> The idea is sometimes expressed that instead of speaking of arithmetical statements as true or false, we should say that they are "true in the standard model" or "false in the standard model". The following comment illustrates:
>> This is the source of popular observations of the sort: if Goldbach's conjecture is undecidable in PA, then it is true. This is actually accurate, if we are careful to add "in the standard model" at the end of the sentence.
> The idea in such comments seems to be that if we say that an arithmetical statement A is "true" instead of carefully saying "true in the standard model", we are saying that A is true in every model of PA. This idea can only arise as a result of an over-exposure to logic. In any ordinary mathematical context, to say that Goldbach's conjecture is true is the same as saying that every even number greater than 2 is the sum of two primes. PA and models of PA are of concern only in very special contexts, and most mathematicians have no need to know anything at all about these things. It may of course have a point to say "true in the standard model" for emphasis in contexts where one is in fact talking about different models of PA.
This is hairsplitting. It's okay to say that there are true yet unprovable sentences without always carrying "true under the standard interpretation" around. It's understood from context.
Regarding the first point, Caplan personally estimates that about 80% of education is signalling and that 30% is the lowest figure one can plausibly maintain.
As for the second point, I wouldn't bet my life on it. But I think the case is strong enough that experimental measures in that direction are warranted.
In brief: there are strong reasons to believe that much of higher education and even large parts of school serve not to teach but to test students. An English major doesn't lead to higher pay because it makes one a substantially better worker. It does so because achieving it signals three primary traits to the potential employer: intelligence, conscientiousness, and conformity.
This isn't education's only function. Some learning undeniably still takes place. But in Caplan's estimation signalling is probably about 80% of the payoff.
This picture is supported by a large number of observations:
- Why do even top schools like Harvard make little to no effort to prevent non-students from attending lectures?
- Why do students cheer when class is canceled?
- Why does ratemyprofessor.com have the measures "overall quality" and "difficulty" but not an explicit "informativeness" measure and why is high difficulty considered bad?
- Why do students cheat on tests and why do teachers make such a large effort to prevent it?
- Why do employers rarely show concern that you might've forgotten what you learned?
- Why do statistics indicate that graduation year has a much greater effect on wages than all the other years?
All these points contradict the "education = learning" viewpoint but are straightforwardly explained with the signalling model.
And once you acknowledge the importance of signalling it puts statements such as
> And, a good education pays off even for less gifted people. Their lives are better, they contribute more to the economy and less to crime.
into a completely new perspective. As Caplan writes:
> The classic example: You want a better view at a concert. What can you do? Stand up. Individually, standing works. What happens, though, if everyone copies you? Can everyone see better by standing? No way. Popular support for education subsidies rests on the same fallacy. The
person who gets more education, gets a better job. It works; you see it plainly. Yet it does not follow that if everyone gets more education, everyone gets a better job. In the signaling model, subsidizing everyone’s schooling to improve our jobs is like urging everyone to stand up at a concert to improve our views. Both are “smart for one, dumb for all.”
As somebody who spent many years in education (as a student and about 1 year as a teacher) I find Caplan's case deeply convincing.
Caplan actually goes through reams of evidence. As an extreme example, in 2008-9 there were 34000 new history graduates in the US. But there are only 3500 historians working in the whole country.
Now, are you trying to argue that history can actually measurably improve productivity in other fields, such as accounting, and it's just the curse of knowledge that prevents us from seeing this?
Also, incompetent homeschooling parents and Scientology schools are hardly the only alternatives to public schooling. One option that Caplan advocates strongly is vocational training. Instead of giving them history classes, let teenagers who chose to do so become apprentice carpenters, there are 900000 of those.
"If T is effectively axiomatized, in the sense that there is an algorithm for deciding whether or not a given sentence is an axiom of T,..."
In Computability: Turing, Gödel, Church and Beyond Martin Davis even makes the definition "T is axiomatizable if it has an axiom set that is computable" on page 42.
And that's also the definition Shoenfield gives on page 125 of Mathematical Logic.
Besides, "effective" has always been a synonym for "computable", "recursive", or "decidable" whenever I've encountered it in this context.
Also, I must insist that "it makes no difference whether we require the axioms to be recursively enumerable, recursive, or primitive recursive." is not what Craig's Theorem says. This is dangerously close to claiming that recursively enumerable sets are recursive which is plainly untrue. If recursively enumerable axioms (not just recursively enumerable theorems) are indeed interchangeable with recursive axioms, then I'd like to hear more about that.
I don't see why there's a theorem required here. Let our formal system be effectively axiomatized. Then by definition its axioms are a recursive set. So I can just check a proof by starting at the top and verifying that every line is either an axiom or follows from a line above via an inference rule. Both of these two checks are recursive. Hence, proof checking is recursive.
Also, the Wikipedia article doesn't say that Craig's Theorem proves recursively enumerable axiomatizations equivalent to (primitive) recursive axiomatizations. I would find it very surprising if this was a consequence. It only says that a recursively enumerable set of formulas (e.g. the provable sentences in Peano arithmetic, not merely its axioms) can be given a (primitive) recursive axiomatization. That's a very much weaker claim.
I'm only assuming that checking whether "does s prove S?" is a recursive property. That's not the same as demanding "is S provable?" to be recursive.
Upon reflection I agree that effective axiomatization isn't actually that difficult to explain. So I've changed the footnote to say: "For the purpose of this discussion every formal system is effectively axiomatized by definition. This basically just boils down to the fact that proofs are computer checkable."
I was initially worried that this might raise the question what non-effective axiomatizations are all about. That's not a can of worms I want to open. But I think this should be fine.
I feel like I didn't argue this as clearly as I could have. Let me make one more addition.
Throughout the discussion I'm making the tacit assumption that there is one standard viewpoint to which we adhere. That's the "normal" world. Numbers are finite and halting programs halt after finitely many steps. It is from this fixed viewpoint that I'm declaring certain claims to be lies. They are not lies in some grand universal sense.
I realize that the word "lie" usually also entails an accusation of deliberate deception. But that's immaterial here. Formal systems don't have intentions. They simply make claims.
You can call it "similar but weirdly different" in the same sense that the people who are subject to propaganda live in similar but weirdly different realities. What is true depends on your viewpoint.
When a formal system says: "this computation halts after some number of steps", then under the default interpretation that means that after say 10000 steps the computation really halts. But in the "similar but weirdly different" reality where transfinite numbers exist the above claim can still be considered true if it runs indefinitely. One simply has to entertain the idea that "some number of steps" might mean a transfinite number of steps.
In other words, yes, we can say that the formal system lies provided we accept that what is and what isn't a lie depends on the viewpoint.
I agree in the cases where I argue "suppose, then..." But there are instances where I argue "either this is the case, or this is not the case." Unless there are multiple laws of the excluded middle, that's an application of it.
In any case, that was merely an example. The central point I was making doesn't depend on this.
Every time I said "either/or" I was making use of the law of the excluded middle. But I didn't add a detailed disclaimer explicitly highlighting this fact either. Should I have? I think it's okay to just take some things for granted.
Though, I admit that I have never spent much thought on non-effective axiomatizations. So I'm open to be educated.
Perhaps Gödel had a vague notion of what he was up to. Nonetheless, the whole concept of universal computation was still half a decade away and higher level languages several decades. So I think the remark is still essentially accurate. He had to work without all the modern cognitive conveniences such as for-loops and if/else branching.