Why is there still a lot of unsolved math problems, despite having really advanced computers?
In: 35
Because some problems, we don’t have ways of computing all possibilities, because there are infinite possibilities. Computers, despite being really powerful, are not infinitely powerful.
And even for some finite problems, it’s really easy to get numbers far bigger than a computer can deal with.
A lot of unproved hypotheses are of the form “There exists no number for which the property X is true” or “Property X, which applies to every known number so far, actually applies to every number.”
For example, the Collatz conjecture is that every integer is part of a hailstone sequence without looping. No matter how strong your computer is, it can’t test whether that is true of every number, individually. It’s an infinite amount of work.
The trick in mathematics is often rephrasing the problem in a different way, so that it becomes a finite amount of work to prove. Maybe computers will one day be advanced enough to help us with that aspect as well, but they aren’t there yet, except for really simple examples.
Computers are generally good for solving problems that you can tell them how to solve by doing a bunch of concretely defined operations. Mathematical proof is a different sort of issue to solve.
In order to solve a problem, you need to show it’s true for **all** cases. Not just many, or most. You could check these one by one, but that would take about infinitely long. Take the Riemann Hypothesis, for instance. Long story short, there is a special function, called the Riemann zeta function, that has been conjectured to only have non-trivial solutions where the real part of the solution is 1/2. So far, we haven’t found a proof. We have found many, many many solutions to this function, but we haven’t found a way to prove all of them have 1/2 as the real part.
Now, that’s not to say that doing it via computer is irrelevant. Sure, you can’t prove it by doing it like that, but it’s possible to disprove it. After all, if there is even one solution where the real part is something else than 1/2, it’s disproven. And that’s as much of a big deal as proving it.
Computers can only do what they’re specifically told how to do. They just do it very quickly.
“Unsolved” math problems are questions where we simply *don’t have a good answer*. A computer can be used to either verify or disprove a possible answer, but you have to come up with an answer first.
The answers so far are a bit off: yes, there are a lot of statements about infinitely many things, so you cannot just test them all; but that isn’t the only thing a computer can do. And the real answer is: our computers and algorithms are simply too slow. So far. Given extremely long run-times (not truly infinite, just absurdly large), a computer could solve all the problems mathematicians can.
So how can a computer theoretically solve every problem? First we have to notice that mathematical “evidence” is all about _proofs_, formal logical reasoning from the given axioms. Those are often presented in a human-readable format (but some authors are worse at this than others), but ultimately, a proof uses only the following constructs:
– Variable and constant names such as a, b, c, e, pi, Hamster or any other name. Restricting them to be somewhat sane is usually a good idea.
– Function names such as f(…), +, -, ·, /, and more, including the information how many entries (and of what type) it wants.
– Symbols for basic constructs such as = (“equal”) or ∈ (“element of”).
– Logical constructs such as “for all” (“∀”), “there exists at least one” (“∃”), “implies” (“⇒”) and “and” or “or”.
– Maybe a few more, but lets end the list here.
The key point is that ultimately there are only finitely many different proofs of a given length. What a computer can now do is literally try one-by-one every possible sequence of such symbols to check if it
(a) makes sense (unlike “apple being ⇒ hamster up down +”),
(b) ends with a formalized version of what we want to prove.
All this is perfectly doable. But even for proving the simplest math from over 2000 years ago, this would fail. It simply takes way too long. Our computers are sooooo many orders of magnitude off to get to the end.
Now obviously that also means this is not what mathematicians do. Indeed, they use their experiences, intuition and knowledge to try those avenues to the goal which are much more likely to actually work. And results show this often works really really well.
Recently and for the first time, we were able to make basic artificial intelligence. Assuming some further advancement, which is likely, we may see the day when computers have a comparably good intuition than us, and thus will be able to tackle real mathematical problems.
These unsolved problems aren’t “find the solution to this equation” but more “we’ve noticed that all numbers that share this one property also seem to share another property, is this always true or have we just not yet found exceptions” or “no numbers seem to satisfy this specific behavior/property, is there actually no number that does this or have we just not checked the right one yet”.
You *can* solve these via brute forcing them if you find an exception that disproves it. [Eg one of the shortest math papers ever published was a *dis*proof.](https://miro.medium.com/max/1400/1*uXaMFryQq43U5ZOjvQPs8g.jpeg) but to prove these true you need to find a way to demonstrate the fundamental behavior of the mathematics behind it, something computers aren’t great at.
Despite the advancements in computer technology and the increased computational power available to us, there are still many mathematical problems that remain unsolved. This is due in part to the fact that mathematical problems can become increasingly complex and abstract as we continue to make progress in the field. Additionally, many unsolved problems may not have a known solution method, or the solutions may be so computationally intensive that even our most advanced computers would require an infeasible amount of time to solve them. Furthermore, many mathematical problems are connected to other areas of science and technology and solving them may require a combination of mathematical and non-mathematical approaches. In summary, the ongoing presence of unsolved mathematical problems is a reflection of the inherent complexity and diversity of the mathematical landscape, and the fact that new and challenging problems continue to be discovered.
One example of an unsolved mathematical problem is the Riemann Hypothesis. This conjecture, first proposed by mathematician Bernhard Riemann in 1859, states that all nontrivial zeros of the Riemann zeta function, which is a complex function that encodes the distribution of prime numbers, have a real part of 1/2. Despite much effort by mathematicians over the past century and a half, a proof or counterexample of the hypothesis has yet to be found. The Riemann Hypothesis is considered to be one of the most important unsolved problems in mathematics, and solving it could have significant implications in number theory and other areas of mathematics.
The meaning of “solving a math problem” is different when you’re learning math in school vs. professional mathematicians advancing the field of math.
For example, let’s consider the technique of long division. This is solving a mathematical problem of “How do you divide two numbers?” and long division is one solution to that problem. If you are asked in school to divide 2348 by 32 and you use long division to get an answer, you are not really solving a math problem. You are carrying out a known solution for a specific example of that problem.
So we don’t have unsolved math problems because they require too much computation for a human to do, and therefore computers could easily solve these problems. These are unsolved because we haven’t yet figured out what computation we need to do. That’s something that computers can’t do, only humans can do that. Once we figure out what to compute, computers are a big help in doing that, but that isn’t really the core of what solving a mathematical problem is.
Let’s say I come up with a hypothesis – I think they an even number multiplied by another number will always result in an even number.
How do I prove it?
I could easily write a computer program to multiple every even number between 2 and 1000 with every other number in that same range and confirm my idea, but does this rule still hold true at number 1002 or 10000002?
You might think – well it’s obvious. Unfortunately it’s not, these hypotheses often fail in certain cases.
We could have computers searching huge spaces of millions of numbers to check if this rule holds true, but that will never PROVE that it holds true for every possible number.
Latest Answers