# What is the role of conjectures in modern mathematics?

Today, I heard of something so called Goldbach's conjecture from my mathematics teacher in the class. This was one of the most interesting things that I have ever heard in mathematics.

This made me curious to study a bit more about conjectures. The definition of conjecture on google says that:

A conjecture is an opinion or conclusion formed on the basis of incomplete information.

Now the question which is stuck in my mind is: What is the use of conjectures in modern mathematics?" Are they used in problem solving as we use theorems/lemma?

If it is the case then how can we use something to solve a problem which is not even known with certainty (Which we can't prove)??

• I think mathematicians conjecture all of the time. Most of the time they prove or disprove their conjectures. Even now, some interesting conjectures appear whose truth hasn't been decided yet. – steven gregory Nov 17 '16 at 12:01
• Not nearly as much would get accomplished if we only proved/disproved statements that we already know are true/false! – David Nov 17 '16 at 18:14
• If you're exploring some new wilderness of ideas, then of course you will constantly be making guesses about what is true and what's not true, trying to piece together a coherent understanding of what's going on. – littleO Nov 17 '16 at 18:25
• Presumably a conjecture stops being a conjecture if a counter-example is found. – joeytwiddle Nov 18 '16 at 4:53
• Everything that is a theorem today was once a conjecture. Things that may become theorems in the future must start as conjectures today. – cobaltduck Nov 18 '16 at 17:51

The word "conjecture" is rather fuzzy and doesn't in itself tell you much. It can be used about just about every statement where

1. Someone whose judgment you respect thinks it is likely to be true,
2. No proof of it is known, but
3. It feels like the kind of statement that ought to be subject to proof if it is true.

Thus, simply being told that "such-and-such is a conjecture" doesn't tell you much useful.

Conjectures play at least two different roles in mathematical research:

• They're goals we set ourselves to have something to strive for. Often these are fairly simple statements that give the mathematician the impression that they ought to have a proof or disproof, but where we simply don't have the tools to attack them. So we set out trying to invent such tools!

Goldbach's conjecture falls into this category, as does, for example, the twin prime conjecture or (until it was proved) Fermat's Last Theorem. These are things that really won't have any particularly important consequences, but it is hoped that searching for techniques that can whack them will also be actually useful for less famous but more practical purposes.

Sometimes these get resolved by proving that they cannot be proved from a reasonable set of assumptions (so condition 3 above is not satisfied). This famously happened to the continuum hypothesis, almost a century after it was first conjectured, when Paul Cohen showed that it doesn't follow from the usual axioms of set theory.

• They're stepping stones towards what we really want to know. This is a matter of division of labor: A community of researchers want to investigate this-or-that, and a respected and experienced person suggests that it ought to be possible to prove such-and-such and then prove that such-and-such implies this-or-that. If the suggestion is accepted, people can now work independently on proving such-and-such and on proving the step from such-and-such to this-or-that, and the Such-and-Such conjecture is now the point that connects these two efforts.

This can sometimes result in the Such-and-Such Conjecture being famous for its own sake, particularly if the step from such-and-such to this-or-that gets completed, but proving such-and-such itself turns out to be hard. (That is, without uncovering evidence that such-and-such is simply false).

Note that the terminology here is not very consistent. Even though it is now common to speak of this general kind of claims as "conjectures", particular named conjectures need not have "conjecture" in their name. Some are named Hypothesis instead (and this doesn't encode any particular different meaning, but is just a historical accident), and Fermat's Last Theorem spuriously had "theorem" in its name for several centuries before it was actually proved.

From wikipedia:

Sometimes a conjecture is called a hypothesis when it is used frequently and repeatedly as an assumption in proofs of other results. For example, the Riemann hypothesis is a conjecture from number theory that (amongst other things) makes predictions about the distribution of prime numbers. Few number theorists doubt that the Riemann hypothesis is true. In anticipation of its eventual proof, some have proceeded to develop further proofs which are contingent on the truth of this conjecture. These are called conditional proofs: the conjectures assumed appear in the hypotheses of the theorem, for the time being.

