If prison was an option, it would have already happened. They will get away with it. Dems aren't going to be able to break this administration. They will just ignore congress. What will congress do?

Even worse they (the GOP) is just lighting everything on fire now and leaving it in a dysfunctional state on purpose. Because they know for the next two years voters will just start blaming the Dems for everything that goes wrong. Trump will say "I would fix it but the radical left Democrats are too busy persecuting me. High gas prices are their fault, and Zelensky's etc"

What's happening in Brazil right now should give people pause about thinking that an interregnum with someone left-ish in power trying to restore normality will reset things to a saner state.

These are people totally willing to burn the crops and salt the fields. Trump could die tomorrow and a thousand heads would pop up looking to benefit from the anger and resent.

Exactly. Almost be better if Dems lost. Now they will be a ignored, and a target for blame, an excuse.

Of course, have to be careful. You might get targeted. People posting online are being targeted by official gov agencies now. Freedom of speech is kind of gone now.

"It's illegal in the United States to form or participate in a mob that interrupts worship service"

What if it is a mob of religious worshipers that are participating in the mob?

A large percentage of Black Mirror episodes have now come true.

that murderous robo-dog will be happening shortly

Exactly this. Deep Seek hasn't caused a lot of knew talk, but it has silenced the talk about slowing down. An absence is harder to notice. But also the news cycle is so fast, its tripping over itself. So who knows.

It is. Maybe this is just executing one criminal to send a message to all the others. If they can take on MS, they can take on anyone.

Pretty tenuous connection to blame mathematicians for everything bad created in the world, that happened to use math.

"community building"

For what?

If math is just about having a community of other mathematicians to hang out with, it still isn't a career. Nobody is paying money you need in order to to eat, just to hang out in a community.

Just like a software engineer, "Well AI can write all my projects now, but I have my local Rust Users Group to hang out with". Nobody is paying me to hang out and hand code Rust.

Not a mathematician. Why not just always use LEAN? Why use natural language at all?

Because it is really hard to read and the level of detail is so high that even lemmas that you can read may have such enormous levels of detail that makes real understanding difficult given that humans have limited working memory.

Why not always write machine code? Why use programming languages at all?

If a programming language compiler isn't guaranteed to be re-producible, then yeah, you'd have to revert to machine code.

The same JavaScript code compiles to different machine representations under different browser engines; yet webdevelopers don't need to learn assembly.

There is an abstract javascript machine which can obviously be translated to hardware instructions. "Obvious" in the mathematical research sense, as in the statement is flat out wrong under closer inspection. Websites regularly crash under memory pressure.

But they are good enough most of the time, and that's the key. With more resource a more correct program can be created, but cost-benefit calculations show a ceiling. A local pet store does not have the money to pay 2000 hours of formal verification work, and usually a WordPress site is good enough.

Mathematicians work with their brains. They have finite time to understand papers. Formalists say that all theorems could be reduced to logical axioms, but most of time it's jumping at a much higher level, because no one has time for minute details. Proofs are theoretically right or wrong, but in practice there are "slightly wrong" proofs, where there are some minute errors that "feels" like can be correcte, and most of the time it can be. This ambiguity is not a problem, but a feature of mathematics, because it means more time can be spent to move faster at a higher level of abstraction, but this would be lost with Lean.

Isn't that the joke with Fermat's last theorem?

That he jotted in a margin that he had just seen a simple proof, and would write it down later. And we spent a hundred years trying to reproduce.

Or some other joke about a mathematician will put "and here we see it simply follows" or some such.

I'm all for abstraction, I get the point.

But for natural language it seems like " I saw my ex boyfriend last night and, yada yada yada, I'm really exhausted this morning".

See, the yada-yada is a bit too much abstraction.

I took the problem like this:

If we treat each human as a compiler of natural language, and there is enough ambiguity that each human arrives at a different result. Then is it really a proof?

Guess I'm missing the point of the paper, I thought it was trying to show that natural language is flawed and formalized systems are better.

And AI is getting tripped up by the human ambiguous natural language.

There are bugs found in programming language implementations all the time. I guess you are switching to machine code, good luck.

Yeah, and if you re-run the same program with the same bug, it will fail the same way each time. It is reproducible. What are you even trying to argue about?

If we treat each human as a compiler of natural language, and there is enough ambiguity that each human arrives at a different result. Then is it really a proof?

> Yeah, and if you re-run the same program with the same bug, it will fail the same way each time.

[Citation needed]

Got to thinking maybe you were still in school and hadn't covered determinism yet. Here is old school stack overflow answer that is short and clear.

The environment does matter. I think in your original objection, you were executing the code in different environments.

https://cs.stackexchange.com/questions/38152/what-is-determi...

And to the point of the original article. Each human interpreting natural language is a new environment. Each brain is a new compiler with different outcomes. Thus the ambiguity. Natural language isn't concrete enough to always have the same result when processed by different humans, or AI.

The top reply is already all you needed:

> the answer with many multi-threaded programs is often "no."

And, nobody would correct them? If a programs output is random, what use is it.

Isn't the original point, that natural language isn't sufficient for proofs? Aren't you agreeing now?

Citation

2 + 2 = 5 -> Error

Every time.

Lean is a write only programming language.

Same reason humans write code not only for a compiler to translate into machine code but also so other humans can understand what we write, learn from it, modify it etc...

A programming language, when compiled, is a guaranteed reproducible result. If you recompile a program, you get the same thing each time.

The point of the article is that natural language is not these things.

Not only that, we also have code comments and standalone documentation.

Because people need to understand what is being proven.

The example in Figure 1 should help understand why... the NL version is much more approachable for humans.

If its ambiguous or wrong, then what are you understanding ?

Is not a natural language (NL) proof a demonstration of mastery and understanding?

If you understand the Lean, then you can create a NL proof. The LLM clearly doesn't understand the Lean code it produced.

I guess I misunderstood the paper, that natural language proofs are not mastery/understanding because they can be ambiguous, miss-interpreted. Natural language can be misinterpreted enough that it is not a given to be correct. And AI was misinterpreting what a human meant in the ambiguous natural language.

I was using a lot of dashes before AI. What do I do now? -- Change?

AI has gotten to the point where there really is no telling.