I am interested in Artificial Intelligence, in formalising and generalizing a human mind, in the limits of the human reasoning and the comprehensibility of the truth:
Can humans solve the halting problem?
Is Church-Turing thesis true?
Can every mathematical proof of the consistency of ZFC (if there is one) as a result of human activity be translated to use the axioms of ZFC? (c.f. Key Assumption of Timothy Chow)
What is the algorithm that could solve efficiently every mathematical problem that any possible smartest human mathematician could ever solve?
Is there an algorithm that could answer metaphysical questions in a human language wisely?
What is the cognitive universe of a human mind? How vast is its observable part? Is there a limit to how much of a human mind can be formalised and defined mathematically and simulated physically?
Can a human mind be simulated by a Turing machine? Can humans solve the halting problem? What is the class of problems solvable by a human mind?
What is the creativity of a human mind? Is it just a very smart algorithm executable by a Turing machine without oracle or with an oracle? What oracle does it need? Quantum randomness, inspiration from the physical cosmos or non-physical means wired in our mind (e.g. Plato’s light and goodness)?
What is the language of ideas? How does a human mind compute and reason with ideas? How are ideas formed? Is a form of an idea definable by a human mind prior to the discovery of a specific idea? Or there are ideas that are new and very different from any idea we comprehend now and such ideas are indescribable with the semantic means we posses now? Is the initial state of the acquisition of an idea accompanied by the observation of the reality in which a human mind is present and by the ability of a human mind to form patterns in a random cosmos of thoughts and associate them with the patterns in the real world?
Does our physical world operate according to the set of rules that give rise to the patterns observable by a human mind? Is the key to the scientific discovery observation of the reality and then the search for the explanation? Is the order of the reality what makes us able to reason about it since a human mind operates according to a certain order as well?
How does the mechanism of a human mind relate to the mechanism of the physical world? Could it be that the two worlds mirror each other and this is the fundamental ingredient realising the full potential a human mind? Does our mind develop its potential in the physical world and is this true because of the response of the physical world to our actions commanded by the reasoning processes in our mind? Is the crucial component of a human mind the capacity for the interaction with the physical world and the reality?
What are the fundamental principles of a human mind?
Taken from the book Computability, part Computability and Unsolvable Problems, chapter The Search for Natural Examples of
Incomputable Sets by Barry Cooper.
Even more divisive is the debate as to how the human mind relates to practical incomputability. The unavoidable limitations on computers and axiomatic theories suggest that mathematics — and life in general — may be an essentially creative activity which transcends what computers do.
To what extent can the human mind be likened to a Turing machine or a large computer?
The basic inspiration for Alan Turing’s computing machines was, of course, the human mind, with things like “states of mind” feeding into the way he described the way his machines worked. Turing made clear in a number of places which side of the argument he was on! On the other, we feel like our mental processes are not entirely mechanical, in the sense that a Turing machine is. And various people have explicated these feelings to a point where it can be argued convincingly that these feelings have more than purely subjective content. For instance, there is the famous and influential book of Jacques Hadamard on The Psychology of Invention in the Mathematical Field , or the philosophically remarkable Proofs and Refutations: The Logic of Mathematical Discovery by Imre Lakatos. In science, Karl Popper effectively demolished the inductive model of scientific discovery — as was accomplished, more debatably, by Thomas Kuhn at the social level. This raises the question of how to model the way theories are hypothesised, via a process which seems neither random nor simply mechanical.
A purely mathematical answer to the question is very difficult. Roger Penrose (in his Shadows of the Mind ) has argued (unsuccessfully it seems) that the overview we have of Gödel’s Incompleteness Theorem for axiomatic arithmetic (see Chapter 8) shows that the human mind is not constrained by that theorem. But it is hard to be clear what it is that the human mind may be doing that Turing machines are incapable of. Obviously it will help to know more about both the physical and the logical structures involved. What is really needed is an alternative mathematical model to that of the Turing machine, and providing this must be one of the main aims of computability theory. Some speculations in this direction are provided in the 2003 paper by myself and George Odifreddi on “Incomputability in Nature”.