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.
I’m going to assume you’re talking about unsolved problems, because your description sounds like it.
Thing is, most of the time they’re not like your normal school problem which is like “which number is the solution to this equation?”
More often it’s like “There is no number that has property XYZ?” People try it with very many numbers (nowadays with a computer) and if they find that the first gazillion numbers don’t have property XYZ it’s a clue that probably no number has. So we think it’s probably true (and call the statement a “conjecture”) but that’s no proof. There might still be number so large we haven’t checked it that has XYZ.
Solving the problem means proving that it’s true for all numbers that exist, which takes an abstract argument because you can’t just check infinitely many numbers.
So how does such a problem come about? By noticing a pattern. Much of mathematics is noticing patterns, then finding out what reason they have — which is sometimes very hard.
I hope that was your actual question, if not, sorry.
If, by unsolvable math problems you mean problems that haven’t yet been solved then these often come from mathematicians playing around with some mathematical concept, noticing something that seems to be true but not being able to actually prove that it’s true yet.
As an example, one of the biggest open problems in maths is the Riemann hypothesis. For an eli5 explanation of what this is, we have a function, called the Riemann zeta function that takes two numbers, does some maths and gives you two numbers back (technically it takes a complex number as an input and an output, but you can just think of those as being made of two real numbers). Mathematicians have noticed that the only way they can make this function give you (0,0) as an output is either to give it (-2, 0), (-4,0), (-6,0) and so on or to give it a half as the first number (to clarify, it’s not (0,0) for any pair of numbers where the first one’s a half, just some pairs). As they experimented with this function, they couldn’t find any other way to make it give you (0,0), so some mathematicians started to suspect these were the only ways to make it give you a zero. But they haven’t worked out exactly why these are the only ways yet and they can’t say for sure that there aren’t other ways that they haven’t thought of yet.
> How did unsolvable math problems ‘start’?
Usually somebody observes are particularity and wonders if that is always true:
– When printing an atlas somebody noticed that four colors are enough to print it such that neighbours never share the same color; is this true for absolutely every possible map?
– Every even integer beyond two and up to a million is the sum of two prime numbers; could this be the case for literally all of them?
– The sum of two cubes (third powers of positive integers) seems to never be a cube again; might that be true in general?
– Factoring numbers seems to be difficult and takes centuries on supercomputers even for most 1000 digit numbers; does an efficient method to factor any integer exist after all?
Then we set out to _prove_ this: find a formal, logical argument why it _must_ be so. Or find a _counterexample_ or _disproof_, something that demonstrates that our guess was wrong after all.
Until that we are uncertain what is truly going on. Maybe we have already checked all cases that we might ever see in real world problems by computer; or maybe we didn’t, as for the factorization example on which a lot of cybersecurity hinges on!
It sometimes even happens that we find questions where we can then prove(!) that it cannot ever be answered. We call those statements _independent_, ones where we cannot ever ascertain with our logic and from our assumptions whether they are correct. Kurt Gödel even proved that any non-silly system of rules (logics plus assumptions) always begets such statements, it is impossible to not have them.
> How can you have an answer but no idea how to get there in maths?
You can’t. Or rather, you then don’t know if the answer is actually correct. You can always take a _guess_ but without evidence in the form of a formal _proof_ it is just that. You don’t have an answer, just the guess.
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