There are multiple definitions of species, and we choose whichever abstraction is most useful for a given approach.

The concept of species exists purely as a human, scientific abstraction: evolution/biology doesn't give us absolute categories that you can unambiguously put any organism into.

Isn't the ability to produce fertile offspring a de facto proof that two organisms are of the same species?

Depends on the context, but generally nope. Consider the example of "ring species"--A can breed with B, B with C, .., Y with Z, and Z with A. However, A can't interbreed with M. What's the species?

That's a somewhat rare version of a general pattern where that abstraction is leaky. Much more common is a temporal version of that. A might beget A1 and B, with A1 begetting A2 and B begetting C, continuing all the way down. And though A can interbreed with A1 and B, and B with C, etc, Z can't interbreed with A25. That's speciation, which happens all the time. The only thing that makes it a clear species is defining the concept species while abandoning time invariance.

Actually, "species" has multiple competing definitions in the scientific literature. For details see: https://en.wikipedia.org/wiki/Species_problem

The biological species concept is probably one of the most popular and very similar to what you are suggesting. In this concept, a species is an interbreeding population that is reproductively isolated from other populations (either due to genetics or geography).

That's a junior high level of explanation tho. 'Species' is ultimately an attempt to impose a quantization of nature, and nature can't always be quantized in a way we'd like.

Exactly. Species is more of a descriptive term than it is prescriptive.

yeah, and things can get complicated: https://en.wikipedia.org/wiki/Ring_species

My impression of defining species wasn't necessarily can they mate, but rather will they mate. Geography and sexual discrimination play as much of a part in separating species as does fertilization.

So me and Taylor Swift are two seperate species?

In all seriousness, species definition is a case of trying to define a black/white classification in a gradient. It might sort of work in higher animals, such as mammals. But it certainly breaks down in bacteria, which is clearly seen since it's possible to compare genomes on population level.

I'm not a biologist, but my understanding is that animals that can interbreed but usually wouldn't are (if at all) divided into subspecies within the same species.

See https://en.wikipedia.org/wiki/Subspecies

"A taxonomist decides whether to recognize a subspecies or not. A common way to decide is that organisms belonging to different subspecies of the same species are capable of interbreeding and producing fertile offspring, but they do not interbreed in nature due to geographic isolation or other factors."

That's an old and very problematic definition of species. In fact, I believe biologists tend to focus on populations rather than species these days, since those at least can be clearly defined.

So dogs and wolves are the same species? I know that different species usually cannot interbreed, but I don't think that's the sole criterion.

Interestingly, yes.

"The domestic dog lost its recognition as a separate species in 1993, and is now considered a subspecies of gray wolf."

http://karmatics.com/docs/evolution-species-confusion.html

"Species" is very clearly defined by "speciation": the point where something becomes too different to interbreed. Usually when two things can interbreed, but have many differences, they are referred to as separate subspecies of the same species.

My understanding is that that particular definition is outdated.

So each infertile individual belongs into their own species?

You can't use such a literal-minded interpretation in the real world. The real world is complex and fuzzy.

For instance, in species with sexual differentiation, two males or two females cannot mate, but are considered the same species.

For a more precise definition, you would have to, say, look at the probability that a pair of organisms of the appropriate sex from two different populations would be able to mate, and produce viable offspring (that is, offspring that could themselves mate and produce more offspring).

Asking about a single organism that happens to be infertile, or a single pair that happens not to be able to mate or produce viable offspring, is not particularly interesting, but at a population level you could use a statistical test to get a probability distribution, rather than a yes or no answer.

See https://en.wikipedia.org/wiki/Species#Difficulty_defining_or... for more information.

Note also that it can be the case that two different species can mate, and produce offspring, but the offspring itself is not viable (cannot mate and produce offspring). This happens in the case of mating two closely related species, such as when mating a horse and donkey to get a mule (https://en.wikipedia.org/wiki/Mule), or a lion and a tiger to get a liger (https://en.wikipedia.org/wiki/Liger).

Not always, but some infertile individuals belong to their own new species, yes. It happens all the time with plant hybrids.

Plants can clone itselves so to be infertile is not the same as to be extinct for them. Infertile also means normally something like "99,9999% unable to produce seeds" but with enough time and a high number of tries, planets will align in the polinisation of two "unfertile" stems of the same plant. At the end seeds will be produced and the new species will exit the 'trap' perfectly (or all DNA of the father will be discarded reverting to the original mother species). Is just a question of time.

One method of speciation is reproductive isolation. One model for this is the Bateson–Dobzhansky–Muller model, where if a species is, say, separated by distance, and then subgroup A has a change at one loci, and subgroup B has a change at another loci, and due to these mutations, individuals from subgroup A can no longer reproduce with individuals fom subgroup B - then it can be said that speciation from reproductive isolation has occurred according to this model.

Each one of an infertile individual's parents belongs to its own species.

Until they get another child that does turn out to be fertile.