It appears like they're trying to make an industry that works outside of conventional bounds. Just like with private military contractors. Not quite privateers like in the 1630s.
I don't disagree with the result but the argument may be flawed. How do you know it's not the other way around, that richness causes having rich friends, rather than rich friends causing richness?
I know this is an implementation bug not a meta-theory bug, but I'd almost consider the fact soundness bugs are possible as a bug in the ideology, or at least a severe drawback. Stuff like this just wouldn't happen in Metamath. In a future where AI is autogenerating formalizations, why not have the AI use a harder but airtight system like Metamath?
That's with the original C verifier only. The actual database is cross-checked by 6 independent implementations. This is the whole point of Metamath: its kernel is so tiny that you can implement a verifier in a weekend.
Metamath isn't a silver bullet in the design space of formal proof tools, but I personally think it just about nails the metatheory we want. Maybe some explicit facility around definitions would be desireable.
As it happens, the Python verifier mmverify.py has an even simpler soundness bug [0], and so far I've reviewed two independent AI-written verifiers that have replicated that bug, since they apparently really like to copy the strategy from mmverify.py. A sound verifier isn't too difficult to write in terms of architecture (I just wrote one myself for differential testing [1]), but it requires some close attention to the details.
That is to say, the Swiss-cheese approach definitely lends authority, but individual implementations are unfortunately not as foolproof as they're made out to be.
Nothing more complex than Hello world is easy to implement in C without risking mistakes in checking boundary conditions if you are not really careful.
The utility of autoformalization is not actually in confirming the correctness of human-checked results. Mathematicians have a pretty good peer review process. I'm not actually aware offhand of any mathematical results that were accepted and later found to be incorrect -- though I'm sure cases exist, it's astoundingly rare.
The mathematics community's motivation for formalizing problems like Maryna Viazovska's sphere packing results in Lean weren't because the results were in doubt -- she won a Fields Medal for it, it's an extremely examined proof -- but because formalizing those results would lead to a lot of interesting and useful mathematical objects needing to be formalized as a prerequisite, which could then be merged into Lean's Mathlib and become useful for anyone working with Lean, particularly students. Having a library of idiomatic proofs available in a formal system capable of checking your work is Really Cool! Working in a proof assistant is a great way to develop mathematical maturity, especially for people who might not have an undergraduate education available to them.
(Unfortunately in the sphere packing case, the research group working on it made the mistake of trusting one of the various "AI for Math" slop companies, who promptly rugpulled them: https://arxiv.org/html/2603.03684v3)
So the short answer is "it depends on what you want". Lean is an eminently usable system for humans and LLMs alike; Metamath is uh. Not. But yes, Metamath seems to have a more trustable kernel wrt. the independent verifiers, so if that's all you're after it would be a better pick. But... there's only so many bugs Lean's kernel can have. At some point, they'll all be found.
(I should note that there is also utility in confirming the correctness of results that are too complex to be adequately verified by humans. The recent result that NF is consistent wrt. ZFC, among others (Peter Scholze's Liquid Tensors project) are examples of this. But that's distinct from "autoformalization" as commonly applied.)
It might not be clearly worded, but 132.5 watts is the power used by 2 incandescent light bulbs, whereas 918 watts is the power present in the light produced by 765 light bulbs. The factor of 50 comes from the fact that incandescent lightbulbs are only ~2% efficient at turning power into light.
An overwhelming number of Christians do not believe this and do not interpret Revelation in this way. The belief that "Jesus will come back and solve everything instantly" is only held by a minority of small 2nd Great Awakening groups and a few outspoken Evangelical groups, and is unique to theological traditions that arose in the United States. The official teachings of the Catholic Church, Orthodox Churches, Anglican and Episcopal communions, Presbyterian denominations, etc and other mainline Christians don't believe this. Before you blame a group for ruining the world or rejecting climate change please actually talk to a representative sample of real people in the group you are making claims about.
Due to the southern strategy that has been cemented into 1/2 of the available political parties for the better part of the last 50 years, outspoken evangelical groups hold an outsized influence on policy at large. Evangelicals are also the largest donors to AIPAC, which heavily influences our society.
It doesn't matter that most Christians don't believe it. Most Muslims don't do violent attacks on civilians, yet they all seem to pay the price either in their home countries or in the US.
It seems world-wide, a hard-right extremist subset of religious believers are ruining things for everyone else.
> An overwhelming number of Christians do not believe this and do not interpret Revelation in this way.
It's actually not that particularly relevant what Christians believe exactly (and they do believe that the world will be destroyed as-is), what's relevant is what the actual crazy book says.
And if you believe the book that says crazy things, or perhaps you cherry pick and interpret the crazy things to your liking and convenience - his point stands.
And the crazy book days say that the non Christians will burn (for eternity?) in flames of hell and that great many Christians won't be able to get through the narrow gate (and also be tortured to death in hot lava).
An overwhelming number of Christians with relevant political and economic power believe this. Orthodox, Anglicans, Episcopalians etc. might as well be a rounding error compared to the influence of American Evangelical and Conservative Christians. Might as well bring up the Amish while you're at it.
You might have a point about Catholics in terms of sheer numbers, but... the only people still listening to Catholics are right wing chuds and Warhammer fans - and even the former aren't listening anymore because the Pope is too "woke" for them. The days when the Pope's word mattered are long, long gone. What happened when he gave his encyclical on AI? It made headlines for a day or two then the world proceeded to ignore the silly hat man as usual.
So I stand by my point. The Christians who matter as far as climate change policy is concerned don't care because as far as they're concerned this world just exists for God to burn anyway.
I know nothing about economics or finance, but how is it that the first $1T dollar company was Apple and that didn't happen until 2018 and now it's a $5T dollar company just 8 years later. Did Apple really become 5x richer in that time?
Apple seems to be well positioned to profit from AI without investing to much. Other major companies are very AI exposed and people like Apple positioning.
Generally lots of money flowing around bidding up the top companies.
Ok but like how has their valuation gone up 5x? I don't see how people betting their money and valuation going up is related. Sorry if this is a stupid question, I really don't know anything about the stock market.
