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Interbreeding
Defining species by the criteria of not being able to breed is problematic:

  • The concept loses its meaning in phylogenetic prospective, since closely related species were able to breed between themselves in past, during the speciation event (an "event" that often lasted hundreds or thousands of generations)
  • The definition is problematic for simpler life forms, such as bacteria exchanging genetic material, or viruses where different species from the same family may differ only quantitatively (e.g., HIV vs. SIV).

(Excellent exposition of this view on the difficulty/impossibility of defining species is presented in this answer, as well as in the answer by @SPr below.)

Emergence
On the other hand, species are not reducible to merely accumulation of genetic variation. When looked from more general prospective (e.g., here), species are an example of an emergent phenomenon (the same inability to interbreed is a qalitative rather than quantitative difference.) We thus should be able to define a critical boundary between what is the same species and not, although this boundary does not necessarily lie in time domain. Inability to interbreed is a vague definition, since it may be due to very different reasons: phisiological, genetic, assortative mating, etc.

Operational concepts
In some domains we could choose to define species based on a criteria of convenience. E.g., considering an ecological community one may define as species the organisms that cannot interbreed, but also those that do not share the same habitat. While such an operational definition may work well for practical studies, it is not unique and subject to change from one study to another.

Are species an emergent property or an ensemble of quantitative differences? Can we give a robust definition of species/speciation? Can we define a domain of applicability of such a definition? (e.g., defining the forms of life for which species can be defined)

Background
When speaking of emergent properties, I tried to skip some techncialities that led me to this question, which is why this part of my question may appears somewhat vague. So I would like to add more details here.

Phil Anderson in his article Broken symmetry, emergent properties, dissipative structures, life: are they related? formulates emergence in mathematically rigorous terms as a kind of phase transition in a non-equilibrium system. Phase transitions are characterized by summetry breaking, i.e., appearance of a completely new property in a system. What I termed previously robustness is, in Anderson's language, called phase rigidity.

The question is thus: whether speciation is a kind of such phase transition (as claimed, e.g., in the Melamede's article, already cited earlier) OR whether it is merely incremental accumulation of changes. (However, in such a formulation question is unlikely to be answered either in physics or biology communities.)

For more context: in his article Anderson criticizes Ilya Prigogine's school on a few technical points, but it is Prigogine and his co-workers, who are at the origin of such thinking. In particular, they have multiple publications on the subject of such disspative structures, and non-equilibrum thermodynamics more generally. Many of their articles are available by googling.

Update
After some discussions in the comments: the crux of the question is whether (or tow hat extent) speciation occurs via abrupt or incremental changes.

  • Organisms differing by abrupt acquisitions, such as presence of a nuclear membrane or an extra chromosome, are unambiguously assigned to different species (and cannot breed or reproduce viable offspring).
  • On the other hand, most ambiguous cases seem to be those where the "species" differ only by a incremental changes (such as the number of nucleotide substitutions/insertions/deletions).

Ring species could be a possible test case: if the species at the beginning and the end of the ring cannot breed, do they always differ only by incremental changes? If one species differed by a number of chromosomes, it likely wouldn't be able to breed with any of the others.

Update 2
Here is a useful answer: species can be defined from many different viewpoints.

