"Species" is a broken definition.

So here's the deal. We want the word to be what mathematicians call an "equivalence relation" but it's not. What does this mean? It means that we want three things to be true.

We want

1. (reflexivity) Every animal to be in it's own species.
2. (symmetry) If animal A is the same species as animal B then animal B is the same species as animal A
3. (transitivity) If animal A is the same species as animal B and animal B is the same species as animal C then animal A is the same species as animal C.

By induction, we can extend the chain in the transitivity requirement as long as we want. That is, we can conclude that an animal D is the same species as A if it's the same species as C by saying that
A is the same as C which is the same as D and so D is the same as A. And so on.

This is where it breaks down. We can use transitivity to conclude that we are the same species as the lobe finned fish which is an ancestor to all tetrapods. We do this by saying. I'm the same species as my mother. My mother is the same species as her mother. I'm the same species as my grandmother. My grandmother is the same species as her mother. I'm the same species as my great grandmother. etc. Eventually, we arrive at a lobe finned fish which is pretty cool.

What's troubling me is that it doesn't break down at a particular point. Every step is valid and yet the whole conclusion is plainly invalid because I am not the same species as any lobe finned fish.

Any thoughts?

There was a typo in my statement of the transitive property. I also approached my OP from the wrong angle.

The reason that we want "species" to define an equivalence relation is that we want it to define a partition on the set of all organisms. Namely, we want the union of all species to be equal to the set of all organisms and we want each organism to belong to precisely one species. It's a mathematical fact that this is possible only if there is a corresponding equivalence relation: in this case, "is the same species as". Essentially, a partition of a set defines an equivalence relation (two elements are equivalent if they lie in the same segment) and an equivalence relation defines a partition (two elements lie in the same segment if they are equal under the equivalence relation).

So the fact that "is the same species as" isn't an equivalence relation means that "species" isn't a partition. That is, there is at least one organism that isn't in any species, or there is at least one organism that is in more than one species.

I find this deeply troubling. I can't think of a good way to get what we want here...

EDIT:
We can trivially resolve this by claiming that every organism is it's own species. This disallows my construction but is not at all satisfying.

EDIT 2:
Also, any organism that gave birth to a member of a different species would break my construction for its descendants. With enough such organisms, the definition of "species" could be restored. That seems very artificial though.

Your model is insufficient. You're ignoring key information by trying to treat species as members of a set.

I'll let you figure out what I mean by that.

Just to clarify, I'm trying to treat organisms as members of a set and species as a partition (or equivalence relation) on that set. So if O is the set of all organisms, then a species would be a subset of O. I'm not sure what the problem is but I'll think about it. I'm pretty sure that there is absolutely no problem with taking the set of all organisms.

Species is really just an arbitrary label we use for convenience.

Let's say we have a property, which I'll call "humanness" for the sake of argument. I'll also define the current generation as having a humanness value of exactly 1.

Now our ancestors from 10 000 years ago might have a value of say, 0.9, which means they would be capable of breeding with current generation humans. Because of this we include them as part of the same species.

Go back further and this value decreases so that breeding to produce viable offspring becomes pretty much impossible. Again, let's say for the sake of argument this occurs at a value of 0.5, at which point we declare them a different species. They can still reproduce with humans with a certain humanness value, but not with modern humans, which is the reference point we're using for what constitutes a human.

Species are just arbitrary points we plot on the evolutionary pathways using arbitrary references. The term does have some value, but there's no point getting too hung up over it. It's just more convenient than referring to a creature as 99.9% [Species A], 0.1% [Species B].

Quote:

---

Originally Posted by cmind

Your model is insufficient. You're ignoring key information by trying to treat species as members of a set.

I'll let you figure out what I mean by that.

---

You're going to have to tell me. As near as I can determine, my model is perfectly sufficient for proving that there is no correct way to define species. Specifically, any partition of the set off all organisms with the property that an organism is in the same segment as both it's parents and its offspring must be the trivial partition. But on the other hand, we certainly expect a "species" to have this property. Actually, either "is the same species as its parents" or "is the same species as its offspring" are equivalent and either breaks everything. This is all utterly trivial.

Quote:

---

Originally Posted by Photolysis

Species is really just an arbitrary label we use for convenience.

