Posted by: Alexandre Borovik | June 12, 2011

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 ${\mathbb{Z}_2}^n$ of the imaginary outcomes (binary nsequences) equals the (normalized) Haar measure that is, moreover, invariant under the permutation group $S_n$.

And now to biology:

In the 1913 paper “The linear arrangement of sex-linked factors in Drosophilaas shown by their mode of associationAlfred 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 $2^L$ 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.

## Responses

1. One of the great delights of studying pure mathematics is of learning of deep and surprising connections between disparate parts of mathematics. Perhaps Monster Moonshine is the most famous example, but in truth these connections are pervasive throughout mathematics. In mathematical statistics, for example, RA Fisher’s non-mathematical (even naive) treatment of statistical tests and estimation turned out to be intimately linked to ideas in the differential geometry of information.

With such metal-level connections between disparate fields, one is tempted to ask where is the meta-meta-level mathematical theory which would explain them? In other words, are we so focused on the parts and their connections that we are ignoring a deeper and more profound theory which would explain why all these connections exist?

2. @Peter:you wrote:

In other words, are we so focused on the parts and their connections that we are ignoring a deeper and more profound theory which would explain why all these connections exist?

I agree, this is one of fundamental flaws of modern mathematics. Foundations of Mathematics continue to exist as a (increasingly marginalised) mathematical discipline, but we need more than foundations: we need a careful study of the structural framework of mathematics.