In this thesis we are concerned with Markov’s principle, a statement that originated in the Russian school of Constructive Mathematics and stated originally that “if it is impossible that an algorithm does not terminate, then it will terminate”. This principle has been adapted to many dierent contexts, and in particular we are interested in its most common version for arithmetic, which can be stated as “given a total recursive function f, if it is impossible that a there is no n for which f(n) = 0, then there exists an n such that f(n) = 0”. This is in general not accepted in constructivism, where stating an existential statement requires one to be able to show at request a witness for the statement: here there is no clear way to choose such an n. We introduce more in detail the context of constructive mathematics from dierent points of view, and we show how they are related to Markov’s principle.

We discuss the hypothesis that the debate about the interpretation of the orthodox formalism of quantum mechanics (QM) might have been misguided right from the start by a biased metaphysical interpretation of the formalism and its inner mathematical relations. In particular, we focus on the orthodox interpretation of the congruence relation, ‘=’, which relates equivalent classes of different mathematical representations of a vector in Hilbert space, in terms of metaphysical identity. We will argue that this seemingly «common sense» interpretation, at the semantic level, has severe difficulties when considering the syntactic level of the theory.

Here we present in a single essay a combination and completion of the several aspects of the problem of randomness of individual objects which of necessity occur scattered in our texbook «An Introduction to Kolmogorov Complexity and Its Applications» (M. Li and P. Vitanyi), 2nd Ed., Springer-Verlag, 1997.

My first contact with paraconsistent logic was a one page article in the French psychoanalysis magazine, L’âne, entitled something like «Paraconsistent logic: a logic for the inconscious». This was in fact an interview with da Costa. It was of quite general nature, paraconsistent logic was presented in a totally informal way, just as a logic violating the principle of contradiction. This was not my case. I was attracted by paraconsistent logic because I was interested in the question What is logic? Traditionally the principle of contradiction is taken as a fundamental pillar of logic. The idea is that reasoning is not possible without it. Paraconsistency goes against this idea. And if paraconsistent logic is rightly a logic, therefore what are the ground principles of logic, if any?

To develop a spacefaring civilization, humankind must develop technologies which enable safe, affordable and repeatable mobility through the solar system. One such technology is nuclear fusion propulsion which is at present under study mostly as a breakthrough toward the first interstellar probes. The aim of the present paper is to show that fusion drive is even more important in human planetary exploration and constitutes the natural solution to the problem of exploring and colonizing the solar system.

This in an introduction to free probability theory, covering the basic combinatorial and analytic theory, as well as the relations to random matrices and operator algebras. The material is mainly based on the two books of the lecturer, one joint with Nica and one joint with Mingo. Free probability is here restricted to the scalar-valued setting, the operator-valued version is treated in the subsequent lecture series on «Non-Commutative Distributions».

From the beginning of the 16th century to the end of the 18th century, there were not less than ten philosophers who focused extensively on Venn’s ostensible analytical diagrams, as noted by modern historians of logic (Venn, Gardner, Baron, Coumet et al.). But what was the reason for early modern philosophers to use logic or analytical diagrams? Among modern historians of logic one can find two theses which are closely connected to each other: M. Gardner states that since the Middle Ages certain logic diagrams were used just in order to teach “dull-witted students”. Therefore, logic diagrams were just a means to an end. According to P. Bernard, the appreciation of logic diagrams had not started prior to the 1960s, therefore the fact that logic diagrams become an end the point of research arose very late. The paper will focus on the question whether logic resp. analytical diagrams were just means in the history of (early) modern logic or not. In contrast to Gardner, I will argue that logic diagrams were not only used as a tool for “dull-witted students”, but rather as a tool used by didactic reformers in early modern logic. In predating Bernard’s thesis, I will argue that in the 1820s logic diagrams had already become a value in themselves in Arthur Schopenhauer’s lectures on logic, especially in proof theory.

El artículo explora el sentido en que tanto Kant como Husserl entienden lo trascendental y la importancia de este concepto para sus respectivas filosofías. El autor aborda esta relación con la intención de mostrar que ambos filósofos parten de una reflexión radical sobre las condiciones de posibilidad de la experiencia, llegando a conclusiones diferentes sobre ésta. Se pasa revista de las consecuencias de establecer una filosofía trascendental y la enorme diferencia que ambos autores establecen sobre el tema de la experiencia como articuladora de lo trascendental en sus respectivas filosofías.

This article discusses how entropy/information methods are well-suited to analyzing and forecasting the four processes of innovation, transmission, movement, and adaptation, which are the common basis to ecology and evolution. Macroecologists study assemblages of differing species, whereas micro-evolutionary biologists study variants of heritable information within species, such as DNA and epigenetic modifications. These two different modes of variation are both driven by the same four basic processes, but approaches to these processes sometimes differ considerably. For example, macroecology often documents patterns without modeling underlying processes, with some notable exceptions.

This operation was somewhat different from our previous operations targeting ISIS, which were unilateral operations in which we targeted the various terrorist groups without facing any response from the enemy. But in this strike against the United States, our perspective was different, and we were almost certain to be attacked. This is why all of our units were on high alert, ready to face the enemy, and at least ready for a limited war; and we had also prepared for a large-scale war. Since the Second World War, there has been no open attack by a State against the Americans or their interests. No formal military action has been taken against the United States since the Second World War. We have clearly demonstrated that those days are over, and that the decision to strike directly at the United States had been taken unanimously within the country and forces of the Resistance Axis.