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An Investigation on the Logical Structure of Mathematics (XIII)*): A Method of Programming of Proofs in Mathematics for Electronic Computing Machines
Published online by Cambridge University Press: 22 January 2016
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In spite of invaluable importance of the fundamental circuit logic of performing the basic logical operators AND, OR, and NOT in achieving complicated computations for electronic computing machines, the research of planning and programming of proving a given mathematical or logical assertion for electronic computing machines is incomparatively far behind. This is of course partly because of little practical demand for solving such a theoretical problem, but there is also the essential reason for it that the proving procedures are much more difficult than the computing ones.
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- Copyright © Editorial Board of Nagoya Mathematical Journal 1960
Footnotes
*)
(Added January 15, 1960) This is the reproduction of the preprints of my talk at the Meeting for the Research of Mathematical Sciences in Tokyo on June 19, 1959, so that this Part (XIII) is quite independent of previous parts. On the contrary, the definition of UL given in Part (I), Hamb. Abh. vol. 22, is repeated with a generalization of defining formulas (see the formula (D*/p) in §5). This generalization is necessary for the purpose of irreducible deductions of some branches of mathematics, which was anticipated in the foot-note 1 in Part (I). However, (D*/p) will not be used in further continuation of this investigation, unless it is explicitly noted to do so.
We do not enter here into actual programming but we note that detailed research of programming along this line is successfully going on by several mathematicians, using the computing machine M-1 in the Electrical Communication Laboratory in Tokyo and TAC in the Faculty of Engineering, University of Tokyo.
Meanwhile, we have come to be accessible to the publications at the International Conference on Information Processing, Paris, 1959, in which several researches on proving mathematical theorems by electronic computing machines are reported. See H. Gelernter: Realization of a geometry theorem proving machine, UNESCO/NS/ICIP/ 1.6.6; A. Newell, J. C. Shaw, and H. A. Simon: Report on a general problem-solving program, UNESCO/NS/ICIP/1.6.8; B. Dunham, R. Fridshal, and G. L. Sward: A non-heuristic program for proving elementary logical theorems, UNESCO/NS/ICIP/1.6.10; P. C. Gilmore: A program for the production of proofs for theorems derivable within the first order predicate culculus from axioms UNESCO/NS/ICIP/6.1.14. See also H. Wang: Toward Mechanical Mathematics, reported in Gelernter’s paper, cited above, to be published in IBM Journal of Research and Development.
In programming of mathematical proofs it is especially necessary to apply heuristic method or strategy (see H. Gelernter’s paper cited above) in order to minimanize the numbers of mechanical trials. The method of using UL has the advantage for this purpose because UL is faithful to the “meaning” of concept formation in mathematics and is eligible for mechanization of proofs (see also §5, mechanization of mathematics, Part (IV), this J. vol. 13).
References
*) (Added January 15, 1960) This is the reproduction of the preprints of my talk at the Meeting for the Research of Mathematical Sciences in Tokyo on June 19, 1959, so that this Part (XIII) is quite independent of previous parts. On the contrary, the definition of UL given in Part (I), Hamb. Abh. vol. 22, is repeated with a generalization of defining formulas (see the formula (D*/p) in §5). This generalization is necessary for the purpose of irreducible deductions of some branches of mathematics, which was anticipated in the foot-note 1 in Part (I). However, (D*/p) will not be used in further continuation of this investigation, unless it is explicitly noted to do so.
We do not enter here into actual programming but we note that detailed research of programming along this line is successfully going on by several mathematicians, using the computing machine M-1 in the Electrical Communication Laboratory in Tokyo and TAC in the Faculty of Engineering, University of Tokyo.
Meanwhile, we have come to be accessible to the publications at the International Conference on Information Processing, Paris, 1959, in which several researches on proving mathematical theorems by electronic computing machines are reported. See H. Gelernter: Realization of a geometry theorem proving machine, UNESCO/NS/ICIP/ 1.6.6; A. Newell, J. C. Shaw, and H. A. Simon: Report on a general problem-solving program, UNESCO/NS/ICIP/1.6.8; B. Dunham, R. Fridshal, and G. L. Sward: A non-heuristic program for proving elementary logical theorems, UNESCO/NS/ICIP/1.6.10; P. C. Gilmore: A program for the production of proofs for theorems derivable within the first order predicate culculus from axioms UNESCO/NS/ICIP/6.1.14. See also H. Wang: Toward Mechanical Mathematics, reported in Gelernter’s paper, cited above, to be published in IBM Journal of Research and Development.
In programming of mathematical proofs it is especially necessary to apply heuristic method or strategy (see H. Gelernter’s paper cited above) in order to minimanize the numbers of mechanical trials. The method of using UL has the advantage for this purpose because UL is faithful to the “meaning” of concept formation in mathematics and is eligible for mechanization of proofs (see also §5, mechanization of mathematics, Part (IV), this J. vol. 13).
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