The Automation of Reason - Some History
Overview
Leibniz dreamed of a universal language and a calculus of reason which would reduce all problems to numerical computation. Unrealisable in his time, it is still today a dream, but one which (subject to qualifications) advances in mathematics, logic and information technology may have brought within our grasp.
Leibniz dreamed of a universal language and a calculus of reason which would reduce all problems to numerical computation. Unrealisable in his time, it is still today a dream, but one which (subject to qualifications) advances in mathematics, logic and information technology may have brought within our grasp.
The calculator which Leibnxiz designed and built was nowhere near adequate for doing anything worthwhile with the calculus ratiocinator which he envisaged. Todays global networks of electronic computers are more like what is needed. It took a long history of innovation to get from the one to the other.
The "calculus", independently invented by Isaac Newton and Leibniz, was the beginning of a new branch of mathematics called "analysis". Analysis developed rapidly over the next hundred years, providing the mathematical techniques necessary for the growth of science and engineering. Success in application made continued development possible despite reservations about the coherence and rigour of the methods which had been adopted. Rigourisation, leading eventually to logicisation began only in the nineteenth century.
Leibniz and the Automation of Reason
Leibniz dreamed of a universal language and a calculus of reason which would reduce all problems to numerical computation. Unrealisable in his time, it is still today a dream, but one which (subject to qualifications) advances in mathematics, logic and information technology may have brought within our grasp.
The Dream of Leibniz
Leibniz conceived of and attempted to design a lingua characteristica (a language in which all knowledge could be formally expressed) and a calculus ratiocinator (calculus of reasoning) such that when philosophers disagreed over some problem they could say 'calculemus' (let us calculate) and agree to formulate the problem in the lingua characteristica and solve it using the calculus ratiocinator.
Problems and Solutions
Leibniz was shackled:
  • possibly by some philosophical weaknesses,
  • by the inevitable incompleteness of science
  • by insufficient rigour in the mathematics of his day,
  • by the limitations of the logic known in his time,
  • by the lack of adequate information technology.
since then, some progress has been made.
From Calculators to Global Networks
The calculator which Leibnxiz designed and built was nowhere near adequate for doing anything worthwhile with the calculus ratiocinator which he envisaged. Todays global networks of electronic computers are more like what is needed. It took a long history of innovation to get from the one to the other.
Calculators
The further development of calculating machines was clearly necessary to realise Leibniz's dream, and Leibniz himself contributed to this development, building the first calculator capable of multiplication and division as well as addition and subtraction. What may not be so obvious is that a fundamentally different kind of calculator was needed, one which is computationally universal. Such a calculator can do the calculations performed by any other calculators, once given a description of the required computation. Nowadays we call this kind of calculator a computer.
Babbage's Universal Engine
Babbage was the first to design a universal calculator, and devoted his life to building one. He didn't succeed, hampered by the limits of the available technologies in the nineteenth century.
Turing Machines
A proper understanding of the fundamental theory of computing did not come until the idea of an effective computation was formalised by mathematical logicians who were enquiring about the limits of computation in the 1930's. At that time convincing formalisations of the notion of effective computation were realised, the conviction strengthened by the appearance of several quite different formalisations of the idea which were shown to be equivalent. These included the notion of recursive function due mainly to Steven Kleene, that of Alonxo Church's Lambda-calculus, systems of "productions" studied by Emil Post, and the Turing machines of Alan Turing.

Of these the one most resembling the digital computers which soon followed was the Universal Turing machine.
Electronic Computers
The availability of new technologies such as the electronic valve, the need for computing machines to help descipher encrypted messages during the second world war, and the by then well understood theory of universal computing machines were among the factors which made the development of the electronic stored program digital computer possible in the years following the second world war.

From the earliest days the development of software to enable digital computers to undertake deductive reasoning was initiated.
Global Networks
A further element in the technological support necessary for the development of the kind of near universal capability envisaged by Leibniz is the networking of computers through the internet. This facilitates the kind of collaborative development on encyclopaedic scales necessary for success. Without this kind of extended man/machine collaboration, overcoming the problems of scale and complexity which prove critical might well be infeasible.
The Logicisation of Analysis
The "calculus", independently invented by Isaac Newton and Leibniz, was the beginning of a new branch of mathematics called "analysis". Analysis developed rapidly over the next hundred years, providing the mathematical techniques necessary for the growth of science and engineering. Success in application made continued development possible despite reservations about the coherence and rigour of the methods which had been adopted. Rigourisation, leading eventually to logicisation began only in the nineteenth century.
The Automation of Analysis

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