In contrast to the solutions of applied mathematics to Zeno’s paradoxes, I focus on the concept of motion and show that, by distinguishing two different forms of motion, Zeno’s apparent paradoxes are not paradoxical...In contrast to the solutions of applied mathematics to Zeno’s paradoxes, I focus on the concept of motion and show that, by distinguishing two different forms of motion, Zeno’s apparent paradoxes are not paradoxical at all. Zeno’s paradoxes indirectly prove that distances are not composed of extensionless points and, in general, that a higher dimension cannot be completely composed of lower ones. Conversely, lower dimensions can be understood as special cases of higher dimensions. To illustrate this approach, I consider Cantor’s only apparent proof that the real numbers are uncountable. However, his widely accepted indirect proof has the disadvantage that it depends on whether there is another way to make the real numbers countable. Cantor rightly assumes that there can be no smallest number between 0 and 1, and therefore no beginning of counting. For this reason he arbitrarily lists the real numbers in order to show with his diagonal method that this list can never be complete. The situation is different if we start with the largest number between 0 and 1 (0.999…) and use the method of an inverted triangle, which can be understood as a special fractal form. Here we can construct a vertical and a horizontal stratification with which it is actually possible to construct all real numbers between 0 and 1 without exception. Each column is infinite, and each number in that column is the starting point of a new triangle, while each row is finite. Even in a simple sine curve, we experience finiteness with respect to the y-axis and infinity with respect to the x-axis. The first parts of this article show that Zeno’s assumptions contradict the concept of motion as such, so it is not surprising that this misconstruction leads to contradictions. In the last part, I discuss Cantor’s diagonal method and explain the method of an inverted triangle that is internally structured like a fractal by repeating this inverted triangle at each column. The consequence is that we encounter two very different methods of counting. Vertically it is continuous, horizontally it is discrete. While Frege, Tarski, Cantor, Gödel and the Vienna Circle tried to derive the higher dimension from the lower, a procedure that always leads to new contradictions and antinomies (Tarski, Russell), I take the opposite approach here, in which I derive the lower dimension from the higher. This perspective seems to fail because Tarski, Russell, Wittgenstein, and especially the Vienna Circle have shown that the completeness of the absolute itself is logically contradictory. For this reason, we agree with Hegel in assuming that we can never fully comprehend the Absolute, but only its particular manifestations—otherwise we would be putting ourselves in the place of the Absolute, or even God. Nevertheless, we can understand the Absolute in its particular expressions, as I will show with the modest example of the triangle proof of the combined horizontal and vertical countability of the real numbers, which I developed in rejection of Cantor’s diagonal proof. .展开更多
数字经济与实体经济融合(以下简称数实融合)已成为中国经济发展的新动力。为探究当前中国数实融合发展的现状和区域差异,首先在剖析数实融合发展内涵及机理的基础上,从融合条件、融合应用、融合效益三个维度构建评价指标体系,其次利用...数字经济与实体经济融合(以下简称数实融合)已成为中国经济发展的新动力。为探究当前中国数实融合发展的现状和区域差异,首先在剖析数实融合发展内涵及机理的基础上,从融合条件、融合应用、融合效益三个维度构建评价指标体系,其次利用纵横向拉开档次法对中国30个省区市2013—2020年数实融合发展水平进行测度,最后结合基尼系数与探索性空间数据分析法(exploratory spatial data analysis,ESDA)研究区域间融合发展的时空差异。实证研究结果表明:近几年中国数实融合发展水平持续上升,整体具有向好的发展态势;区域间发展差异明显,发展水平从东向西依次递减,融合发展水平最高的是广东,最低的是青海;在发展过程中,融合应用能力不足是制约发展的关键原因且地域间缺乏融合互动,在空间上表现出明显的正向集聚特征。本研究从战略指引、应用强化和区域合作等方面提出对策建议,可为各部门制定数实融合发展的相关计划提供参考。展开更多
文摘In contrast to the solutions of applied mathematics to Zeno’s paradoxes, I focus on the concept of motion and show that, by distinguishing two different forms of motion, Zeno’s apparent paradoxes are not paradoxical at all. Zeno’s paradoxes indirectly prove that distances are not composed of extensionless points and, in general, that a higher dimension cannot be completely composed of lower ones. Conversely, lower dimensions can be understood as special cases of higher dimensions. To illustrate this approach, I consider Cantor’s only apparent proof that the real numbers are uncountable. However, his widely accepted indirect proof has the disadvantage that it depends on whether there is another way to make the real numbers countable. Cantor rightly assumes that there can be no smallest number between 0 and 1, and therefore no beginning of counting. For this reason he arbitrarily lists the real numbers in order to show with his diagonal method that this list can never be complete. The situation is different if we start with the largest number between 0 and 1 (0.999…) and use the method of an inverted triangle, which can be understood as a special fractal form. Here we can construct a vertical and a horizontal stratification with which it is actually possible to construct all real numbers between 0 and 1 without exception. Each column is infinite, and each number in that column is the starting point of a new triangle, while each row is finite. Even in a simple sine curve, we experience finiteness with respect to the y-axis and infinity with respect to the x-axis. The first parts of this article show that Zeno’s assumptions contradict the concept of motion as such, so it is not surprising that this misconstruction leads to contradictions. In the last part, I discuss Cantor’s diagonal method and explain the method of an inverted triangle that is internally structured like a fractal by repeating this inverted triangle at each column. The consequence is that we encounter two very different methods of counting. Vertically it is continuous, horizontally it is discrete. While Frege, Tarski, Cantor, Gödel and the Vienna Circle tried to derive the higher dimension from the lower, a procedure that always leads to new contradictions and antinomies (Tarski, Russell), I take the opposite approach here, in which I derive the lower dimension from the higher. This perspective seems to fail because Tarski, Russell, Wittgenstein, and especially the Vienna Circle have shown that the completeness of the absolute itself is logically contradictory. For this reason, we agree with Hegel in assuming that we can never fully comprehend the Absolute, but only its particular manifestations—otherwise we would be putting ourselves in the place of the Absolute, or even God. Nevertheless, we can understand the Absolute in its particular expressions, as I will show with the modest example of the triangle proof of the combined horizontal and vertical countability of the real numbers, which I developed in rejection of Cantor’s diagonal proof. .
文摘数字经济与实体经济融合(以下简称数实融合)已成为中国经济发展的新动力。为探究当前中国数实融合发展的现状和区域差异,首先在剖析数实融合发展内涵及机理的基础上,从融合条件、融合应用、融合效益三个维度构建评价指标体系,其次利用纵横向拉开档次法对中国30个省区市2013—2020年数实融合发展水平进行测度,最后结合基尼系数与探索性空间数据分析法(exploratory spatial data analysis,ESDA)研究区域间融合发展的时空差异。实证研究结果表明:近几年中国数实融合发展水平持续上升,整体具有向好的发展态势;区域间发展差异明显,发展水平从东向西依次递减,融合发展水平最高的是广东,最低的是青海;在发展过程中,融合应用能力不足是制约发展的关键原因且地域间缺乏融合互动,在空间上表现出明显的正向集聚特征。本研究从战略指引、应用强化和区域合作等方面提出对策建议,可为各部门制定数实融合发展的相关计划提供参考。
