“Super-humans” is usually to be linked to Nietzsche or to Heidegger’s criticism to Nietzsche, or even to the ideology of Nazism. However, they can be properly underlain by philosophical and scientific anthropology as that biological species who will originate from humans eventually in the course of evolution. There is a series of more or less well-established facts in anthropogenesis, which would be relevant to the philosophical question about the “super-humans”: bipedalism, cooling by sweating, specific hair or its lack, omnivorous-ness, thumb opposition and apposition, vocal system of speech production, human brain, long childhood; our species is evolutionary young (about 200 000 years old), but it is the last survived descendant being genetically exceptionally homogenous (<00,01% genetic differences) of the genus “homo” (about 6 000 000 old). All this generates a few main features of our population: society, technics, language, and mind, which guarantee the contemporary absolute domination of mankind. The society has reached a natural limitation of earth. The technics depends on how much energy is produced. The mind is restricted by its carrier, i.e. by the brain. Thus only the language seems to be the frontier of any future development inducing a much better use of the former three. The recent informational technologies suggest the same. Language is defined as symbolic image of the world doubling it by an ideal or virtual world, which is fruitful for creativity and any modeling of the real world. Consequently, a gap between the material and the ideal world produces language. The language increases that gap in turn. Furthermore, the ideal world is secondary and derivative from the material world in origin and objectivity: Language serves for the world to be ordered. Thus language refers to the philosophical categories of ‘being’ and ‘time’. Any “super-language” should transcend some of those definitive borders of language and be a generalization. The involving of infinity can extend the language. Any human language is finite and addresses some finite reality. Thus the gap between reality and any model in language can be seen as that between infinity and its limitation to any finite representation: Finite representations dominate over society, technics, and the mind use. A “super-language” as an “infinite language” can be approached in a few reference frames: Husserl’s “Back to the things themselves!” if “phenomenon” in his philosophy is thought as the ‘word’ of the language of consciousness; the semantic and philosophical theory of symbol: from consciousness and language to reality; the concept of infinity in mathematics and its foundation: set or category theory; quantum mechanics and information: the coincidence of the quantum model and reality; quantum computer. Mankind is approached the problem of infinite language as the language of nature

Donald Capps suggested the hypothesis that “the Nash equilibrium is descriptive of the normal brain, whereas the game theory formulated by John van Neumann, which Nash’s theory challenges, is descriptive of the schizophrenic brain”. The paper offers arguments in its favor. They are from psychiatry, game theory, set theory, philosophy and theology. The Nash equilibrium corresponds to wholeness, stable emergent properties as well as to representing actual infinity on a material, limited and finite organ as a human brain.

The paper discusses a few tensions “crucifying” the works and even personality of the great Georgian philosopher Merab Mamardashivili: East and West; human being and thought, symbol and consciousness, infinity and finiteness, similarity and differences. The observer can be involved as the correlative counterpart of the totality: An observer opposed to the totality externalizes an internal part outside. Thus the phenomena of an observer and the totality turn out to converge to each other or to be one and the same. In other words, the phenomenon of an observer includes the singularity of the solipsistic Self, which (or “who”) is the same as that of the totality. Furthermore, observation can be thought as that primary and initial action underlain by the phenomenon of an observer. That action of observation consists in the externalization of the solipsistic Self outside as some external reality. It is both a zero action and the singularity of the phenomenon of action. The main conclusions are: Mamardashvili’s philosophy can be thought both as the suffering effort to be a human being again and again as well as the philosophical reflection on the genesis of thought from itself by the same effort. Thus it can be recognized as a powerful tension between signs anа symbol, between conscious structures and consciousness, between the syncretism of the East and the discursiveness of the West crucifying spiritually Georgia.

A historical review and philosophical look at the introduction of “negative probability” as well as “complex probability” is suggested. The generalization of “probability” is forced by mathematical models in physical or technical disciplines. Initially, they are involved only as an auxiliary tool to complement mathematical models to the completeness to corresponding operations. Rewards, they acquire ontological status, especially in quantum mechanics and its formulation as a natural information theory as “quantum information” after the experimental confirmation the phenomena of “entanglement”. Philosophical interpretations appear. A generalization of them is suggested: ontologically, they correspond to a relevant generalization to the relation of a part and its whole where the whole is a subset of the part rather than vice versa. The structure of “vector space” is involved necessarily in order to differ the part “by itself” from it in relation to the whole as a projection within it. That difference is reflected in the new dimension of vector space both mathematically and conceptually. Then, “negative or complex probability” are interpreted as a quantity corresponding the generalized case where the part can be “bigger” than the whole, and it is represented only partly in general within the whole.

