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.
Highlights
The question of the foundations of mathematics has been discussed many times since its inception
The proposed mathematics is constructive in the sense that any mathematical object exists if it is physically realized
All mathematical and logical constructions exist only because they are based on physical media with certain laws of interaction
Summary
The question of the foundations of mathematics has been discussed many times since its inception. Previous works [1,2] show that the acquisition of new knowledge is contradictory, since the recognized image is not new, and the unrecognized image is not useful To solve this problem, previous works [1,2] suggest that all behavioral programs are innate. As a mechanism for the operation of such programs, nontrivial quantum effects of interaction between biologically important molecules are proposed [3,4,5] This question is closely related to the foundations of mathematics and logic. LimThis question is there closely related number to the foundations of mathematics ited physical system (e.g., computer, brain) can store only a finite number of objects in if all the objects with which we work are innate, the same applies to mathememory and work with them. This property should play an important role in the construction of the foundations of mathematics
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