There's no such thing as "mathematically impossible" in the physical world. If it's impossibility is evident in a mathematical model, that's still physical impossibility.
I think that this is not quite true: there is such a thing as mathematical impossibility in the real world, but it applies only to mathematical models; that is, models cannot exhibit the behaviour. This doesn't prevent the real world from exhibiting that behaviour, but it does mean that, if the real world exhibits that behaviour, then the model is inaccurate.
These caveats might make the notion seem useless (as you seem implicitly to be arguing), but I'd argue that they are quite useful. The whole pursuit of quantum gravity comes from the realisation that relativistic and quantum mechanics are mathematically incompatible. That fact alone doesn't tell us which one, if either, is correct, but it certainly tells us that they can't both be correct in all regimes, and that's indisputable information about the real world that we wouldn't have without idealising it through mathematical models.
By the way, did you edit your post? I thought the post to which I responded only had the sentence above, but on preview I saw:
> If it's impossibility is evident in a mathematical model, that's still physical impossibility.
This sentence I don't understand at all. I would argue instead that there is no such thing as physical impossibility; the real world will do what it likes, and we may not forbid it, only observe that certain things haven't happened yet. Any impossibility we observe in a mathematical model is, I would say almost by definition, a mathematical impossibility, even if we deduce physical consequences from it.
No it is. It is quite true that mathematical impossibility is mathematical and has nothing to do with the real world.
> ... but it applies only to mathematical models; that is, models cannot exhibit the behaviour. This doesn't prevent the real world from exhibiting that behaviour, but it does mean that, if the real world exhibits that behaviour, then the model is inaccurate.
You are talking about the validity of a mathematical model (with respect to a real world phenomena). That is a completely different notion.
Mathematically impossible means that, using the rigors of a mathematical proof, a contradiction is reached starting from the original mathematical assumptions. You started with math and ended with math. No real world involved.
> [quantum mechanics and gravity]
First of all, you're not making your life easier by defending your point using the argument of incompatibility of quantum mechanics and gravity.
And second of all, again you're mixing validity of a mathematical model as applied to observed phenomena, with the notion of mathematical impossibility. If the theory of quantum mechanics and the theory of gravity are incompatible and each on its own is unable to explain all of the observed physical world, that means both are likely inaccurate (rather incomplete) models of reality and we need a new model (maybe string theory) to explain the physical world.
Every math-based prediction about the real world will be based on two parts:
1) The world behaves like this model [up to our measurement accuracy].
2) The model implies this prediction.
Their claim would only be speaking to 2) -- that, under the model, you cannot reach absolute zero. The model can still be wrong.
I would argue (as I did in a sibling thread https://news.ycombinator.com/item?id=13878107) that it is this that can't happen. Mathematics has no power over the real world, only over mathematical models (optionally of the real world). A proof of mathematical impossibility does not constrain the real world; it only allows us to deduce that, if the real world does the mathematically impossible, then our models are wrong (or at least incomplete).
Mathematical proof is mathematical. It has nothing to do with the physical world.
P.S.: Note that 'mathematically shown to be physically impossible' still allows the possibility that the mathematical model was invalid. E.g., what if, after you have mathematically shown it to be physically impossible, you conduct an experiment and the "physically impossible" happens? That means you just provided an evidence that the mathematical model is invalid.
P.S.: There are tons of resources on this topic [1]. Read and make up your own mind.
[1] https://www.google.com/search?q=mathematical+proof+of+physic...