The origin of the idea of rationality as a fundamental model of human behaviour (the concept of homo oeconomicus) that was introduced during the neoclassical revolution in economics and that implied the mathematization of economics thereby adopting the mathematical scheme of and borrowing some fundamental conception from classical mechanics, as well as the crisis of this rationality model that followed in the latest decades are here discussed. The origin of the least action principle in 18th century is first described. The least action principle was the basis that lead to the development of Lagrangean dynamics in classical mechanics that, on its turn, entered economics in 19th century as a fundamental method to describe the behaviour of a postulated rational agent and was the basis for the development of mainstream economics in 20th century. The epistemologically unjustified transfer of concepts from mechanics, in particular in Walras? work, is stressed. In the second half of 20th century experimental psychology slowly entered economics and several paradoxes in the real agents? choices were shown giving strong evidence of rationality?s axioms violation in the real behaviour of people. By mid-20th century, quantum mechanics had already developed a highly effective mathematical apparatus to describe the wave-particle behaviour of quantum objects and in particular the phenomenon of probability interference that is clearly shown in the double-slit experiment with single electrons, as well as with photons. Probability interference is peculiar to quantum mechanics and is extraneous to the Kolmogorov axiomatic theory of classical probability. The recent idea of applying the quantum mechanics concepts and formalism, including quantum probability interference and superposition of states, to the description of mental processes is discussed. The aim of this new field of research, which is developing along the lines of the centuries-old question of the mind-body relation, is to create a general framework where cognition might be modeled. Quantum cognition is therefore introduced as a possible answer to the rationality violation paradoxes shown by experiments on ambiguous figures recognition and on the effects brought about by conjunction and disjunction of concepts.
Abstract The classic game of Battleship involves two players taking turns attempting to guess the positions of a fleet of vertically or horizontally positioned enemy ships hidden on a 10×10 grid. One variant of this game, also referred to as Battleship Solitaire, Bimaru or Yubotu, considers the game with the inclusion of X-ray data, represented by knowledge of how many spots are occupied in each row and column in the enemy board. This paper considers the Battleship puzzle problem: the problem of reconstructing an enemy fleet from its X-ray data. We generate non-unique solutions to Battleship puzzles via certain reflection transformations akin to Ryser interchanges. Furthermore, we demonstrate that solutions of Battleship puzzles may be reliably obtained by searching for solutions of the associated classical binary discrete tomography problem which minimise the discrete Laplacian. We reformulate this optimisation problem as a quadratic unconstrained binary optimisation problem and approximate solutions via a simulated annealer, emphasising the future practical applicability of quantum annealers to solving discrete tomography problems with predefined structure.
Quantum mechanics is our most successful physical theory. It predicts the behavior of atoms, molecules, and light to ten decimal places. No experiment has ever contradicted it. It also contains a crack at its foundation. The theory’s central equation—the Schrodinger equation—describes a smooth, continuous, perfectly predictable evolution of the quantum state. But whenever anyone performs a measurement, they get a single, definite result. Which result they get appears to be random. The transition from smooth evolution to random outcome is called the “collapse of the wavefunction,” and after nearly a century of analysis, there is no agreed-upon explanation for what causes it, when it occurs, or whether it occurs at all. This is the measurement problem. The measurement problem is not a minor puzzle at the margins of the theory. It is a failure to explain the most basic operation in science: looking at the result of an experiment. Around it cluster other puzzles. Light behaves like a wave in some experiments and like a particle in others—two contradictory descriptions. Two particles that have interacted remain correlated across arbitrary distances, as though communicating faster than light. The probabilities that govern measurement outcomes appear as a postulate—the Born rule—inserted by hand with no deeper justification. And the theory offers no unified picture of reality: quantum mechanics, general relativity, and the standard model of particle physics speak incompatible mathematical languages. A century of effort has produced many interpretations of quantum mechanics. Copenhagen. Many-worlds. De Broglie-Bohm. Objective collapse. Quantum Bayesianism. None has achieved consensus. The measurement problem remains. This document proposes that the source of these puzzles is not quantum mechanics itself. It is a single, invisible assumption made at the very beginning of the theory in 1900—an assumption so deeply embedded that nobody noticed they were making it. Change that assumption, and the puzzles do not get solved. They dissolve. They stop being puzzles. They become predictable consequences of geometry.