What is the role of conjectures in modern mathematics?

Question

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)??

Answer 1

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).

Answer 2

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.

Answer 3

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.

Answer 4

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.

Answer 5

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.

Answer 6

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.

Answer 7

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.