I notice that no-one has chosen to dispute or otherwise comment on my observation that the human brain gets things done, not just by virtue of being "organismic" (or embodied or fleshy or corporeal), but because its constituent neurons are arranged so as to perform elaborate and highly specific transformations of input to output, which correspond to specific cognitive functions like learning and memory, and which, at the mathematical level of description, fall squarely within the scope of the subfield of theoretical computer science which studies algorithms.
Under other circumstances, I'd be happy to have a freewheeling discussion about the subjective constitution of imputed intentionality in the practice of programming, or the right way to talk about the brain's "computational" properties without losing sight of its physicality, or exactly why it is that consciousness presents a challenge to the usual objectifying approach of natural-scientific ontology.
But however all that works out, and whatever subtle spin on the difference between natural and artificial intelligence best conveys the truth... at a crude and down-to-earth level, it is indisputable that the human brain is full of specialized algorithms, that these do the heavy lifting of cognition, and that such algorithms can execute on digital computers and on networks of digital computers.
That is why you can't handwave away "artificial intelligence" as a conceptual confusion. If you want to insist that the real thing has to involve consciousness and the operation of consciousness, and that this can't occur in digital computers, fine, I might even agree with you. But all that means is that the "artificiality" of AI refers to something a little deeper than the difference between being manufactured and being born. It does not imply any limit on the capacity of machines to emulate and surpass human worldly functionality.
My point is not that our intelligence is "just" embodied, but that it is indispensably so, and in ways that bedevil especially the hopes of those Robot Cultists who hope to code a "friendly" sooper-parental Robot God, or to "migrate" their souls from one materialization to others "intact" and quasi "immortalized."
That you can find maths to describe some or, maybe -- who now knows? (answer: nobody and certainly not you, whatever your confidence on this score, and also certainly not me) -- even much of the flavor of intelligence would scarcely surprise me, inasmuch as maths are, after all, so good at usefully getting at so much of the world's furniture.
I am happy to agree that it may be useful for the moment to describe the brain as performing specialized algorithms, among other things the brain is up to, and it is surely possible that these do what you call the "heavy lifting" of cognition. But that claim is far from "indisputable," and even if it turns out to be right that hardly puts you or anybody in a position to identify "intelligence" with "algorithms" in any case, especially if you concede "intelligence" affective dimensions (which look much more glandular than computational) and social expressions (which look far more like contingent stakeholder struggles in history than like beads clicking on an abacus).
Inasmuch as all the issues to which you allude in your second paragraph -- subjective imputation of intention, doing justice to the materiality that always non-negligibly incarnates information, to which I would add uninterrogated content of recurring metaphors mistaken in their brute repetition for evidence -- suffuse the discourse of GOFAI dead-enders, cybernetic totalists, singularitarians, and upload-immortalists I do think you better get to the "other circumstances" in which you are willing to give serious thought to critiques of them (my own scarcely the most forceful among them) sooner rather than later.
You will forgive me if I declare it seems to me it is you who is still indulging in handwaving here. As an example, in paragraph three, when you go from saying, harmlessly enough, that much human cognition is susceptible to description as algorithmic then make the point, obviously enough, that digital and networked computers execute algorithms, you hope that the wee word "such" can flit by unnoticed, un-interrogated, while still holding up all the weight of the edifice of posited continuities and identities you are counting on for the ideological GOFAI program and cyber-immortalization program to bear its fruits for the faithful. You re-enact much the same handwave in your eventual concession of the "something a little deeper" between even the perfect computers of our fancy and the human intelligences of our worldly reality, which may indeed be big enough and deep enough as differences go to be a difference in kind that is the gulf between the world we share and the techno-transcendence Robot Cultists pine for.
You know, your "colleague" Giulio Prisco likes to accuse me of "vitalism" for such points -- which to my mind would rather be like a phrenologist descrying vitalism in one who voiced skepticism about that pseudo-science at the height of the scam. So far you seem to be making a comparatively more sophisticated case, bless you -- we'll see how long that lasts -- but the lesson of Prisco's foolishness is one you should take to heart.
I for one have never claimed that intelligence is in any sense supernatural, and given its material reality you can hardly expect me to deny it susceptibility of mathematical characterizations. It's true I have not leaped on futurological bandwagons reducing all of intelligence to algorithms (or the whole universe to the same), seeing little need or justification for such hasty grandiloquent generalizations and discerning in them eerily familiar passions for simplicity and certainty (now amplified by futurologists with promises of eternal life and wealth beyond the dreams of avarice) that have bedeviled the history of human thought in ways that make me leery as they should anybody acquainted with that history.
But I am far from thinking it impossible in principle that a non-organismic structure might materially incarnate and exhibit what we would subsequently describe as intelligent behavior -- though none now existing or likely soon to be existing by my skeptical reckoning of the scene do anything like this, and I must say that ecstatic cheerleading to the contrary about online search engines or dead-eyed robotic sex-dolls by AI ideologues scarcely warms me to their cause. Upon creating such a differently-intelligent being, if we ever eventually were to do as now we seem little likely remotely capable of, we might indeed properly invite such a one within the precincts of our moral and interpretative communities, we might attribute to such a one rights (although we seem woefully incapable of doing so even for differently materialized intelligences that are nonetheless our palpable biological kin -- for instance, the great apes, cetaceans).
That such intelligence would be sufficiently similar to human intelligence that we would account it so, welcome it into our moral reckoning, recognize it the bearer of rights, is unclear (and certainly a more relevant discussion than whether some machines might in some ways "surpass human... functionality" which is, of course, a state of affairs that pervades the made world already, long centuries past, and trivially so), and not a subject I consider worthy of much consideration until such time as we look likely to bring such beings into existence. I for one, see nothing remotely like so sophisticated a being in the works, contra the breathless press releases of various corporate-militarist entities hoping to make a buck and certain Robot Cultists desperate to live forever, and in the ones who do one tends to encounter I am sorry to say fairly flabbergasting conceptual and figurative confusions rather than much actual evidence in view.
Indeed, so remote from the actual or proximately upcoming technodevelopmental terrain are such imaginary differently-materialized intelligences that I must say ethical and political preoccupations with such beings seem to me usually to be functioning less as predictions or thought-experiments but as more or less skewed and distressed allegories for contemporary political debates: about the perceived "threat" of rising generations, different cultures, the precarizing loss of welfare entitlements, technodevelopmental disruptions, massively destructive industrial war-making and anthropogenic environmental catastrophe, stealthy testimonies to racist, sexist, heterosexist, nationalist, ablest, ageist irrational prejudices, all mulching together and reflecting back at us our contemporary distress in the funhouse mirror of futurological figures of Robot Gods, alien intelligences, designer babies, clone armies, nanobotic genies-in-a-bottle, and so on. I suspect we would all be better off treating futurological claims as mostly bad art rather than bad science, subjecting it to literary criticism rather than wasting the time of serious scientists on pseudo-science.
Be all that as it may, were differently-materialized still-intelligent beings to be made any time soon, whatever we would say of them in the end, the "friendly" history-shattering post-biological super-intelligent Robot Gods and soul migration and cyberspatial quasi-immortalization schemes that are the special "contribution" of superlative futurologists to the already failed and confused archive of AI discourse would remain bedeviled by still more logical and tropological pathologies (recall my opening paragraph), and as utterly remote of realization or even sensible formulation as ever.