Passionflower said:
Producing infinities is nothing unusual in quantum theory, and mathematical methods have been devised to get rid of them.
And those methods totally fall apart when you do gravity.
The problem is that the way that you deal with infinities in QM is to basically expand things out into a power series. At each stage, as you add more terms, you rescale (i.e. renormalize) so that when you do the infinite series you end up with finite values.
(Actually, you sort of cheat, and figure out that all you need are the first few terms, and you sweep the rest under the rug.)
This doesn't work with gravity. The problem is that gravity produces gravity. If you work with EM and imagine two electrons exchanging a photon, there is a tiny correction as that photon generates more photons, but it's small enough so that you can sweep under the rug.
With gravity, this doesn't work, because gravity generates gravity which generates more gravity which generates more gravity, and soon you have infinities popping up all over the place.
This is a terribly oversimplified version of what happens, and corrections are appreciated if I got something wrong.
So I think that does not exclude the possibility that we have not devised a mathematical method to get rid of those infinities if we try to combine GR and QM.
Most people think that no such method exists, and the reason we have all sorts of infinities while the world works is that some point in the real world you can chop off the power series because the rules change.