
“Let Us Calculate”: Leibniz, Llull, and the Computational Imagination - benbreen
https://publicdomainreview.org/2016/11/10/let-us-calculate-leibniz-llull-and-computational-imagination/
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olooney
This essay is pretty sensationalized and doesn't really paint an accurate
picture. Leibniz did build the first 4-operation calculator:

[https://en.wikipedia.org/wiki/Stepped_reckoner](https://en.wikipedia.org/wiki/Stepped_reckoner)

[https://en.wikipedia.org/wiki/Leibniz_wheel](https://en.wikipedia.org/wiki/Leibniz_wheel)

[http://history-
computer.com/MechanicalCalculators/Pioneers/L...](http://history-
computer.com/MechanicalCalculators/Pioneers/Lebniz.html)

And he did demonstrate it in England and Paris - but framing it as "faltering
through live demonstrations" ignores the fact that this was in the 17th
century and the only similar thing they would have seen was Pascal's
calculator which was limited to addition and subtraction. A device which could
also multiply and divide was seen as a huge advance. While it does seem likely
that early versions did not work very well (seeing as Liebniz would go on to
commission a brass version from a master clockmaker after lamenting, "If only
a craftsman could execute the instrument as I had thought the model!") these
early demonstrations were still successful enough to illustrate the principle
and in fact brought Leibniz considerable fame: to quote Wikipedia, "This
'stepped reckoner' attracted fair attention and was the basis of his election
to the Royal Society in 1673."

[https://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz#Comp...](https://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz#Computation)

[http://history-
computer.com/MechanicalCalculators/Pioneers/L...](http://history-
computer.com/MechanicalCalculators/Pioneers/Lebniz.html)

While the story of an early version of the machine being lost and recovered is
true, this is pretty much just a piece of trivia. The important ideas were
never forgotten because Liebniz wrote several books about his machine. His
design was well known and many successors worked on their own versions of it.
In fact, for the next two centuries, the Liebniz wheel and the pinwheel
calculator (another Liebniz design) would dominate the field and most machines
produced in this time derived from one of Liebniz's designs.

[https://en.wikipedia.org/wiki/Mechanical_calculator#The_18th...](https://en.wikipedia.org/wiki/Mechanical_calculator#The_18th_century)

Two centuries later, the Liebniz wheel would form the basis of the first
commercially successful mechanical calculator:

[https://en.wikipedia.org/wiki/Arithmometer](https://en.wikipedia.org/wiki/Arithmometer)

The essay's insinuation that "the machine never worked as intended" seems to
be based on this statement: "Burkhardt reported that, while the gadget worked
in general, it failed to carry tens when the multiplier was a two- or three-
digit number." We know that Burkhardt only had access to an early version of
the machine, but even assuming the problem persisted in later versions, this
is extremely binary thinking. You don't dismiss an innovative and
fundamentally successful design because the proof-of-concept implementation
has a minor bug.

I did learn one thing from the essay though - Liebniz could not catch a break.
I knew that Voltare had viciously satirized him as Professor Pangloss in
Candide, but I didn't realize that Swift had also mocked him. I did know
attacking formalists as somehow damaging the "creative" side of thinking has a
long history and the argument was still being leveled at Frege and Boole
centuries later. Just as you have to define the rules of chess before you can
get a Bobby Fisher, you have to have a trivial procedure for _verifying_ a
given proof before mathematicians can fully exercise their own creativity in
inventing and proving new theorems. Liebniz's "art of infallibility" is of
course about the proof verification/checking, as the full quote makes clear:

"if controversies were to arise, there would be no more need of disputation
between two philosophers than between two calculators. For it would suffice
for them to take their pencils in their hands and to sit down at the abacus,
and say to each other (and if they so wish also to a friend called to help):
Let us calculate."

If two mathematicians have arrived at opposite conclusions on the same
theorem, then they simply need to check their work and formally verify each of
their proofs to decide who is right.

~~~
nanna
Swift's satire of Leibniz wasn't premised on science, it came down to them
each occupying opposing political stances at the time in respect of the Whigs
and Torys. Remember they were both heavily invested politically.

Wiener argues that Swift also had a go at Leibniz through his satire of the
little people who tie up Gulliver, which can be read as mocking his concept of
infinitely divisible substances. Had a go looking for the refference but can't
find. Might be in the manuscript to his last unpublished lectures...

What really damaged his reputation for centuries was not Swift though, but the
debacle with Newton over the invention of calculus.

~~~
olooney
That's not surprising. Voltaire's caricature of Leibniz was politically
motivated too, or religious, which was all tangled together during the
enlightenment. Basically a liberal/atheist criticizing a
conservative/religious thinker. And the Leibniz/Newton debate seems to me to
have been primarily political too, dividing neatly on national lines. Politics
is always with us, it seems.

Regardless of who came first, Leibniz's work on calculus was excellent. He
knew the chain rule, product and quotient rule, linearity, and how to
differentiate powers of x; he defined both integration and differentiation and
was aware of (but did not prove) the fundamental theorem of calculus (that
integration and differentiation are opposites) We still use his notation
today: sigma f(x) dx for integration, and dx/dy for differentiation. In fact,
even the word for function, in the sense "f is a function of x", seems
original to Leibniz and came directly out of his work on calculus. Leibniz
could walk into a classroom today and lecture on his calculus and it wouldn't
look much different than a modern Intro to Calculus class. If it hadn't been
for the climate of German/English intellectual rivalry it wouldn't have even
been an issue.

[https://www.maa.org/book/export/html/641727](https://www.maa.org/book/export/html/641727)

~~~
nanna
Wish I understood calculus as well as you obviously do. I guess the one thing
I'd add is that it was a far broader art than just for maths. The classroom
today would be pretty surprised to hear how its continuous with metaphysics
too.

Regarding Leibniz/Newton, not exactly the case that it divided on national
lines, because the lines didn't exist in the same way as today. He worked as
archivist for the Elector of Hanover George Ludwig, who would be brought over
to the UK to become King George I towards the end of Leibniz's life. To
Leibniz's consternation, George I would leave him behind in Hanover, where he
would die. Lots of theories about why this was the case, but one reason is
that Newton and his supporters via the Royal Academy ensured he wouldn't be
brought over. Their spat (with Clarke as Newton's mouthpiece) was in the
general context of that, hence its rancorous tone.

Norbert Wiener at least considered the two centuries of suppression of Leibniz
and his calculus and its superior notation in Britain to have held back the
British and American sciences.

