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?

866 views

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

20 Answers

Anonymous 0 Comments

Mathematics is a Hydra. Everytime we solve a problem, we see at least 2 new rising up from that.

Anonymous 0 Comments

The best explanation I ever heard was this:

Imagine you’re brought into huge, dark mansion. It’s impossible to see so you fumble around the main entrance before finding a light switch. Ah, ok. There is a hall before you that stretches off into darkness. You decide to open the first door you can see and begin searching this new room. More darkness. You carefully walk inside, trying to use the little light from outside that you can. You hit your knee on strange furniture, reach out into the blackness for anything to steady yourself on. After a long time, you find the switch and AH HA! The room looks almost how you imagined it in the dark.

You keep doing this for hours, days, weeks… years. The mansion is unending.

The rooms and branches of the mansions are like branches of mathematics. Many are already known, studied, and understood. You can continue to analyze these if you wish, but there are more rooms, more branches, that are barely understood. Some are still unknown. Good luck.

Anonymous 0 Comments

Your question could be replaced by “if alphabet, vocabulary and grammar are so well defined, how are there new books written every year?”.

Anonymous 0 Comments

Math is the study of what’s true. You start with a few obvious things that you know are true, and from them you prove new things, which are also true. You can use those new truths to prove more things, and so on. It never stops, unless you want it to.

Anonymous 0 Comments

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

Anonymous 0 Comments

Generally speaking, most high-level math problems are proofs.

Proofs are far less like finding solutions and more so like discovering the laws of mathematics itself. Solving one, depending on it’s implications, is akin to discovering Newton’s laws of physics. We don’t tend to say we have solved a proof. Instead we prove it.

Hence many are unsolved and are, for lack of a better word, hypothesis waiting to proved right.

Anonymous 0 Comments

Different types of math can or cannot be “solved” completely.

For example, calculating PI, which is half of the number of radii that could be fitted into the circumference of a circle, if the radii were squished out to the sides, can’t be solved to perfect precision.(The answer isn’t 3.0, it’s 3.141<a bunch of decimals>.) This would be an example of something that can be solved, but not completely. You can always be more correct with effort and time.

On the other hand, calculating the number of apples in a bag if Percy the horse eats 2 is easy.

a=apples

p=percy’s share

x=how many apples we have

x=a-p

If we know percy ate 2 and we bought 10, x=10-2 therefor x=8

I think this is called “discrete math” but I’m no mathematician so I may be corrected on that.

I hope this helps. Sorry if it doesn’t.

Anonymous 0 Comments

Here is a fun concept for you. Gödel’s incompleteness theorems basically prove that there are things which are mathematically true which can never be proven to be true.

What a concept! We know that there things where not only does no proof exist, but no proof can exist, despite them being true. Given this, there will always be things we can never prove to be true or false.

Anonymous 0 Comments

A great YouTube channel for starting to think about math in a more general way and appreciate the beauty in the field is 3Blue1Brown. Highly recommend.

Anonymous 0 Comments

Surprised nobody has mentioned Goedel’s incompleteness theorem: https://en.m.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems

It was established almost 100 years ago that there must be at least some mathematical theorems which are true but unprovable.