Mechanization of Mathematics

W. G. BICKLEY
1950 Nature  
The mechanization of mathematics refers to the use of computers to find, or to help find, mathematical proofs. Turing showed that a complete reduction of mathematics to computation is not possible, but nevertheless the art and science of automated deduction has made progress. This paper describes some of the history and surveys the state of the art. Before Turing In this section we review the major strands of thought about the mechanization of mathematics up to the time of Turing. The major
more » ... res in this history were Leibniz, Boole, Frege, Russell, and Hilbert. The achievements of these men have been discussed in many other places, most recently in [39] , and twenty years ago in [38] . Therefore we will keep this section short; nevertheless, certain minor characters deserve more attention. Gottfried Leibniz (1646-1716) is famous in this connection for his slogan Calculemus, which means "Let us calculate." He envisioned a formal language to reduce reasoning to calculation, and said that reasonable men, faced with a difficult question of philosophy or policy, would express the question in a precise language and use rules of calculation to carry out precise reasoning. This is the first reduction of reasoning to calculation ever envisioned. One imagines a roomful of generals and political leaders turning the crank of Leibniz's machine to decide whether to launch a military attack. It is interesting that Leibniz did not restrict himself to theoretical speculation on this subject-he actually designed and built a working calculating machine, the Stepped Reckoner. He was inspired by the somewhat earlier work of Pascal, who built a machine that could add and subtract. Leibniz's machine could add, subtract, divide, and multiply, and was apparently the first machine with all four arithmetic capabilities. 8 Two of Leibniz's Stepped Reckoners have survived and are on display in museums in Munich and Hanover. George Boole (1815-1864) took up Leibniz's idea, and wrote a book [26] called The Laws of Thought. The laws he formulated are now called Boolean 7 This is the fine print containing the disclaimers. In this paper, "mechanization of mathematics" refers to getting computers to find proofs, rather than having them check proofs that we already knew, or store proofs or papers in a database for reference, or typeset our papers, or send them conveniently to one another, or display them on the Web. All these things are indeed mechanizations of mathematics, in a broader sense, and there are many interesting projects on all these fronts, but we shall limit the scope of our discussions to events in the spirit of John Henry and Big Blue. Moreover, we do not discuss past and present efforts to enable computer programs to make conjectures, or to apply mechanized reasoning to other areas than mathematics, such as verification of computer programs or security protocols, etc. 8 The abacus does not count because it is not automatic. With Leibniz's machine, the human only turned the crank.
doi:10.1038/1661087a0 fatcat:67maf3rrvjg73mivang5isihza