We don't understand why? Reality is mathematical. As evidenced by lawful physics.

I have never heard this theory that mathematics is finishable before.

Arguably language itself I a cognitive bias.

There is a shorter proof but since thinking ossified in the 20th century we won't be sociologicaly ready to accept it at this time. Much of math is playing according to arbitrary culturally enforced rules that are not natural in the sense of being minimum logical requirements. Take the axiom of infinity or the axiom of choice for example. Fundamental math need not be based on zfc but that is what we have chosen as our foundation because we elevated continuity, infinity to ontological higher status than distinguishability. In the past similar cultural barriers were present in math for example imaginary numbers are so called because the name originated as derision. It seems unlikely to suggest that math today is not similarly culturally constrained in certain areas and some things we find confounding are more so due to our choice of foundation than their intrinsic nature.

I don't understand what you are trying to say. Which of the following is it, (or is it something else entirely)?

1. There is a much shorter proof that would also be accepted by lean, we just aren't thinking about the problems in the right way so we can't find it.

On one level this is obviously true, Anthropic did not put any effort in to minimising the length of the proof during its development or afterwards.

2. There is a much shorter proof if we took different axioms instead of the ones built into lean.

I find this much harder to believe, unless your new axiom is basically just FLT. Otherwise all reasonable axioms are not too hard to show as equivalent to each other (in terms of what they prove in PA anyway), so such an equivalence proof would be a small portion of the 13 million lines of lean.

https://en.wikipedia.org/wiki/Proof_theory is pretty much about this.

What is that shorter proof and how does it work? Is there a layman-accessible version?

Someday. It is has to do with degrees of freedom and information encoding in terms. Stop assuming operations are external but consider them as relational degrees of freedom of a logical statement. Different complexity statements can support different complexity results.

Muggles? You me unskilled overpaid llm prompters who are amazed by the world of real work?

We can't all be wizards at everything, as evidenced by the electrical "surprises" I keep uncovering in this house I own that was built by a plumber.

> you got to figure slope and elevations

I mean, you should... but there's still a lot of wiggle room on the way from doesn't meet modern code / never met code / shouldn't be done / is unsanitary or unsafe / will not work.

Residential electric is easier to do in ways that work but are either not to code or not to common best practice. The handier the previous owner was, the more new ways to do things you can learn. If a previous owner was in the trades and also heavily involved in the initial build, expect lots of weird repairs. :p

Again electric is not complex.I would say modern residential wiring is pretty fool proof. The majority of the work involves stripping wires and tightening screws. 120v is quite low vs street or transmission line voltage. Yes mis-wiring can kill you or burn down your house but it really takes some effort + stupidity. Nail into a wire, arc fault breaker trips, similiar for other faults. The electric code is far more nanny than plumbing code. Sure if you run a 40 amp dryer off 14 gauge romex with a 40 amp breaker you may have a fire but more likely the wire gets hot maybe melts the insulation and shorts and trips the breaker regardless. everything in wiring is over engineered to prevent fire and death even in the case of improper installation or damage. Meanwhile let's say you plumb a gas line wrong and gas leaks into the house and blows up or boiler explodes from lack of functioning pressure relief valve. Electric has defense in depth as it has layers of fail safes.

Oh... You're talking about plumbing gas lines? Gotta be real careful with the slopes on those, yeah?

Did see the coast? Pray tell where outside of Russia did vikings live in a non martime environment? Have you seen a map of scandanavia?

I live in Denmark. So yes. I have seen a map of Scandinavia. People didn't travel. Even here in Denmark which is tiny compared to Norway and Sweden, if you only ever go in a 10-15km radius of your home then there is a good chance you won't see the ocean.

This is true. I concede. However I am not sure that the majority of people were as imobile as you say even in mideveal times there were religious pilgrimages, fairs. Maybe the lowest stratum of non free society lived in a land locked mannor and never saw the sea in Denmark but I wonder if it was the majority of people in a lifetime?

This site is a fantastic demonstration of what crap AI is when operated by nit wits. Notably the drawing is not in line with the actual population distribution over time. Pre 1500 years are vastly over represented. If the point is draw a random year without regard to population distribution then the site is pretty pathetic. I would be embarrassed to post this on HN.

"I drew a woman born in 715 CE in the Yangtze Basin. It claims that 96% of women were married, the average age of marriage was 18, and 44% of women died before age 15. Clearly these facts can't all be true simultaneously." Umm yeah they can be true. Dead people are not counted. The straightforward interpretation is 96% of women living to 18 years or greater marry at some point in life. However the app has bigger problems, namely it does not seem to select from the distribution correctly, pre 1500 lives seem to be vastly over represented.

Yes, if a human told me this collection of facts I would assume they failed to properly explain their methodology. Given that the LLM is just making up numbers (see the "sources" page, where these are listed as "attributed",) there isn't really a methodology to explain in this case.

I am almost certain this is miscoded. I drew about 20 lives and I think 17 of them were before 1500. Meanwhile the vast majority of the integral of population over time is right of 1500.

Fundamentally physics is what logic permits. You can't falsify ontology from the inside. Kind of like proving a negative. These evolutionary universe theories take analogy too literally. You can not have selection outside of time. What you can have and what our lawful, conservation abiding physics suggests is a natural (mathematical) fixed point outside of which alternatives are zero measure. Liebniz was right this is in a sense the best of possible worlds. It is funny to me people have no problem accepting mathematical constants as natural (e.g. e, or pi) but assume that natural constants are not actually mathematical equilibria but instead selected contingencies. What in the world would be doing the selection? This is god of the gaps applied to science: we can't accept there is nothing more to reality than persistence of non contradiction so we reach for mystery or brute facts. Meanwhile in another area stuff like the mandelbrot set or even the distribution of primes show that great complexity can arise from simple rules or constraints This may not be falsifiable, as no view from no where is attainable but it is logical and parsiminous. What can persist across scales (renormalization) is physics is math is the world.