---

Right. I don't see the "convenience" in using words to describe a system that cause us to think unclearly about that system. :P According to my spell checker, I just had to make up the word unclearly.

Quote:

---

Originally Posted by Photolysis

Let's say we have a property, which I'll call "humanness" for the sake of argument. I'll also define the current generation has having a humanness value of exactly 1.
...

The term does have some value, but there's no point getting too hung up over it. It's just more convenient than referring to a creature as 99.9% [Species A], 0.1% [Species B].

---

This is getting in the right direction but you're still using "species". I think that what we want to do is to define, between any two organisms related by descent (a descends from b or b descends from a) the set of ordered pairs (α, 1 - α). α will of course only take discrete values. This is just like the equation for a line but here, we haven't defined a vector space structure so we can't write αa + (1 - α)b. Also, it's only really defined if a and b are related by descent. We could get the number α by setting α' to the number of mutations between a and the organism we're considering. Then we can count all the mutations between a and b (call it c) and normalize by that. So α = α'/c. This is really crude though because it counts all mutations as being equally important. It seems like I'm going to have to go wading into genetics here...

Quote:

---

Originally Posted by PhilosopherStoned

Right. I don't see the "convenience" in using words to describe a system that cause us to think unclearly about that system. :P According to my spell checker, I just had to make up the word unclearly.

---

The main purpose of using such labels in the first place is for the ease of communication. It's not best suited for situations like these where we're dealing with many shades of grey, but labels are useful in general, and besides, a lot of people feel the need to categorize things.

As another example, consider other labels such as colour. If I want (or need) to be really precise with my explanation of what the colour was, then I could create an absorption/transmission spectroscopy. Or I could give you an RGB value. If I want to give you a vague idea of what I saw, then I could instead say "I saw an orange-red [noun] the other day". The vague "orange-red" label is more useful in day to day usage rather than trying to picture in your head exactly what #B7500B looks like, or what shade you would see based on a spectroscopy graph.

We'll continue to use the term species for a similar reason we use colours: for conveying general and/or relative information. Unless there's a need for a conversation to be so precise, we can safely stick to using labels, even if it had the downside of confusing understandings about various systems and phenomenon.

Quote:

---

This is getting in the right direction but you're still using "species".[..] It seems like I'm going to have to go wading into genetics here... .

---

If you want to ditch the term species completely, then you'll need to go to the genetic level.

Species is unambiguous at any given time, it's worth pointing out. There is no way of using transitivity to make any living bonobo make fertile offspring with a chimpanzee.

I'm surprised I haven't pointed that out. Of course it's because the last common ancestors of bonobos and chimpanzees aren't alive so the "female parent of X" operator will only build a chain 2 or 3 elements long and you can have a non-trivial partition on the organisms alive at any given time.

Quote:

---

Originally Posted by Xei

Species is unambiguous at any given time, it's worth pointing out. There is no way of using transitivity to make any living bonobo make fertile offspring with a chimpanzee.

---

It's also worth pointing out that even with the at-any-given-time clause, the common definition only applies for species that reproduce sexually. Which seems pretty odd.

To be honest, I don't think there exists an elegant definition of species that does all the things we'd ideally like it to do. Although I'd be interested to explore definition space with all of you in search of one anyway.

Yes, I'm quite unsatisfied with the notion that a species is defined by the potential to breed with another organism and produce fertile offspring. If we really wanted to be pedantic, than this would exclude non-fertile members of a species as well. If a woman couldn't have fertile offspring for some reason, then according to that definition, she wouldn't be a member of any species. Of course we "know" that she's still homo sapiens but we have to bend the definition to make it work.

There is no single definition of species. I'm aware of ten definitions/concepts that are regularly used, the biological one (based on reproductive characteristics) being only one of them.
Furthermore, organisms that fit into the same species according to one definition, do not fit together under the other. Funny thing is that this issue hasn't really stopped or interfered with progress in biology.

The question whether species is an artificial concept or if it represents something physical is also under debate. John S. Wilkins has written about this, and some of his work is available for free on the internet, if anyone is interested.

Species concept (regardless of the operational definition) will probably always break down when extend over long enough time.