in 1959 ernst mayr ( figure 1 ) flung down the gauntlet   at the feet of the three great population geneticists ra fisher , sewall wright and jbs haldane ( figure 2 ) : " but what , precisely , " he said , " has been the contribution of this mathematical school to the evolutionary theory , if i may be permitted to ask such a provocative question ? " his skepticism arose in part from the fact that the mathematical theory at the time had little to say about speciation , mayr 's major interest . a particular focus was the simplification that he called " beanbag genetics " , in which " evolutionary genetics was essentially presented as an input or output of genes , as the adding of certain beans to a beanbag and the withdrawing of others . " . photograph reproduced with permission from the archives of the ernst mayr library of the museum of comparative zoology , harvard university . ( a ) r a fisher ( 18901962 ) , ( b ) sewall wright ( 18891988 ) photograph reproduced with permission from the capitol times , and ( c ) j b s haldane ( 18921964 ) . mayr was , however , criticizing textbook simplifications , rather than the actual work of the three pioneers . far from treating gene frequency changes as analogous to the consequence of beans jostling at random in a bag , both fisher and wright considered gene interactions in detail . fisher ( figure 2a ) showed that , despite interactions between genes , natural selection acts on the additive component of the genetic variance . the beanbag criticism was particularly inappropriate for wright ( figure 2b ) , who specifically devised his ' shifting balance ' theory as a way for a population to go from one harmonious gene combination ( mayr would say " integrated genotype " ) to another when intermediates were disadvantageous . who was to answer mayr 's criticism ? fisher was already dead , and in any case preferred attack to defense , and wright was too gentle  though admittedly not always when mayr was involved : returning from italy where he had received the prestigious balzan prize in 1984 , wright told me that the value of the prize was considerably diminished when he discovered that mayr had won it the year before . in the event , however , it was haldane ( figure 2c ) who took up the challenge . this was haldane at his best  witty , spirited , informed , interesting and convincing . but the larger question remains : what indeed has been the contribution of mathematical theory to evolution ? mathematics is not central to evolution in the way it has been in theoretical physics . solid advances have been made without using mathematics , much being due to mayr himself . and these continue . yet , i shall argue that mathematical ideas have made important , and often essential , contributions , and still do . many concepts that are now established were arrived at mathematically , although their origins have since been forgotten . for example , the idea that polymorphisms become stabilized in populations because heterozygotes are at an advantage is now found in elementary textbooks , but fisher was the first to formulate it . loss of heterozygosity with inbreeding is also textbook knowledge , but it was not clear until wright developed the theory and invented a simple algorithm for quantifying it . similarly , the idea that the impact of mutation on the population depends on the mutation rate rather than the magnitude of the mutant effect is now taken for granted , but that was not known until haldane showed it mathematically . one final example is the inheritance of the abo blood groups , which was in doubt from the time of their discovery at the turn of the twentieth century until bernstein 's mathematical population analysis in 1924 . all of these applications used only elementary methods , and they must have been known to mayr . often , concepts that were developed mathematically were later explained in intuitive , non - mathematical ways . it 's a lot easier to find an intuitive explanation when you already know the answer . 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 . partly this is due to skilled mathematicians entering the field and bringing new techniques with them ; especially noteworthy are stochastic processes . an additional influence is the explosive growth of molecular data , which lend themselves to mathematical treatment . in the first half of the twentieth century , population genetics and evolution had a beautiful theory , but there were very limited opportunities to apply it . what are some of the newer developments in evolution that are owed to mathematical theory ? here are a few . one striking result in the post - mayr period was motoo kimura 's neutral theory , independently developed in 1968 by him and by jack king and thomas jukes . these writers shocked the biological world by arguing that the bulk of molecular evolution is due to selectively neutral mutations driven by the mutation process rather than selection . i think it would please mayr that the general idea  that the rate of evolution in the population is equal to the rate of mutation in a single individual  can be derived by simple reasoning using school mathematics . yet , in order to apply the idea , we need to know how long a time period must be observed . this depends on how long it takes for a lucky new mutant to increase in frequency and completely replace its predecessors . when selection and migration are taken into account , the theory is much more complicated . one contribution of the neutral theory has been to provide a rationale for a molecular clock . essentially , all our estimates of evolution rates depend on the assumption that the molecular changes used in constructing the clock are mutation - driven . fortunately , enough of the dna does not have an obvious function and can reasonably be supposed to be evolving by neutral kinetics , or near enough so that the neutral theory can be used in practice . and a second important attribute of the neutral theory is that it supplies a natural null hypothesis for the study of selection . and yet another outgrowth of the neutral theory is the view that much of the molecular polymorphism in natural populations is effectively neutral . this is especially useful now that variation in the frequencies of single - nucleotide polymorphisms ( snps ) is easily observed . the various measures that are used to quantify genetic variability are outgrowths of population genetics theory . one striking result of such theory is the realization that all of the worldwide human population is descended from