The article argues that in Turing there is an unresolved tension between his approach to the theory of computation and his approach to mathematics. While Turing's approach to the theory of computation is based on a rational process, his approach to mathematics essentially depends on intuition, of which no rational account can be given. It is remarkable that Turing uses two different approaches to the theory of computation and mathematics, because the theory of computation is commonly considered part of mathematics.
This article investigates the computational foundations of Immanuel Kant's transcendental logic and its relevance to artificial intelligence and machine consciousness. We demonstrate that while Kantian analytic judgments map to decidable computations on standard deterministic Turing machines, synthetic a priori judgments require extended models, specifically, Turing machines equipped with oracles, to capture their non-algorithmic, self-positing nature. Building on Johann Gottlieb Fichte's formulation of consciousness as a self-negating loop ("I = not-I"), we formalize a consistent syllogistic system that accommodates non-trivial self-reference. By interpreting this system topologically via spanning simplicial complexes, we show that reflexive reasoning remains both logically consistent and polynomial-time decidable. Furthermore, we introduce quantitative evaluation frameworks, including a Turing Simulation Test and a Reflexion Loop Consciousness Test, to assess whether large language models exhibit minimal conscious-like behavior, such as escaping semantic stagnation and integrating contextual grounding. Thus, we argue that transcendental logic constitutes a viable, computationally tractable framework for modeling self-aware, synthetic reasoning in artificial agents.
We introduce and elaborate a new, evolutionary worldview, as a “Third Story”, intended to replace and complement the earlier religious (First) and mechanistic (Second) worldviews. The confusions created by a world that is ever more volatile, uncertain, complex and ambiguous have eroded people’s “sense of coherence”, i.e. the degree to which they experience the world as comprehensible, manageable and meaningful. The First Story provides meaning and values, but its descriptions no longer provide an accurate understanding of the universe. The Second Story, which sees the universe as a clockwork mechanism governed by the laws of nature, provides more accurate predictions that allow us to build powerful technologies. However, it does not provide meaning or values. The Third Story sees the universe as self-organizing towards increasing complexity and consciousness, subsequently producing matter, life, mind and society. It understands the fundamental mechanism of evolution as mutual adaptation between interacting systems, thus generating synergetic wholes that in turn interact, so as integrate into even more complex wholes. Its implicit value is the search for fitness and synergy, thus inviting individuals to work towards a further evolutionary integration into the noosphere, i.e. the planetary superorganism formed by humanity, its technological extensions, and the ecosystem.
The article argues that in Turing there is an unresolved tension between his approach to the theory of computation and his approach to mathematics. While Turing’s approach to the theory of computation is based on a rational process, his approach to mathematics essentially depends on intuition, of which no rational account can be given. It is remarkable that Turing uses two different approaches to the theory of computation and mathematics, because the theory of computation is commonly considered part of mathematics.
Calculating longitude as precisely as possible is important for navigation and requires resolving issues in relation to position and measurement while the ship is moving. The names of Euler, Bezout, Laplace, and other distinguished mathematicians are linked to studies on this problem. It can be said that in past times, these theoretical and practical problems were as important as they are today. Now cutting-edge technologies and computers are used. The determination of longitude (Throughout this paper, “latitude” and “longitude” referring to the ship’s position are understood as geographical coordinates. When discussing the position of celestial bodies, the corresponding celestial reference system (mainly ecliptic coordinates) is explicitly indicated.) by astronomical means is mathematically a seemingly simple problem of spherical trigonometry. However, its effective resolution required theoretical developments and the design of special instruments for observation and calculation. Within the group of mathematicians who were interested in these problems at the turn of the eighteenth to the nineteenth century, the figure of Joseph de Mendoza y Ríos stands out. This article is concerned with the achievements of this mathematician from Seville who contributed substantially to the solution of the problem of calculating longitude for which he received one of the great prizes awarded by the Board of Longitude in London. His contributions in this area were manifold and cover a broad spectrum of the problems of nautical astronomy. All these elements: design of instruments to capture astronomical information, development of an analytical method, formulation of an approximate calculation procedure, and preparation of auxiliary (supportive) calculation instruments allowing skipping stages in the transition from observation to position data. Mendoza y Ríos In this work, the authors explain the mathematical calculations of determining longitude by means of lunar distances that are due to Joseph de Mendoza y Ríos.
