| 摘要: |
| 数学是抽象的, 它使用的符号和概念, 以及研究的对象可以完全和实际的物体无关. 但是数学家和计算机都是物理的实体, 受物理规律的约束. 这个无法逃避的基本事实会影响数学的发展. 通过仔细分析图灵机, 本文指出图灵和他同时代科学家忽略了一个基本的物理可能, 信息的载体可以是量子系统. 这个疏忽使得他们提出的计算机模型只能处理经典信息, 能力受到极大限制. 哥德尔不完全定理则反映了这样一个事实: 数学家和计算机是由有限多的原子和分子构成, 他们只能从有限的假设或公理出发, 利用有限的符号和字母, 完成有限步的推导. 因此即使加上未来所有的数学家和计算机, 他们能证明的数学定理一定是可数无穷多的. 但是世界上有不可数多的数学命题, 这样总是存在很多数学命题你既无法证明也无法证伪. 朗道尔(Landauer) 曾经说, 信息是物理的; 在同样的意义上, 数学是物理的. |
| 关键词: 图灵机, 量子计算机, 对角论证法, 动力学系统, 哥德尔不完全定理, 停机问题 |
| DOI： |
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| 基金项目: |
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| Mathematics Is Physical |
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WU Biao1,2
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| (1..International Center for Quantum Materials , School of Physics , Peking University , Beijing 100871 , China;2.Wilczek Quantum Center , School of Physics and Astronomy Shanghai Jiao Tong University , Shanghai 200240, China) |
| Abstract: |
| The world of mathematics is often considered abstract, with its symbols, concepts, and topics appearing unrelated to physical objects. However, it is important to recognize that the development of mathematics is fundamentally influenced by a basic fact： mathematicians and computers are physical objects subject to the laws of physics. Through an analysis of the Turing machine, it becomes evident that Turing and his contemporaries overlooked a physical possibility： information carriers can be quantum systems. As a result, computing models like the Turing machine can only process classical information, limiting their computing power. G?del's incompleteness theorem highlights the basic fact that mathematicians and computers are made up of finite numbers of atoms and molecules. They can only start with a finite number of axioms, use a finite number of symbols and deduction rules, and arrive at theorems with a finite number of steps. While the number of proofs may be infinite after including all future mathematicians and computers, they must still be enumerable. In contrast, the number of mathematical statements is uncountable, meaning that there will always be mathematical statements that cannot be proved true or false. Just as Landauer claimed that information is physical, mathematics is also physical, limited or empowered by the physical entities that carries it out or embodies it. |
| Key words: quantum dot, single |
