Ive read a few courses on statistical mechanics, and while their textual explanations and example choices differ, the flow of information from microscopy to macroscopy seems the same, and reading between the lines you can see some mathematical construct. Has statistical mechanics been formalized in the sense that say analysis has been formalized (down to quantifiers, sets, functions,...) more rigorously? Where can one find a formal axiomatic approach to statistical mechanics as opposed to an introductory descriptory approach?
Thermodynamics today is subsumed as the continuum limit of statistical mechanics. For statistical mechanics, the closest to an axiomatic deduction of the laws is Jaynes's approach, detailed in a series of papers starting in the 1950s. The basic law is that for every conserved quantity, you have a thermodynamic conjugate, and the statistical ensemble is the maximum entropy consistent with the thermodynamic conjugate values, if you don't fix the conserved quantity, or the maximum entropy distribution consistent with the value of the conserved quantity.
The philosophy behind this is that statistical mechanics is really a calculus regarding our knowledge of the microscopic state of a macroscopic body. It is in many ways a rigorous completion of the formalism of 19th century thermodynamics. It has been discussed here before--- you can find three classic reference (freely available--- thank you Physical Review) linked in Jaynes's Wikipedia article
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All valid statements in the equilibrium thermodynamics of standard systems can be deduced from the following deﬁnition.
7.1.2 Deﬁnition. (Phenomenological thermodynamics)
(i) Temperature T, pressure P, and volume V are positive, mole numbers $N_j$ are nonnegative. The extensive variables H, S, V, $N_j$ are additive under the composition of disjoint subsystems. We combine the $N_j$ into a column vector with these components.
(ii) There is a convex system function ∆ of the intensive variables T, P, µ which is monotone increasing in T and monotone decreasing in P. The intensive variables are related by the equation of state
∆(T, P, µ) = 0.$~~~~~~~~~~~~~~~~~$ (7.1)
The set of (T, P, µ) satisfying T > 0, P > 0 and the equation of state is called the state space.
(iii) The Hamilton energy H satisﬁes the Euler inequality
H ≥ TS − PV + µ · N $~~~~~~$ (7.2)
for all (T, P, µ) in the state space.
(iv) Equilibrium states have well-deﬁned intensive and extensive variables satisfying equality in (7.2). A system is in equilibrium if it is completely characterized by an equilibrium state.
This is the complete list of assumptions deﬁning phenomenological equilibrium thermodynamics for standard systems; the system function ∆ can be determined either by ﬁtting to experimental data, or by calculation from a more fundamental description, cf. Theorem 9.2.1. All other properties follow from the system function. Thus, all equilibrium properties of a material are characterized by the system function ∆.
This is from the beginning of Part II of Classical and Quantum Mechanics via Lie algebras.
Later comes statistical mechanics proper, in a similar, but more technical style.
The usual "axioms" of statistical mechanics are that all microstates are equiprobable and maximum entropy principle. The distribution of macrostates is the maximum entropy distribution consistent with the known statistics of the macro variables. This is enough to derive the Boltzmann distribution: It is the maxent distribution of energies given a fixed mean energy. There is a nice derivation of the Boltzmann distribution in Susskind's online Stat Mech lectures The most relevant are lectures 3 and 4.