

面对一道难题时，不沿单一路径硬推，而是轮流用不同「透镜」观察它。每个透镜是一种改变看问题方式的策略；切换透镜常能暴露新结构、新不变量或新证明入口。

## 透镜清单

1. **公理化**：显式列出全部假设与结论，追问哪条假设是必要的、去掉会怎样。识别最小充分条件。
2. **对偶**：把对象换成对偶对象——向量与泛函、图与补图、命题与逆否命题、范畴与反范畴。
3. **对称**：寻找作用在对象上的对称群，利用对称性约化问题；不变量与商对象常是钥匙。
4. **谱**：把问题转成特征值/特征函数语言，用谱分解、谱半径、主特征值刻画结构。
5. **变分**：把要证明的结论改写成某个泛函的极值问题，用变分原理或极值条件。
6. **极端**：考虑极端情形——最大/最小、极值点、退化对象、取等情形，极端处常最脆弱也最易算。
7. **归纳**：对规模做数学归纳、强归纳、结构归纳；寻找可归纳的规模参数。
8. **计数**：两边计数同一集合、生成函数、双射、容斥原理，把存在性换成计数。