Do Goedel's incompleteness theorems set absolute limits on the ability of the brain to express and communicate mental concepts verifiably?

Classical interpretations of Goedel's formal reasoning imply that the truth of some arithmetical propositions of any formal mathematical language, under any interpretation, is essentially unverifiable. However, a language of general, scientific, discourse cannot allow its mathematical propositions to be interpreted ambiguously. Such a language must, therefore, define mathematical truth verifiably. We consider a constructive interpretation of classical, Tarskian, truth, and of Goedel's reasoning, under which any formal system of Peano Arithmetic is verifiably complete. We show how some paradoxical concepts of Quantum mechanics can be expressed, and interpreted, naturally under a constructive definition of mathematical truth.Comment: 73 pages; this is an updated version of the NQ essay; an HTML version is available at http://alixcomsi.com/Do_Goedel_incompleteness_theorems.ht

REASONING OVER STRINGS AND OTHER UNBOUNDED DATA STRUCTURES

Ph.DDOCTOR OF PHILOSOPH

Biases in human behavior

The paper shows that biases in individual’s decision-making may result from the process of mental editing by which subjects produce a “representation” of the decision problem. During this process, individuals make systematic use of default classifications in order to reduce the short-term memory load and the complexity of symbolic manipulation. The result is the construction of an imperfect mental representation of the problem that nevertheless has the advantage of being simple, and yielding “satisficing” decisions. The imperfection origins in a trade-off that exists between the simplicity of representation of a strategy and his efficiency. To obtain simplicity, the strategy’s rules have to be memorized and represented with some degree of abstraction, that allow to drastically reduce their number. Raising the level of abstraction with which a strategy’s rule is represented, means to extend the domain of validity of the rule beyond the field in which the rule has been experimented, and may therefore induce to include unintentionally domains in which the rule is inefficient. Therefore the rise of errors in the mental representation of a problem may be the "natural" effect of the categorization and the identification of the building blocks of a strategy. The biases may be persistent and give rise to lock-in effect, in which individuals remain trapped in sub-optimal strategies, as it is proved by experimental results on stability of sub-optimal strategies in games like Target The Two. To understand why sub-optimal strategies, that embody errors, are locally stable, i.e. cannot be improved by small changes in the rules, it is considered Kauffman’ NK model, because, among other properties, it shows that if there are interdependencies among the rules of a system, than the system admits many sub-optimal solutions that are locally stable, i.e. cannot be improved by simple mutations. But the fitness function in NK model is a random one, while in our context it is more reasonable to define the fitness of a strategy as efficiency of the program. If we introduce this kind of fitness, then the stability properties of the NK model do not hold any longer: the paper shows that while the elementary statements of a strategy are interdependent, it is possible to achieve an optimal configuration of the strategy via mutations and in consequence the sub-optimal solutions are not locally stable under mutations. The paper therefore provides a different explanation of the existence and stability of suboptimal strategies, based on the difficulty to redefine the sub-problems that constitute the building blocks of the problem’s representation