I'll have to disagree that those notions of entropy are disjoint. I'll try to explain my view.

In Statistical Mechanics entropy is defined in terms of accessible regions in phase space. It is the logarithm of this volume times a constant. In the process of deriving this formula starting from the number of accessible configurations it is postulated that all configurations must be equally accessible. This postulate is called the Ergodic Hypothesis. Since you're a mathematician I think you're probably familiarized with the term ergodic: it is a system whose evolution preserves measure (in our case, Liouville measure, which is Lebesgue's measure on phase space). Now, not every system is ergodic. Even though, estimates can be carried out and point that in a general gas, which has a huge number of particles, non-ergodicity would result in an extremely small error in Physics measurements (Laudau does that in his first volume on Thermodynamics). Even though, systems like spin glasses are canonical examples of non-ergodic systems where usual Statistical Mechanics is not applicable.

You see that the ergodic hypothesis is a key assumption in Statistical Mechanics. But what does chaos mean in Classical Mechanics? Well, it means your trajectories will cover your whole phase space. If you take a chaotic system (which is not only ergodic but also strongly mixing), the particle's trajectory will cover each and every bit of phase space accessible to it, bounded by energy conservation laws.

The conclusion is that if you assume Statistical Mechanics as being applicable, this is the same as assuming you cannot predict trajectories in phase space, either because you have too many initial conditions or because you can't track each and every trajectory. This is intrinsically connected to the notion of chaos in Classical Mechanics.

In Thermodynamics I think no one really understood what entropy meant, so I can't elaborate on that. It only gets clear in Statistical Mechanics.

I'll have to disagree that those notions of entropy are disjoint. I'll try to explain my view.

In Statistical Mechanics entropy is defined in terms of accessible regions in phase space. It is the logarithm of this volume times a constant. In the process of deriving this formula starting from the number of accessible configurations it is postulated that all configurations must be equally accessible. This postulate is called the Ergodic Hypothesis. Since you're a mathematician I think you're probably familiarized with the term ergodic: it is a system whose evolution preserves measure (in our case, Liouville measure, which is Lebesgue's measure on phase space). Now, not every system is ergodic. Even though, estimates can be carried out and point that in a general gas, which has a huge number of particles, non-ergodicity would result in an extremely small error in Physics measurements (Laudau does that in his first volume on Thermodynamics). Even though, systems like spin glasses are canonical examples of non-ergodic systems where usual Statistical Mechanics is not applicable.

You see that the ergodic hypothesis is a key assumption in Statistical Mechanics. But what does chaos mean in Classical Mechanics? Well, it means your trajectories will cover your whole phase space. If you take a chaotic system (which is not only ergodic but also strongly mixing), the particle's trajectory will cover each and every bit of phase space accessible to it, bounded by energy conservation laws.

The conclusion is that if you assume Statistical Mechanics as being applicable, this is the same as assuming you cannot predict trajectories in phase space, either because you have too many initial conditions or because you can't track each and every trajectory, and after an infinite time they'll also cover the whole phase space. This is intrinsically connected to the notion of chaos in Classical Mechanics.

In Thermodynamics I think no one really understood what entropy meant, so I can't elaborate on that. It only gets clear in Statistical Mechanics.