Mathematics Is Physical (2306.03766v2)

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\"odel'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 carry it out or embody it.