Leibniz On Computers

Leibniz's first great mathematical success was his creation of a
calculating machine in the early 1670's.  It was apparently the first
machine capable of multiplication, division, and root extractions,
and was impressive enough to get Leibniz elected to the Royal Society
of London in 1673.  He later (in a 1685 manuscript) gave the following
account of the moment of inspiration for this invention:

    When, several years ago, I saw for the first time an
    instrument which, when carried, automatically records
    the number of steps taken by a pedestrian, it occurred
    to me at once that the entire arithmetic could be
    subjected to a similar kind of machinery so that not
    only counting, but also addition and subtraction,
    multiplication and division could be accomplished by
    a suitably arranged machine easily, promptly, and with
    sure results.

    The calculating box of Pascal was not known to me
    at that time...  When I noticed, however, the mere name
    of a calculating machine in the preface of his "posthumous
    thoughts"...I immediately inquired about it in a letter 
    to a Parisian friend.  When I learned from him that such 
    a machine exists I requested the most distinguished 
    Carcavius by letter to give me an explanation of the 
    work which it is capable of performing.

This pattern of discovery seems to have been charactistic of Leibniz. 
He was a sponge for knowledge and ideas, and he was tireless in 
ferreting it out of friends, acquaintences, and strangers alike.  
He tells us that, once he had gotten all the information he could 
about Pascal's machine, he set himself the task of making an even 
better machine.  However, notice that he begins the discussion by 
recalling the earlier moment of inspiration when he saw the step-
counting machine, and how "it occurred to me AT ONCE that the entire
arithmetic could be subjected to a similar kind of machinery".  In 
other words, this was the moment when his invention of the calculating
machine actually occurred, although he didn't act on the inspiration 
at the time.  It was only later, when he saw the words "calculating 
machine" in some papers of Pascal, that he was prompted to actually
design such a machine - after studying the design and capabilities 
of Pascal's machine.

There are some interesting parallels between this sequence of events
and Leibniz's development of the calculus.  In January of 1673, soon
after taking up the serious study of mathematics (enlisting Christian
Huygens as a tutor!), he visited London and made the acquaintance of 
Heinrich (Henry) Oldenberg, the secretary of the Royal Society of 
London.  In April of that year Leibniz received from Oldenberg a 
report drawn up by John Collins on the state of mathematics in England.
A prominent part of this report was a list of problems (many involving
infinite series) that could be solved by an unspecified method 
possessed by a secretive man at Cambridge named Isaac Newton.  As
Leibniz later remembered it, he himself had already had the original
inspiration for calculus, prior to seeing any of the reports on the
work of Barrow, Gregory, and Newton, and there was no actual description
of calculus in the 1673 report, but it's possible to imagine that, as
with the appearance of the "mere name" of a calculating machine in 
Pascal's papers, the mere mention of a wonderful general method for 
handling a wide range of mathematical problems of this kind was enough
to perk up Leibniz's ears and set him working in ernest on the task of
developing his own ideas for the calculus.

By 1675 (or so) he had a method that was, for practical purposes, at 
least the equal of Newton's fluxions (which Newton, builing on the
work of Isaac Barrow, had developed in the mid-1660's), although his
philosophical justification for it was perhaps less sophisticated than
Newton's.  In that year, and on into 1676, there was an exchange of
letters between Newton and Leibniz (via Oldenberg) in which Newton,
although still very reticent about revealing general methods, gave
even more explicit hints of calculus, including even an anagram 
stating the inverse relationship between differentiation and 
integration.  At this point it's clear that Leibniz was already in 
possession of his own version of calculus.  Thus the two men, each 
having made the same great discovery, neither having published on the
subject, were corresponding with each other, and we can imagine that
they were each trying to figure out exactly how much the other knew, 
without revealing too much of what he himself knew.  These letters 
became infamous during the subsequent priority dispute, but (in my
opinion) it was the Collins report of 1673 that (analagous to the mere
mention of the words "calculating machine") represents the crucial 
influence of the British mathematicians on Leibniz's discovery of 

In any case, Leibniz was certainly an enormously creative and brilliant
man, and his roles in the development of both the calculus and the
artificial computer were tremendously beneficial.  Toward the end of
his 1685 computer manuscript, with the characteristically Leibnizian

   "Machina arithmetica in qua non additio tantum et subtractio
    sed et multiplicatio nullo, divisio vero paene nullo animi
    labore peragantur"

he wrote about the value of the computer:

  "...the astronomers surely will not have to continue to 
   exercise the patience which is required for computation. 
   It is this that deters them from computing or correcting 
   tables, from the construction of Ephemerides, from working 
   on hypotheses, and from discussions of observations with 
   each other. For it is unworthy of excellent men to lose 
   hours like slaves in the labour of calculation which could 
   safely be relegated to anyone else if machines were used."

It's interesting that one of the persistent themes in all of Leibniz's
work was the idea of reducing thought processes to automatic and/or
mechanical operations that could "safely be relegated to anyone".
This was true not only in his development of the computer, but also
in his reduction of calculus to a set of straightforward algorithms
on a well-defined set of symbols.  Likewise his work on the "universal
characteristic", which we would now call symbolic logic, was intended
to reduce all thought to a set of definite rules, or, as he wrote,

 "...a general method in which all truths of reason would be 
  reduced to a kind of calculation.  At the same time, this
  would be a sort of universal language or script, but infinitely
  different from all those imagined previously, because its
  symbols and words would direct the reason, and errors - except
  those of fact - would be mere mistakes in calculation..."

Leibniz was, after all, originally trained as a lawyer, so it may be
understandable that he yearned for some automatic way of settling
disputes.  The youthful optimism of Leibniz in this regard was later 
satirized by Voltaire in the play Candide ("Come, let us calculate").
On the other hand, is it too farfetched to see in Leibniz's view that
we inhabit the "best of all possible worlds" some hint of Maupertius'
principle of least action?

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