Robotics Reference
In-Depth Information
For all inquiries which depend on reasoning would be performed
by the transposition of characters and by a kind of calculus,
1
which
would immediately facilitate the discovery of beautiful results. . .
Moreover, we should be able to convince the world what we should
have found or concluded, since it would be easy to verify the cal-
culation either by doing it over and over or by trying tests sim-
ilar to that of casting out nines in arithmetic. And if someone
would doubt my results, I should say to him: “let us calculate,
Sir”, and thus by taking to pen and ink, we should soon settle the
question. [2]
In some of his later writings Leibniz described his method as combin-
ing logic with algebra (“algebraico-logical synthesis”). Thus, in 1666,
at the age of 19, Leibniz wrote his
Dissertio de Arte Combinatoria
,in
which he attempted to formulate a
Mathesis universalis
, a sort of sci-
entific language which would permit any two disputants to settle their
differences with pen and ink, as he describes above. One problem with
Leibniz' and Llull's thinking is that mankind's ideas of what is right and
wrong are constantly changing and, therefore, cannot always be treated
as scientific facts that are subject to logical proof. And even if meth-
ods of logical proof
were
applicable to their ideas, the twentieth-century
Austro-Hungarian mathematician Kurt Godel has since shown that it is
not possible to settle all differences of opinion by Leibniz-like proof.
2
How Logic Machines Work
It was not until almost sixty years after Leibniz' death that the first ma-
chine of any kind was constructed that was able to solve problems in
logic. Most of the early (mechanical) logic machines were based upon
the same logical principles, employed in a process called the
method of
elimination
. The first stage of this process consists of enumerating all the
possible statements that can apply to the matter in question. For exam-
1
A calculus is a mathematical system of calculation using symbols.
2
In 1931 Godel demonstrated that, within any given branch of mathematics, there would always
be some propositions that could not be proven either to be true or to be false using only the rules
and axioms of that branch of mathematics itself. For example, it might be possible to prove every
conceivable statement about numbers
within
a system by going
outside
that system to find new rules
and axioms, but by doing so one only succeeds in creating a larger system with its own unprovable
statements. The implication of Godel's work is that
all
logical systems of any complexity are, by
their very definition, “incomplete”. In other words, each of them contains, at any given time, more
true statements than it can possibly prove according to its own defining set of rules.
