# Is there any math conjecture that would cause a lot of damage if disproven?

As a plot device I need some kind of conjecture that is used in lot proofs and widely assumed true. Something like many papers in computer science start with "Assuming P != NP ...".

In my story for dramatic purposes my heroine disproves it.

It doesn't matter how, or is it plausible in real world, I just need a lot of esteemed work and many careers in tatters. The story is basically about a complicated proof, and those who could understand it due to working in same or nearby fields must choose between the truth or their prestige.

I would prefer something from pure math, so the damage would be contained to academia. Breaking cryptography would make three letter agencies very interested. Anything related to relativity and or quantum mechanics would gain too much attention.

• When Marie Currie discovered nuclear reactions, that paved the way for the atom bomb. Think of that. Prior to her works it was generally assumed that atoms could not be broken nor become atoms of other elements through fission or fusion. – Renan Nov 12 '19 at 22:21
• It is commonly believed that finding the prime factors of large numbers is hard. For example, when your browser displays a lock in the address bar and says that the connection is secure it implicitly believes that finding the prime factors of large numbers is hard. There is actually no reason to believe this, other that we haven't ever found an easy way to do it; and if disproven it would lead to a lot of turmoil in the field of internet security. – AlexP Nov 12 '19 at 22:47
• "...must choose between the truth or their prestige." Disproving a longstanding conjecture will get you tons of prestige in pure math, so this isn't really doable. Some of the big ones that could shake things up (Riemann, P = NP, Navier-Stokes) also come with \$1M prizes, giving even more incentive to divulge. – eyeballfrog Nov 13 '19 at 7:02
• How about something like a proof that the ZFC axioms for set theory can be used to derive a contradiction? Godel's theorem means we can't prove their consistency, and our intuitions about infinite sets of various cardinalities generally seem less reliable than assumptions about something like arithmetic, so this doesn't seem as implausible as deriving a contradiction from the Peano axioms for arithmetic. – Hypnosifl Nov 13 '19 at 8:32
• @Whitecold "All of the fundamentals math can be used to prove each other, so pick your axioms, and you can prove the others." This isn't true-- some systems are stronger than others and some aren't even comparable. You can build a model of peano arithmetic within ZFC but you can't construct a model of ZFC within peano arithmetic. The main reason is that when you construct peano within ZFC you don't need the power set axiom so within peano there's no real way to get to all of the crazy infinite cardinals that pop up in ZFC. – el duderino Nov 13 '19 at 14:02

The Riemann hypothesis is what you're looking for.

Basically everyone in number theory assumes it to be true (although no one can prove it). Variants of it have been proven in other settings. Many results, including entire theories of math, are conditional on its truth; these would all collapse if it was shown to be false. The discovery of even one nontrivial zero off the critical line would wildly throw off widely-accepted heuristics concerning the random-like behavior of primes.

Your plot idea isn't actually new. The ancient Pythagoreans embraced the doctrine that whole numbers were the basis for all things in the universe. For Greek geometry, this meant that all lengths and volumes should be ratios of whole numbers (i.e. $$\frac{p}{q}$$ where $$p$$ and $$q$$ are integers).

It turns out that $$\sqrt{2}$$ cannot be expressed in this way (i.e. the diagonal of a square of side length 1). This completely demolished the Pythagorean worldview from its very foundations.

The first person to discover this was a Pythagorean around 500 BC who caused a great scandal among his peers by the revelation. Surviving sources tell us that he perished at sea as punishment from the gods for his impiety.

Legend has it that his colleagues may have given the gods some help by throwing him overboard.

• Comments are not for extended discussion; this conversation has been moved to chat. – Monty Wild Nov 28 '19 at 12:33

This definitely wouldn't work in pure math, as there's nothing so "destructive" you could prove that wouldn't make you a celebrity among mathematicians. Finding a contradiction in ZFC would get you a Fields medal. Disproving the Riemann hypothesis would get you a Fields medal and a million dollars. There would never be any question about whether to publish, only how quickly to do so.

Now, if one wants to get a little silly, a world where P = NP and NP has efficient algorithms is one very unfriendly to mathematicians. In such a world, proof finding (that thing mathematicians pride themselves on) is no more difficult than proof verifying (that thing that mathematicians spent a lot of effort to get machines to do for them), so they'd mostly be either out of jobs or monitoring the automatic proof-finding machines. Perhaps this would be a reason mathematicians wouldn't want to publish such a result, as making your entire job redundant is widely considered a bad career move. Though, since that result also solves a huge number of practical problems, breaks all public-key cryptography wide open, and wins you a million dollars, I can't imagine it staying secret for long.

