This line of reasoning feels similar to that which says AGI will never be possible.
It's also a bit unclear to me, how one can justifiably hold this position with such confidence?
It's also a bit unclear to me, how one can justifiably hold this position with such confidence?
I only meant that the computers of today are tools we use to explore math, but that the work today is very human. The "computer" that transcends that will look very different. What Alan Turing showed us was that you cannot turn all of "mathematics" into an algorithm. The beauty and elegance and human experience of the exploration of mathematics are important. Some day we will probably be able to teach a machine those traits.
The title reads more like someone wrote an app that solved math, which is silly. The article itself is fortunately much better.
How do we humans identify and solve the halting problem?
Not quite. Turing proved that no single algorithm cannot solve the halting problem for all machines; it has always been trivial to show that we can answer the halting question for many machines we care about, often because they are specifically designed to always halt.
Chaitin elaborated this into a detailed theory of quantified incomputability, showing (roughly) that any N-bit axiomatic system could decide the halting problem only for machines composed of strictly less than N bits of information (Kolmogorov complexity).
So the question is either, depending on your philosophy of mathematics, how we locate ontologically special cases where halting proofs are possible, or how we obtain the information we put into our axioms that allows us to write halting proofs for increasingly complex programs.
But, I do believe the original commenter misunderstood my question. Why is it that we humans, when armed with how a computer works, can identify halting problems? Why do we not get stuck in a loop when studying a potential halting problem?
What allows us to make the intuitive leap past that? And to follow up, what are the implications about "brains are 10^10 200Hz computers in parallel" hypothesis?
Well, given my current understanding, I'd say it's because human beings reason inductively. We pick up new axioms by observing our environment, which includes things like equations and programs. Since we reason inductively, we're only bound by incompleteness theorems for phenomena too random/complex for our current understanding. We "catch up" eventually to be able to solve computability problems, at least to some degree.