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DOI: 10.6084/m9.figshare.4956353

It presents a problem theory, which is an intuitionist set theory where the resolving subject is a computing device.

A *problem* is a set defined by intension,
*its set of solutions* is the same set but defined by extension,
and to *resolve* a problem is to calculate its set of solutions,
though most times it would be enough to calculate one solution,
that is, it would be enough to *solve* the problem.

Following Turing (1936), computing is the model for the resolutions.
As a resolver implements a resolution, we model a resolver as a Turing machine.
Then we define the *range of a resolver* as the set of problems that the resolver solves.
If the range of resolver A is a superset of
the range of resolver B, then resolver A is better (at solving) than resolver B.
This allows us to construct a series of improving resolvers.

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,
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, but 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.

So we build a problem theory by fusing set and computing theories. 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 framework for
the evolution of cognition.
Then the answer to our question would be:
* to solve most problems*.
By the way, the problem theory
defines adaptation, perception, and learning,
and it shows that
there are just three ways to resolve any problem:
routine, trial, and analogy.
And, most importantly, this theory demonstrates
how problems can be used
to found mathematics and computing on biology.

The series of improving resolvers has five elements: mechanism, adapter, perceiver, learner, and subject. The best resolver is a Turing complete subject, which is computationally equivalent to a universal Turing machine.

Link to the page of my problem theory in figshare, and direct link to the pdf file.

A second version is in the arXiv, with its pdf file.

And there is another version in The Internet Archive.

External references used in this page:

- Turing (1936): "On Computable Numbers, with an Application to the Entscheidungsproblem", doi: 10.1112/plms/s2-42.1.230

Última actualización: **2019-08-10**.

© Ramón Casares 2019