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I don't get conjectures. Does it mean they are widely believed to be true, just unproven? If so, why, where does this confidence come from? If not, what makes t

by smegel 13y ago

I don't get conjectures. Does it mean they are widely believed to be true, just unproven? If so, why, where does this confidence come from? If not, what makes them interesting in the first place?

kruhft 13y ago
The definition of conjecture says they are basically 'guesswork': http://www.thefreedictionary.com/conjecture http://www.thefreedictionary.com/conjecture. Someone will say 'I think that there are no values of x,y and z can solve x^n+y^n=z^n for n other than 2' but doesn't prove it. This can be due to lack of time, space, or a supreme insight on the part of the conjecturer[1]. The conjecturer has said some important and correct things in the past, so many people will take this experience and look at the conjecture and say 'this is probably right because this guy was right before and we see no errors with this conjecture so we will use it in as a theorem', which is to say they will use it without proof because it makes sense. Then, sometimes 100's of years later, somebody does prove (or disproves) the theorem and get's some press, and for some very important conjectures, prizes. If the conjecture is disproven, then all the math based on it falls down. Conjectures interesting because they are puzzles left by people that they can't answer but they want them to be true, so it is up to future generations with their advanced minds, tools and insights to prove them, leading to progress in the field with the advancements of techniques use to prove these conjectures and make them actual theorems. [1] http://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem http://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem
- ky3 13y ago
  Not all guesswork is dignified by the term conjecture though. Part of math school is to learn making and breaking guesses about things mathematical. As noted elsewhere ITT, a conjecture becomes one when there's a community-acknowledged aura of moral rightness about it. So at the very least, a bunch of people have tried to disprove it and failed.
revelation 13y ago
I guess the confidence comes from having tried it with a bunch of numbers. I'm sure there are people that have written programs that just tried to show the identity holds for all numbers up to n with n very large. Of course the problem is you can't show this by exhaustion. And you also can't currently prove it, hence why its a conjecture. But it's interesting because it means you might be missing some integral property that would prove the conjecture.
mrbrowning 13y ago
The confidence comes from two things, really: 1. The conjecture has been found to hold for a large number of trial cases, and 2. The conjecture has an intuitive aura of rightness. A good example is the 3n + 1 conjecture: http://en.wikipedia.org/wiki/Collatz_conjecture http://en.wikipedia.org/wiki/Collatz_conjecture Anyway, obviously point two is extremely subjective, but I think it's the crux of why long-standing conjectures are interesting. Why should something feel correct without necessarily being so? If it does turn out to be correct, what about its nature put people on its scent trail? Mathematics so often seems to be a kind of hermetic universe unto itself, so part of the beauty of conjectures is that they're kind of a contact point between that universe and human congnition.
- Someone 13y ago
  Where's the intuitive arena of rightness for the Collatz conjecture? I would put an 'or' between #1 and #2 (with #1 being less preferred; why would anybody start checking whether P holds for a large number of trial cases without having a hint of #2? (How did Collatz came up with this conjecture?)), and add a #3: the conjecture isn't easily disproven ('easily' being subjective, but the other two are subjective, too)
  - danbruc 13y ago
    While working on some problem or just playing around (with numbers) you could discover something by accident. Noticing the Collatz conjecture for only two numbers is probably enough to cause some curiosity and then you try some more examples and not much later you have a conjecture without any idea why it could be true.
lotharbot 13y ago
A conjecture is something that a mathematician thinks is probably true, but has not been able to prove nor disprove. There are some widely-known conjectures, but most are just something a mathematician ran into while thinking about some other problem and then ran into a dead end while trying to prove/disprove it. Often, a conjecture will be invented and then proved or disproved a short time later (I remember my wife disproving one her thesis adviser had proposed in class earlier that day), but occasionally one will remain unproven either way for a long time. With many of the "big" conjectures, this gives people confidence they're true -- they've been tested for millions, billions, or even trillions of cases over the course of decades or centuries, but nobody has found a counterexample. But they also haven't come up with an airtight proof; maybe they'd fail on the ten trillionth try, or maybe they'd hold true forever. That uncertainty is part of what makes them interesting.

