For example, there is a simple deterministic algorithm for determining whether any given number is a perfect square. But why don’t we know how to solve things like the p vs np problem or the collatz conjecture?
Goddag Hr. Nilsson, Hr. Nielsen here with a comment.
Imagine you wish to change the spark plugs on your Saab 93 but all you have is a screwdriver. the problem is hard because you lack the correct tool. One way to look at the hardness of things like the p-vs-np problem is we lack the tools powerful enough to get the plugs out of the head.
You can also imagine an engine with an infinite number of spark plugs so that even if you had a plug wrench it would require infinite time to do the job- and a different approach would be needed.
Med venlig hilsen,
Gödel once thought in a somewhat Kantian way that human reason would be fatally irrational if it asked questions it could not answer since essentially all problems are formed from human being's mind and not set in stone or printed in the sky, unless some problem exits on their own in some Platonic realm independent of human's mind.
For those hard problem which we currently don't know how to solve within current framework, Grothendieck once suggested you may imagine them like a rusted nub to be opened. Usually we can use a hammer to strike it hard, but sometimes it won't help much or will get even worse. Another general approach is you can try to immerge it into some lubricant and just be patient to wait. Eventually the nub may get loose and screwed or hammered out...
Here's maybe two arguments:
The world of mathematics is infinite. An finite set of Axioms is finite and can only ever talk about a subset of all of mathematics. I am limited; I am not infinite in spacetime, so I can only ever work with a finite set of Axioms. I can not explore the entire infinite world of mathematics from my vantage.
How do you ask about a system if it is consistent, if you are within that system and there are no other systems? If you don't know a system is consistent, you wouldn't ask it to produce the answer. We are stuck in a single universe with universal laws of physics. Perhaps there are questions about the universe we can't be sure of the answer. This is basically a more general version of Godels Incompleteness
*Inspired from arguments by Gregory Chaitin and Tim Maudlin from The Limitation of Mental and Physical Reality Video
The question is plain and simple, it is better to approach this with clarity of definitions in mind. One could only distinguish, what an unsolvable problem is, compared to the undecidable problem that are often present, and are confuse about. So here’s the definitions:
An unsolvable problem is one for which no algorithm can ever be written to find the solution.
An undecidable problem is one for which no algorithm can ever be written that will always give a correct true/false decision for every input value. Undecidable problems are a subcategory of unsolvable problems that include only problems that should have a yes/no answer (such as: does my code have a bug?).
As what @Double Knot says, it is irrational to pertain question that our mind itself couldn’t not answer, since all the fundamentals of reality as we know it lies within our mind. As intuitionist believe that: “mathematics is just an invention created by our imagination”, unless there are grounding for which the laws of logic, and the laws of mathematics, or in general the laws of thought is entirely independent of human mind.
In essence, solutions must follow established philosophical principles, for example they should start from agreed axioms and they should not rely on erroneous logic. However, there are no such rules regarding what problems we might pose. Furthermore, while it is intuitively acceptable to put forward a problem without suggesting a solution, it’s not normal to offer a solution with no corresponding question, and so the number of unsolved problems must be greater than or equal to zero.