“Let Us Calculate”: Leibniz, Llull, and the Computational Imagination
This essay is pretty sensationalized and doesn&#x27;t really paint an accurate picture. Leibniz did build the first 4-operation calculator:<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Stepped_reckoner" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Stepped_reckoner</a><p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Leibniz_wheel" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Leibniz_wheel</a><p><a href="http:&#x2F;&#x2F;history-computer.com&#x2F;MechanicalCalculators&#x2F;Pioneers&#x2F;Lebniz.html" rel="nofollow">http:&#x2F;&#x2F;history-computer.com&#x2F;MechanicalCalculators&#x2F;Pioneers&#x2F;L...</a><p>And he did demonstrate it in England and Paris - but framing it as &quot;faltering through live demonstrations&quot; ignores the fact that this was in the 17th century and the only similar thing they would have seen was Pascal&#x27;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, &quot;If only a craftsman could execute the instrument as I had thought the model!&quot;) these early demonstrations were still successful enough to illustrate the principle and in fact brought Leibniz considerable fame: to quote Wikipedia, &quot;This &#x27;stepped reckoner&#x27; attracted fair attention and was the basis of his election to the Royal Society in 1673.&quot;<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Gottfried_Wilhelm_Leibniz#Computation" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Gottfried_Wilhelm_Leibniz#Comp...</a><p><a href="http:&#x2F;&#x2F;history-computer.com&#x2F;MechanicalCalculators&#x2F;Pioneers&#x2F;Lebniz.html" rel="nofollow">http:&#x2F;&#x2F;history-computer.com&#x2F;MechanicalCalculators&#x2F;Pioneers&#x2F;L...</a><p>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&#x27;s designs.<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Mechanical_calculator#The_18th_century" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Mechanical_calculator#The_18th...</a><p>Two centuries later, the Liebniz wheel would form the basis of the first commercially successful mechanical calculator:<p><a href="https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Arithmometer" rel="nofollow">https:&#x2F;&#x2F;en.wikipedia.org&#x2F;wiki&#x2F;Arithmometer</a><p>The essay&#x27;s insinuation that &quot;the machine never worked as intended&quot; seems to be based on this statement: &quot;Burkhardt reported that, while the gadget worked in general, it failed to carry tens when the multiplier was a two- or three-digit number.&quot; 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&#x27;t dismiss an innovative and fundamentally successful design because the proof-of-concept implementation has a minor bug.<p>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&#x27;t realize that Swift had also mocked him. I did know attacking formalists as somehow damaging the &quot;creative&quot; 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 <i>verifying</i> a given proof before mathematicians can fully exercise their own creativity in inventing and proving new theorems. Liebniz&#x27;s &quot;art of infallibility&quot; is of course about the proof verification&#x2F;checking, as the full quote makes clear:<p>&quot;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.&quot;<p>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.