These "proofs", however, would fall apart if it turned out that the hypothesis was false, so there is considerable interest in verifying the truth or falsity of conjectures of this type. Instead, a conjecture is considered proven only when it has been shown that it is logically impossible for it to be false.

But on the other hand, not every conjecture ends up being proven true or false. The continuum hypothesis, which tries to ascertain the relative cardinality of certain infinite sets, was eventually shown to be undecidable (or independent) from the generally accepted set of axioms of set theory. It is therefore possible to adopt this statement, or its negation, as a new axiom in a consistent manner (much as we can take Euclid's parallel postulate as either true or false).

• Wait a minute, you're right. Undecidable is used in this context (and Wikipedia lists my objection as a disambiguation remark). – The Vee Nov 17 '16 at 11:07
• And note that the RH has been used to prove certain theorems by first assuming it's true, and then assuming it's false, and winding up with the same results with both assumptions. – Turambar Nov 17 '16 at 16:35
• Conjectures become important when they are shown to be equivalent to conjectures in various branches of mathematics. – Keith McClary Nov 18 '16 at 18:23
• In 2005, assuming the truth of Goldbach's Conjecture, I proved that Bertrand's Postulate must be true. (See proofwiki.org/wiki/Goldbach_implies_Bertrand .) – PolyaPal Nov 22 '16 at 18:57
• Another user has drawn our attention to the fact that you have plagiarized Wikipedia verbatim. Care to explain yourself? – Jyrki Lahtonen Jan 11 '17 at 13:43

A conjecture is an unproved theorem. Since it's unproved, you can't use a conjecture to prove a theorem or solve a problem. But some conjectures are famous enough to get their own names so that they can be referred to easily.

More importantly, conjectures often represent an area of research that a community of mathematicians will work on. Many differential geometers worked on the Poincaré conjecture until it was proven by Grigori Perelman. Andrew Wiles and Richard Taylor's proof of Fermat's Last Theorem was actually a proof of the Taniyama-Shimura-Weil conjecture. The Langlands program is a set of conjectures that has directed number theory for decades. So conjectures serve as goals for mathematicians to work towards.

It can happen that conjectures are proven false as well, which is significant too.

• Baffled by the downvote here, the post is informative and interlinks with my post but includes links to the TSW conjecture etc. For what it's worth Matthew, +1. – Kevin Nov 17 '16 at 10:00
• @Bacon: Thanks for the extra comment. The most baffling downvote I got was from someone on a personal vendetta against trigonometric substitution. But I don't worry about it. – Matthew Leingang Nov 17 '16 at 14:27
• "a personal vendetta against trigonometric substitution" - had to laugh at that! – Kevin Nov 17 '16 at 16:00
• – Matthew Leingang Nov 17 '16 at 16:10
• "Since it's unproved, you can't use a conjecture to prove a theorem or solve a problem" -- It seems to me that this is too strong, because one of the ways conjectures take on significance is when people prove things by assuming them (for instance, if the Riemann Hypothesis is true, then lots of things about prime numbers). And yes, the consequences of a conjecture have not been fully proven, but it still represents progress in the field. – zwol Nov 17 '16 at 21:49

Once upon a time there was a conjecture known as The inner function conjecture for several complex variables.

Now, an inner function $f$ for the unit disc $\mathbb{D}=\{z\in\mathbb{C}: \,|z|<1\}$ is an analytic function such that $|f(z)|<1$ for all $z\in \mathbb{D}$, that is $f:\mathbb{D}\to\mathbb{D}$ and such that the radial limit towards the boundary is $1$ almost everywhere, i.e. $$\lim_{r\nearrow1}|f(re^{i\theta})|=1,\qquad\textrm{for almost all \theta}.$$ These functions are very useful complex analysis and there are fine factorization results on e.g. the Hardy space. For example, the zeros can be factored into a Blaschke product (which is an inner function).

The inner function conjecture asserts there are no non-constant inner functions on $\mathbb{B}^n$, the unit ball of $\mathbb{C}^n$ ($n>1$). That is, if $f:\mathbb{B}^n\to\mathbb{D}$ is analytic then it is constant. (I think it was conjectured by W. Rudin in 1966).

