1
$\begingroup$

Imagine that alien civilization contacted you and offered to answer one math question. This should be a Yes/No question (so, you cannot ask for a million-digit binary string encoding the answers to a million questions), and the answer will be "Yes", "No", or some impossibility statement like "Independent from ZFC". The answer will come together with a proof (if it exists).

So, what would you ask?

Obvious candidates are millennium prize problems like P vs NP. However, it seems that it is better to ask "Is public-key cryptography possible?" The answer is believed to be "Yes", and if so, it implies that P is not equal to NP, and much more. However, there may be an even better questions in this area.

If you work in number theory, you can ask about Riemann hypothesis, which would give you a lot of information about prime numbers distribution (and more!), but an obvious better question is generalized Riemann hypothesis. If we talk about prime numbers then what is a well-believed conjecture that implies the most information about their distribution? Ideally, it would be best to have the result in the form "If property P satisfies some minimal sufficient conditions, and is true in this random model of primes, then it is true for primes". This would resolve all main problems about primes including Landau's problems at once, but I am not aware about any formal conjecture in these lines.

Of course, the above areas (computational complexity and number theory) are just examples, and questions from all areas of mathematics are welcome.

Let us put aside discussion whether any hint from aliens is good for the development of mathematics. What I am asking is a single Yes/No question that gives us most. If we ask question such that only one answer (say, "Yes") have strong consequences, then we should have all the reasons to believe that the true answer is indeed "Yes". Alternatively, it may be a win-win question such that both Yes and No answers are very informative.

Improve this question
$\endgroup$
3
  • 2
    $\begingroup$ Is the GRH really better than RH? It could be false, and disproving it just requires finding a single (edit: off axis) root of a single L-function, which provides very little information. GRH could still be true for infinitely many L-functions, which is often all one needs. $\endgroup$
    – Harry Wilson
    Commented May 4, 2022 at 12:37
  • 3
    $\begingroup$ No I can't stop wondering what makes that alien civilization so confident that they will be able to answer any math question we might possibly ask... :-) $\endgroup$
    – Jochen Glueck
    Commented May 4, 2022 at 13:02
  • 3
    $\begingroup$ @JochenGlueck This might be the best question to ask them, really. $\endgroup$
    – Emil Jeřábek
    Commented May 4, 2022 at 13:46

2 Answers 2

Reset to default
8
$\begingroup$

We could encode several interesting YES/NO questions into one YES/NO question with a suitable function, for example XOR:

Is it true that the number of YES answers to the following questions is even? 1. Is P=NP? 2. Is the Riemann hypothesis true? 3. Is Goldbach's conjecture true? ... 100. Is $\pi$ normal?

Since we are promised a proof, it would seem we would get a lot of proofs for the price of one (reminiscent of the fairy-tale trick "I wish for 100 more wishes"). It is difficult to see how one could prove that number to be even (or odd) without proving all sub-questions.

Not sure if the alien civilization would accept this trick. Maybe we should first ask if this is allowed. (Oops, there went our one question.)

Improve this answer
$\endgroup$
5
  • 5
    $\begingroup$ You run a considerable risk that at least one of these 100 questions is independent of ZFC, in which case very likely the whole XOR is independent, and the answer will give you no proof whatsoever. $\endgroup$
    – Emil Jeřábek
    Commented May 4, 2022 at 13:43
  • $\begingroup$ @EmilJerabek Could be that e.g. P=NP and the Riemann hypothesis are both independent, but are actually equivalent. $\endgroup$
    – Ville Salo
    Commented May 4, 2022 at 15:34
  • $\begingroup$ I see you wrote "very likely", and covered this possibility. $\endgroup$
    – Ville Salo
    Commented May 4, 2022 at 15:35
  • $\begingroup$ @Emil, good point. Mitigating that risk in a systematic fashion would be an interesting task of metamathematics, careful wording ("Is it provable in XXX" vs. "Is it true that..."), and probabilistic reasoning (assessing the probabilities, risks and utilities associated with each proposed subquestion). If we really care about getting the most out of that single question (and if we believe we are going to get the answer as promised), we would better set up a multi-year, well-funded project to design the question. Besides mathematicians, we might need xenolinguists and xenolawyers in the project! $\endgroup$
    – Jukka Kohonen
    Commented May 5, 2022 at 7:24
  • 1
    $\begingroup$ This is a clever idea. However, it is just barely conceivable that the aliens could prove that 1 and 2 are equivalent, that 3 and 4 are equivalent, etc., and therefore proving that the answer to the question is YES, without giving away how to prove or disprove any individual statement. (Though even in this case, we would presumably learn a lot of valuable math.) $\endgroup$
    – Timothy Chow
    Commented May 6, 2022 at 3:52
1
$\begingroup$

A general strategy:

Since you state that the yes/no answer will come with a proof, I presume the proof will be understandable by humans, so it will need to contain much background material. I would argue that the most informative question we can ask is the one that would require the aliens to teach us the largest amount of new math for an answer. A simple numerical counter-example (an off-axis root of the Riemann zeta function) is unlikely to provide much new math. Asking whether P is equal to NP seems a better candidate.

Improve this answer
$\endgroup$
7
  • 1
    $\begingroup$ Really? You don't want to weight by the likelihood the answer is "yes" (or "no")? $\endgroup$
    – mathworker21
    Commented May 4, 2022 at 9:10
  • 1
    $\begingroup$ @mathworker21 --- not if the answer would be informative either way: if P != NP the proof is likely profound and will teach us much new mathematics; if P = NP then the proof may well be simple, but the consequences will be far reaching (there would be a way to solve NP-complete problems in polynomial time). $\endgroup$
    – Carlo Beenakker
    Commented May 4, 2022 at 10:22
  • $\begingroup$ Huh? If we thought a proof of RH would be $1000$x more informative than a proof or a disproof of P = NP, and we thought RH was true with probability $\ge 0.99$, then we should ask the aliens about RH. $\endgroup$
    – mathworker21
    Commented May 4, 2022 at 11:07
  • 2
    $\begingroup$ my fear with RH would be to receive as an answer a single number --- like "42" --- en.wikipedia.org/wiki/… $\endgroup$
    – Carlo Beenakker
    Commented May 4, 2022 at 11:23
  • 1
    $\begingroup$ @CarloBeenakker If we were handed a proof of a contradiction from the assumption that there does not exist a positive integer $d$ such that SAT is solvable in time $O(n^d)$, but with no effective upper bound on the value of $d$, then that might not be so informative. $\endgroup$
    – Timothy Chow
    Commented May 4, 2022 at 22:28

Not the answer you're looking for? Browse other questions tagged .