Mathematics
AI Models and the
Entscheidungsproblem
AI models are becoming an approximate solution of the Entscheidungsproblem by David Hilbert.
The Past
In the 20th century, David Hilbert proposed 23 problems in mathematics that he considered of utmost importance, some of them still open, however the one that we are interested here is the Entscheidungsproblem which states that:
Is there an algorithm that considers an inputted statement and answers “yes” or “no” according to whether it is universally valid, i.e., valid in every structure.
Alan Turing was interested in this question and in fact he solved it by creating the concept of a Turing Machine, which would subsequently form the modern computer that we have nowadays. This by itself was remarkable, nevertheless, Turing proved that there could not be such a Turing Machine, and therefore there could not be such a computer.
However, this story did not stop there and in fact in this story we will learn that it is more alive than ever before.
The Present
Jumping to nowadays, we have AI systems that are getting better and better at solving mathematics problems and therefore getting near to the mathematical truth, so my question is:
Is it possible that a computer might not solve all mathematics problems, but instead approximately solve all problems ?
I do not see this question often asked, and I believe there are grounds to believe that such is possible if we look at the benchmark for Mathematics Competition
or the problems for non research mathematics problems
or for the research mathematics problems
We can see that these models are approximating the capacity to say mathematical truth more and more at the first try ! This is astonishing !
In my view, we are currently witnessing an ever better approximation solution to the Entscheidungsproblem. If we continue like this we will never have a computer that will solve 100% of the problems, however we will have a computer that will solve the Entscheidungsproblem in 95% of cases, and this has huge consequences mathematically.
The Future
It is not so farfetch to think that in a not so far tomorrow there might be a day when all human mathematical research will become trivial very quick, and we will enter the era of AI knowledge.
After all, imagine we have an AI that scores 90% in the FrontierMath benchmark, are really bothered in solving or are more like understanding what the AI is doing and asking it more questions?
In a way mathematical research will never be more the same, the best researchers will not be the ones that are agile in solving, but the ones who are best in asking the right questions.
The future will be a place where asking questions and having critical thinking in conjunction with knowing how to integrate AI systems in our workflow will unleash the next era of mathematical discovery and humanity.
Indeed, Hilbert’s Entscheidungsproblem has a negative solution, however it seems that an approximate solution is coming ever closer as time passes, and this is equally exciting as if the Entscheidungsproblem is true!
England, Newcastle Upon Tyne.
