Existence of negative temperatures and the definition of entropy How negative temperatures can be possible has been treated on StackExchange before (several times in fact), but in light of some recent academic discussion, most of these answers seem to be possibly wrong or incomplete. The literature I am referring to is Dunkel & Hilbert, Nature Physics 10, 67 (2014) arXiv:1304.2066, where as I understand it, it is shown that negative temperatures are an artefact of choosing an incorrect definition of entropy. The Wikipedia article on the matter has also been amended to reflect this. 
This was later challenged by similarly well-known scientists, in arXiv:1403.4299 where it was, among other things, pointed out that this argument is actually decades old (citing Berdichevsky et al., Phys. Rev. A
43, 2050 (1991)). The original authors quickly countered the arguments made in the comment, by what seems to be a rigorous treatment of the matter, arXiv:1403.6058. The first arXiv comment (arXiv:1403.4299) has been updated since and it still reads that "Obviously severe points of disagreement remain".
What I am asking, then, is whether someone on StackExchange might be able to shed some light on the matter as to how there can be a disagreement about something that seems should be a mathematical fact. I would also be interested in hearing whether changing the definition of entropy from that of Boltzmann to that due to Gibbs might potentially change any other results. Might for example the Wang-Landau algorithm be affected seeing that it does use the density of states and that you can never simulate infinite systems (although as I understand it, even in the present context with finite scaling you should be able to get consistent results)? 
EDIT: An update on the matter for those who might care. arXiv:1407.4127 challenged the original paper and argued that negative temperatures ought to exist. They based their claims on their earlier experiments in Science 339, 52 (2013). A reply was offered in arXiv:1408.5392. More physicists keep joining in, arguing for arXiv:1410.4619 and against arXiv:1411.2425 negative temperatures.
 A: 
What I am asking, then, is whether someone on StackExchange might be able to shed some light on the matter as to how there can be a disagreement about something that seems should be a mathematical fact.

The main disagreement seems to be about which definition of the word "entropy" in the context of statistical physics is "correct". Definition is an agreement on choice that seems preferable but is not necessitated by facts. Different people regard different things more useful, so there should be no surprise that they are lead to use different definitions in their work. There should be no objection as long as this leads to some new knowledge that is in a sense independent of the choice made.
The surprising thing is the authors of the paper claim that their definition is the definition of entropy and proclaim its superiority.
I did not find any convincing argument in their paper to convince me that there is any problem with the standard formula $S = k_B\log \omega(U)$ for entropy and that their formula $S' = k_B\log \Omega(U)$ should replace it.
The two formulae lead to almost the same value of entropy for macroscopic systems, for which the concept of entropy was originally devised. This is because their difference is negligible due to high magnitude of the relevant number of states. Consequently, the standard rules that use entropy lead to the same conclusions for such systems whether one uses $S$ or $S'$.
For "strange" systems with constant or decreasing density of states $\omega(U)$ like particle in a 1D box or 1D harmonic oscillator, their definition leads to very different value of entropy for given energy $U$ and also to a different value of temperature, since $\partial U/\partial S'|_{V=\text{const}} \neq \partial U/\partial S|_{V=\text{const}}$. The authors say that positiveness of so calculated temperature is a virtue of their entropy $S'$. 
But such strange systems cannot be in thermodynamic equilibrium with ordinary systems when they have the same $\partial U/\partial S'|_{V=\text{const}}$. Why? When ordinary system is connected to such strange system, the most probable result is that the strange system will give as much energy to the normal system until its energy decreases to a value at which its density of states equals density of states of the normal system (or there is no transferable energy left). According to the principle of maximum probability, the average energy $U_1$ of the first system in equilibrium is such that the number of accessible states for the combined system is maximum. Let us denote total energy of the first system $U_1$, of the second system $U_2$ and of the combined isolated system $U$   (constant). If density of states is differentiable, we are lead to the condition
$$
\frac{d}{dU_1}\left(\omega_1(U_1)\omega_2(U-U_1) \Delta U^2\right) = 0
$$
$$
\omega_{1}'(U_1)\omega_2(U_2) = \omega_{2}'(U_2)\omega_1(U_1)
$$
$$
\frac{\omega_{1}'(U_1)}{\omega_1(U_1)} = \frac{\omega_{2}'(U_2)}{\omega_2(U_2)}
$$
and this implies the condition
$$
\frac{\partial U_1}{\partial S_1}  = \frac{\partial U_2}{\partial S_2}~~~(1)
$$
where $S_1 = k_B\log \omega_1(U_1)$ and $S_2=k_B\log \omega_2(U_2)$. The principle of maximum probability does not lead to the condition
$$
\frac{\partial U_1}{\partial S'_1}  = \frac{\partial U_2}{\partial S'_2}.~~~(2)
$$
where $S_1' = k_B\log \Omega_1(U_1)$  and $S_2' = k_B\log \Omega_2(U_2)$.
