Also, it doesn't make sense to price homes in absolute terms. Surely relative to wage would be a better indication of the health of the market. Unless we can bring the wealthy to heel of course.
Can you come up with some conception of recursion that doesn't involve symbols referring to themselves, directly or indirectly? Ie what is left of recursion when you remove the linguistic component?
I agree that there's enormous value in carving out mathematics from other linguistic reasoning, but I don't see defining as something as mathematic rather than linguistic is generally useful. You use the same skills to look for incoherency in both situations, but human language is generally expected to be incoherent on some level.
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.
> We don't call this recursive... it's just incrementing x
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.
That's fair. I agree that there's more to problem solving than just linguistic ability, so I rescind my claim that they're indistinguishable, but I still think there's a deep relationship between the two.
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.
There are two ways that recursion intersects with language that are relevant here:
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.
Recursive functions are just a subset of all possibly recursive concepts. In the case of human spoken language, the recursion exists in our characterization of the grammar. You could just as easily frame this in "iterative" terms just like you can make any recursive function iterative, but that's less convenient for analysis.
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.
If you define recursion as a symbol referencing itself, either directly or indirectly, and if you define language as a system of relating symbols to each other, recursion is a linguistic concept, it is a concept that describes a relationship between symbols. There are good reasons to define each concept differently, but if you identify recursion empirically, recursion won't "actually" exist outside of the description of the process. It's our characterization of the process that reveals the recursive structure, even if that characterization doesn't actually exist outside of language.
To be clear, the symbols themselves don't bother me so much as trying to refer to them in spoken english. I have no particular beef with the use of symbols in code, which can be quite readable.
> That problem solving skills are relevant is pretty obvious, but language less so.
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.
I don't really know what you mean by "conjecture", but I thought apriori was implied by positing it as a linguistic construct. "Fundamental" doesn't imply empiricism at all. All of apriori knowledge for a language is a set of all sets of coherent statements: the outer set represents a set of implied axioms required to make the statements cohere. Recursion just broadens the complexity of the statements you can express, but it's fundamentally a concept that arises from language and can be evaluated for coherency (like all other apriori concepts).
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.
I'd argue none of that math is really necessary. While I have used most of my classes at least once, it was never a barrier to advancement in my career. Hell you could say the same thing about any of the theory. Like yea it's cool I know what a "merkle tree" is but it ultimately is a distraction from most of the skills you need to work with git.
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.
I can't speak for other forms of FP, but symbol operators make communicating about haskell very annoying. Outside of that FP seems to be doing fine, IMO.