Problem Theory

The Turing machine, as it was presented by Turing himself, models the calculations done by a person. This means that we can compute whatever any Turing machine can compute provided we have enough time and memory, and therefore we are Turing complete. The question addressed here is why, Why are we Turing complete? Being Turing complete also means that somehow our brain implements the function that a universal Turing machine implements. The point is that evolution achieved Turing completeness, and then the explanation should be evolutionary. Nevertheless, our explanation is mathematical. The trick is to introduce a mathematical theory of problems, under the basic assumption that solving more problems provides more survival opportunities. Then we construct a series of resolvers, where each resolver is defined by its computing capacity, that exhibits the following property: all problems solved by a resolver are also solved by the next resolver in the series if certain condition is satisfied. The last of the conditions is to be Turing complete. This series defines a resolvers hierarchy that could be seen as a first and broad description of the evolution of cognition, where achieving Turing completeness is the final step. Then the answer to our question would be: to solve most problems. By the way, the problem theory defines adaptation and learning, and it shows that there are just three ways to resolve any problem. And, most importantly, this theory demonstrates how problems can be used to found mathematics and computing on biology.