This is an intriguing idea.

How do you set up the optimization problem, though?

The theory is that if an investor is under no compulsion to purchase shares that entitle them to no say in governance, so if they do so, they only have themselves to blame.

So much for theory!

Can you explain why your view, which is the opposite of the OP's, is correct?

"RL training directly raises the success probability , by spending a lot of compute on searching for successful paths and then training the model to reproduce them. The gains we credit to “thinking longer” come from the search we paid for during training, rather than at inference."

Not mutually exclusive. LLMs are kind of like a self-fulfilling prophecy. Once some technique is discovered, it'd be part of the training process and becomes more efficient.

Oh yeah? Please tell us which federal statute preempts Utah's gambling laws.

What is the unit for productivity here, though? Time required to complete this one task, which was never attempted before and never will be again? That's fine, but economically uninteresting.

Ultimately, the economic analysis needs to use monetary units. How much faster can return on investment be realized, and at what scale? This analysis has proven elusive because people believe that the very definition, not just the execution, of a task can be delegated to an LLM (which in some trivial sense must be true because means and ends are never sharply distinguished).

I'm more interested to know if LLMs lead to more intelligent decisionmaking than whether they boost "productivity" by any measure.

Because everyone in the world doing more of the wrong things faster isn't necessarily a good thing.

"Many of them have switched to gig work like Uber or Doordash, which makes them considered "employed" in employment surveys."

That's true for the classic "U-1" unemployment rate. But the U-6 captures the "marginally attached" as well, and it hasn't shot upward.

https://fred.stlouisfed.org/series/u6rate

Is that going to be mathematically useful?

It's nice to know that they didn't miss anything. The natural language proof is 10,000 pages and they have missed groups in the past.

Indeed, this formalization effort spotted a flawed argument in one of the original publications and worked around it to repair the result.

How do we know the formalization itself doesn't have any errors? AFAIK it's very far from trivial to formalize something.

Well, the point of the _computer_ formalization is that it's checkable.

An error in the formalization would be a bug in lean. Which, as we know, certainly seems plausible (I've mused on if these giant proofs are incentivizing LLMs to find lean bugs to exploit), but at least it's a smaller surface area.

An error in the formalization could also be that you formally prove something different from what you think you proved.

All that needs to be checked to avoid that is the statement of the main theorem. Any divergence further inside means you proved the main theorem differently than you thought you did -- but you still proved it.

This assumes the Lean kernel itself isn't bugged, but getting it correct is a matter that affects all uses, not just this one, so the cost of making it correct is spread over all those uses.

In general checking that the definitions are correct isn't that hard and assuming at least one of the lean kernels you verify with is correct (which tbf is an assumption, the implementations are based on each other so could have a common bug) that's all you need to verify.

In this case the statement is unusually complicated but still much much easier to verify than the proof itself.

So, the specification of the theorem we are fairly solid on, formalising what a group is, and a simple group, have been done a long time and we are confident in the correctness of.

Assuming you do trust those, then assuming the formalisation passes lean (and lean has no bugs!), then the whole idea is you've got a proof of the theorem.

The papers describing lean and it's core operations are deliberately small. The kernel of a theorem prover is meant to be such that a human can check it for correctness. The kernel does not include things like type inference or execution. It literally checks the proof and that's all and everything needs to match exactly. It's fairly trivial to write such a thing.

Humans have to establish trust in this.

After that, the rest of the language is written. Yes it is possible the type inference engine is broken but in that case the kernel will reject it

The proof was of such complexity and length there were lingering doubts. This dispels them.

More so if Lean uncovers logical errors. Note also there is already an ongoing 2nd generation proof.

I think the OPs argument (which I am not endorsing) is that they in fact enjoy a higher quality of life even if their "standard of living" (measured in purchasing power) is higher.

It's all subjective though. Asking most in a 'developed' country whether they'd trade their high rent for a dirt floor/house would result in an affirmative no. Same for health/insurance - "Would you trade your high premium for 'the nearest so called doctor is a days walk away'" is going to be a resounding no.

But then it's the same the other way around "Would you trade your dirt floors for linoleum if it meant you hard to do hard labor 40 hrs a week?" and "Would you trade your witch for the luxury of owing the doctor thousands of hours of labor?" would both also be resounding nos.

I know there is a growing contingent of those moving off of the grid and spurning modernity for log cabins built by hand, and the converse OLPC in dirt huts thing but for the most part everyone everywhere wants to have their cake and eat it too.

It does very poorly in detecting Dutch vs. German for this Flemish-accented speaker.

"The Electric Monk was a labour-saving device, like a dishwasher or a video recorder. Dishwashers washed tedious dishes for you, thus saving you the bother of washing them yourself, video recorders watched tedious television for you, thus saving you the bother of looking at it yourself; Electric Monks believed things for you, thus saving you what was becoming an increasingly onerous task, that of believing all the things the world expected you to believe." - Douglas Adams