Part of the trouble is that the species an individual belongs to depends on more than the individual.
Consider a ring species. I'll use domestic dogs as a convenient example, assuming for the sake of argument that teacup poodles and great Danes can't mate (for obvious reasons). If they can, the argument is still solid, and other examples would work, but I'd have to use a lot of Linnaean nomenclature and talkorigins links.
Canis familiaris is a single species. This means that, in principle, a mutation in any one individual could spread through the entire population. EG, if the teacup has a mutation, it can breed with a chihuahua, the chihuahua with a Jack Russell, ... hound ... Dane. Eventually all dogs would have the mutation.
But if all breeds went extinct except for teacups and great Danes, they would be reproductively isolated, and would have to count as separate species; there's no way a mutation could go from one to the other.
To summarize: the question, "are teacups and Danes the same species" cannot be answered without more knowledge. If things are as they are now, the answer is "yes, same species, different variety"; if the mid-size ones go extinct, the answer is "no, different species but closely related".
Incidentally, this inherent "fuzziness" of the species concept, first explicitly stated by Darwin, is one of the reasons for disputes under the Endangered Species Act. Is the Mt. Graham squirrel a species, or a more-or-less distinct variety of a more widely-distributed species?Does it ever mate with squirrels from other mountains? If it had the chance, would it? Should we interfere in an ongoing speciation event?
But is the ability to interbreed what constitutes a species of animal, in the broader philosophical sense?
Huh? what does a "broader philosophical sense" possibly have to do with biology?
The basic definition is that a species is all the individuals in a common gene pool. It's not at all clear in many cases whether we have a single gene pool with a "sandbar" making two subpools, or whether there are two pools. This can change back and forth over time.
Consider that a child apprehends the nature of "squirrel" upon seeing a squirrel for the first time, without knowing anything about the interbreeding of species or even whether other squirrels exist.
Utterly irrelevant. The kid may, or may not, be able to differentiate species of squirrel. Squirrel is a genus.
If after 10,000 years, the "squirrel" was to develop stubby wings, while still maintaining the ability to interbreed, would the "squirrel" still be a squirrel? It seems not, because were it possible for the same child to travel to the future, he would not apprehend the same species.
It would still be a single species, whatever it's called then. At some point, a biologist will say something to the effect, "we'll call it a "gliding squirrel" now, because it's sufficiently different from the older, hopping ones."
Now consider a line of 10,000 of these animals [assuming one generation per year]. They fade imperceptibly from the hopping to the gliding form. There is no, repeat, no, clear dividing line between the species. Some biologist will have to come up with a more-or-less arbitrary criterion.
An example of this is the reptile -> mammal-like reptile -> reptile-like mammal -> true mammal fossil sequence. The criterion is the inner ear bones.
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Obviously a child with a birth defect is a human being.
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Your example of "minor change" is a squirrel developing wings! I think most people consider this major enough to call it a new species. In fact, I'd bet there are plenty of CRIDers saying it's too major to happen in 10,000 years.
...we apprehend the species of an individual that differs dramatically [child with hydrocephaly] from other members of its species as a member of the same species...
An individual with a fatal birth defect cannot pass its genes along. So this has nothing to do with novel species
The mind then apprehends species in a manner at times antagonistic to biological methods of categorization.
Welcome to Science 101! You're intuitions and naive ideas are often wrong or misleading.
Think of how much better a reputation Aristotle would have now if he'd had the gumption to do a few of Galileo's experiments!
The difference between species at each instant would have to be almost immeasurable. Yet it would be necessary for the mind to apprehend each stage in development as a distinct species, since the mind apprehends diverse species regardless of biological means of classification...
See above, the line of squirrels.
How is it possible for the mind to recognize a hydrocephalic child as a member of the species human, yet differentiate between extremely fine gradations of (hypothetical) missing links?
the child is obvious, and IMO, a red herring; we don't need to make these very fine distinctiions. In fact, they are imperceptible from generation to generation - the average wing size gradually gets larger, but at all times there are individuals with wings above and below the average size.
How is it possible for the mind to recognize...
To give a short answer, through study, training and observation, which is how one learns what is and isn't important in reasoning about individuals, populations, species, and genera.
The mind of a child, or of an armchair philosopher, is simply not up to the task.
The terms "genus" and "species" were borrowed from Aristotle's Categories. The term "species" is synonymous with Aristotle's notion of form.
The notion of "species" in the biological sense is largely nominal, and of less metaphysical value.
Who cares? Scientists should, because scientific research is predicated on the abstraction of universals from particulars.
Since you don't seem to know what I'm talking about, it's pointless to proceed. But the problem of universals is of paramount importance for scientists and laymen alike. Most modern scientists are de facto Nominalists, not realizing that their position is incoherent and ultimately solipsistic.