The Brain and the New Foundations of Mathematics

Many concepts in mathematics are not fully defined, and their properties are implicit, which leads to paradoxes. New foundations of mathematics were formulated based on the concept of innate programs of behavior and thinking. The basic axiom of mathematics is proposed, according to which any mathematical object has a physical carrier. This carrier can store and process only a finite amount of information. As a result of the D-procedure (encoding of any mathematical objects and operations on them in the form of qubits), a mathematical object is digitized. As a consequence, the basis of mathematics is the interaction of brain qubits, which can only implement arithmetic operations on numbers. A proof in mathematics is an algorithm for finding the correct statement from a list of already-existing statements. Some mathematical paradoxes (e.g., Banach–Tarski and Russell) and Smale’s 18th problem are solved by means of the D-procedure. The axiom of choice is a consequence of the equivalence of physical states, the choice among which can be made randomly. The proposed mathematics is constructive in the sense that any mathematical object exists if it is physically realized. The consistency of mathematics is due to directed evolution, which results in effective structures. Computing with qubits is based on the nontrivial quantum effects of biologically important molecules in neurons and the brain.

Hypothesis of the law of material satisfaction and its perception by human

The research topic is aimed at solving the problem of achieving the stable development of society, so it is relevant. The problem is that, according to the existing axiom of the economy, the material needs of a man are limitless. Thus, this axiom justifies and stimulates the pursuit of enrichment, which is achieved even by breaking laws. An insatiable effort to be financially successful has led to inequalities between people and countries. In spite of all the efforts, inequality is increasing rapidly. The question is, what pleasure does a person enjoy in achieving material goals? The study of human satisfaction with material benefits has been widely reported in the literature, however, a mathematical model of human satisfaction with material goods and a unit of satisfaction has not been developed. The absence of a mathematical model makes it impossible to find out the regularities of this phenomenon - human satisfaction in the process of obtaining material goods. The paper hypothesizes that the law of human satisfaction with material goods is expressed by a logarithmic function, which is shifted along the abscissa axis so that at zero abscissa the function equals zero. A unit of satisfaction, called comfort, has been introduced. The feeling of satisfaction is expressed as the first derivative of the logarithmic function. It is known that this function approaches zero as the argument grows. The introduction of the human satisfaction law from material goods in the form of a mathematical model gives grounds to make fundamental conclusions. Since the increase in satisfaction with the increase in material goods is slowing down, and the feeling of satisfaction in general approaches zero, it is not necessary to follow the basic axiom of the economy of a capitalist society and spend exertion to satisfy material needs, especially since the feelings of it will become zero. Scientific research proves that a person is able to absorb new ideas, able to learn and apply the acquired knowledge in everyday life. This gives confidence that due to the qualitative motivational implementation of the proposed law, people will realize it as a personal life strategy, and the development of society will become more stable.

Bounded Rationality in C. elegans

Rational choice theory assumes optimality in decision-making. Violations of a basic axiom of economic rationality known as “Independence of Irrelevant Alternatives” (IIA), have been demonstrated in both humans and animals, and could stem from common neuronal constraints. We developed tests for IIA in the nematode Caenorhabditis elegans, an animal with only 302 neurons, using olfactory chemotaxis assays. We found that in most cases C. elegans make rational decisions. However, by probing multiple neuronal architectures using various choice sets, we show that asymmetric sensation of odor options by the AWCON neuron can lead to violations of rationality. We further show that genetic manipulations of the asymmetry between the AWC neurons can make the worm rational or irrational. Last, a normalization-based model of value coding and gain control explains how particular neuronal constraints on information coding give rise to irrationality. Thus, we demonstrate that bounded rationality could arise due to basic neuronal constraints.

On J M Keynes’s Rejection of the Moscow School of Probability’s Limiting Frequency Approach to Probability and Kolmogorov’s Axiom of Additivity (Countable Additivity ): Non –Additivity was the fundamental, basic axiom upon which all of the Economics of Ke

<p>J M Keynes was an acknowledged, world renown, and internationally recognized expert in probability and statistics in the 1930’s based on his A Treatise on Probability (1921). . Keynes had been selected by statistics journals to serve as a referee during the 1930’s. It is, therefore, no surprise that he was selected as the referee by the League of Nations to review Jan Tinbergen’s work on business cycles that used an econometrics approach based on The Law of Large Numbers, the Central Limit Theorem, and the Gaussian (Normal) Distribution .The fundamental axiom used by Tinbergen was additivity . Kolmogorov and the Moscow School of Probability’s main innovation was to go from the axiom of additivity to the axiom of countable additivity. However, Keynes rejected additivity except in the special case that the weight of the evidence, w, which measured the relative completeness of the evidence ,defined on the closed unit interval [0,1],equaled 1 , approached 1,or approximated 1. Keynes also accepted goodness of fit tests, such as the Lexis –Q test, and exploratory data analysis as evidence that could be used to support using a particular probability distribution.</p>

A New Approach

We need waste disposal methods that allow the human race to live on this planet in harmony with nature, preserving our resources and habitat and leaving a legacy for our children and grandchildren that does not deprive them of opportunities. These changes will not come easily; they will require resolve and foresight. Just as a mathematician develops the proof to a mathematical theorem, we must start from a basic axiom, and step by step, following a logical progression, we must build a practical framework for waste management. We started this task in chapter 2, where we derived three general principles from the axiom of sustainable development. Can we apply these general principles to develop practical guidelines—first, to overcome the shortcomings of existing landfills, and second, to find other, innovative disposal methods that will conform with sustainable development? Let us look at each of the three principles in turn. Human health and the environment must be protected, both now and in the future. This principle is fundamental and places important constraints on the siting and design of disposal facilities, and also on the form of the waste. In particular, the final four words, “and in the future,” are very important. This principle can be satisfied in two ways: by reducing the toxicity of the wastes so they pose minimal risk, or by containing wastes so that they cannot escape and cause harm. In some cases, the latter method includes controlled leakage at a rate that the environment can assimilate without long-term degradation. Wastes must be managed so that no burden is placed on future generations and they are not deprived of the opportunities we have had. In other words, our grandchildren should not have to spend their valuable resources to solve our waste problems, nor should they be denied resources because our generation has depleted them. Neither should their health and environment be placed at risk because of our actions. The main impacts of landfills on future generations are the requirement to provide ongoing guardianship and maintenance; the loss of valuable land; and impairment of groundwater, surface water, and the atmosphere.

Cognitive Dissonance in the Treatment of Paranoid Schizophrenia

Paranoid thinking is often well organized, systemized and detailed so that treating a paranoid patient is a difficult challenge. It is suggested that cognitive dissonance, once created, may cause a person to change his cognitions in order to lower or eliminate it. The treatment of a paranoid schizophrenic is presented wherein by introducing strong group pressure the patient accepted an axiom, which created dissonance with his paranoid content. This axiom was “A wise man sees more than two alternatives in any given situation.” Therefore, by stepwise pacing, alternative evaluations of affect “free” situations were made by the patient using the accepted basic axiom. This technique was gradually used with the patient’s paranoid content, and later on with basic characteristics underlying the paranoid system. The patient exhibited curiosity and a willingness to investigate his paranoid inner state according to the new accepted axiom, and a significant symptomatic relief was seen within a few days, in contrast to past hospitalizations where only partial response to antipsychotic medication was noted. Research is needed in order to examine the effectiveness of this mode of treatment.