This paper arose out of a study of the notes that Joseph Agassi and Czeslaw Lejewski took at Karl Popper's seminar on Logic and Scientific Method (1954–1955).1 It ponders on a basic logical ...distinction Popper had made: between sound inference (valid inference with sound premises) and proof (a collection of inferences that show that a given sentence follows from any premise). The difference between sound inference and proof seems crucial to Popper's epistemology, especially to his emphasis on the distinctness of epistemology and methodology. In this paper, (1) The distinction is explained; (2) The difference is presented as the basis for Popper's view of the history of logic; (3) Some modern hesitations about all this are discussed.
As we have seen so far, biological cells are complex systems containing many different molecular species that interact with one another to form molecular complexes or entirely different molecular ...species. Biomolecular interactions may be conveniently described as chemical reactions, and, in fact, the cell itself can be regarded as a complex biochemical reactor, in which many reactions occur simultaneously. Some examples have already been introduced in previous chapters (see, e.g., the self-association of amphiphiles into micelles and membranes), with others yet to follow.
In the next section, we will lay down the classical framework for describing reaction kinetics. We will first consider that biochemical reactions take place in an aqueous solution (e.g., the cell cytosol), assumed to be homogenous, and that the chemical reaction of interest does not interfere with others taking place simultaneously in the same cellular volume. Many of these approximations do seem to break down under most circumstances in biological cells. In the second part of this chapter, therefore, we will relax some of the approximations, and will make use of fractal concepts to incorporate deviations of biological systems from the Euclidian geometry of smooth objects, which may impinge on the reaction kinetics inside the cell.
Consider Kant’s famous acknowledgment in the opening remarks to his Prolegomena: Hume’s critique of induction, he famously notes, had awakened him of his dogmatic slumber. Is it not a little ...puzzling? Hume explicitly criticized Locke’s philosophy of science: he portrayed the limits of Locke’s empiricism. Kant (even in his deepest of dogmatic slumbers) was no advocate of Locke’s philosophy of science. He was certainly no empiricist. Rather, to the extant that such matters can be determined at all, he was a Leibnizian (or, perhaps more accurately, a Wolffian). Leibniz, let me stress, was the greatest critic of Locke until Hume came around. So what exactly had impressed Kant in Hume’s criticism of a philosophy he did not advocate, to the point of shattering his old (completely opposite) philosophy?
Importantly, Kant deserves here credit which he does not take, and one for which he is hardly ever credited: his distinctive understanding of Hume’s critique is one of his great contributions to philosophy. He understood Hume to have implied the bankruptcy of Leibniz’s project, the project of rationalizing the conflation of science and traditional logic (by means of the explicit founding of the former on the latter). Hume, let me stress, simply did not speak in those terms.
From David Hume1 to Karl Popper2, science theoreticians have struggled with the problem of induction: Scientists usually are convinced that conclusions deduced from scientific methodology, if they ...have proven to be correct over and over in the past, will continue to hold true in the future. Philosophers, in contrast, insist that according to the principles of formal logic one can never be sure about the validity of an empirically deduced scientific fact at a later time or another place. The best one can say is that among the many possible theories that can be deduced from empirical evidence, the one that has proven the most successful in the past is likely to continue being successful in the future3. As a way out of the dilemma, Popper has suggested that scientists should refrain from declaring their theories, even the most successful ones, absolute facts. They should rather accept them as long as they are successful, and be prepared to drop them if they fail to deal successfully with upcoming recalcitrant evidence4. As discussed in Part I, subsequent theoreticians have realized that science does not work that way, and so the lack of a logical foundation of inductive knowledge remains a dilemma.
In this letter, we determine the bounds on covering radius of repetition codes, simplex codes and MacDonald codes over Formula Omitted with respect to the Chinese Euclidean distance.
Abstract
In this paper, we research the cyclic inequalities of generalized width-integral in convex geometry analysis. Based on previous studies, we use Hölder’s inequality and inverse Hölder’s ...inequality to study further the generalized width-integral inequalities of the convex body in Euclidean space, establish a series of cyclic inequalities, and give the condition of equality sign.
In this paper, we study the probability distribution of eigenvalues of the Euclidean random matrix whose entry is of the form for some real function f. Random points 's are independently distributed ...in the ellipsoid or on its surface in including the unit sphere , the simplex, the ordinary ellipsoid and the hyper-cube. Here is allowed to be any real number which includes the two most interesting cases and . The limits of the empirical distribution of its eigenvalues are derived in two high dimensional settings: and as both n and N go to infinity. By taking to be suitable functions, we also give the explicit limiting spectral distributions for some distance matrices whose entries are based on the Euclidean distance and the geodesic distance.
The discovery of novel topological states has served as a major branch in physics and material sciences. To date, most of the established topological states have been employed in Euclidean systems. ...Recently, the experimental realization of the hyperbolic lattice, which is the regular tessellation in non-Euclidean space with a constant negative curvature, has attracted much attention. Here, we demonstrate both in theory and experiment that exotic topological states can exist in engineered hyperbolic lattices with unique properties compared to their Euclidean counterparts. Based on the extended Haldane model, the boundary-dominated first-order Chern edge state with a nontrivial real-space Chern number is achieved. Furthermore, we show that the fractal-like midgap higher-order zero modes appear in deformed hyperbolic lattices, and the number of zero modes increases exponentially with the lattice size. These novel topological states are observed in designed hyperbolic circuit networks by measuring site-resolved impedance responses and dynamics of voltage packets. Our findings suggest a useful platform to study topological phases beyond Euclidean space, and may have potential applications in the field of high-efficient topological devices, such as topological lasers, with enhanced edge responses.
This study aims to find the proper distance calculation method that will be applied to the Sidoarjo on Hands (SoH) application. This study was conducted by comparing three distance algorithms namely ...Euclidean Distance, Manhattan Distance, and Haversine Formula. The results showed that the Euclidean Distance method was the proper method because this method had has the smallest Mean Absolute Deviation (MAD) with 1.71 in amount.
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