ALBERT

Description: To explore the extent of embeddability of Leibnizian infinitesimal calculus in first-order logic (FOL) and modern frameworks, we propose to set aside ontological issues and focus on procedural questions. This would enable an account of Leibnizian procedures in a framework limited to FOL with a small number of additional ingredients such as the relation of infinite proximity. If, as we argue here, first order logic is indeed suitable for developing modern proxies for the inferential moves found in Leibnizian infinitesimal calculus, then modern infinitesimal frameworks are more appropriate to interpreting Leibnizian infinitesimal calculus than modern Weierstrassian ones.

Description: The aim of science is the explanation of complicated systems by reducing it to simple subsystems. According to a millennia-old imagination this will be attained by dividing matter into smaller and smaller pieces of it. The popular superstition that smallness implies simplicity seems to be ineradicable. However, since the beginning of quantum theory it would be possible to realize that the circumstances in nature are exactly the other way round. The idea “smaller becomes simpler” is useful only down to the atoms of chemistry. Planck’s formula shows that smaller extensions are related to larger energies. That more and more energy should result in simpler and simpler structures, this does not only sound absurd, it is absurd. A reduction to really simple structures leads one to smallest energies and, thus, to utmost extended quantum systems. The simplest quantum structure, referred to as quantum bit, has a two-dimensional state space, and it establishes a cosmological structure. Taking many of such quantum bits allows also for the construction of localized particles. The non-localized fraction of quantum bits can appear as “dark matter”.

Description: Human language has the characteristic of being open and in some cases polysemic. The word “infinite” is used often in common speech and more frequently in literary language, but rarely with its precise meaning. In this way the concepts can be used in a vague way but an argument can still be structured so that the central idea is understood and is shared with to the partners. At the same time no precise definition is given to the concepts used and each partner makes his own reading of the text based on previous experience and cultural background. In a language dictionary the first meaning of “infinite” agrees with the etymology: what has no end. We apply the word infinite most often and incorrectly as a synonym for “very large” or something that we do not perceive its completion. In this context, the infinite mentioned in dictionaries refers to the idea or notion of the “immeasurably large” although this is open to what the individual’s means by “immeasurably great.” Based on this linguistic imprecision, the authors present a non Cantorian theory of the potential and actual infinite. For this we have introduced a new concept: the homogon that is the whole set that does not fall within the definition of sets established by Cantor.

Description: The use of mathematics in economics has been widely discussed. The philosophical discussion on what mathematics is remains unsettled on why it can be applied to the study of the real world. We propose to get back to some philosophical conceptions that lead to a language-like role for the mathematical analysis of economic phenomena and present some problems of interest that can be better examined in this light. Category theory provides the appropriate tools for these analytical approach.

Description: In 1975, two experimental groups have independently observed the \(4\pi \) -symmetry of neutrons’ spin, when passing through a static magnetic field, using a three-blade interferometer made from a single perfect Si-crystal (analogous to the Mach Zehnder interferometer of light optics). In this article, we provide a complete analysis of the experiment, both from a theoretical and conceptual point of view. Firstly, we solve the Schrödinger equation in the weak potential approximation, to obtain the amplitude of the refracted and forward refracted beams, produced by the passage of neutrons through one of the three plates of the LLL interferometer. Secondly, we analyze their passage through a static magnetic field region. This allows us to find explicit expressions for the intensities of the four beams exiting the interferometer, two of which will be interfering and show a typical \(4\pi \) -symmetry, when the strength of the magnetic field is varied. In the last part of the article, we provide a conceptual analysis of the experiment, showing that a neutron’s phase change, when passing through the magnetic field, is due to a longitudinal Stern–Gerlach effect, and not to a Larmor precession. We also emphasize that these experiments do not prove the observability of the sign change of the wave function, when a neutron is \(2\pi \) rotated, but strongly indicate that the latter, like any other elementary “particle,” would be a genuinely non-spatial entity.

