Recursive definitions are possible, but they must still be founded on a base level that does not reference itself. Each other level can then be defined in a way that is not self-referential.
I believe usually it is also required that the number of steps (levels) to reach the non-recursive base should always be finite (e.g. recursion via a countable set).
Yeah, but in this case Annoyance is right. You need to find two extensional mammals such that their last common ancestor is the LCA of everything you want to call a mammal, then the definition is complete. As it stands it’s like a factorial without the base case.
EDIT: Stupid me.. I should have remembered that I was dealing with Annoyance, and been a bit more incredulous at the prospect of him being right, since Annoyance actually does use his intelligence in order to deliberately arrive at wrong answers.
You have to be careful—the LCA of those is not the LCA of all mammals, because those all happen to be placental mammals—and the splits leading to marsupial mammals and monotremes came before then.
That’s true. I addressed that originally by saying all monophyletic clades were natural groups, and they have a natural sequence (in order of increasing exclusivity).
So while we can debate which group should be called “mammals”—whether placentals, or placentals+marsupials, or something bigger yet—we can all agree that “mammals” is a group whose border lies somewhere near the placentals-marsupials joining point in the evolutionary sequence. Dolphins are nestled very deeply in the group, and so dolphins are definitely mammals by any definition you may use. (That’s what the original debate was about: I argued that “mammals without dolphins” is not a natural group.)
It’s not a question of fact, but of definition—to what group do we refer as “mammals”? So it makes little sense to argue over it.
All groups of animals defined via shared characteristics have fuzzy borders. The fact that all extant species either clearly do or clearly don’t lactate is purely an evolutionary accident, since all the intermediaries once existed. Even today we have species that lactate but don’t have localized nipples (the platypus exudes milk from a wide area of skin). Farther out, we have various fishes, amphibians, etc. that exude specialized non-milk substances from their skin for their young to eat. If you throw away the categories, “feeding the young on substances released from skin pores” is a wider category than just mammals.
Also, this whole thread started with people saying that “common people” wouldn’t know if dolphins are mammals, even though they do lactate.
Yeah, but in this case Annoyance is right. You need to find two extensional mammals such that their last common ancestor is the LCA of everything you want to call a mammal, then the definition is complete. As it stands it’s like a factorial without the base case.
In what way does your comment differ from Dan Armak’s original comment? how is Annoyance right in this case?
Which is why the next sentence after the one you quoted explained: “In practice, to avoid circularity, it is sufficient to take the MRCA of a few indisputable mammalian groups such as primates, rodents, carnivorans, ungulates, etc. to include all mammals.”
IOW, start from a few groups everyone agrees on calling mammals, and you have a precise rule stating whether any given animal is a mammal or not.
“The definition of a mammal is simple: descent from the most recent common ancestor of all mammals.”
Valid definitions cannot reference themselves.
You need to be a little more careful about such absolute statements. The definition of factorial(.) as
factorial(n) = n*factorial(n-1)
factorial(0) = 1
references itself and is valid.
Recursive definitions are possible, but they must still be founded on a base level that does not reference itself. Each other level can then be defined in a way that is not self-referential.
I believe usually it is also required that the number of steps (levels) to reach the non-recursive base should always be finite (e.g. recursion via a countable set).
Indeed. I’m just asking for a little precision, e.g., valid definitions cannot just reference themselves.
Some definitions which reference themselves plus something else are also invalid. :P
I presume that’s why he said “e.g.”, not “i.e.”
Yeah, but in this case Annoyance is right. You need to find two extensional mammals such that their last common ancestor is the LCA of everything you want to call a mammal, then the definition is complete. As it stands it’s like a factorial without the base case.
EDIT: Stupid me.. I should have remembered that I was dealing with Annoyance, and been a bit more incredulous at the prospect of him being right, since Annoyance actually does use his intelligence in order to deliberately arrive at wrong answers.
My comment did provide a base case (primates, rodents, carnivorans, ungulates). Annoyance didn’t quote it.
You have to be careful—the LCA of those is not the LCA of all mammals, because those all happen to be placental mammals—and the splits leading to marsupial mammals and monotremes came before then.
That’s true. I addressed that originally by saying all monophyletic clades were natural groups, and they have a natural sequence (in order of increasing exclusivity).
So while we can debate which group should be called “mammals”—whether placentals, or placentals+marsupials, or something bigger yet—we can all agree that “mammals” is a group whose border lies somewhere near the placentals-marsupials joining point in the evolutionary sequence. Dolphins are nestled very deeply in the group, and so dolphins are definitely mammals by any definition you may use. (That’s what the original debate was about: I argued that “mammals without dolphins” is not a natural group.)
I don’t think anyone debates whether monotremes and marsupials are mammals.
Both groups produce milk through mammary glands to feed their young—and both groups have long been recognised as being mammals.
It’s not a question of fact, but of definition—to what group do we refer as “mammals”? So it makes little sense to argue over it.
All groups of animals defined via shared characteristics have fuzzy borders. The fact that all extant species either clearly do or clearly don’t lactate is purely an evolutionary accident, since all the intermediaries once existed. Even today we have species that lactate but don’t have localized nipples (the platypus exudes milk from a wide area of skin). Farther out, we have various fishes, amphibians, etc. that exude specialized non-milk substances from their skin for their young to eat. If you throw away the categories, “feeding the young on substances released from skin pores” is a wider category than just mammals.
Also, this whole thread started with people saying that “common people” wouldn’t know if dolphins are mammals, even though they do lactate.
Oh, I didn’t see that.
Voted down Annoyance’s original, then. (Argh! Can’t believe I fell for his old tricks!)
In what way does your comment differ from Dan Armak’s original comment? how is Annoyance right in this case?
Annoyance is almost right—his criticism is just a little too inclusive, that’s all.
Which is why the next sentence after the one you quoted explained: “In practice, to avoid circularity, it is sufficient to take the MRCA of a few indisputable mammalian groups such as primates, rodents, carnivorans, ungulates, etc. to include all mammals.”
IOW, start from a few groups everyone agrees on calling mammals, and you have a precise rule stating whether any given animal is a mammal or not.