How can a math problem be unsolvable?
In: Mathematics
Mathematician here. The amount of bad mathematics happening in this thread is *extremely unsettling*.
There definitely do exist problems that are fundamentally unsolvable (not just unsolv**ed**). Do not listen to anyone who tells you otherwise. Godel’s incompleteness theorems state this explicitly.
To answer the OP:
It might help to understand that there are math problems that look *very* different from anything you would have seen in your high school (and potentially even early University) math classes. In high school pre-calculus and university intro calculus you learn how to do a bunch of different kinds of calculations. From 1+1=2 all the way up to finding the rates of change and computing integrals.
Mathematicians typically don’t do very many calculations. Instead, our interest focuses on a thing called *theorems*. A theorem is 1) a statement that is true and 2) requires mathematical proof. You may have heard about Pythagoras’s theorem: for a right angle triangle, the hypotenuse^2 = width^2 + height^2. This is a statement that is true. How do know its true? Someone proved it mathematically!
To prove something is true, you have to start with some other information that we know is also true. This could be theorems or a thing called *axioms*. Axioms are statements that we *assume* to be true, but we don’t require proof. Remember drawing points on graph paper in class? In geometry on flat paper (this is called Euclidean geometry), one axiom is that you can identify a point on the sheet of paper. Another axiom is that, if you have two points then you can connect them with a line. These statements are true, but we don’t really need to prove them, we just assume them.
So if you follow the chain of logic, all theorems are constructed using the basic axioms we assume. So the axioms we pick are really important. For a while, mathematicians were concerned with the minimum number of axioms needed to construct all of modern mathematics. How few assumptions can we make and still build all of the math we’re used to? (Spoiler alert: 10 gets you there. Although the 10th one is controversial). One set of assumptions that allows us to build all of math is called Zermelo-Fraenkel Set Theory (optionally including the Axiom of Choice, in which case we call it ZFC). If you look at the Wikipedia page for ZFC, it’ll show you the 9 axioms and then mention the 10th. These axioms are enough to be able to build all of modern math. Set Theory is usually a 4th year honours level university course, so you can imagine how much work goes into translating these axioms into everything else!
Now to answer your question! Godels incompleteness theorems (2 statements which are true and have been proven, mathematically) state: if an axiomatic system assumes enough axioms to be able to do arithmetic then there are statements in that system that 1) are true and 2) cannot be proven using those axioms.
In any field of math, we have axioms and do arithmetic. So in *every* field of math, there are statements that are fundamentally true but cannot be proven. This is what we mean when we talk about unsolvable problems. Mathematicians want to come up with things we think are true and then prove it. Sometimes the true things aren’t provable.
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