## Some Derivations and Criticisms |
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$\backslash sum\_i\{p\_i^2\}$
with
$p\_i$ =
$n\_i/\backslash sum\_i\{n\_i\}$
and*
$n\_i$* is
the abundance of species
$i$

*derivation*: the probability to choose a specific species
twice is
$p\_i*p\_i$ and
thus the probability that any two chosen specimens belong both to
the same species
$i$ is the sum
over all species
$i$.

*criticism*: none, it is easy to understand and, as a true
probability, bounded between 0 and 1. Of course we shouldn't
interprete the "choosing" of a specimen as catching one (this has
definitely a different probability), but rather we'll have to say
that the sample is representative for the population, what is
still questionable but difficult to avoid.

$$

$$$N$

$entropy\; =\; -1*\backslash sum\_i\{p\_i*\backslash ln\backslash left(\{p\_i\}\backslash right)\}$,
again with
$p\_i$ =
$n\_i/\backslash sum\_i\{n\_i\}$
and*
$n\_i$* is
the abundance of species

or as entropy change per mole $\backslash Delta\; \backslash hat\; S\_i\; =\; -p\_i*R*\backslash ln\backslash left(\{p\_i\}\backslash right)$. To get the total entropy of mixing we simply sum up over all species: $\backslash Delta\; \backslash hat\; S\; =\; -R*\backslash sum\_i\; p\_i*\backslash ln\backslash left(\{p\_i\}\backslash right)$, which differs from the more metaphorical use above only by the ideal gas constant $R$ which is set to 1.

criticism

In the 70ies and 80ies of the last century we had a broad
discussion (mostly in the mass media) about the biological species
concept, which left as it's only trace today that this topic is
now "mega-out". But there has been no solution and I don't think
it's a good idea to ignore this point. To me it sometimes appears
as if taxonomy and the species concept is like a black hole buried
deep inside of biology, which - once it becomes clear that there
is no species concept - could lead to the collapse of many fields
of modern biology.

In the most simple (i.e. mathematical) terminology a species is
an equivalence class, which is created by an equivalence relation
R. Such a relation needs just 3 properties: aRa is true
(reflexivity), aRb entails bRa (symmetry) and aRb together with
bRc entails aRc (transitivity). (A simple example for a transitive
relation is "smaller than" or "<": a < b and b < c
entails a < c.) These three properties are sufficient to ensure
that any set on which an equivalence relation is defined, can be
partitioned into disjoint subsets (or classes): in our case the
species.

The problem in taxonomy is: we don't have such an equivalence
relation. What we use instead is the similiarity relation - and
this is not transitive.

As an example let's talk about the most famous species
definition: 2 specimens from a population belong to the same
species if they can produce fertile progeny (- we skip the sex
discussion). Let's imagine the population is somehow ordered (due
to it's gene-pool or whatsoever) in a plane, with the more similar
specimens situated in the middle and the less similar ones farther
outside on the margins. Of course the specimens from the middle
will all belong to the same species. Furthermore let's assume that
specimens from the left margin belong to the same species than the
middle ones, as well as the specimens from the right margin. But
this does not entail, that the specimens from left and right
margin are still similar enough to produce fertile offspring.

More precisely this means that if specimen a belongs to the same
species as H (the holotype) and H belongs to the same species as
b, this does NOT entail that a and b belong to the same species!

And we are not talking about exceptions to a rule (the typical
excuse of biologists), we talk about a concept which is self
contradictory - until we don't find a proper equivalence relation.