A commentator to this blog pointed me to a remarkable paper by Michael Gromov: Mendelian Dynamics and Sturtevant’s Paradigm. In: Geometric and probabilistic structures in dynamics. Contemporary mathematics – American Mathematical Society 469 (2008), 227-242. A few quotes:
the “theory of coin tossing” derives its mathematical beauty and the (probabilistic) predictive power not from such “deﬁnitions” as “the probability is a measure of uncertainty” but from the assumption that the probability distribution on the space of the imaginary outcomes (binary nsequences) equals the (normalized) Haar measure that is, moreover, invariant under the permutation group .
And now to biology:
In the 1913 paper “The linear arrangement of sex-linked factors in Drosophila, as shown by their mode of association” Alfred Sturtevant, long before the advent of the molecular biology and discovery of DNA, has deduced the linearityof the arrangement of genes on a chromosome from the statistics of simultaneous occurrences of particular morphological features in generations of suitably interbred Drosophila ﬂies. Thus he obtained the world’s ﬁrst genetic map, i.e. he determined relative positions of certain genes on a chromosome, where he used his ideas of linearity and of gene linkage.
And a striking conclusion:
On the mathematics side, Sturtevant’s reasoning may seem to be limited to the banal remark saying that if in a ﬁnite metric space the triangle inequality reduces to equality on every, properly ordered, triple of points then the metric is linear, i.e. inducible from the real line. But this is not exactly what is truly needed as the Sturtevant’s linearity is more about the order or, rather the ”between” relation, than about metrics.
More interestingly, the idea of Sturtevant suggests the following, novel even from the to-days perspective, way of thinking of geometric structures on a set L that are, according to this point of view, encoded by probability measures µ on the set of all subsets of L or by something similar to such measures.
I would not quote this ate length if it was not so concordant with my own feeling about mathematics.