gottheUIblues

I think lean can verify it implements its own rules, but not that its own rules are sound (you can't prove a contradiction). If you are just prepared to trust that the rules are sound, then you would be able to trust leans implementation (if you had that Lean proof that it implements its own rules)

I think people on here tend to somewhat fixate on the determinism issue. Even with a deterministic LLM - stabilising the floating point arithmetic, and choosing from the distribution by a fixed method, or just save the random seeds - there is still a kind of a chaotic unpredictability that can exist between its inputs and outputs. However maybe that is a price that needs to be paid to get creativity.

Did any mathematics done by G H Hardy turn out to be quite useful in the end?

Ironically of course the answer is yes.

Yes indeed, it's about the time sound takes to travel 4mm

Bill 8 months at $8.33 and 4 months at $8.34?

No they don't - eg https://en.wikipedia.org/wiki/A_History_of_Britain_(TV_serie...

The dichotomy is more 'human understood' vs 'not human understood', I think

That was exactly my thought - taking out the problems that PhDs and early stage researchers work on kills the pipeline of developing mathematicians

It's an article written by a guest contributor - not an editorial written as the voice of the newspaper

Is the word "nonautological" autological ?

This is only the "problem solving" side of mathematics. Completely missing the theory builders who have completely reshaped the world of mathematics (and far beyond). AI is a long way from matching original thinkers of the calibre of Euclid, Al-Khwarizmi, Newton, Leibniz, Euler, Galois, Riemann, Cantor, Hilbert, and Grothendieck. Or Turing, Gödel and Von Neumann?

The normal term for the logic that has two truth values, the law of non-contradiction, and the law of the excluded middle is 'Classical Logic' which dates back centuries before George Boole. Non-classical logics which deny one or more of three properties that I've mentioned also exist

Rather COBOL is a living fossil? And today's Fortran is the FORTRAN family with horizontal gene transfer from the Algol lineage of programming languages.

Err? Peano Arithmetic is provably consistent in ZFC, but it is not in itself (if PA is consistent). Therefore if PA is consistent it is not equivalent to ZFC (regardless of whether ZFC is consistent or not)

"Embarrassingly" considered harmful?

To the contrary (as summarised by Gemini):

Gödel showed that arithmetic cannot prove everything about itself.

Turing showed that computers cannot predict everything about themselves.

Rice showed that we cannot automatically verify what programs will do.

Chaitin showed that mathematics is full of random, unprovable facts.

Lawvere showed that they are all failing for the exact same structural reason!

These are not fringe issues. They define the absolute boundaries of human and machine intelligence.

I don't think of 9 as really being a signal to the process at all, more of an instruction to the OS kernel to terminate the process

If that theory holds - have to ensure that the models have not been trained on any code that is licensed incompatibly with the GPL, in which case the models could not be distributed at all

So specified .. that it can actually prove it can't be completely specified by any single specification

Right so strictly speaking C++ could do anything here when passed a null pointer, because even though assert terminates the program, the C++ compiler cannot see that, and there is then undefined behaviour in that case

Shouldn't that be the "dam spelling" then?

Film score composers are quite famous for borrowing from 12 tone serialism - quite a bit of discussion on it available by Googling or using your favourite chatbot

Money is a social construct, not some kind of physical quantity subject to conservation laws, and can be and is introduced into the economic system all the time. The real question is really would introducing more money or a UBI cause social disruption by e.g. disrupting price signalling by high inflation or changing incentives to work so less goods and services that people actually value are produced.

A trit is log(3)/(8log(2))=0.19812031259014 of a byte

The verifier doesn't need to be deterministic, just to output a proof artifact that can be independently validated for correctness.

There are infinitely many different p-adic completions of the rationals for each prime p, so we have 2-adics, 3-adics, 5-adics etc, all different.

It's topography not topology! Would be funny if a country has a government department dedicated to the mathematical field of topology!

Except protium, deuterium and tritium

Netscape open sourced to try and head off Internet Explorer as a 'last throw of the dice'? Chrome based originally on WebKit from Safari which was forked off KHTML from KDE's konqueror, came along years later.

I wrote as my comment kind of as a rhetorical question, but perhaps in retrospect the connection between the string comparison problem the pigeonhole principle is not so surprising in the end.