How to know if one problem is more difficult than another one?

Question

I have seen in many places (books, lectures, ...) that people say that some unsolved problem is more (or much more) difficult than another one or sometimes they point some problem as the most difficult one. How they know the difficulty of reaching an answer when they don't know the answer? Especially, as quoted by Andrew Wiles:

"You enter the first room of the mansion and it’s completely dark. You stumble around bumping into the furniture but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark...

As an example of unsolved problems consider The Millennium Prize Problems.

And to mention a few of [the contradictory?] facts about it :

1- Clay Mathematics Institute had considered Poincaré Conjecture among the others but it is solved just after 2 years the list was published (or at most 7 years if we consider Perelman's starting time of focus on the problem).

2- Michael Atiyah states Yang–Mills existence and mass gap to be the most difficult one because it requires more fields of knowledge to work on, as a criterion. While hearing from people in maths when they come to say (or say metaphorically) the most difficult unsolved problem they usually point to Riemann hypothesis. I have also seen several written texts about that claim for example the quote by M. C. Gutzwiller :

"The zeta function is probably the most challenging and mysterious object of modern mathematics ..".

I don't think deciding which one is more difficult is some personal naive opinion because for example almost all the great mathematicians in history [esp. Euler, Gauss and Hilbert] have refused to spend long time on the Fermat's Last Theorem because they knew they won't be successful at it. But how they knew it? I would like to mention Hilbert's quote on Fermat's Last Theorem on why he didn't try to solve it : "Before beginning I should put in three years of intensive study, and I haven't that much time to squander on a probable failure." How he knew that before trying it for a long time?

Answer 1

There's no definitive answer, but there are a few ways you can guess how difficult a problem might be.

The most obvious sign of a tough problem is fame. I know that the Riemann Hypothesis is difficult because thousands of very intelligent people have worked on it for a couple hundred years and it's still unsolved. It's still possible that a simple proof exists that no one has ever thought of, but it seems unlikely.

Another sign of a difficult problem is that it implies a lot of things that are also unsolved. Thus, the Generalized Riemann Hypothesis is probably harder than the Riemann Hypothesis, although it's still possible the first proof of RH will be as a corollary of GRH.

Another sign is the lack of meaningful progress towards the problem. The Twin Prime Conjecture, while probably hard, has recently began yielding to attacks on it by Terry Tao and others - they have proved weakened forms of it, like the statement that there are infinitely many prime pairs that are a distance of less than $246$ apart. So I would not rate the Twin Prime Conjecture as being as difficult as RH.

On the other end of the spectrum, there are some conjectures that nobody has any clue how to tackle. My impression is that the $P$ vs. $NP$ problem is the hardest of all the famous problems, as there have been no attempts on it that have yielded anything remotely close to a solution (as far as I know.) A proof would immediately imply a plethora of corollaries that are all also believed to be very difficult.

Answer 2

I will try answering this question from a math student point of view :)

While studying a subject Mathematical Analysis II- Part 2 We have learned stuff the old fashioned way (1900's theory as my new professor says) But after failing the subject for so many times (a usual thing for that subject for so many students) A new professor came with a new curriculum. New theory.New way of introducing contour and surface integrals with differential forms of higher orders. New way of introducing old things. Harder way of introducing new things. But when some time has passed i realized the new theory is more Abstract. It targets multiple things with one hit. Instead of proving Stokes Theorem you can prove General Stokes and target higher dimension spaces.

Was that harder to prove? Maybe, maybe not. The basis needed is harder. The knowledge needed to just understand the theorem is larger. But people saw these theorems, looked at them, analyzed them and figured out a pattern. A mathematician needs to be very gifted to recognize patterns that occured but even if you recognize it , it might be just your imagination if you cannot describe that pattern in the language of mathematics.

All in all there are ways to "get a feeling" about certain theorems. The way that they are declared. The amount of basis needed to approach and understand the theorem and its components.

Its exactly as Hilbert put it "Before beginning I should put in three years of intensive study, and I haven't that much time to squander on a probable failure." . Three years to understand the theorem and its components. Sure any dummy can understand a $x^n + y^n = z^n$ But what is behind that. If you were to start now where would you start? Hilbert knew that and the answer is a theory he did not want to tackle. And also since tackling a certain problem is subjective :) . Some problems appeal to some people while not to others.

And for the last point . Some problems are simply to valuable if they are proven or disproved. That said - you know allot of top minds in the field are tackling these problems as we speak. Imagine knowing the distribution of Primes. Every popular cryptographic system would fail. Banks would be in danger. That said there is always a possibility some government knows that but refuses to share :) Though highly unlikely.

All those fine points and details result in a personal feeling of a Theorem. Hard unsolved VS. Less Hard (again unsolved). And even that doesn't mean that the Hard wont be proven first.