请问HN：业余爱好者能对纯数学或理论物理做出贡献吗？
我一直对理论物理（特别是量子力学）和纯数学（特别是数论）充满兴趣。然而，这些并不是我想要作为职业发展的领域。实际上，作为一名兼职业余爱好者，我是否有可能对这些领域做出有意义的贡献？也就是说，我没有正式的培训，除了可能修过一些本科课程，其他方面都是自学的。
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I've always been fascinated by theoretical physics (specifically, quantum mechanics) and pure math (specifically, number theory). However, they're not subjects I'd like to make a career out of. Realistically, might I be able to make meaningful contributions do these fields as a part-time amateur? That is, someone who has no formal training beyond possibly undergraduate courses and is otherwise self-taught.