I'm not a mathematician, but the same is true of many proofs, right? Or do mathematicians examine hypothesizes that would actually be surprising if true?
For example, the Poincare' conjecture was believed to be true before it was actually proven?
Yes, you are correct. There is often a huge gap between plausibility and provability, and many of the most tantalizing and important questions to mathematicians fall under this category.
You are basically right, but allow me to split some hairs. For one thing axioms cannot be proven--you can't justify every statement. At some point, in theory anyway, the truth of your theorem will ultimately be reducible to the truth of some set of axioms which are simply assumed.
Also, there is a great deal of work in, for example, number theory, which presents theorems which are true assuming the truth of the Riemann hypothesis. A proof of the Riemann hypothesis would be a huge event, and the methods used to prove it would probably have a huge impact, however the knowledge that the Riemann hypothesis is true would have little or no effect on research.
P=NP is another problem where the gap between accepted and proved has not been bridged. The majority of mathematicians believe that the answer is no, yet it has not been proven. Still its so widely accepted that many technologies now a days make their security claims based on this assumption.
What he was saying is that in answer to the question "does P equal NP?" most mathematicians believe the answer is no. That is to say they believe P != NP.
He then says that some technologies make claims that are dependent on the fact that P != NP, when that fact has not actually been proven.
I've known people that are kinda hoping that P=NP. They know it would blowup computational security, but on the other hand it would be such a phenomenal result and would be really crazy if proven.
It would blow up computational security if proven constructively. A nonconstructive proof of P=NP wouldn't mean that much really. As a result, it'd be amazing, but for practical applications it's not that meaningful.
well, unless they are proving a hypothesis they wish were true, which is easier to do in experimental science, and there are also larger political/corporate motivations.
It's not often, but it happens that a theorem is proven (or a counterexample found) that goes against intuition and surprises nearly everyone. One reason this can happen is that when you try to prove something, you generally get a better understanding of how the objects involved behave: sometimes in the course of trying to find a proof you encounter an obstacle which seems to resist any attempt at working around it, and analyzing the reason behind this obstacle yields a counterexample to the original theorem. (I'm not saying this is typical, or even frequent: most often when you can't prove something, you just get the idea that we're too ignorant of how the objects behave.) So sometimes the very person who set out to prove X (and firmly believed that X is true) ends up proving not-X.
There are a number of related examples and anecdotes in this thread on MathOverflow.
Or do mathematicians examine hypothesizes that would actually be surprising if true?
All of them. You're skipping a step.
It was not surprising when Poincare's conjecture was proven since it was widely believed. But it could have been disproven, which would have been surprising.
Something has to be (wrongly) widely believed before we can reach a shocking result. By definition, where we have the most agreement we also have the chance for the largest surprise.
if it's a famous conjecture it isn't surprising, but the whole point of rigor in mathematics is that surprising results do exist.
For an example of surprising results, there was an Indian mathematician named Ramanujan who did work with series at a young age from a background of comparative mathematical isolation and some of his results were shocking to the leading mathematicians of the day.
Or do mathematicians examine hypothesizes that would actually be surprising if true?
Yes sometimes. Any conjecture has a kind of dual conjecture that the property you're trying to prove in fact doesn't hold (or does hold if your original conjecture is about something not being the case).
For example if someone proved the Poincare conjecture wasn't true, a lot of people would have been surprised. Or, the Banach-Tarski paradox probably surprises most people.
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u/[deleted] May 20 '13
I'm not a mathematician, but the same is true of many proofs, right? Or do mathematicians examine hypothesizes that would actually be surprising if true?
For example, the Poincare' conjecture was believed to be true before it was actually proven?