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
Background： So far, studies of avian space use are mostly realized in 2D, with the vertical dimension ignored. We propose here a new, relatively simple and computationally reasonable method for the estimation of volu...Background： So far, studies of avian space use are mostly realized in 2D, with the vertical dimension ignored. We propose here a new, relatively simple and computationally reasonable method for the estimation of volumetric （i.e. 3D） avian home ranges.Methods： Through accurate GPS data-loggers, we collected 25,405 GPS points on Lesser Kestrels＇ （Fdlco noumonni） space use during the nestling period in one main colony in Italy.We applied our 3D home range estimator to the whole GPS dataset, and also separately to diurnal and nocturnal GPS points.Results： The 3D colony home range resulted equal to 28.12 km3. By considering daytime and night-time separately, the volumetric home ranges resulted considerably different.Conclusions： Our 3D home range estimator, because of its intuitive and straightforward properties, can easily capi-talize on the datasets offered by modern biotelemetry （data-loggers, light detection and LIDAR sensors） and enhance conservation strategies for mitigating anthropogenic impacts on bird species. Its applications embrace, but are not limited to, more accurate estimates of collision risk with power lines, aircrafts and wind farms, and increased knowledge of birds＇space requirements in order to persist in their distribution areas.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. .
基金supported by Lipu UK and the Alta Murgia National Park
文摘Background： So far, studies of avian space use are mostly realized in 2D, with the vertical dimension ignored. We propose here a new, relatively simple and computationally reasonable method for the estimation of volumetric （i.e. 3D） avian home ranges.Methods： Through accurate GPS data-loggers, we collected 25,405 GPS points on Lesser Kestrels＇ （Fdlco noumonni） space use during the nestling period in one main colony in Italy.We applied our 3D home range estimator to the whole GPS dataset, and also separately to diurnal and nocturnal GPS points.Results： The 3D colony home range resulted equal to 28.12 km3. By considering daytime and night-time separately, the volumetric home ranges resulted considerably different.Conclusions： Our 3D home range estimator, because of its intuitive and straightforward properties, can easily capi-talize on the datasets offered by modern biotelemetry （data-loggers, light detection and LIDAR sensors） and enhance conservation strategies for mitigating anthropogenic impacts on bird species. Its applications embrace, but are not limited to, more accurate estimates of collision risk with power lines, aircrafts and wind farms, and increased knowledge of birds＇space requirements in order to persist in their distribution areas.
