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DOI: 10.6084/m9.figshare.4956353
This paper presents a problem theory, which is an intuitionist set theory where the resolving subject is a computing device.
There are two ways of defining a set:
Whenever we have a set defined by intension, a set defined by extension, and a proof that both are equal, we will say that both sets are the same set, but defined differently. For formalism and other objectivisms, a set defined by intension and a set defined by extension are either the same set or they are not the same set. However, for intuitionism and other subjectivisms, there is a third possibility: that the subject does not know whether they are the same or not. Only by allowing this third possibility can problems exist, because a problem is a state of ignorance.
The most abstract way of seeing a problem is as the list of requirements that anything has to fulfill to be a solution to that problem. Therefore, we will say that a set defined by intension is a problem, and the same set but defined by extension is its set of solutions. And we will say that to resolve a problem is to calculate its set of solutions, though most times it will be enough to calculate one of its solutions to solve the problem. In any case, in subjectivism, set theory has three parts: A problem, which is a state of need, a solution, which is a state of satisfaction, and a resolution, which is a transition from uncertainty to certainty.
Solutions are elements, as in set theory. Problems are predicates on the set of elements that are true for solutions and false for everything else. And resolutions, what are resolutions? Resolutions have to model the calculating capacity of the subject doing mathematics, and therefore we have to choose something that models our human calculating capacity. The answer to this seemingly impossible question was given by Turing (1936), who postulated that a problem is unresolvable if and only if there is not any Turing machine that can calculate its solution. That is, he posited that ‘resolvable’ is synonymous with ‘calculable by a Turing machine’, which is nowadays synonymous with ‘computable’. Therefore, for Turing, 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 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.
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 key point in this discussion is that ‘intuition’ would refute Turing’s thesis, see §4.3.1, because if there were ‘intuitive’ resolutions, then we could effectively calculate what is not computable. Turing’s thesis is not a theorem, and we follow Post (1936) in considering Turing’s thesis to be a law of nature that states a limitation of our own species calculating capacity, see Casares (C), by which we are bound to see ourselves as a final singularity, see Casares (T). Summarizing: If Turing’s thesis were eventually false, then this problem theory would be about computable problems. But, while Turing’s thesis remains valid, the problem theory is about problems, the set of effectively calculable functions is countable (§4.3.1), universal computers are the most capable computing devices (§4.3.4), everything is an expression (§4.3.10), resolving is computing (§4.3.12), and the problem theory is complete (§5.7.3).
Link to the page of my problem theory in figshare, and direct link to the pdf file.
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Última actualización: 2022-03-03.
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