Could a Computer Solve All Mathematical Conjectures and Revolutionize Science?

[1MileCrash]

I wish I could take credit for the question, but I was asked this.

Imagine a computer that could do the following:

- Check if any proposed provable conjecture in mathematics is true

- If it is true, write the most elementary proof possible of the conjecture and spit it out

- If it is not true not, give a counterexample

No limitations at all. E.G. press a button, and the Goldbach Conjecture immediately solved. Press a button, a proof of Poincaire comes out that makes Perelman's proof look overly complex and round about (if one exists, somehow, some way).

Would you like for this machine to exist?

 

[lendav_rott]

If you have a car that can drive on its own, what's the point of you? What happens if you have to take the wheel, but haven't driven a car before?

Such a program should be used only by the people who understand through and through what exactly they are asking. E.g, I have no use for a chef who memorises recipes, but doesn't know how to prepare the orders.

 

[Enigman]

Isn't this the same as P vs. NP problem (here P=NP)?

 

[1MileCrash]

Isn't this the same as P vs. NP problem (here P=NP)?

Not at all.

Do you value the information provided by math more or do you value the thrill of searching for truth in math more? Certain theorems in math go unsolved for centuries, countless of mathematicians work on them over that span of time. Then one day, maybe it is solved by a profoundly insightful individual, and it is celebrated. With this machine, that would never happen again, but we would immediately know all truths in mathematics.

This question has nothing to do with P vs NP, which is related to whether or not this machine could exist. I am asking if you would personally want this machine to exist. Pretend it is magic. Pretend it is a genie. P vs NP doesn't matter.

 

[1MileCrash]

If you have a car that can drive on its own, what's the point of you? What happens if you have to take the wheel, but haven't driven a car before?

Such a program should be used only by the people who understand through and through what exactly they are asking. E.g, I have no use for a chef who memorises recipes, but doesn't know how to prepare the orders.

Alright, so you are saying that the knowledge should be available at the press of the button, but it is worthless to someone who does not understand that knowledge, and so should be limited to them.

What if you spent 10 years trying to prove the Goldbach conjecture yourself though? Let's say you deeply understand it and have gotten closer than anyone else, but have still never come up with successful proof. Would you press the button on this machine yourself and view the result?

I don't feel like I could do it!

 

[micromass]

http://en.wikipedia.org/wiki/Decidability_(logic)
http://en.wikipedia.org/wiki/Decision_problem

Anyway, I would absolutely like such a machine to exist. I would still do mathematics and I would still enjoy it, even if all the questions are solved.

Like Einstein said " I want to know Gods thoughts; the rest are details". Likewise, I want to know mathematics, it doesn't matter if I don't find it myself.

 

[micromass]

A somewhat different question: let's say such a machine exists, but for some reason you could only press the button once. What theorem would you choose the machine to solve?

 

[1MileCrash]

Like Einstein said "I want to know God's mind, all the rest is irrelevant". Likewise, I want to know mathematics, it doesn't matter if I don't find it myself.

Yeah, in my case it is not so much finding anything myself that I am worried about. Humanity's collective search is also important to me.

I honestly don't care at all if the Goldbach conjecture is true or not. But, I do care if a person manages to prove it. I would not care if a machine automatically proved it.

I don't know, I just view it all as a big game, and this machine as cheat codes.

 

[Enigman]

A somewhat different question: let's say such a machine exists, but for some reason you could only press the button once. What theorem would you choose the machine to solve?

Prove that P=NP.
And then of course use it to make a dozen more machines.
()

 

[micromass]

Yeah, in my case it is not so much finding anything myself that I am worried about. Humanity's collective search is also important to me.

I honestly don't care at all if the Goldbach conjecture is true or not. But, I do care if a person manages to prove it. I would not care if a machine automatically proved it.

I don't know, I just view it all as a big game, and this machine as cheat codes.

Well, personally, I don't care about the Goldbach conjecture. To me it is just a random statement about numbers. I doubt humanity would profit much if such a conjecture were ever proven.

That said, there are mathematical things which would benefit humanity a lot. Or at least: there are things that I personally think would be really cool if solved. In such case, I would absolutely press the button. The possibly advance of humanity should not be halted by the pride of some mathematician who really wants to do it himself.

 

[micromass]

Prove that P=NP.
And then of course use it to make a dozen more machines.
()

Ah, but you assume that if you've proven P=NP then you can make efficient algorithms for many problems. Sadly enough, the proof could be nonconstructive. Or the involved constants may be way too large to allow for an efficient computation! http://en.wikipedia.org/wiki/P=NP#P_.3D_NP

And of course, even if the above is not true and you could actually make efficient algorithms, that doesn't mean that you could actually build a machine like in the OP. In fact, it's been proven that you can't build it

 

[1MileCrash]

.. don't care about the Goldbach conjecture. To me it is just a random statement about numbers. I doubt humanity would profit much if such a conjecture were ever proven.

That said, there are mathematical things which would benefit humanity a lot. Or at least: there are things that I personally think would be really cool if solved. In such case, I would absolutely press the button. The possibly advance of humanity should not be halted by the pride of some mathematician who really wants to do it himself.

I don't really know of any math conjectures that, if proven, would have any effect on the quality of life or technology for humans. Did proving any of the millenium prize problems do anything? Maybe P=NP I suppose, as already mentioned.

In your question, sure, I'm sure I could come up with something that I'd just have the computer solve for the benefit of humanity. But in the question posed in the OP, there is no way to filter who would press the button and what they would press it for.

 

[micromass]

I don't really know of any math conjectures that, if proven, would have any effect on the quality of life or technology for humans. Did proving any of the millenium prize problems do anything? Maybe P=NP I suppose, as already mentioned.

Well, you don't need to cure cancer in order to do something for humanity. Solving some mathematical questions are already interesting enough to count as helping humanity. It doesn't need to have practical applications.

That said, even if such a machine were to exist, I still don't think mathematics would be over. Surely, many people would be out of a job and mathematics would be made significantly easier. But still, there is work to be done. For example, the machine will not construct theories and models for you. For example, Dirac in his work on physics came up with the weird Dirac delta function which is nonsense mathematically. If you want to make this mathematically rigorous, then you can't just ask the machine "make the Dirac delta function rigorous" since the machine can only solve true/false questions. So it is still up to the humans to provide a good framework for the Dirac delta function. Same with all the math currently used in QFT and the like, a machine like in the OP will not help us with that.
Also, it is important to ask the right questions. Something like "classify all groups" would not be accepted by the machine. So we still need to come up with some way to classify all groups. The machine can then prove that this actually works.

It is often said that there are two kind of mathematicians: problem solvers and structure builders. The machine would definitely put the problem solvers out of a job. But the structure builders might still have work to do.

 

[AlephZero]

I don't really know of any math conjectures that, if proven, would have any effect on the quality of life or technology for humans. Did proving any of the millenium prize problems do anything? Maybe P=NP I suppose, as already mentioned.

Sorting out the foundations of computational fluid dynamics would be a good bet. Right now, nobody really knows for sure if any CFD calculation is right, except by comparing it with an experiment. For extra credit, also sort out turbulence modeling.