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At the risk/certainty of both sounding super ignorant and talking out of my arse, I have always wondered why there is some big mystery about why there are contradictions between the predictions arising from QFT and GR.

I know almost knowing about either but (after) being a math student in university I do believe that the mathematics of GR require local differentiability except at points (maybe), and that the empirical evidence from quantum mechanics observations shows that there is a "minimum" distance required to ignore quantum effects.

Then to believe GR is right, you have to conceive an almost-everywhere-locally-differentiable function who derivative is necessarily defined piece-wise. To believe QFT is right, you just have to believe that GR is incomplete but very close. (Right?)

If someone could shed some light on this and why people bother believing GR is "the" description of gravitation (and consequently that we live in this locally smooth universe) I would love to be introduced to some of what I don't know. For a layperson like me, having an opinion on this is pretty much useless, because I don't know enough to have a solid intuition about it. But I can see that having an opinion is super important for a theoretical physicist because a wrong choice could mean a lifetime largely squandered, trying to square the circle.

tacos_tacos_tacos
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    This isn't a full answer, but you should be careful to distinguish between properties of objects in spacetime and properties of spacetime itself. In particular, it is the fields of QFT that are quantized, not their domains, and indeed QFT lives on whatever spacetime we give it. It's not clear to me the smoothness you find surprising in GR isn't there in QFT as well. –  Oct 14 '15 at 22:28
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    possible duplicate of Does the Planck scale imply that spacetime is discrete? – Timaeus Oct 14 '15 at 22:52
  • Related questions about the need to quantize gravity: http://physics.stackexchange.com/q/6980/2451 and http://physics.stackexchange.com/q/10088/2451, and http://physics.stackexchange.com/q/52211/2451 – Qmechanic Oct 14 '15 at 23:10
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    I'm not sure any authorities in fundamental physics believes that GR is right or "the" description of gravity at all scales, although it is of course a superb description of gravity on large scales. – innisfree Oct 15 '15 at 01:17

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