Abstract
Complexity in our universe, Herbert Simon once noted, generally takes a hierarchical, nearly decomposable form. If our purpose as biologists is to "carve Nature at the joints," then the quantitative biologist's pattern questions must embody some tentative claim of where the explanatory joints are—only after meaningful qualifications can notions of variance and covariance make sense. In morphometrics, specimens and variables alike can be "carved at the joints," with a correspondingly great gain in explanatory power in both versions. Simon's advice is that the competent biologist's measurements should lie entirely within a single organismal component or else deal entirely with one of the joints. In either context, our best contemporary rhetorics of explanation in biology may resemble morphometrics in their frank combination of carefully (i.e., qualitatively) supervised parallel quantifications that, taken together, result in new qualifications, leading in turn to new quantifications, and so on. In short, the relation between qualitative and quantitative in the organismal biological sciences is not an opposition but a complementarity, and the modern biometrical statistics of organismal form may be a particularly apposite praxis for exploring it.
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Notes
The reader with an eye for mathematical aesthetics may find the explicit formula for this distribution to be pleasingly compact, in analogy to the likewise aesthetically pleasing formula \({1\over {\sigma \sqrt{2\pi }}}e^{-{1\over 2}x^2}\) for that Gaussian distribution. The probability distribution of S actually goes as \(C \vert S\vert ^{(N-p-2)/2} e^{-{1\over 2} {\rm{tr}}(NS)}\) where N is sample size, S is the sample covariance matrix as above, p rows by p columns, \(\vert \cdot \vert \) is the determinant function that you probably were introduced to in high school but have since forgotten, tr is the trace operator (the sum of the diagonal elements of a square matrix), and C is a complicated constant guaranteeing that the probability distribution integrates to 1. The probability density is taken with respect to the measure dS that is a little \({{p(p+1)}\over 2}-\)dimensional cube with one edge for every independent entry \(s_{ij}\) of the covariance matrix, \(i\ge j.\)
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In memory of Werner Callebaut, philosopher, colleague, friend.
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Bookstein, F.L. No Quantification Without Qualification, and Vice Versa. Biol Theory 10, 212–227 (2015). https://doi.org/10.1007/s13752-015-0221-3
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DOI: https://doi.org/10.1007/s13752-015-0221-3