• Comments are not for extended discussion; this conversation has been moved to chat. – Monty Wild Nov 28 '19 at 12:34

A very well known conjecture (hopefully anybody with a CS degree would recall this, for example) is that there does not exist an efficient algorithm to take the discrete logarithm, in the most general case. This fact is sometimes used in cryptopgraphy. There are enough cases in which it is computationally tractable that it is plausible, at least, that your character could have with found an algorithm that covers many commonly used cases, or just solved it completely.

• "Somewhat well known"... This is related to the problem of factoring large numbers -- solving one would solve the other. Proving that factoring large numbers is easy would instantly break SSH, SSL / TLS (and thus HTTPS) etc. – AlexP Nov 12 '19 at 22:49
• I meant among people generally. If you walked up to a person in the street and asked them about any of this kind of stuff, they would look at you weird probably. – Zwuwdz Nov 12 '19 at 23:49
• Well, the chances of a random person in the street understanding what a mathematical conjecture is are not great in the first place. – AlexP Nov 12 '19 at 23:50
• Fair enough, I'll change it. – Zwuwdz Nov 12 '19 at 23:51

The following statement is unproven but widely assumed to be false: e+π is rational.

A rational number is a number you can write as a fraction. For instance "0.25" is rational because it can be written as "1/4". "7" can be written as "7/1", "14/2" or whatever. We are absolutely sure that π and e are not rational, but mathematicians only assume that this goes for the expressions "e+π", "π^π" and "e^e". Nobody has ever proven it.

The "damage" dealt is more subtle than say, proving "P=NP": If it was proven that the above sum is in fact rational, it would not unravel loads of previously watertight theorems. It would however unravel the mathematicians! It flies in the face of the intuition of everyone in the field. An intuition they rely on to guide them towards new results. It undercuts their belief that they "know math", because they "know" that the above sum is irrational in the same sense that they "know" the sun will rise tomorrow.

Idea courtesy of mathematician Alon Amit. See 1,2,3

• Sure, but at the same time, i^i is a real number, so stranger things have happened (value is approx 0.04). ;) – Yakk Nov 13 '19 at 19:04
• So that would be deeply surprising, but how would it shake the foundations of mathematics or "do a lot of damage"? – josh314 Nov 13 '19 at 19:25
• Wait, how important is the algebraic independence of e and pi? I'm not actually aware of any major theorems that require it. – eyeballfrog Nov 13 '19 at 20:22
• As far as I know, this would fall into "mildly annoying, but with no actual consequences". – Mark Nov 15 '19 at 3:38
• @yakk. i^i has many values, one of them is a real number. – Dast Nov 15 '19 at 13:28

I find it hard to believe that a proof or disproof of any mathematical statement could cause "many careers in tatters".

• If professional mathematicians around the world all made the same mistake, it hardly reflects poorly on any individual mathematician. Moreover, when a mathematician makes a significant mistake, it almost always is for some non-trivial reason, and the disproof is highly interesting. E.g. Euler's conjecture relating to orthogonal Latin squares (Wikipedia) is still studied hundreds of years later, despite it being wrong in every single case Euler did not prove himself (i.e., he couldn't have been more wrong).

• If someone disproves e.g. the Riemann Hypothesis, the likely result would undoubtedly be intense mathematical interest in the Riemann Hypothesis. Mathematicians studying the Riemann Hypothesis would likely have a massive boost to their careers as people update their theorems with the new knowledge.

If you want a plausible scenario where a disproof of a theorem could result on some egg on mathematician's faces, I suggest the Classification of finite simple groups. This theorem is proved, but the proof is so long that it's plausible it contains an error somewhere. Again though, it's not going to destroy anyone's career.

But I think there's a better idea...

### Automated theorem proving

If you want mathematicians to "choose between the truth or their prestige", I recommend looking into automated theorem proving, i.e., computers automatically generating proofs. In fact, the computer can be used to make conjectures, and subsequently prove them too.

Mathematicians instinctively hate "proofs by computer" and consider them inferior because they don't give human intuition. Thus it's within the realm of plausibility that mathematicians don't want to accept the truth. As proof of concept, see for example the drama surrounding the computer proof of the Four Color Theorem.

This result was finally obtained by Appel and Haken (1977), who constructed a computer-assisted proof that four colors were sufficient. However, because part of the proof consisted of an exhaustive analysis of many discrete cases by a computer, some mathematicians do not accept it. (Mathworld)

It is within reason that "what a mathematician does" changes radically because of some brilliant ideas in the area of automated theorem proving: many mathematicians are quickly rendered obsolete, replaced by automation (like factory workers previously).

