Conjectures vs. Hypotheses
Date: 01/12/99 at 03:29:08 From: Ernest Lau Subject: Number theory What is the difference between the terms 'conjecture' and 'hypothesis'? For example, could the Riemann hypothesis be called the Riemann conjecture? What happens to a conjecture after it has been proven? Will it still be called a conjecture? For example, if the Taniyama-Shimura conjecture is proven in full, will it still be called that or something else?
Date: 01/12/99 at 09:59:05 From: Doctor Ezra Subject: Re: Number theory Dear Ernest, Thanks for writing to Dr. Math. Let's take your questions in order: A conjecture is a guess. Someone says, "I believe the following statement is true." Examples include the Twin Prime Conjecture (there are infinitely many primes p such that p+2 is also a prime), the Goldbach Conjecture (every even number can be written as a sum of two primes), and the Odd Perfect Numbers Conjecture (there are no odd perfect numbers). A conjecture is a statement for which someone thinks that there is evidence that the statement is true. The main thing about a conjecture is that there is no proof. The word "hypothesis" has two meanings. The first is the "if" part of a theorem and the conclusion is the "then" part. (Example: In calculus, there is a theorem that states that if f is differentiable at a point c, then f is continuous at c. The hypothesis of this theorem is "f is differentiable at c" and the conclusion is "f is continuous at c.") In addition, mathematicians often use "hypothesis" as a substitute for "conjecture" - hence, the Riemann Hypothesis, which is really a conjecture. So, you're right: it ought to be the Riemann Conjecture! Once a conjecture has been proven, it becomes a theorem. For example, the Four-Color Conjecture was a conjecture until 1976, when Appell and Haken constructed a proof. It is now the Four-Color Theorem. What all of this means is that Fermat's Last Theorem should have properly been called the Fermat Conjecture (but Fermat's Last Theorem had rather a nice ring to it...). It should now be known as the Fermat-Wiles Theorem; however, I suspect that people will refer to it as Wiles' proof of Fermat's Last Theorem. Finally, as you said, once there's a full-blown proof of the Taniyama-Shimura Conjecture, it will become a theorem. What people will call it is open to, well, conjecture! Hope this helps. - Doctor Ezra, The Math Forum http://mathforum.org/dr.math/
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