S S symmetry Article The Brain and the New Foundations of Mathematics Alexey V. Melkikh Institute of Physics and Technology, Ural Federal University, Yekaterinburg 620002, Russia; [email protected] or [email protected] Abstract: 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. Keywords: Banach–Tarski paradox; axiom of choice; solution of Smale’s 18th problem; quantum mind; brain hidden symmetries; quantum computations Citation: Melkikh, A.V. The Brain and the New Foundations of Mathematics. Symmetry 2021, 13, 1002. https://doi.org/10.3390/sym13061002 1. Introduction The question of the foundations of mathematics has been discussed many times Academic Editors: Danko D. Georgiev since its inception. A number of scientists, for example, considered set theory as such a and José Carlos R. Alcantud foundation. However, at the present time, issues such as the following remain unresolved: Received: 29 April 2021 - Why can we prove anything at all within the framework of mathematics? Accepted: 2 June 2021 - Where do new mathematical concepts and structures come from? Published: 3 June 2021 - Why do we consider inductive inference to be correct? - How do we manage to work with infinities in mathematics? Publisher’s Note: MDPI stays neutral Mathematics itself cannot answer such questions. It is fundamentally important to with regard to jurisdictional claims in consider the structure of the system (brain or computer) that operates with mathematical published maps and institutional affil- concepts. On the other hand, mathematics is a part of thinking as such and must obey iations. some general laws. Traditionally, human thinking is partially explained on the basis of a neural network paradigm, according to which a large number of neurons, connected with each other in a complex way, allow humans and animals to process information, recognize images and Copyright: © 2021 by the author. make decisions. At the same time, one of the problems of human and animal thinking is Licensee MDPI, Basel, Switzerland. the acquisition of new knowledge. Previous works [1,2] show that the acquisition of new This article is an open access article knowledge is contradictory, since the recognized image is not new, and the unrecognized distributed under the terms and image is not useful. To solve this problem, previous works [1,2] suggest that all behavioral conditions of the Creative Commons programs are innate. It also agrees with the Chomskian innate, generative grammar. As a Attribution (CC BY) license (https:// mechanism for the operation of such programs, nontrivial quantum effects of interaction creativecommons.org/licenses/by/ between biologically important molecules are proposed [3–5]. 4.0/). Symmetry 2021, 13, 1002. https://doi.org/10.3390/sym13061002 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, x FOR PEER REVIEW 2 of 27 Symmetry 2021, 13, 1002 2 of 26 This question is also closely related to the foundations of mathematics and logic. In- deed, if all the objects with which we work are innate, then the same applies to mathemat- ical objects.This question However, is there also closelyare an infinite related number to the foundations of mathematical of mathematics objects, and and any logic.lim- itedIndeed, physical if all system the objects (e.g., with computer, which we brain) work ca aren store innate, only then a finite the same number applies of objects to mathe- in memorymatical objects.and work However, with them. there This are property, an infinite as numberwell as the of mathematicalpresence of a number objects, of and other any contradictionslimited physical and system paradoxes, (e.g., computer,entails the brain)need to can revise store the only foundations a finite number of mathematics. of objects in memoryThis andarticle work is devoted with them. to the This formulation property, asof wellnew as foundations the presence of ofmathematics a number of based other oncontradictions the concept of and innate paradoxes, programs entails of behavior the need and to revise thinking. the foundations of mathematics. This article is devoted to the formulation of new foundations of mathematics based 2.on The the Main concept Results of innate of the programs Theory ofof behaviorInnate Behavioral and thinking. Programs In previous studies [1,2,6], a theory of innate behavior programs is proposed, within 2. The Main Results of the Theory of Innate Behavioral Programs the framework of which the main problem of thinking is solved—the problem of knowledgeIn previous acquisition. studies The [1, 2main,6], a results theory of the innate theory behavior can be programs described is as proposed, follows: within the framework of which the main problem of thinking is solved—the problem of knowledge 1. All human and animal behavior is innate. When we learn something or understand acquisition. The main results of the theory can be described as follows: something, it means that we are using existing programs. 2.1. SuchAll humaninnate programs and animal allow behavior behavior is innate. to respond When flexibly we learn to something changes in or the understand environ- ment.something, it means that we are using existing programs. 2. Such innate programs allow behavior to respond flexibly to changes in the environment. Thus, if the information is new, then it is useless; if it is useful, then it is not new Thus, if the information is new, then it is useless; if it is useful, then it is not new (Figure 1). (Figure1). 3. Behavioral programs are formed on the basis of genes in the process of ontogenesis. Image Objects ? Operation 1 Operation 2 Operation 3 ? Figure 1. Recognition and actions [1]. FigureControl 1. Recognition of complex and actions behaviors [1]. requires much more information than can be encoded in genes. An additional information resource can be the quantum effects of interactions 3.between Behavioral biologically programs important are formed molecules, on the which basis alsoof genes take in place the duringprocess theof ontogenesis. formation of the brain,Control as of well complex as other behaviors organs (see, requires for example, much more [4,7 ]).information than can be encoded in4. genes.As aAn result additional of the registration information of theresource signal can of the be external the quantum environment effects byof theinteractions receptors, betweenit is biologically memorized important and recognized. molecules, which also take place during the formation of the brain, as well as other organs (see, for example, [4,7]). Memorizing the temporal and spatial ordering of the received signals is used by the 4.organism As a result to launch of the behavioral registration programs of the signal that areof the most external appropriate environment for a given by the situation. recep- If thetors, received it is memorized signal is notand recognized recognized. (the image is new), then any a priori program, adequate for the given situation, cannot be launched. Symmetry 2021, 13, 1002 3 of 26 5. In the presence of uncertainty in the environment, innate programs do not start immediately, but after several repetitions, which makes it possible to reduce errors. Another response to environmental uncertainty is trial and error, which allows the organism to find the most appropriate innate program. 6. The transfer of experience from one organism to another occurs if the innate programs of behavior in both organisms are the same. Thus, learning of animals and humans is only a method of finding the innate program of behavior that is most suitable for a given situation. New behavioral programs cannot arise in the learning process. An important aspect of thinking is that we are aware of only part of the brain’s work. We are not aware of a significant part of the work associated, for example, with the work of ion channels, neuroreceptors and other intra- and inter-neuronal events. This property should play an important role in the construction of the foundations of mathematics. 3. Computer Proofs Before considering the foundations of mathematics, let us consider how a computer (i.e., a system, the structure of which we know) can operate with mathematics. Naturally, regarding the human brain, we cannot say that we are fully aware of its structure and patterns of work. One of the most important concepts in mathematics is a proof. What is a proof? One of the definitions is (see, for example, [8]): A proof is a sequence of formulae each of which is either an axiom or follows from earlier formulae by a rule of inference. At first glance, based on proofs in mathematics, new knowledge and new useful information arise.