[论文解读] Introduction to Projective Arithmetics

Mark Burgin|arXiv (Cornell University)|Oct 15, 2010

Computability, Logic, AI Algorithms参考文献 15被引用 11

一句话总结

本文引入了投影算术（projective arithmetics），这是一类通过可逆函数重新定义加法与乘法的非丢番图算术（non-Diophantine arithmetics），从而推广了传统的（丢番图算术）算术。通过使用函数 f 参数化运算，该框架能够构建一致的算术系统，以建模各种自然与社会现象，当 f(x) = x 时，丢番图算术即成为其特例。其主要贡献在于为替代算术提供了严谨的数学基础，从而解决了科学与日常推理中的不一致问题。

ABSTRACT

Science and mathematics help people to better understand world, eliminating many inconsistencies, fallacies and misconceptions. One of such misconceptions is related to arithmetic of natural numbers, which is extremely important both for science and everyday life. People think their counting is governed by the rules of the conventional arithmetic and thus other kinds of arithmetics of natural numbers do not exist and cannot exist. However, this popular image of the situation with the natural numbers is wrong. In many situations, people have to utilize and do implicitly utilize rules of counting and operating different from rules and operations in the conventional arithmetic. This is a consequence of the existing diversity in nature and society. To correctly represent this diversity, people have to explicitly employ different arithmetics. To make a distinction, we call the conventional arithmetic by the name Diophantine arithmetic, while other arithmetics are called non-Diophantine. There are two big families of non-Diophantine arithmetics: projective arithmetics and dual arithmetics (Burgin, 1997). In this work, we give an exposition of projective arithmetics, presenting their properties and considering also a more general mathematical structure called a projective prearithmetic. The Diophantine arithmetic is a member of this parametric family: its parameter is equal to the identity function f(x) = x. In conclusion, it is demonstrated how non-Diophantine arithmetics may be utilized beyond mathematics and how they allow one to eliminate inconsistencies and contradictions encountered by other researchers.

研究动机与目标

• 挑战‘传统算术是自然数唯一有效系统’的误解。
• 将投影算术形式化为基于可逆函数的一类非丢番图算术。
• 证明丢番图算术是该更广泛参数化框架中的特例。
• 展示投影算术如何解决科学与实际推理中的不一致问题。
• 为非丢番图算术在纯数学之外的应用建立基础。

提出的方法

• 通过一个可逆函数 f 定义投影预算术（projective prearithmetic），将标准运算转换为新运算。
• 通过 f 构造新系统中的加法与乘法：a ⊕ b = f⁻¹(f(a) + f(b)) 与 a ⊗ b = f⁻¹(f(a) × f(b))。
• 证明当 f 足够光滑且可逆时，所得结构满足交换环的公理。
• 表明丢番图算术对应于 f(x) = x 的情况，从而将其嵌入更广泛的家族之中。
• 利用参数 f 生成适用于不同应用领域的多样化算术系统。
• 分析所得系统的代数与结构特性，包括封闭性与分配律。

实验结果

研究问题

• RQ1 能否为自然数定义除丢番图算术外的一致算术系统？
• RQ2 投影算术如何通过函数变换推广传统算术？
• RQ3 函数 f 需满足何种条件，才能使所得系统成为有效的算术结构？
• RQ4 丢番图算术如何作为投影框架中的特例出现？
• RQ5 非丢番图算术在哪些方面能解决科学与实际推理中的不一致问题？

主要发现

• 投影算术是由可逆函数 f 定义的非丢番图算术参数族，当 f(x) = x 时对应于丢番图算术。
• 投影算术中的加法与乘法运算通过 f⁻¹(f(a) + f(b)) 与 f⁻¹(f(a) × f(b)) 一致定义，从而保持代数结构。
• 当 f 光滑且可逆时，所得系统为交换环，确保数学一致性。
• 非丢番图算术可建模自然界与社会中传统算术失效的多种现象。
• 该框架通过提供适用于特定领域的算术系统，能够解决科学研究中遇到的矛盾与不一致。
• 投影预算术被识别为更一般的结构，投影算术是其子类。