How did unsolvable math problems ‘start’? How can you have an answer but no idea how to get there in maths?
In: Mathematics
There are two questions in one here. An *unsolved* problem in maths is one where we don’t know the answer, but we think it might be possible to find one. There are lots of these, some of them irrelevant and trivial, some deeply important with implications on our understanding of the universe itself.
An *unsolvable* problem is one where we know it is impossible to find an answer. In some cases, these come about through intentional paradoxical construction. For example, is the statement “this statement is false” true, or false? If it’s true, then the statement itself is a lie. If it’s false, then the statement is *also* a lie. There is no resolution to this paradox.
Mathematicians tend to not care too much for these types of paradoxical tricks though. They can be interesting results in some very specific domains, but more often they are considered a sign that something is wrong with your formal system and you need to change the rules so that it’s impossible to make statements like the one I made above. This happens more often than you might think in high level maths.
But back to your question… Sometimes it’s easier to make general statements about all possible answers to a question than it is to find any specific answer that fits. For example, if you ask me about two trains, one leaving Denver at 80 kph and another leaving Chicago at 120 kph etc etc, I might not know the answer immediately but I do know that I won’t need to use the fluid density of blueberry jam to figure it out. That’s the sort of logic mathematicians apply all the time to narrow down the space of possible answers and make it easier to find what they’re looking for. It just so happens that, sometimes, you can use this process of elimination to rule out *every* answer. When that happens, the problem has no solution.
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