If math is a such a definite subject with solid answers, how are there still unsolved math problems? How do people even come up with them?

Question

If math is a such a definite subject with solid answers, how are there still unsolved math problems? How do people even come up with them?

In: Mathematics

Answer 1

Somebody just asked the $64 bazillion question. I love this, and it’s so far beyond ELI5 (and I’m late to the game) that I’d like to take a long-winded stab at it!

Let’s start with a simple example. “What does 1 + 1 equal?” You might emphatically say “two!” It’s obvious, right? Not so fast!

It turns out we have to define 1 (One) first. Like, philosophically. It might seem obvious what “1” is but we must remember that in all the time that the concept of “1” has existed, *most of the time the number 0 had not been conceptualized*. That point aside, we must now, if we wish to define “1 + 1”, decide what “+” means. This is called an “algebra” and was philosophically pioneered by some brilliant folks quite a while ago. We *could* create an algebra wherein 1 + 1 = 5, and that’s been done before. But it turns out that it’s not very useful. It turns out that 1 + 1 = 2 is correct only because it *works* and is *proved* to work.

Something like 50 pages of *Principia Mathematica* (a philosophical treatise) are dedicated just to establishing through logical proof that 1+1=2. So, please feel free to go read through that rigorously (I haven’t done that myself, by the way, I just take its word for it).

Now, that being said, next comes say, 2 + 2 = 4. Based on 1 + 1, can we prove that 4 is correct? How so? If we multiply all three numbers by 2, does everything work out? It does! That’s cool!

Can Pythagoras prove that a^2 + b^2 = c^2? Moreover can we say that a^n + b^n = c^n if and only if n = 2?

At each branch, more questions come up, more proofs are needed, and more discoveries can be made. Algebras and Calculi have cropped up for various purposes.

And here I am. I made it through Calc 2 and Linear Algebra, and when friends of mine have talked about their Doctorate theses I just smile and nod because I have no idea how n-polytopes tesselate in parabolic n-spaces.

Or, for that matter, why 1+1=2