Plato on the foundations of Modern Theorem Provers

Abstract

IntroductionAccording to Platonism, a mathematical proof is the metaphysical view that there are abstract mathematical objects whose existence is independent of human language, thought, and practices. A full-blooded Platonism or platitudinous Platonism (FBP) asserts that it is possible for human beings to have systematically and non-accidentally true beliefs about a platonic mathematic realm - a mathematical realm satisfying Existence, Abstractness and Independence. Could there be such proof? Could a proof be objective and completely understood, independently of the possibilities of our knowing of truth or falsity?Plato is one of the great contributors to the foundations of mathematics. He discussed, 2400 years ago, the importance of clear and precise definitions as fundamental entities in mathematics. In the seventh book of his masterpiece, The Republic, Plato states "arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and rebelling against the introduction of visible or tangible objects into the argument". In the light of this thought, I will discuss the status of mathematical entities in the twentieth first century, an era where it is already possible to demonstrate theorems, construct formal axiomatic derivations of remarkable complexity with artificial intelligent agents - the modern theorem provers. A computer-assisted proof is written in a precise artificial language that admits only a fixed repertoire of stylized steps. It is formalized through artificial intelligent agents that mechanically verify, in a formal language, the correctness of the proof previously demonstrated by the human mind.In contrast, calculi are exactly the kind of feature, which make it appealing for mathematicians. There are two reasons for this: (i.) it can be studied for its own properties and elegance of pure mathematics; and (ii.) can easily be extended to include other fundamental aspects of reasoning. According to Hofstadter, (1979), a proof is something informal, that is, a product of human thought, written in human language for human consumption. All sorts of complex features of thought may be used in proofs, and, thought they may "feel right", one may wonder if they can be logically defended. This is really what formalization is for.1. Plato's conception of ArithmeticAccording to Plato, at the end of the sixth book of The Republic, mathematicians' method of thinking is not a matter of intelligence, but rather a matter of Siavoia, which means understanding. This is a definition by Plato that seems to etimologically imply Sia (between), voua (intelligence) and SoCa (opinion), as if understanding would be something in between opinion and inteligence.In 525a, Plato considers the concept of number, as a non-limited unity trough plurality, since "this characteristic occurs in the case of one; for we see the same thing to be both one and infinite in multitude" (525a). Also, with this conception of plurality as much as unity "thought begins to be aroused within us, and the soul perplexed and wanting to arrive at a decision asks 'What is absolute unity?' This is the way in which the study of the one has a power of drawing and converting the mind to the contemplation of reality." (525a).As reported by Plato, the reality of calculus is a pure contemplation since in reasoning about numbers there are no visible bodies:"Plato - Now, suppose a person were to say to them, Glaucon, 'O my friends, what are these wonderful numbers about which you are reasoning, in which, as you say, there are constituent units, such as you demand, and each unit is equal to every other, invariable, and not divisible into parts,' - what would they answer?Glaucon -They would answer, as I should think, that they were speaking of those numbers which can only be realized in thought, and there is no other way of handling them." (Republic, 526a).This means that arithmetic compels the soul to reach the pure truth trough intelligence. …