I am using a digital computer to write this message. Such a machine has a property which, if you think about it, is actually quite remarkable: It is one machine which, if programmed appropriately, can perform any possible computation.
Of course, calculating machines of one kind or another go back to antiquity. People have built machines which for performing addition and subtraction (e.g., an abacus), multiplication and division (e.g., the slide rule), and more domain-specific machines such as calculators for the positions of the planets.
The striking thing about a computer is that it can perform any computation. Any computation at all. And all without having to rewire the machine. Today everybody takes this idea for granted, but if you stop and think about it, it's kind of amazing that such a device is possible.
I have two actual questions:
When did mankind figure out that such a machine was possible? Has there ever been any serious doubt about whether it can be done? When was this settled? (In particular, was it settled before or after the first actual implementation?)
How did mathematicians prove that a Turing-complete machine really can compute everything?
That second one is fiddly. Every formalism seems to have some things that cannot be computed. Currently "computable function" is defined as "anything a Turing-machine can compute". But how do we know there isn't some slightly more powerful machine that can compute more stuff? How do we know that Turing-machines are the correct abstraction?