Description: Although the present paper looks upon the formal apparatus of quantum mechanics as a calculus of correlations, it goes beyond a purely operationalist interpretation. Having established the consistency of the correlations with the existence of their correlata (measurement outcomes), and having justified the distinction between a domain in which outcome-indicating events occur and a domain whose properties only exist if their existence is indicated by such events, it explains the difference between the two domains as essentially the difference between the manifested world and its manifestation. A single, intrinsically undifferentiated Being manifests the macroworld by entering into reflexive spatial relations. This atemporal process implies a new kind of causality and sheds new light on the mysterious nonlocality of quantum mechanics. Unlike other realist interpretations, which proceed from an evolving-states formulation, the present interpretation proceeds from Feynman’s formulation of the theory, and it introduces a new interpretive principle, replacing the collapse postulate and the eigenvalue–eigenstate link of evolving-states formulations. Applied to alternatives involving distinctions between regions of space, this principle implies that the spatiotemporal differentiation of the physical world is incomplete. Applied to alternatives involving distinctions between things, it warrants the claim that, intrinsically, all fundamental particles are identical in the strong sense of numerical identical. They are the aforementioned intrinsically undifferentiated Being, which manifests the macroworld by entering into reflexive spatial relations.

Description: Over the past few decades the notion of symmetry has played a major role in physics and in the philosophy of physics. Philosophers have used symmetry to discuss the ontology and seeming objectivity of the laws of physics. We introduce several notions of symmetry in mathematics and explain how they can also be used in resolving different problems in the philosophy of mathematics. We use symmetry to discuss the objectivity of mathematics, the role of mathematical objects, the unreasonable effectiveness of mathematics and the relationship of mathematics to physics.

Description: Very large databases are a major opportunity for science and data analytics is a remarkable new field of investigation in computer science. The effectiveness of these tools is used to support a “philosophy” against the scientific method as developed throughout history. According to this view, computer-discovered correlations should replace understanding and guide prediction and action. Consequently, there will be no need to give scientific meaning to phenomena, by proposing, say, causal relations, since regularities in very large databases are enough: “with enough data, the numbers speak for themselves”. The “end of science” is proclaimed. Using classical results from ergodic theory, Ramsey theory and algorithmic information theory, we show that this “philosophy” is wrong. For example, we prove that very large databases have to contain arbitrary correlations. These correlations appear only due to the size, not the nature, of data. They can be found in “randomly” generated, large enough databases, which—as we will prove—implies that most correlations are spurious. Too much information tends to behave like very little information. The scientific method can be enriched by computer mining in immense databases, but not replaced by it.

Description: Axiomatization is uncommon outside mathematics, partly for being often viewed as embalming, partly because the best-known axiomatizations have serious shortcomings, and partly because it has had only one eminent champion, namely David Hilbert (Math Ann 78:405–415, 1918 ). The aims of this paper are (a) to describe what will be called dual axiomatics , for it concerns not just the formalism, but also the meaning (reference and sense) of the key concepts; and (b) to suggest that every instance of dual axiomatics presupposes some philosophical view or other. To illustrate these points, a theory of solidarity will be crafted and axiomatized, and certain controversies in both classical and quantum physics, as well as in the philosophy of mind, will be briefly discussed. The upshot of this paper is that dual axiomatics, unlike the purely formal axiomatics favored by the structuralists school, is not a luxury but a tool helping resolve some scientific controversies.

Description: 〈h3〉Abstract〈/h3〉 〈p〉The Vitruvian Man of Leonardo da Vinci is one of the most famous and most studied drawings over the world as well as one of the most reproduced ones, e.g. in coins (Italian euros), space suit patches (NASA), books and movies. The aim of the present work is to discuss the Vitruvian Man as a figurative representation of the Leonardo’s scientific method. Our analysis is based on scientific elements both present in the drawing and provided by Leonardo in his approach to this drawing. Our thesis is that the square symbolizes the measurable physical world and that the man inscribed within the square refers to the physics measurement process based on the operational definition of quantities, including the measurement unit system and the quantities conversion factors. Therefore, the measurement process is fundamental for the Leonardo’s approach to the scientific knowledge, albeit, the drawing also suggests that this latter does not correspond with the true knowledge. The circle, which has a different center with respect to the square, symbolizes the truth, to which the man inscribed in the square yearns, without ever achieving it, the truth being reachable only by the man inscribed within the circle.〈/p〉