So that is kind of the point of studying maths right?
Why something in unsolvable or undecidable can be as important as the output of a theorem.
Questions like these, fields medal level problems or Karp’s 21 NP-complete problem are problems working mathematicians are interested in.
Will LLMs help as an human assistant in the future? Probably.
Will LLMs answer these questions themselves, provide insights and bounds to these new mathematics and teach other mathematicians why this new math they create is true?
Will these models have phds and take candidates teaching them how to apply and think about the maths problems they are interested in?
For myself it was learning what a limit is in calculus, then learning about vector spaces, then learning about metric spaces and then learning about different topological spaces.
Then I had to relearn how a limit worked.
From a proof with epsilon delta inequalities.
To a proof with showing for some n dimensional metric spaces that has all the properties needed to converge does in-fact converge. Finally to a proof that for any space that is metric there is an isometric function into that metric space that also converges.
And that does touch measure theory, functional analysis or set theory. So there’s still so so much more for me to learn.
At the same time if you imagine a machine that can associate different maths. Would said machine encounter undecidable statements more frequently?
Would the rules of said machine have statements they themselves cannot prove by parameters set in their ‘programmed(by humans, machines, or other machines)’ assumptions?
It’s not that. Consider the definition of the limit. The idea existed for a long time. Newton/Leibniz had the idea.
That idea wasn’t formally defined until 134 years later with epsilon-delta by Cauchy. That it was accepted. (I know that there were an earlier proofs)
There’s even arguments that the limit existed before newton and lebnitz with Archimedes' Limits to Value of Pi.
Cauchy’s deep understanding of limits also led to the creation of complex function theory.
These forms of creation are hand-wavy not because they are wrong. They are hand wavy because they leverage a deep level of ‘creative-intuition’ in a subject.
An intuition that a later reader may not have and will want to formalize to deepen their own understanding of the topic often leading to deeper understanding and new maths.
If you are in the US. Proportionate representation stopped completely with the Reapportionment Act of 1929.
Subsequently the tail end of the gilded age and enacted in June 18, only 5months before the crash of oct 1929.
Constitutionally the size of the US government was expected to scale proportionally with population and 3/5ths of slaves.
This is why your vote ‘feels’ meaningless. We have been under a state of corporate capture for coming up to 100years. Last time there was push back from congress we got the Powell memo. That memo reinforces and defends corporate power in American politics.
There’s a bit of a double edged sword here. Removal of taxes leads to more coupled private/government relationships. Where external interests fund politicians to protect their own monopolistic interests. That money needs to come from somewhere. Wealthy people don’t want to pay taxes. But they sure do like to pay money to members of the government to further their own interests. Remember it was about taxation without representation. Wealthy people pay a lobbying ‘tax’ and get representation. In this way they are just paying a tax in a different way, where they get benefits and you get very little to nothing.
Yeah that’s the trade off of this implementation. Lobste.rs already uses this implementation
https://lobste.rs/about#invitations
The comments are considerably better. I’m not even a member but get more out of reading those comments than hn, and I’ve worked at multiple YC’s. This place is not what it used to be.
Compilers can never be error free for non trivial statements. This is outlined in Rices theorem. It’s one of the reasons we have observability/telemetry as well as tests.
There are some numbers that are uncomputable in lean. You can do things to approximate them in lean however, those approximates may still be wrong. Leans uncomputable namespace is very interesting.
There are other bounds here at play that are often not talked about.
Ai runs on computers. Consider the undecidability of Rices theorem. Where compiled code of non trivial statements may or may not be error free. Even an ai can’t guarantee its compiled code is error free. Not because it wouldn’t write sufficient code that solves a problem, but the code it writes is bounded by other externalities. Undecidability in general makes the dream of generative ai considerably more challenging than how it’s being ‘sold.
LLMs are bounded by the same bounds computers are. They run on computers so a prime example of a limitation is Rices theorem. Any ‘ai’ that writes code is unable (just like humans) to determine if the output is or is not error free.
This means a multi agent workflow without human that writes code may or may not be error free.
LLMs are also bounded by runtime complexity. Could an llm find the shortest Hamiltionian path between two cities in non polynomial time?
LLMs are bounded by in model context:
Could an llm create and use a new language with no context in its model?
There still maybe some variance at temperature 0. The outputted code could still have errors. LLMs are still bounded by the undecidable problems in computational theory like Rices theorem.
LLMs and its output are bounded by Rices theorem. This is not going to ensure correctness it’s just going to validate that the model can produce an undecidable result.
Why something in unsolvable or undecidable can be as important as the output of a theorem.
Questions like these, fields medal level problems or Karp’s 21 NP-complete problem are problems working mathematicians are interested in.
Will LLMs help as an human assistant in the future? Probably.
Will LLMs answer these questions themselves, provide insights and bounds to these new mathematics and teach other mathematicians why this new math they create is true?
Will these models have phds and take candidates teaching them how to apply and think about the maths problems they are interested in?