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

Just because a solid answer exists (which may not always be true) doesn’t mean we have the methods to find it or more importantly prove it. There are a lot of problems which are very easy to formulate but we don’t yet have the techniques to solve.

Unsolved math problems aren’t just difficult equations that you can solve with algebra. They are questions that require creativity to solve. A well known unsolved problem is the Goldbach Conjecture: prove that every even whole number over two is the sum of two prime numbers. People have been working on that one for 250 years.

There’s a big difference between solving a math equation and solving a generalized math problem

If you have 2 + X = 7 you can solve for X this one time and know that right here, right now, it must be 5

But the unsolved problems are wayyy harder than that. Fermat’s Last Theorem was unsolved for a few hundred years it goes “For any integer n>2, the equation a^n + b^n = c^n has no integer solutions”

You’re probably already familiar with the case of n=2, that’s a^2 + b^2 = c^2 or Pythagoras’s Theorem. But how do you prove that for n>2 there are no integer solutions? You could try brute forcing it but what if it works out when n=51,437? You’d have to try literally every combination of numbers which is, by definition, infinite

Its problems like these that you can’t just set a computer to and crush through the numbers, you have to fall back onto the basic properties of math and other postulates and theorems to show that there is no way that any n>2 results in a, b, and c all being integers. These are the hard ones that require people and hundreds of sheets of paper to prove.

My favorite unsolved problem, because it’s so easy to understand, is the Collatz Conjecture. We have a game that goes like this:

1) Pick any number and check if it’s even or odd.

2) a) If it’s even, divide by 2

2) b) If it’s odd, multiply by 3 and add 1

3) Take your new number and go to step 1.

For example, if you start with 10:

10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 -> 4 -> 2 -> 1…

Starting with 10, you end up going to 1 and then getting stuck in a loop between 4, 2, and 1. If you start with 9, it takes 20 steps (and goes all the way up to 52 at one point), but it also goes to 1. The conjecture is:

**Every positive whole number eventually reaches 1 when using this pattern.**

To _disprove_ this, all you have to do is find some starting number that gets stuck in a different loop. We’ve tried that, though – we’ve tried every number up to 20 digits long, and they’ve _all_ hit the 4,2,1 loop. To _prove_ this is true, though, you’ll need to come up with some creative insight about the way that numbers relate to each other.

A pattern you’ll notice with a lot of the examples given in this thread: often times the trouble is with **infinity**. If you ask for example, does the Fibonacci sequence contain any square numbers besides 144 (12×12)? I can write out the first couple numbers in the sequence, or have a computer generate the first billion – and each one is trivial to check if it’s a square – but it’s fundamentally impossible to check ALL of them, because the sequence is infinite.

The only way to solve such a thing is come up with a mathematical argument – a proof – that employs some clever logic to prove something about an infinite set. As a very simple example, consider the question, “are there any even prime numbers besides 2?”. We can answer this by saying, *suppose there were such a number. Then since it’s even, it can be divided by two – and since it can be divided by 2, it can’t be a prime!* So we have proven something about ALL numbers, even though we never had to check them individually. A slightly harder problem in this vein, *is there a biggest prime number?*

Problems like this arise all the time when mathematicians are just playing around – exploring patterns, asking questions, finding neat arguments that then lead to other natural questions. Some of the most famous unsolved problems are famous because, if we knew the answer, it would unlock truths about a lot of other related questions. (An example is the [“P vs NP” problem](https://en.wikipedia.org/wiki/P_versus_NP_problem) in computer science)

One thing to consider is that all the math you’ve learned had to be discovered by people who came before you. At some points in the past, people knew how to do some geometry but hadn’t figured out the quadratic formula, for example.

We have a similar situation now. There are things we know already and things we’ll discover in the future and things that will never be discovered.

The math most of the world learns and the math that academics studying math study are completely different maths. The vast, vast majority of us learn “how” math works. We learn the methods and formulas that explain the world around us, the ones we’ve known and understood for, for the most part, hundreds of years. It’s what we need to navigate a 3 dimensional world with an economy. Sometimes we learn slightly newer fun tricks you can do with math, without really going deeper than the surface level of the trick.

Academics are studying “why” math works. They look into the rules governing math and what it takes to break them, and what breaking the rules tells us about math. Unsolved problems in mathematics aren’t the same as the algebra homework you forgot to do. They’re things that either work or don’t, and we’re still trying to figure out *why.*

You are thinking about math as the set of tools needed to solve a specific problem that has to do with numbers. That’s an accurate description of the math that people are taught in school.

However, when you are doing math in university (“real math”, if you will), you understand that the scope of the subject is very different. One thing worth noting – most mathematicians will disagree on what even is a proper definition of mathematics. For what it’s worth, I will give you my own definition: math is the process of deriving properties from axioms and definitions. In more ELI5 terms: math is about creating rules and definitions, and seeing what interesting consequences follow from those rules when applied to those definitions.

If you think about it in these terms, then you can see how open ended the subject is. You can come up with your own definitions or rules, see how they fit in the existing rules or definitions that other people agree upon, and see if using your own stuff creates any interesting results. As an example of this happening in real life – mathematicians used to think that infinity was just an absolute concept. But Cantor showed that if you looked at two different infinite sets and tried to match their items one by one, you could come up with some sets that would have infinitely many “unmatched” items left over even after you ran out of items on the other side. So he came up with a definition for two different types of infinity, based on whether you could match items of different sets with one another without running out of items on either side. So then lots of questions crop up – can you find some properties that only one type of infinite sets have, but the other one doesn’t?

I hope this gives you a sense of how and why the subject is open ended – mathematicians can come up with interesting new definitions and ideas, and then as they apply existing rules to them there is a whole host of questions that crop up about what general statements can be made.

While you might think that this is a simple question because you don’t understand math, you have actually just stumbled on one of the most incredible and complex mathematical topics to exist!

See, back in the day, starting from some of the original Greek mathematicians, there was an idea of how to solve math problems. Not just one or two math problems, but all of them. The idea is that you start with a few base assumptions that you know are true but aren’t really provable called axioms. A proof is then built up out of some combination of these axioms. The idea is that you could go through every combination of these axioms to find every possible proof out there and solve everything that can be solved. This concept is called completeness and was embraced by many, if not most, mathematicians.

However, as recently as 1931, the mathematician Godel proved that mathematics was not complete. In other words, Godel’s incompleteness theorem mathematically proves that you cannot prove everything that is true in mathematics. So not only are there still unsolved math problems, but there will always be unsolved math problems, even with infinite computing power.

On a related note, soon after that, an impossible math problem was also found. It was proven that you cannot build a program to detect whether a program has an infinite loop in it!

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.

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.

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.

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.

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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.

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

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.

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

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.

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

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.

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

Sometimes, even if there is a solid answer, we don’t have the ability to get to it.

Take Chess for example. What’s the best move at the start? There has to be an answer, even if that answer is “multiple opening moves lead to a draw/win”. However, we do not have the computing power to go through all the possible moves that could prove the answer to this question. That’s an unsolved problem.

We can come up with many of these very easily, just by asking “given a set of rules, is a statement true?” And if that question hadn’t been asked and answered, then presto, you have just found an unsolved math problem.