The fact that you can recognize Arabic and English as being different languages, with practical implications, means that a substantive distinction exists.
That's a really simplistic way of looking at it. To most people, red and blue are very distinct colors. No one would argue that they're the same. But are crimson and pink different colors, or different shades of the same color? What about #f00 and #f11? It's both arbitrary and contextual.
Categorization is easy when a difference is so large as to be irrefutable. But thinking in terms of continuum is much more useful when looking at small differences.
Language is simply uncountable. House and haus are pretty much the same word to me, so English and German overlap. If they are not disjunct, it doesn't make sense to count them one by one. That would be like counting sand by the corns, and by extension, coast lines by the sand. Language is recursive (a rose is a rose is a ...), what is its Housedorff dimension!? If you take a point in a topological space and give it a closed cover, you can calculate the area of the disc, but ... ok, I have close to no idea of complex analysis (didn't expect it would come in helpful for linguistics). The point is, if two arbitrarily large set's of Language are easier to compare by what they have in common, than where they are mutually exclusive, than you could colloquially call those different sets of language. If you can't even parse the speech, you won't be able to do that, so a natural speaker will perceive it as different. Book keeping Linguists are more interested in, or even bound to writing, as is clear from them calling spoken syntax "grammar" (which is from gramm~graph~scrapho writing, drawing, unless I'm missing something).
Well, it's countable in the sense that a countable number of humans have existed, with a countable (or at least bounded) number of idiolects per human. That's pretty much the "point in discrete color space" point of view I was getting at on one extreme. That's not a very useful level of granularity for much else other than upper bounding. At the lower bound, maybe we all just speak and sign dialects of some inherent Chomsky-esque universal grammar. In between, there are all sorts of ways you can draw borders on the map to tesselate it into languages, but few of these schemes can be defined without resorting to idiosyncratic decisions.
Language is a social construct. We're colloquial beings, and 99% of the time we all agree on what's what. What irritates me are the folks that think that the remaining 1% can be decided in some kind of objective way. Is Scots its own language? Depends on who you ask, and in what context. There are valid arguments either way. Embrace the ambiguity.
