I’d like to point readers to James Crow’s article in the open access Journal of Biology. Titled, “Mayr, mathematics and the study of evolution,” it’s a brief summary of some of the important results from mathematical genetics – sort of a follow-on to Haldane’s “A defense of beanbag genetics”.
Coming fifty years after Haldane’s effort, Crow has been able to include a lot more stuff – in particular the consequences of the mathematical development of neutral theory, and the effects of computers, permutation tests, and molecular clock models in phylogenetics.
I cannot help but quote this passage, which is direct:
Ironically, Mayr himself unwittingly provided an especially compelling argument for mathematical analysis. His theory of genetic revolutions assumed that from a well integrated population, genetic drift in a small founder offshoot will sometimes produce a population with a new set of genotypes integrated in a new way. Intuitively, a small founder population seemed a particularly unlikely place to find a new favorable gene combination, and this was indeed shown to be the case in a very detailed mathematical analysis by Barton and Charlesworth . If Mayr had had more respect for mathematical population genetics, he never would have made what most theorists regard as the mistake of proposing that small founder populations are a likely source of major evolutionary changes by genetic drift (Crow 2009:13.2).
Lest you think this is an argument against the role of chance, Crow later describes the more au courant view of speciation:
Until recently, mathematical theory had contributed little to the study of speciation. Mayr emphasized allopatric speciation and the prevailing model, due to Dobzhansky and Muller , prevailed. Recent mathematical studies  support it and favor the view that speciation genes correspond to normal genes, selected for their effects within the species. Furthermore, there is evidence that these genes evolve rapidly. Thus, hybrid incompatibility is a by-product of ordinary selection in geographically isolated populations (Crow 2009:13.4).
This model of speciation recognizes chance and contingency, but not mainly from stochastic fluctuations in allele frequency (drift). Instead we have the stochastic processes of mutation and environmental change and the (possibly complex) epistatic interactions among selected alleles.
There’s more in the essay. Crow does refer to human evolution – the out of Africa scenario and Neandertal genetics make appearances not entirely to my taste, but he notes that selective sweeps – dear to my heart – are an important feature of the recent landscape of mathematical genetics as well.
Crow could have included a number of other mathematical developments, particularly the Price equation, Hamilton’s contributions, and Maynard Smith’s “evolutionarily stable strategies”, all of which share his theme of the mathematical derivation coming first, and the non-mathematical descriptive formulations only coming later.
Crow JF. 2009. Mayr, mathematics and the study of evolution. J Biol 8:13. doi:10.1186/jbiol117