by smegel 13y ago

I don't get conjectures. Does it mean they are widely believed to be true, just unproven? If so, why, where does this confidence come from? If not, what makes them interesting in the first place?

kruhft 13y ago

The definition of conjecture says they are basically 'guesswork': http://www.thefreedictionary.com/conjecture http://www.thefreedictionary.com/conjecture. Someone will say 'I think that there are no values of x,y and z can solve x^n+y^n=z^n for n other than 2' but doesn't prove it. This can be due to lack of time, space, or a supreme insight on the part of the conjecturer[1]. The conjecturer has said some important and correct things in the past, so many people will take this experience and look at the conjecture and say 'this is probably right because this guy was right before and we see no errors with this conjecture so we will use it in as a theorem', which is to say they will use it without proof because it makes sense. Then, sometimes 100's of years later, somebody does prove (or disproves) the theorem and get's some press, and for some very important conjectures, prizes. If the conjecture is disproven, then all the math based on it falls down. Conjectures interesting because they are puzzles left by people that they can't answer but they want them to be true, so it is up to future generations with their advanced minds, tools and insights to prove them, leading to progress in the field with the advancements of techniques use to prove these conjectures and make them actual theorems. [1] http://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem http://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem

ky3 13y ago
Not all guesswork is dignified by the term conjecture though. Part of math school is to learn making and breaking guesses about things mathematical. As noted elsewhere ITT, a conjecture becomes one when there's a community-acknowledged aura of moral rightness about it. So at the very least, a bunch of people have tried to disprove it and failed.

revelation 13y ago

I guess the confidence comes from having tried it with a bunch of numbers. I'm sure there are people that have written programs that just tried to show the identity holds for all numbers up to n with n very large. Of course the problem is you can't show this by exhaustion. And you also can't currently prove it, hence why its a conjecture. But it's interesting because it means you might be missing some integral property that would prove the conjecture.

mrbrowning 13y ago

The confidence comes from two things, really: 1. The conjecture has been found to hold for a large number of trial cases, and 2. The conjecture has an intuitive aura of rightness. A good example is the 3n + 1 conjecture: http://en.wikipedia.org/wiki/Collatz_conjecture http://en.wikipedia.org/wiki/Collatz_conjecture Anyway, obviously point two is extremely subjective, but I think it's the crux of why long-standing conjectures are interesting. Why should something feel correct without necessarily being so? If it does turn out to be correct, what about its nature put people on its scent trail? Mathematics so often seems to be a kind of hermetic universe unto itself, so part of the beauty of conjectures is that they're kind of a contact point between that universe and human congnition.

Someone 13y ago
Where's the intuitive arena of rightness for the Collatz conjecture? I would put an 'or' between #1 and #2 (with #1 being less preferred; why would anybody start checking whether P holds for a large number of trial cases without having a hint of #2? (How did Collatz came up with this conjecture?)), and add a #3: the conjecture isn't easily disproven ('easily' being subjective, but the other two are subjective, too)
- danbruc 13y ago
  While working on some problem or just playing around (with numbers) you could discover something by accident. Noticing the Collatz conjecture for only two numbers is probably enough to cause some curiosity and then you try some more examples and not much later you have a conjecture without any idea why it could be true.

lotharbot 13y ago

A conjecture is something that a mathematician thinks is probably true, but has not been able to prove nor disprove. There are some widely-known conjectures, but most are just something a mathematician ran into while thinking about some other problem and then ran into a dead end while trying to prove/disprove it. Often, a conjecture will be invented and then proved or disproved a short time later (I remember my wife disproving one her thesis adviser had proposed in class earlier that day), but occasionally one will remain unproven either way for a long time. With many of the "big" conjectures, this gives people confidence they're true -- they've been tested for millions, billions, or even trillions of cases over the course of decades or centuries, but nobody has found a counterexample. But they also haven't come up with an airtight proof; maybe they'd fail on the ten trillionth try, or maybe they'd hold true forever. That uncertainty is part of what makes them interesting.