• Obsolete mathematicians, what a thought! Somehow this conjures up the image of ragged figures holding begging bowls and signs proclaiming "Will prove theorems for food". This undoubtedly reveals a dark region of my soul, but I find it appealing. An excellent & sensible answer. Well done! – a4android Nov 14 '19 at 23:43
• If the Riemann Hypothesis is proven false, the destruction of careers will be less a matter of "The Riemann Hypothesis is false! You suck at math!" and more "The Riemann Hypothesis is false. That branch of mathematics you've been studying for 30 years? It doesn't actually exist." – Mark Nov 15 '19 at 3:46
• “[some] Mathematicians don’t want to accept truth” is plausible because they are humans, and we already have evidence in politics, medicine, and other fields that humans are quite capable of vicious attacks on others who question their beliefs. – WGroleau Nov 15 '19 at 16:29

Peano Arithmetic (PA) is a very elementary theory of arithmetic. By Gödel's theorems, if it is consistent then it can't be proved to be consistent using methods which can be formalized in the theory itself. Nevertheless, it can be and has been proved to be consistent in stronger theories, so much so that the most mathematicians would regard the consistency of PA as an known result. There are some dissident mathematicians known as ultrafinitists who regard the notion of actually infinite sets as nonsensical. One of the best known was the Princeton mathematician Edward Nelson. Somewhat famously, he once claimed to have found a proof that PA was inconsistent. An error was found in the proof and Nelson retracted the claim. He died just a few years later.

If your heroine fixes the error in Nelson's proof and shows that PA is inconsistent, the effect would be profound. It would call the coherence of much of pure mathematics into question. Most working mathematicians regard ultrafisitism as a hobbled approach to mathematics, so a result that suggests that ultrafinitism is actually true would be viewed as a hobbling result. (I hedge a bit because, narrowly speaking, a proof that PA is inconsistent is just that. By itself, it wouldn't prove that the ultrafinitist views on the foundations of mathematics are correct).

• Peano Arithmetic is a formalization of ordinary integer arithmetic. If it's inconsistent, the problems extend far beyond the realm of pure mathematics. – Mark Nov 15 '19 at 3:51
• @Mark There are even weaker theories of arithmetic that people like Edward Nelson worked on. He wrote an entire book called Predicative Arithmetic which developed one in detail. Nelson would of course agree that Peano Arithmetic is a formalization of ordinary integer arithmetic, but would disagree that it is the formalization of ordinary arithmetic. From an ultrafinitist point of view, PA is in a sense a lazy theory that doesn't carefully spell out what finite beings can do when they do arithmetic, but instead engages in a sort of infinite hand-waving (full-fledged induction schema). – John Coleman Nov 15 '19 at 11:40
• Even more interesting would be a proof that PA is ω-inconsistent. That means nonstandard numbers exists, period. Foundational consequences would be immense, though probably no practical changes. – Radovan Garabík Nov 15 '19 at 17:51

That large number factoring is computationally expensive.

If You find a way to factor large numbers in a quick way, You've broken most of the encryption used in the world. This means no more banks wiring money to other banks, no more secure websites, no software validation, digital signatures are out... Basically assymetric cryptography as we know it is dead.

• Naah. We'd just switch to ECDSA. The certificate authorities would be busy for a bit, but that's about it. – Martin Bonner supports Monica Nov 14 '19 at 17:55
• Well, a lot of hardware has RSA included in it. So not only the software, but the hardware would have to be replaced. This isn't the end of the world, Elliptic curves are interesting (although not proven mathematically to be secure and a bit young) but the damage (as required by mc_cubed) is quite substantial. I would say until everyone switches over and changes all their passwords (remember, that you can decode past messages as well) the whole internet is going to have a bad time. – Gensys LTD Nov 15 '19 at 8:23
• @MartinBonnersupportsMonica: Factorization and discrete logarithms are closely related problems... – AlexP Nov 15 '19 at 12:32
• @AlexP True, but even if discrete logs were solved in general, there's still the quantum resistant algorithms. – Martin Bonner supports Monica Nov 15 '19 at 12:38

The requirements for such a thing are:

1. Appears to very be strongly correct allowing it to have worked its way into many things, but...
2. the kinks have not been entirely ironed out, which leaves room for it to be disproven.

There are three (technically four) candidates I can think of:

1. The Riemann Hypothesis, because because many other proofs are contingent upon it being true.

2. Theory of Relativity and Quantum Mechanics.

Discovering the appropriate discrepancies would indicate that we aren't just missing a piece in our understanding of things, but that our understanding is fundamentally wrong.

3. Evolution might be included in this but it's tougher to disprove because it's inherently obfuscated due to deep time which leaves lots of gaps to fill in while also having much of the physically available evidence support it. You could only really disprove it at this point by having a perfect record of deep time.

