Four Definitions of Mathematics and the Logic Machine Generating Mathematics

Mathematics is an instance of "the Dao generates one, one generates two, two generates three, three generates the ten thousand things," in which the method of "generating" is rigorous logical reasoning. Therefore mathematics is the only discipline that can be generated by logical machines, whereas physics has experiments providing new physics and mathematics has none. This paper gives four definitions of mathematics: mathematics is the various kinds of equalities and inequalities; mathematics is generated from "nothing," or from the empty set, by rigorous logical reasoning; old mathematics defines new mathematics, or mathematics is the only timeless discipline; the correctness standard of mathematical proof is the highest, or mathematics is the only correct discipline. The four definitions point to one conclusion: mathematics is silent, or provides no information; mathematics grows by rigorous logical reasoning; and all of mathematics is contained in its starting point. Today's computer is in fact a logic machine: its 0 and 1 and its computation are all realized by logic gates. Today's artificial intelligence is likewise based on 0 and 1 and on gate circuits, and therefore differs from a calculator in no essential respect, only in quantity; what it lacks is not computing power but the machine's own language of mathematics, namely the expression of the empty set and its subsets. Just as arithmetic can be generated by binary machines, the new mathematics defined by old mathematics also has its own starting point — for instance, mathematics can be expressed by the empty set and its sets. Because of the translation and one-to-one correspondence between binary and decimal, the former has already been realized by the Turing machine and has surpassed humans in arithmetic; the translation and correspondence between the latter's expression by the empty set and its sets and the human language of mathematics remains to be realized. Current artificial intelligence treats mathematics by an imitation route, searching within formal languages designed by humans, its space bounded by human-given languages and rules; the informational closure of mathematics makes another route feasible in theory — direct generation from the empty set and the axioms. The question is whether translation and correspondence are needed between mathematics expressed in machine language and the human application of mathematics — after all, machines have no difficulty reading human mathematics, whereas humans have difficulty reading machine mathematics, just as humans reading binary.