If (1) holds, in most cases (2) won't. Since in equilibrium thermodynamic temperatures are the same, the statistical definition of temperature is better given by $\frac{\partial U}{\partial S}$ rather than by $\frac{\partial U}{\partial S'}$.
When the strange system is isolated and has energy such that density of states decreases with energy, the temperature thus obtained is negative. This is well, since ascribing it any positive value of temperature would be wrong: the system won't be in equilibrium with ordinary systems (those with density of states increasing with energy) of positive temperature.
A: We have a perfectly unambiguous definition of temperature for canonical ensembles, and this temperature may be negative in bounded-energy systems. This kind of negative temperature is indisputable, and some would argue it has been realized in spin-inversion experiments.
The problem is that there are two decent but imperfect definitions for the entropy of a microcanonical ensemble. This ensemble is believed by some* to be useful for describing some physical situations.
One definition (surface entropy/Boltzmann entropy) is more popular, and can be commended for its intuitiveness. The other definition (volume entropy/Gibbs entropy/Hertz entropy) is less popular, and less intuitive, however mathematically speaking its properties are more convenient and it is more thermodynamically accurate in some ways. Each yields a distinct definition of temperature when used in the formula $T^{-1} = dS/dE$. Which one is correct? The surface entropy gives negative temperatures in many weird systems (and not just in bounded-energy systems), whereas the volume entropy never gives negative temperatures.
This surface/volume argument is more than 100 years old, and it will never be settled. This is because neither definition is perfect, so it is more a matter of taste which you use. Gibbs in his 1902 Elementary Principles in Statistical Mechanics discussed the merits and problems of both approaches, at depth. I am not sure if any literature since then has added anything of value besides repetition.
Essentially, the problem is this: Ensembles that are not canonically distributed are not so simple that we can pretend they are thermodynamic. So, we should not really be trying to name a temperature. Well, we can go ahead and do it, and indeed we can name "temperatures" that work in some ways. However, these "temperatures" will never quite live up to all the properties we expect from thermodynamics. As a concrete example, we intuitively expect that if we thermally connect two equal-temperature systems then nothing should change. But neither surface nor volume "temperature" have this property!
*: In my opinion, it is questionable whether the microcanonical ensemble describes any physical situation. It is often said that isolated systems are described by microcanonical ensembles, however being isolated is not sufficient. Microcanonical ensembles must also have total energy that is exactly known, and this is never the case in any experiment. Rather the strange isolated systems often discussed (such as spin-inversion experiments) are not canonical, nor microcanonical, but some complicated beast inbetween.
A: I have contributed this issue with this arXiv:1411.2425 and previous works. I stress that Gibbs entropy is not "defined" but rather "constructed".
It is constructed on the expression of thermodynamic forces in the microcanonical ensemble, and is constructed in such a way that it reproduces them always and exactly. The construction is actually unique.  Accordingly, if any other expression is employed, e.g. Boltzmann's, you mis-calculate those thermodynamic forces, e.g., the magnetisation. The error you might incur with the Boltzmann entropy can sometimes be very large, see arXiv:1411.2425. So, in general, Boltzmann expression does not coincide with thermodynamic entropy. Accordingly, its derivative, does not generally coincide with inverse temperature.
That's all.
A: Simply the thermodynamical quantities used in the original paper were not suitable for that problem. 
They in particular calculated $T=\frac{\partial U}{\partial S}$ where $ U$ is the internal energy and $S$ the entropy. 
However a wrong definition of entropy has been used. Mathematicians has proved that the use of that specific entropy was wrong and that using the correct entropy would lead to finite positive temperature.
It is my opinion that these days people really like to use misleading terminology just because it looks cool, and because it helps to sell their paper.
Moreover negative temperature is intuitively wrong as it should be in a way related to the thermal excitations of a system. 
So it is my opinion that those authors failed not only with mathematics but as physicists as well as they used a wrong mathematical definition and they published results which have no physical sense.
I must say I feel the same with respect to that article about the Dirac monopole.
PS
In support of negative temperature have a look at these euristic arguments by Immanuel Bloch. He admits it is just a convenient choice but in any case it is Still nonsensical to me.
