Why no foundations of maths in Mellienium problems?

[pivoxa15]

Why isn't there a problem on the foundations of maths in the Mellinium problems?

If there was one which one would it be?

 

[Kummer]

Why isn't there a problem on the foundations of maths in the Mellinium problems?

If there was one which one would it be?

In order to be a Mellinium problem it must have important consequences. For example, my favorite, the Birch-Swinnerton-Dyer Conjecture would allow us to compute ranks of elliptic curves efficiently. However, the Prime-Twin Conjecture (which is a Mellinium problem, literally [>2000 years old]) does not lead to any important consequences. Thus, it is not not among them. For the same reason Goldbach's Conjecture is not included. It appears to me that problems from Mathematical Logic are not going to lead to big consequences ever since the works of Godel and Cohen.

 

[matt grime]

Perhaps the important ones have been answered? Like is CH independent of ZF etc.

I disagree with Kummer. Principally, because it is dangerous to make sweeping statements about mathematics - they almost always turn out to be wrong. However, there are certainly indications that delicate questions about set theory have important consequences in algebraic geometry, topology and representation theory. Who knows what the (increasingly important) work in o-minimal structures will tell us about sheafs, derived categories, and possibly (for those who care about such things) string theory or quantum gravity. I remember reading Jon Baez comment that it might be time for the logicians to look more closely at some of the underpinnings of mathematical physics.

 

[pivoxa15]

This might be off topic but the fact that foundations of maths isn't very popular in maths departments, does it mean it is harder for set theorists to find jobs as mathematicians?

Since Godel showed that mathematics cannot be reduced to axioms without encountering problems, why have people continued to do reserach into fuondations of maths? There can't be absolute perfection can there?

 

[matt grime]

It is no harder for a set theorist to find a job than any other mathematician - your premise would appear to be that there are as many set theorists as other kinds of mathematician.

Who says mathematics cannot be reduced to axioms without encountering problems? Certainly not Goedel (try finding out what the precise statements of his theorems are).

 

[Kummer]

There can't be absolute perfection can there?

I disagree, you got Godel's theorems wrong. One of the reason I chose mathematics many years ago is because I realized it is the only thing out there that is complete perfection.

I am not well-versed in mathematical logic but I can tell you that Godel showed. "A consistent system cannot be complete". Hence the name Incompleteness Theorem. Basically, this means if you have a consistent system (where eveything is perfect, that is the best way to put it) it cannot be complete, meaning not everything can be proven. For example, (classical example), the postulates I,II,III,IV are independent from V (of Euclid). Now the mathematical system composed of I,II,III,IV forms a consistent system. However it is not complete because V can be both true or false without leading to problems within the system (when V is false it is called non-Euclidean geometry).

This means in mathematics there is always something we cannot prove (because it is not within our system). But it does not mean math has flaws. To say something like that is completely insulting.