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I will point out that while there are some things that can never be computed (the last digit of Pi or e come to mind) there are also classes of these problems that can be calculated "close enough for all practical purposes". If it was not for that consideration, we would be missing a lot of great engineering feats.
Overall it was a very good and fun article. I liked the interspersing of math, logic and computers.
It seemed to be very USA focused, and missed a lot of the work done in other countries, especially the work of Konrad Zuse and things like the ACE computer in England.and the EDSAC (the first computer to store its program in its own memory) designed by my friend, Sir Maurice Wilkes..but to cover all that would probably go on to fill tomes if printed out.
You also missed out on the Colossus Computer, designed and built by Tommy Flowers at Bletchley Park as part of the famous code breaking operation of WW II. I mention it because it used vacuum tubes to give a great boost to code breaking operations over the electro-mechanical BOMBAs designed and built by Alan Turing.
You also did not mention the Mark I and Mark II computers built by Howard Aiken and IBM to do ballistics in WW II. It was on the Mark II computer that Lieutenant Grace Murray Hopper (later to rise to the rank of Rear Admiral) located the first "bug" in a computer, a moth that flew in through an open window and was trapped in the contacts of a relay.
Lt. Hopper went ton to develop FLOMATIC, the first "high level" language, which led to the design of COBOL. Lt. Hopper also worked on the ENIAC, so there also is a connection there. Dr. Hopper was another friend of mine, and an inspiration to me and many other computer people.
I do quibble about the statement that the transistor seemed to be an "evolution" over the tube, but the integrated circuit was a "revolution". I doubt that we would have been able to build, power and cool computers of the size and complexity of the IBM mainframes in the late sixties if we had to rely on tubes to run them, and certainly it would have been more difficult to build integrated circuits out of tubes. Perhaps you could concede that both transistors and integrated circuits were revolutionary...it is ok to have two (or more) revolutionary items, just as it is possible to have more than one hero.
Actually Charles Babbage was not trying to disrupt the industry of printed logarithmic tables, he was trying to print accurate tables. His difference engine included a mechanism to transfer the calculated tables directly to a print plate so there would be no transcription errors between the calculated numbers and going to the printer.
Babbage's work on the engine started when he was working with another engineer doing calculations in parallel as was often done in those days. They did one set of calculations and got different answers. They retried the calculations and each got their same answer, but again they were different. Then they looked at the values in the tables and realized that the two books had two different numbers in the table. This frightened Babbage because he realized that if they had been using the same (wrong) book, they would not have discovered the error. So he set out to create a machine that would calculate the tables correctly every time and create the print plate so transcription errors would not happen.
The Difference Engine #2, built for the CHM to Babbage's plans, created these print plates perfectly.
As a side note, in 2008 Linus Torvalds was inducted to the Hall of Fellows for the CHM. The only way I (who had nominated him for the Fellowship) could get Linus to attend the ceremony was to tell him he could turn the crank of the Difference Engine. And it was so.