• Sorry for not specifying anywhere beside the tag, I would like something in the math field. Anything related to relativity, quantum mechanics and especially evolution would be horde of journalists. – mc_cubed Nov 12 '19 at 22:46
• @mc_cubed Oh, you were just asking about math. Well, I doubt I would have noticed it anyways even if you did. Well, lots of math in quantum mechanics and relativity too so not too off the mark. – DKNguyen Nov 12 '19 at 22:48
• I know but it moves my story from the world of academia and math enthusiasts to being on front page of every pop science channel – mc_cubed Nov 12 '19 at 22:53
• @mc_cubed You might get more esoteric answers on math stack exchange. Just don't word it as a world-building question. Ask for a conjecture or hypothesis that is widely used assuming that is true despite never actually having been proven or disproven. Riemann is just the most famous and most important one. – DKNguyen Nov 12 '19 at 23:11
• Physical theories like QM or GR are known to be not end-all truths, but just useful models. It's generally accepted that they are not “correct”, but will eventually disproven by experiments at some length- and/or energy scale. And at least GR is mathematically really waterproof. QM is pretty solid too, though QFT (specifically, the Feynman path integral) do have some things that could mathematically be shaken up yet. – leftaroundabout Nov 13 '19 at 7:15

Seems like there’s a lot of room in data compression theory. If you were to disprove the minimum space needed to store information (I.e. find a way for two bits to encode all the values zero through five) then you’d change a lot of fields. Data storage and communication would be revolutionized, obviously. But since so much of quantum mechanics is about information entropy, I suspect such a math breakthrough would lead us to some novel material science discoveries. Depending upon how the physical expression of the mathematical compression was expressed, we might have a something-from-nothing energy generator.

• Can you explain more what you have in mind? For me a bit is something with states {0, 1}, so in two bits there is, by definition, only four states. No conjecture to be found. An author claiming someone found a way to express five values in two bits would have me roll my eyes. – kutschkem Nov 13 '19 at 7:19
• It's already easy to prove that two bits can't store the values 0 through 5. So I guess if you prove that they can, then you've proven that mathematical logic is BS, which would certainly cause a lot of damage... – user253751 Nov 13 '19 at 10:25
• If you can compress five values into two bits, you can compress arbitrary amount of data into the same two bits by repeating the procedure. Nine bits fit comfortably in four pentits, which your algo squeezes into eight bits. Then you just shove one bit at a time into the accumulator, then smush it together with a fixed-size encoding of the original bit size (at least nine bits - leading bit indicates whether to unsmush the size to twice the size or stop). The decompressor then just unsmushes the original size and then unsmushes that many bits from the accumulator. It even parallelizes easily. – John Dvorak Nov 13 '19 at 12:25
• @JohnDvorak I have a compression algorithm like that. The decompression doesn't quite work all that well yet, though. – Luaan Nov 13 '19 at 13:28
• I'm skeptical that there's much room, here. Disproving the pigeon-hole principal is never going to happen, so there are always going to be practical limits on compression efficiency. – Brian Nov 13 '19 at 14:09

How about going the other way? Proving that PI is 3.1415... instead of 3.2 or 4. You can cook up convincing enough proof that PI=4 for the average joe, and then everyone in academia remain silent because of the ridicule if they go against established truth.

Though that has some implications beyond academia, as other industries rely on it.

• Pi is between 3 and 4 depending on the metric, 3.14 is just the minimum when using lp metrics youtu.be/ineO1tIyPfM – mc_cubed Nov 13 '19 at 10:21
• @mc_cubed If the metric is not given by linear norm, then it's not limited to the range of 3-4, but that's all beside the point. – Spoki0 - Reinstate Monica Nov 13 '19 at 10:34
• I'm a fraid I don't understand how proving pi 4 helps me. People who care about pi, like anyone who makes circular things like aircraft engines would use 3.1415... If costs rize 21% due to increase in material management would be up in arms. I need something that is only important for mathematicians – mc_cubed Nov 13 '19 at 11:48
• I suggested going the other way, from 4 to 3.1415... You could also have the industry use 3.1415... but without know it is PI. Same as how carpenters can use Pythagoras without knowing it. They measure out a triangle with sides 3, 4 and 5, and then know the angle is 90 degrees. – Spoki0 - Reinstate Monica Nov 13 '19 at 11:54
• @mc_cubed I'm amused that your problem will be management complaining about a 21% increase in material, rather than aircraft engines being the wrong shape. If pi is 4 then what shape are all the aircraft engines built with 3.14159? Is there a mass hallucination? "Oh yeah, now that you mention it, they've always been a bit on the triangular side, but we didn't want to cause any fuss..." – user253751 Nov 14 '19 at 14:55