Why I do not believe in fish or zebras
I am sure that this title confuses you. I am sure that some of you are saying "Matan, I know you love gefilte fish, and to do that, you must think it exists."
I am sure that each creature we call a fish or zebra exists, I just don't think those categories are valid.
Because they are not ideals, and so therefore are not ideal.
An ideal is a subset of a POSet, where if two elements are in it, their least common ancestor is also in it, and all descendants are in it.
This might be mathematically pretty, but that mathematical prettiness is actually biologically useful.
If you make a tree of vertebrates, and label what is or isn't a 'fish,' then you will see that it excludes a big chunk for tetrapods (amphibians, reptiles (which also don’t exist), mammals, etc.). There is no way to make an ideal with the 3 species we call 'zebras' without including something we don't think of as a zebra. We are imposing a bias on how evolution worked, that some creature stopped being something else.
Now, subgroups are still valid in this, since you can gain a new feature, and it groups organisms by shared descent, meaning shared characteristics.
This asymmetry is a good thing, because it allows change to make a new group, without destroying previous structures.
It also defeats bizarrely imposed hierarchies, the whole "order, class, phylum" stuff, because it is much simpler, and does not distinguish them.
It's simplicity also allows us to actually handle the problems given by ring species. In a ring species, every population is definitely part of the same "species" as its neighbors, but too different from populations on the other side.
With ideals, since they don't care about hierarchy, we can talk about portions of it or the whole thing, and not have any problems.
And this all comes from properties of ideals on POSets which we want categories of organisms to have.
Edit: while some people have claimed zebras are polyphyletic, it turns out that some recent molecular evidence points to yes, zebras might be monophyletic, so my classification might admit they exist.