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  • $\begingroup$ Ew "closely related species were able to breed between themselves in past", it's also the case in the present. See for instance the development of the "coywolf" in eastern North America: en.wikipedia.org/wiki/Coywolf $\endgroup$
    – jamesqf
    May 5 at 18:50
  • $\begingroup$ @jamesqf sure, horse and donkey can breed, but two breeds of dogs sometimes can't. This underscores the insufficiency of this definition. But the point I was making was dufferent - if you have suggestion of more precise formulation, I am listening. $\endgroup$ May 5 at 19:52
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    $\begingroup$ @Vadim my point was that they can if you take the sperm from one and use it to fertilize the other, it's just they they don't (not for lack of trying I am told) because things don't fit, or can't reach... They are definitely not separate species. Horse and donkey can interbreed but offspring are usually sterile - mule and hinny respectively, so definite separate species. But the boundaries between other organisms aren't so clear. $\endgroup$
    – bob1
    May 5 at 21:40
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    $\begingroup$ @bob1: Yes, just about everyone regards dog breeds as varieties of a single species, yet the "don't fit" problems ensure that some breed pairings don't produce offspring that are viable without human assistance. (Indeed, I've read that some show breeds like bulldogs need human assistance.) OTOH, dogs, wolves, and coyotes are regarded as different species, yet can interbreed and produce viable offspring. $\endgroup$
    – jamesqf
    May 6 at 4:27
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    $\begingroup$ Another example of different species that freely interbreed are trout. Particularly within the US Great Basin, where the drier climate after the Pleistocene created a lot of isolated stream/lake systems, which often evolved their own trout species. Then eastern brook & rainbow trout were introduced, and freely bred with the native species, and with each other, producing viable hybrids such as the tiger trout: ndow.org/Species/Fish/Tiger_Trout $\endgroup$
    – jamesqf
    May 6 at 16:53
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Can we give a robust definition of species?

  • No.

Species constantly evolve, diverge, converge, interbreed, and mix and shuffle and trade and spread genes. To draw a box at any particular point in time around a population or lineage, and to say, "all contained within are members of such-and-such species" fails to appreciate that this would only work for that particular case, and arbitrarily so; it is obviously futile to have a sound microtaxonomy that would be generalizable to all animals, or microbes, or plants, or viruses for that matter. What concensus there might be for (e.g.) beetles is a poor yardstick for judging whether two alike bacteria ought to be lumped together as a species. Also, life, genes, individuals and populations are dynamic over time, so whatever box we draw will by definition require adjustment, and you can see that this adjustment will have to be arbitrary in many respects. This is an old problem, and it is more philosophical and ever-debatable, rather than something to be settled by empirical science triumphantly. From Nicholson's 1872 A manual of zoology (two centuries ago now):

"No term is more difficult to define than "species," and on no point are zoologists more divided than as to what should be understood by this word."

And to echo all that has been said... from Piglucci (2003) Bioessays:

"First, the species problem is not primarily an empirical one, but it is rather fraught with philosophical questions that require — but cannot be settled by — empirical evidence."

Which also summarizes a few major biological definitions, some of which do not align:

enter image description here

And also a few philosophical issues that are worth considering when discussing the species problem:

enter image description here

This is not to say that conventions don't exist, or that there is a lack of an operational and specific definition of a species in each respective field; however, there is no overarching definition that applies robustly, hence the answer at the top!

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    $\begingroup$ Remi's answer here is excellent: biology.stackexchange.com/a/39669/27148 $\endgroup$
    – Bryan Krause
    May 4 at 16:46
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    $\begingroup$ @Vadim What does "emergence of life" have to do with "species"? Who is the "we" that implies abrupt qualitative change? I think Remi's answer there makes clear that "species" is just a classification we draw for convenience when what we are really dealing with is "lineage" which can't be separated except by (relatively) arbitrary means. $\endgroup$
    – Bryan Krause
    May 4 at 17:02
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    $\begingroup$ @Vadim Dug a bit further and did find the article.. so, okay, this seems to me like a lot of word salad and not a lot of substance, at least with a casual glance. The author also seems to be mixing the words "species" referring to biological species and "species" in the context of "chemical species", the more generic "kind/sort", which has little relationship to the concept in biology. I wouldn't recommend taking it too seriously. In any event, I don't see any reference to anything "abrupt", if there's a particular quote that is relevant you should include it in your question. $\endgroup$
    – Bryan Krause
    May 4 at 17:26
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    $\begingroup$ @Vadim - check out ring species as an example of the difficulties in determining what is a species and where. The problem is that you are approaching it in a computational manner, where things have exact definitions. Species do not have exact definitions precisely for the reasons S Pr outlined. Biology is messy! An example of the difficulty is cancer research, where cell lines evolve in the lab so that results from one lab are not reproducible in another, even though both are working on the same cell line. $\endgroup$
    – bob1
    May 5 at 20:43
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    $\begingroup$ @Vadim - both is the answer. Abrupt jumps are seen in some circumstances, incremental in others. $\endgroup$
    – bob1
    May 5 at 21:44

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