Leibniz and the Automation of Reason
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 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.
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.
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.
The Dream
The Lingua Characteristica
Leibniz engaged in four kinds work related to his proposed lingua characteristica, (or universal characteristic).
  • Encyclopaedia - he sought the collaborative development of an encyclopaedia in which would be presented in non-symbolic form all that was so far known. This was to provide a basis for the lingua characteristica in which the knowledge could be formally expressed. This enterprise was not completed, but beneficial side effects were the foundation of new academies and of the journal Acta Eruditorum.
  • Universal Language - he promoted the development of a language universal in the sense of being spoken by all. There have been many such projects of which the best known today is Esperanto
  • The lingua characteristica - a formal language universal both in being understood by all, and in encompassing all knowledge.
  • The calculus ratiocinator - a method of computing the truth value of a proposition in the lingua characteristica
The lingua characteristica was to be a language in which predicates were numerically encoded in such a way as to render the truth of subject predicate proposition (and Leibniz considered all propositions to have this form) could be obtained by arithmetical computation.
The Calculus Ratiocinator
This is roughly how he proposed to do it. He believed that every predicate was either simple or complex and that complex predicates could be analysed into simple constituents. He proposed to assign prime numbers to each simple predicate and then represent a complex predicate by the product of the primes representing its simple constituents (complex predicates are therefore invariably conjunctions of finite numbers of simple predicates). He also believed that every proposition had subject predicate form, and that in a true proposition the predicate was contained in the subject, i.e. the set of simple predicates from which the predicate was composed was a subset of the set from which the subject was composed. This can be sorted out by numerical computation, you just check whether the predicate divides the subject without remainder.

His main difficulty in this was in discovering what the simple predicates are. Leibniz thought the complete analysis beyond mere mortals, but believed that a sufficient analysis (into predicates which are relatively primitive) for the purposes of the calculus ratiocinator would be realisable. From this it appears that Leibniz expected his language and calculus to be not completely universal.
Some Other Relevant Work
Calculators
Calculators capable of addition and subtraction had already been developed before Leibniz. Leibniz designed a calculator to do multiplication and division, and made innovations which later became standard, for example, the stepped reckoner (or "Leibniz wheel"), which had cogs of varying lengths. He presented a model of one of his calculators to the Royal Society on a visit to London in 1673.
The Differential and Integral Calulus (Analysis)
Leibniz was an independent inventor of the calculus (the other was Newton), which is of central importance to any attempt to automate reasoning about the real world. The branch of mathematics which this inaugurated is now called "analysis". It is extensively used in science and engineering and is in that way the basis for most applications of mathematics to the real world.

This work was however, controversial, from its inception, doubts being raise (for example by the Irish philosopher Berkeley) about its coherence (mostly because of its use if infinitesimal or infinitely small) quantities). One of the major problems in realising Leibniz's dream is to make it possible to reason rigorously about this kind of mathematics. This involved the rigourisation, arithmetisation and logicisation of analysis, the development of adequate logical systems for this kind of mathematics, the invention of the digital computer and the implementation of software supporting formal reasoning in the new logics, and by that means in arithmetic and analysis.
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.
Some Philosophical Issues

It might be thought that Leibniz's view that all truths are necessary caused him to overestimate the scope of logic and the domain in which formal techniques can yield algorithmic decision procedures, Our knowledge of the essential incompleteness of arithmetic does tell us that any reliable algorithmic oracle must be incomplete. However, this might in principle be an inconsequential incompleteness. Many people believe that the incompleteness of strong set theories such as ZFC is exclusively in questions which have no bearing on its applications in empirical science.

The Incompleteness of Science

Though God might know enough to realise Leibniz's dream, it is surely doubtful that human knowledge of empirical matters will ever be complete. This, like consequences of the essential incompleteness of formal deduction, would seem to place definite limits on how complete in empirical matters an oracle devised by humans could be. Again, this may or may not be a significant limitation. An oracle which can speak with authority on any matter which is decided by the science of the day would be a highly attractive approximation to the calculus which Leibniz sought.

Mathematical Rigour

In the time of Leibniz applicable mathematics was not known to a standard of rigour which would permit its formalisation. This is an impediment to algorithmic resolution of problems which has now been overcome.

Logical Revolutions

Even if mathematics had been more rigorous, it could not have been reduced in Leibniz's time to formal deductive logic, since the logic of his time was not adequate to the task. This impediment has also been overcome, and many logical systems are known whose strength suffices for the derivation of mathematics (within generous limits).

Information Technology

Leibniz worked upon the design and construction of calculators, which however did not approach the capabilities (even neglecting questions of complexity) of modern digital electronic computers. It is a commonplace that information technology has now been doubling in speed every eighteen months for decades. Though the technology available to Leibniz was clearly inadequate to his task, that of today is closer. It is easy to believe that a useful approximation to Leibniz's oracle could be implemented on today's hardware if only we knew how to write the software.


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