The original conception of atomism suggests “atoms”, which cannot be divided more into composing parts. However, the name “atom” in physics is reserved for entities, which can be divided into electrons, protons, neutrons and other “elementary particles”, some of which are in turn compounded by other, “more elementary” ones. Instead of this, quantum mechanics is grounded on the actually indivisible quanta of action limited by the fundamental Planck constant. It resolves the problem of how both discrete and continuous (even smooth) to be described uniformly and invariantly in thus. Quantum mechanics can be interpreted in terms of quantum information. Qubit is the indivisible unit (“atom”) of quantum information. The imagery of atomism in modern physics moves from atoms of matter (or energy) via “atoms” (quanta) of action to “atoms” (qubits) of quantum information. This is a conceptual shift in the cognition of reality to terms of information, choice, and time

Pattern recognition is represented as the limit, to which an infinite Turing process converges. A Turing machine, in which the bits are substituted with qubits, is introduced. That quantum Turing machine can recognize two complementary patterns in any data. That ability of universal pattern recognition is interpreted as an intellect featuring any quantum computer. The property is valid only within a quantum computer: To utilize it, the observer should be sited inside it. Being outside it, the observer would obtain quite different result depending on the degree of the entanglement of the quantum computer and observer. All extraordinary properties of a quantum computer are due to involving a converging infinite computational process contenting necessarily both a continuous advancing calculation and a leap to the limit. Three types of quantum computation can be distinguished according to whether the series is a finite one, an infinite rational or irrational number.

The paper addresses the problem, which quantum mechanics resolves in fact. Its viewpoint suggests that the crucial link of time and its course is omitted in understanding the problem. The common interpretation underlain by the history of quantum mechanics sees discreteness only on the Plank scale, which is transformed into continuity and even smoothness on the macroscopic scale. That approach is fraught with a series of seeming paradoxes. It suggests that the present mathematical formalism of quantum mechanics is only partly relevant to its problem, which is ostensibly known. The paper accepts just the opposite: The mathematical solution is absolute relevant and serves as an axiomatic base, from which the real and yet hidden problem is deduced. Wave-particle duality, Hilbert space, both probabilistic and many-worlds interpretations of quantum mechanics, quantum information, and the Schrödinger equation are included in that base. The Schrödinger equation is understood as a generalization of the law of energy conservation to past, present, and future moments of time. The deduced real problem of quantum mechanics is: “What is the universal law describing the course of time in any physical change therefore including any mechanical motion?”

Husserl’s phenomenology is what is used, and then the conception of “bracketing reality” is modelled to generalize Peano arithmetic in its relation to set theory in the foundation of mathematics. The obtained model is equivalent to the generalization of Peano arithmetic by means of replacing the axiom of induction with that of transfinite induction. A comparison to Mach’s doctrine is used to be revealed the fundamental and philosophical reductionism of Husserl’s phenomenology leading to a kind of Pythagoreanism in the final analysis

The present first part about the eventual completeness of mathematics (called "Hilbert mathematics") is concentrated on the Gödel incompleteness (1931) statement: if it is an axiom rather than a theorem inferable from the axioms of (Peano) arithmetic, (ZFC) set theory, and propositional logic, this would pioneer the pathway to Hilbert mathematics. One of the main arguments that it is an axiom consists in the direct contradiction of the axiom of induction in arithmetic and the axiom of infinity in set theory. Thus, the pair of arithmetic and set are to be similar to Euclidean and non-Euclidean geometries distinguishably only by the Fifth postulate now, i.e. after replacing it and its negation correspondingly by the axiom of finiteness (induction) versus that of finiteness being idempotent negations to each other. Indeed, the axiom of choice, as far as it is equivalent to the well-ordering "theorem", transforms any set in a well-ordering either necessarily finite according to the axiom of induction or also optionally infinite according to the axiom of infinity. So, the Gödel incompleteness statement relies on the logical contradiction of the axiom of induction and the axiom of infinity in the final analysis. Nonetheless, both can be considered as two idempotent versions of the same axiom (analogically to the Fifth postulate) and then unified after logicism and its inherent intensionality since the opposition of finiteness and infinity can be only extensional (i.e., relevant to the elements of any set rather than to the set by itself or its characteristic property being a proposition). So, the pathway for interpreting the Gödel incompleteness statement as an axiom and the originating from that assumption for "Hilbert mathematics" accepting its negation is pioneered. A much wider context relevant to realizing the Gödel incompleteness statement as a metamathematical axiom is consistently built step by step. The horizon of Hilbert mathematics is the proper subject in the third part of the paper, and a reinterpretation of Gödel's papers (1930; 1931) as an apology of logicism as the only consistent foundations of mathematics is the topic of the next second part.