africa , and moreover from a small area within africa . the evidence for this striking conclusion is that molecular variance is greater in african peoples than elsewhere . the molecular clock can be used as one measure of the time taken during various human migrations and , of course , homo sapiens is not the only species that can be studied in this way . a new favorable allele arises by mutation , spreads through the population and becomes fixed at a rate that is determined mainly by how favorable it is . a consequence of this fixation is that neutral or weakly selected alleles linked to the locus are swept along with it . because of this , there is a region on either side of the selected locus that is deficient in genetic variability . such regions of reduced variability are footprints of a selective sweep in the past and , remarkably , provide evidence for events that occurred long ago and which can no longer be observed . although the basic idea is simple and requires no mathematics , an assessment of how much the variability is reduced and the linkage distance over which the reduced variability occurs depend on mathematical theory . an area of biology in which mathematics , and especially computers , have become absolutely essential is systematics , ernst mayr 's own field . formerly , assessing species relationships and building phylogenetic trees based mainly on morphological differences was a matter of intuition and judgment . a mammalian dna sequence supplies billions of bits of information , thus for the first time providing an opportunity for a procedure independent of personal judgments . in recent decades the preferred procedures , such as fisher 's maximum likelihood , required a great deal of computation , and for a while this meant that large phylogenies were out of computer range . it is now de rigueur to do statistical tests of significance of the tree structure and parts thereof . many of these involve permutation methods , which have the merit of requiring minimum assumptions . they are computation - intensive , but with modern computers this is no longer an impediment . one striking example from such studies , which came as a complete surprise to classical systematists , is the close relationship of the elephant to the shrew . molecular analysis of dna sequences , using the newly developed theory , has shown that our closest relatives are chimpanzees . furthermore , that we and the chimpanzees are 99% identical at the dna level came as a surprise to many . equally surprisingly , we share some 90% of our dna with mice , rabbits , dogs , horses and elephants . these numbers are fully consistent with expectations based on mutation rates and the times involved . finally , there is now help available in the form of computer programs that can work out phylogenies and display the information graphically ( see ) . these not only eliminate a lot of tedious work , but place advanced methods in the hands of relative novices . finally , there has been a major theoretical advance , coalescent theory . instead of looking forward in time any two alleles or homologous nucleotides are ultimately derived from a single one ; that is , looking backward , they coalesce . one problem for which coalescent theory provided at least an approximate answer is the question of whether there was any mating between our ancestors and contemporary neanderthals . small amounts of admixture are not ruled out , but coalescent theory has shown that any substantial intermating is very unlikely , as discussed by john wakeley . my other examples have been relatively simple , but this one is n't , as is apparent from this discussion . another example , also given by wakeley , is evidence for a selective sweep in drosophila simulans . until recently , 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 . thus , hybrid incompatibility is a by - product of ordinary selection in geographically isolated populations . there is no evidence that random drift plays an important part , so mayr 's ' genetic revolution ' and similar ideas have little support . yet it is important to point out that , aside from this , mayr has usually been right . the field of mathematical studies of speciation is barely started ; it will surely increase . i have given only a few examples of the part that mathematical theory has played in evolution studies . there are many more , but these , i hope , constitute a convincing sample of the importance of mathematics in population genetics and evolution . i do not intend to imply that all evolutionary study need be mathematical and theory - driven . for example , ' evo - devo ' studies , looking at changes in development during evolution , have produced exciting results while largely ignoring population genetics . another non - mathematical example is horizontal gene transfer brought about by transposable elements , which is especially important in the evolution of microorganisms . there is also abundant evidence for increases in genetic complexity by the accumulation of small duplications . and , as always , a lot of morphological and behavioral evolution is interesting in and of itself . yet , my guess is that as these subjects become more quantitative , population genetic theory will play an increasing role . the rise of molecular methods has led to an increase in the importance of mathematics in population genetics and evolution . the abundance of data that require mathematical analysis has greatly increased . at the time of mayr 's challenge now the situation is reversed : data appear faster than existing theory can deal with them . that mathematics will play i think these examples show not only that mathematical theory is helpful , but that it is often essential . he might have had a change of mind , but i doubt it . knowing how much he enjoyed arguing , i suspect he would be quite critical of much that i have written . unfortunately , although he lived to be 100 , he was not immortal and died in 2005 . were he still alive , i would surely hear from him and whatever his opinions , he would not keep them to himself . he would have enjoyed an argument , preferably over a glass of sherry . and so would i. i am indebted to bret payseur for reading the manuscript and offering some very useful suggestions .
in 1959 ernst mayr challenged the relevance of mathematical models to evolutionary studies and was answered by jbs haldane in a witty and convincing essay . 
 fifty years on , i conclude that the importance of mathematics has in fact increased and will continue to do so .