Zeno and the Wrong Understanding of Motion—A Philosophical-Mathematical Inquiry into the Concept of Finitude as a Peculiarity of Infinity

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. .

Relative importance of tidal flats and artificial habitats for two spoonbill species and related interspecific differences

Artificial/seminatural environments,such as aquacultural ponds,saltpans,and croplands,have recently been acknowledged as important habitats for coastal waterbirds.Although coastal waterbirds tend to use artificial habitats around tidal flats as roosting sites during high-tide,it remains unclear whether the importance of surrounding habitats relative to tidal flats varies among landscape types,seasons,species,or tidal conditions.The Black-faced Spoonbill(Platalea minor)and Eurasian Spoonbill(P.leucorodia)are two closely related sympatric species in East Asia with narrow and wide distribution ranges and habitat requirements,respectively.We therefore expect that both species will use surrounding artificial habitats across seasons at high tides,but Blackfaced Spoonbills will use them less frequently than Eurasian Spoonbills.Here,we address these hypotheses in the Imazu tidal flat and its surrounding environments in southern Japan.We investigated the habitat use and behavioral patterns of both species through route and behavioral surveys during the fall migration and wintering seasons in 2021.We found that both species used surrounding habitats including artificial ones more frequently than the tidal flat regardless of the tidal condition or season,but spoonbills used these habitats more frequently in winter than in autumn.We also found that Eurasian Spoonbills foraged in surrounding artificial habitats more frequently than Black-faced Spoonbills.These results not only demonstrate how coastal waterbirds exploit surrounding habitats relative to tidal flats but also suggest that the importance of surrounding habitats varies among species and seasons.Our study thus emphasizes that valuing and managing surrounding habitats in addition to tidal flats are key to conserving globally declining waterbirds.

Distinct metamorphic evolution of alternating silica-saturated and silica-deficient microdomains within garnet in ultrahigh-temperature granulites: An example from Sri Lanka

Here we report the occurrence of garnet porphyroblasts that have overgrown alternating silica-saturated and silica deficient microdomains via different mineral reactions. The samples were collected from ultrahigh-temperature(UHT) metapelites in the Highland Complex, Sri Lanka. In some of the metapelites, garnet crystals have cores formed via a dehydration reaction, which had taken place at silicasaturated microdomains and mantle to rim areas formed via a dehydration reaction at silica-deficient microdomains. In contrast, some other garnets in the same rock cores had formed via a dehydration reaction which occurred at silica-deficient microdomains while mantle to rim areas formed via a dehydration reaction at silica-saturated microdomains. Based on the textural observations, we conclude that the studied garnets have grown across different effective bulk compositional microdomains during the prograde evolution. These microdomains could represent heterogeneous compositional layers(paleobedding/laminations) in the precursor sediments or differentiated crenulation cleavages that existed during prograde metamorphism. UHT metamorphism associated with strong ductile deformation, metamorphic differentiation and crystallization of locally produced melt may have obliterated the evidence for such microdomains in the matrix. The lack of significant compositional zoning in garnet probably due to self-diffusion during UHT metamorphism had left mineral inclusions as the sole evidence for earlier microdomains with contrasting chemistry.

Synchronous Changes of Nitrogen and Carbon Isotopic Ratios and Nannoplankton Assemblage in Marine Sediments off Peru at 250 ka: A Role of Phytoplankton in Primary Ocean Productivity

We report the mechanism controlling changes of δ15Norg and δ13Corg values of marine organic matter, based on the change of calcareous nannoplankton assemblage during the last 500,000 years in core samples from ODP Hole 846B off Peru in the equatorial Pacific Ocean. The δ15Norg values fluctuate in a range from 2.1‰ to 6.7‰, giving an abrupt increase since about 250 ka with the averages of 3.8‰ and 5.0‰ during the older and younger periods, respectively. The δ13Corg values change in a range from ?23.5‰ to ?20.1‰ in an inverse correlation with δ15Norg values, describing an increase of average values at 250 ka as well as those of δ15Norg values. The total organic carbon content also shows averages of 0.6% and 0.3% during the younger and older periods, respectively. The numerical increase of deep dwelling species (Florisphaera profunda) of calcareous nannoplankton during the younger period in comparison with the older period indicates that these chemical and isotopic jumps synchronized with nannoplankton assemblage changes in marine sediments are caused by activity of deep dwellers in photic zone more stratified by an abrupt decline of trade wind strength on this sea area since about 250 ka. A study coupling δ15Norg and δ13Corg values and nannoplankton assemblage can be a useful method for evaluating the extent of stratification of photic zone and the roles of surface and deep dwellers of phytoplankton in producing primary organic matter.