Besides, a lot of what people mean when they say they're bad at math is that they're bad at arithmetic, which is honestly understandable.

That's not a recursive function as it's written, but you could certainly consider it a form of symbolic recursion. This just isn't a very useful characterization in an iterative/imperative context. You could frame incrementing as recursive, though—this is just peano axioms/church encoding.

I have a very difficult time trying to extract the difference between "linguistic ability" and "critical thinking", though:

1. The core difference between "critical thinking" and "uncritical thinking" is the ability to discern incoherency from coherency.

2. Coherency is evaluated at the linguistic level: do the terms bind in meaningful ways to the problem? Do any statements contradict each other?

3. The remaining aspect is "creativity": can you come up with novel ways of approaching the problem? This is the hardest to tie to linguistic ability because it sort of exists outside our ability to operate within an agreed context.

So while I agree these are distinct skills, I still have difficulty identifying what remains in "critical thinking" after linguistic ability is addressed.

1. Our spoken and especially written grammar is recursive. We do handle this unconsciously. This is not related to our ability to reason about recursion at a high level, and recursive grammars are not necessary to do so. This is not a skill in the normal sense and we have only (very) limited ability to improve our capacity to interpret deeply nested grammars. However, this is still a useful illustration of what recursion IS, which is why I brought it up.

2. Language also introduces the ability to semantically reason about recursiveness. This is still a linguistic thing—you need a symbol and relations among symbols in order for recursion to be meaningful—but this is a skill and is likely very related to linguistic skill. This is the part that really helps you to program: ultimately, you're just reasoning about symbols and looking for incoherency.

So in this case, "recursive function" would be "clause" or something like that; I'm no linguist. But clauses can embed clauses which can embed further clauses, etc.

I think your usage of recursive functions is just high-level logic—you're describing an inductive proof. We also frame a lot of our social games as recursive processes. But these are conscious processes that we can evaluate consciously; the recursion in spoken language is largely unconscious and very shallow.

To me, problem solving ability is precisely the same as the ability to articulate the problem and a solution. I don't see a major difference.

If you can solve a problem but you can't articulate what the problem is or why the solution will address it, I wouldn't call you a good problem solver. If you can articulate the problem well but not come up with a solution, you're already doing better than a lot of programmers in the world, and I'd probably prefer working with you over someone who presents the solution without "showing their work".

In fact, what is problem solving without such articulation? It's hard to even grasp what the skill means in a raw sense. Arguably creativity in this context is just the ability to reframe a problem in a more approachable manner. Many times, if not most times, such framing implies some obvious solution or sets of solutions with clear tradeoffs.

Edit: added a definition of apriori knowledge.

Edit2: to put this another way, nobody is arguing that recursion doesn't exist. Or that it is empirically-derived. No, it's a useful construct to show certain relations.

Edit3: added a sentence

Edit4: The extent to which our own grammars are inherently recursive vs this being culture or technology is irrelevant to identifying the concept of recursion as an apriori, linguistic concept.

Edit5: i suppose you might also be referring to the idea that we naturally process recursion. I mean, we clearly, evidently do; whether or not that's inherent to being human is a separate question entirely. Hell in the free software world there's a whole recursive acronym meme that taps into some part of our brain and tickles it.

Anyway, both computation and math are grouped under "apriori" knowledge. Any semantic distinction is ultimately silly. But we could just as easily be teaching programming as a craft in the context of the real world—I think this is closer to how it's done outside the US. I am not at all convinced the American style is what people ought to be paying for.

Oddly, I also use spatial intuition when thinking about stuff like stacks and the shape of data structures.

He discusses how motivations fundamentally change after profit margins peak and argues that profit after this point necessarily produces less economically efficient processes (in terms of the flow of demanded goods and services, not the shareholder, who does receive further productivity gains). In this context, undesired market advantage (legal or illegal) is just a symptom of profit, but it can be analyzed like any other sort of inefficiency. The core prescription is to nationalize the process or otherwise remove the profit motive around peak profit (which is, to be clear, not always easy or possible to identify... but in the worst case, this would open further opportunity for private capital to invest in the next generation of improvements).

He also discusses how failed investment cycles can resonate into market crashes faster than capital can rationally reallocate. Both of these above observations seem very very relevant to our current situation in the US today, and should cause everyone to look askance at people who aren't concerned about how healthy our political economy actually is.

> fundamentally undermined by science and technology creating a new market that upends the old one.

The kicker here is that there's no reason to expect either to continue yielding the same rewards. Some industries have projectable, plannable, investable growth patterns; others do not. Almost all the industries with predictable growth rely on consumption driven by yields from the lucrative exploitation of frontiers, mostly technological recently as you point out. I suspect that the market is going to get incredibly volatile as capital sees this frontiers dry up and adjusts expectations. ideally by cannibalizing itself and not eating us, but it might need a little help in that regard....