On the shoulders of giants
A short introduction to the computer.
Table of Contents
A computer is something that computes (calculate; solve for). It took the place of people that could compute, who were also called “computers.” These people solved equations using fixed rules. They were applying algorithms. Multiple teams would do the same problems to check each other’s results. They were checking their errors. That is all a computer is. Everything else is an implementation detail, something done to improve some characteristic of the computer’s operation. It may make it cheaper, or ’faster,’ or somehow more efficient to do these calculations.
1. Computability: a limit to what we may compute
What is mathematics?
Maybe the math reflects what the real world is, which reflects an ideal world created by a being. This view is sometimes called Platonism. Its adherents were called the Scientific Revolution.
Formalism is a view that math is ’symbol manipulation.’ We may define an arbitrary set of axioms and objects. As long as the consequences following these are consistent, we may use the resulting rules. They are “true” for our purposes. Perhaps these axioms and objects should be defined for physical reasons, but that is not required.
Constructivist views state that things in math only exist if they may be constructed.
Finitism is a belief that all things are countable; there are no infinities.
Computability and definability are concepts introduced by among others, Markov. https://plato.stanford.edu/entries/computability/ These are all important concepts because