Unable to prove the theorem, people found that if there were inner functions then they would behave very strangely.

In 1981 Alexandrov found a way to construct inner functions, thus disproving the conjecture.

Lesson learned: Complex analysis of several variables can be hard.

• +1 just for the last line. Complex analysis of several variables is far more difficult than the nice single-variable case would lead one to expect. – anomaly Nov 17 '16 at 17:40
• Why doesn't $f(z_1,z_2,\ldots,z_n)=z_1$ count? – Henning Makholm Nov 17 '16 at 22:19
• @HenningMakholm You are right, I forgot to mention the most crucial part of the definition, thanks! – AD. Nov 18 '16 at 9:00

Firstly, it is very difficult to overestimate the importance of proofs in mathematics. If you have a conjecture then the only way that you can safely be sure that it is true, is by presenting a valid mathematical proof.

The efforts required to prove a conjecture, will require a deeper understanding of the theory in question. A mathematician that will try to prove something may gain a great deal of understanding and knowledge, even if his/her efforts to prove that conjecture will end with failure, it does with on many an occasion. I firmly believe that the role of the conjecture in (modern) mathematics is centrepoint to a lot of new mathematics, even if those do not become theorems.

For a delve into this idea, for many years the Taniyama-Shimura theorem was a conjecture, it took the work of Andrew Wiles et al on elliptic modular functions in the late 80s early 90s to upgrade this to a theorem, Wiles' central idea was to assume (or conjecture) something about elliptic modular functions that would hitherto change the Taniyama-Shimura conjecture to a theorem.

In short, it is vitally important to conjecture in mathematics, as they are the (if you allow the cliche) building blocks of theorems and there are, therefore, many more conjectures within mathematics that than theorems.

• Given that errors are discovered in proofs, I'm not sure how truly safe that is. – djechlin Nov 17 '16 at 21:56

re your edit #9 (deleted again in #11): I think that the answers do answer the formulation in your title: “what is the role of conjectures …?”, and I shall not add to those.

You, however, would also like an answer to your question: “how can we use something to solve a problem which is not even known with certainty (Which we can't prove)??”. This is a misunderstanding: while we may show that we could use a conjecture, were it true, to prove a theorem, we should never claim that this provides a complete proof of the theorem. Such results are, however, still useful:

• If the conjecture is later proven, we suddenly do have proofs of these conclusions.
• If one of these conclusions is later disproved, we can conclude that the conjecture is also false.
• If there are good reasons to be confident of the conjecture, we can also be confident of the conclusions – that could encourage us to continue a certain line of attack on a problem.
• Knowing the implications can improve our understanding of one or more areas of maths.
• Conceivably, someone else may show that the falsehood of the conjecture also implies the same conclusion.

By publishing such results, a mathematician can therefore help others in various ways.

A conjecture is a statement that has not been proven, but has been tested extensively with no failures found. If you use it in a proof as if it is true, then the result of your proof is another conjecture. For example, Goldbach's strong conjecture about every even number greater than 2 being the sum of two primes. It has been tested beyond belief and no counterexample had been found. So the best we can say is that it MIGHT be true.

I had a math prof a long time ago whose hobby was collecting "first failures". These were statements (mostly in number theory) that were true for numbers up to some large value, but then suddenly and unexpectedly failed. His collection was large.

• I am very interested in simple statements that turn out to be true up to a very large number and then, out of the blue, fail (see for example my question [1]). Do you think there is away to gain access to that collection you mention? [1]: math.stackexchange.com/questions/1516260/… – Anguepa Dec 12 '16 at 12:30
• I wish I had a time machine that could go back fifty years and contact Dimitri Thoro at San Jose State. Alas, that might violate axioms of causality. Advice: keep everything, throw nothing away. That is easy these days, with flash drives. But in fifty years, there will be no machine capable of reading a flash drive, as there is now no machine that can read a floppy disk or a zip disk. – richard1941 Dec 15 '16 at 17:46
• I'll keep that advice (not in a flash drive) – Anguepa Dec 17 '16 at 22:00