物理学和数学的一致理论：理论与实验的联系
Towards a Coherent Theory of Physics and Mathematics: The Theory-Experiment Connection
摘要：
数学和物理在基础层面上如何联系的问题是非常有趣的。一种方法是努力建立一个物理学和数学的连贯理论。在这里，我们首先通过考察理论与实验的联系，在这个方向上采取一些步骤。一个隐含的理论层次结构和使用计算机比较理论和实验的作用进行了说明。本文的主要思想是通过将物理理论作为C（复数）上的数学结构，更接近实验测量和计算中实际完成的内容，来加强理论与实验的联系。该方法用C_{n}代替C，C_{n}是n个有理数在某个基上的对R_{n}，I_{n}的集合.这些数字的属性基于表示连续变量的测量结果的数字类型。讨论了基于R_{n}的时空模型。该模型是尺度不变的，恒定步长的区域被指数跳跃中断。一种方法，采取限制n到无穷大，以获得局部平坦的连续空间和时间的概述。可能最有趣的结果是基于R_{n}的空间在尺度变换下是不变的，尺度变换对应于相对于平坦背景的空间的膨胀和收缩。同样，作为时空奇点的原点的位置在这些变换下也不会改变。本文简要地研究了基于C_{n}和基于R_{n}空间的量子力学的一些性质。
英文摘要：
The problem of how mathematics and physics are related at a foundational level is of much interest. One approach is to work towards a coherent theory of physics and mathematics together. Here steps are taken in this direction by first examining the theory experiment connection. The role of an implied theory hierarchy and use of computers in comparing theory and experiment is described. The main idea of the paper is to tighten the theory experiment connection by bringing physical theories, as mathematical structures over C, the complex numbers, closer to what is actually done in experimental measurements and computations. The method replaces C by C_{n} which is the set of pairs, R_{n},I_{n}, of n figure rational numbers in some basis. The properties of these numbers are based on the type of numbers that represent measurement outcomes for continuous variables. A model of space and time based on R_{n} is discussed. The model is scale invariant with regions of constant step size interrupted by exponential jumps. A method of taking the limit n to infinity to obtain locally flat continuum based space and time is outlined. Possibly the most interesting result is that R_{n} based space is invariant under scale transformations which correspond to expansion and contraction of space relative to a flat background. Also the location of the origin, which is a space and time singularity, does not change under these transformations. Some properties of quantum mechanics based on C_{n} and on R_{n} space are briefly investigated.
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