Second order vagueness, identity, and biological species: Part I

Previously on this blog, I've discussed how sometimes whether an organism is a member of a category is vague, and that it may even be vague whether something like a virus is alive. This was vagueness about the classification of a certain life-form. There is also vagueness about the species category itself: exactly how many species are there on the Earth? This is not a question that admits of a clearly correct, precise numerical answer. Such is second order vagueness about species.

The Stanford Encyclopedia of Philosophy contains a nice summary of current models of biological species, and concludes, it seems to me, on an uncertain and potentially paradoxical note, by noting Darwin's writings on the species concept. Says SEP:

Thus far it has been suggested that Darwin doubted the existence of the species category because he doubted the distinction between species and varieties. What about those taxa[1] called ‘species’ by competent naturalists, are they real taxa for Darwin? It seems that Darwin was a realist when it comes to taxa. A passage at the start of the Origin's chapter on classification, Chapter 13, confirms this. Darwin writes that “[f]rom the first dawn of life, all organic beings are found to resemble each other in descending degrees, so that they can be classed in groups under groups. This classification is evidently not arbitrary like the grouping of the stars in constellations” (1859[1964], 411). Those taxa (“groups”) identified by competent naturalists can be real. And classifications of groups within groups, if properly constructed, reflect the hierarchical arrangement of taxa in the world. Thus, Darwin's skepticism of the species category did not extend to taxa and those taxa called ‘species.’

Thus, from Darwin's view, biological taxa of the type considered species are definitely species. Yet most classifications of species overall may be vague or dubious, since there is no abrupt dividing line between varieties of a species and different species.

Let us now consider a set

Set S_taxa = { the set of all taxa considered species by Darwin's criteria }. This an ordinary, non-vague or crisp set, with well defined members.

Now, compare set S_taxa to set

S_all = { the set of all species }, which is then a vague or poorly defined set.

Now, we stipulate that S_taxa is a proper subset of S_all. Then S_all is a vague set, but it has at a minimum the members of S_taxa, plus a vague number of other species. If we now consider that S_all cannot be larger than the number of individual living things, at the largest, we can establish an upper bound on the members of S_all[2]. We then can use the methods of mathematics to investigate S_all, the set of all species, as a rough set.

[1] Taxa is here defined as a well tested, working classification unit for organisms in biology. An example would be a federally defined endangered taxa and species such as Dermochelys coriacea, the leatherback turtle.

[2] I don't pretend to know the exact upper bound here, but to create a definite upper limit we can certainly choose a definite integer bound.