So there are currently tons of unsolved math problems such as The Collatz Conjecture, The Riemann Hypothesis, Goldbach’s conjecture and so on… I get that they are so hard that being good at mathematics isn’t enough, but why can’t computers solve them? Or at least solve some parts of the problem, getting a chunk of the work done for the mathematicians that work on them?
Will computers be able to eventually solve this problems in the future as we’ll develop better technology?
In: Mathematics
This is actually being done and there are a number of famous math problems which have been solved by computers. One things that can work to prove some problems is to just use computers to search for numbers to prove or disprove the theory. But this means that there needs to be a solution within the numbers that even our computers can search. And a lot of these problems might require a computer to search more then is physically possible in our universe. Another approach which have been researched a bit is to have computers do the logic calculations that is involved in some of these long proofs. A computer can apply logic reasoning much faster then a human and can find logical fallacies much faster. So you can program the rules and definitions of mathematics and logic into the computer and have it search through until it can find a logical reasoning that either proves or disproves a theorem. This technique have been used a few times and have produced extremely long logical reasoning for the proof it presents. A computer is able to write hundreds of pages of new theories and mathematical proofs that will eventually end up with a conclusion. The problem is that these are generally very trivial on their own as the computers is not able to come up with completely new concepts and techniques. The algorithms are being made better and better but they still lack the same abilities for critical thinking that humans have.
And although the techniques being developed for these computer generated proofs can be very useful on their own, for example being able to give a computer a program and then ask it to prove that it works or eventually how it can fail. They kind of miss the point of these math problems. They are usually not interesting on their own and have no real world applications. However in order to solve them we need to come up with new ideas and concepts which themselves are very important. It is not the destination that is the goal but the friends we make on the way. For example the discoveries within exponents, prime numbers and eventually elliptic curves that were done while trying to prove Fermat’s last theorem is the foundation of modern asymmetric encryption that allows us to communicate safely and securely today. A computer could not have come up with all these new concepts on its own.
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