[seqfan] Re: The Busy Beaver Game Illuminates the Fundamental Limits of Math | Quanta Magazine

[Allan Wechsler]

I share Jack Grahl's frustration. The Busy Beaver problem is indeed a sort
of holographic summary of a wide variety of mathematical results, some of
which are currently unknown. So if you had a magic oracle, or a
super-mathematical technique, that could give you the value of the Busy
Beaver S and Sigma functions (maximum number of steps, and maximum number
of 1's left on the tape, respectively), you could use that data to resolve
(in principle) many vexing problems. But what it really means is that we
won't know the values of S and Sigma *until* we resolve all those problems.
Finding the magic oracle is strictly harder than solving the individual
conundra.

What we *should* be doing (as Scott Aaronson, a very smart guy, points out)
is digging up little tiny Turing Machines, with, say, six to ten states,
whose behavior is enigmatic, and mining them for interesting mathematical
conjectures. It seems very likely, for example, that the Collatz Conjecture
is equivalent to the halting problem on some six- or seven- state TM;
finding that TM would be interesting, but having other enigmatic machines
suggest other conjectures would be even more interesting!

On Sun, Dec 13, 2020 at 5:01 AM Jack Grahl <jack.grahl at gmail.com> wrote:

> I enjoyed the article, but found it frustrating that yet again the Busy
> Beaver problem is presented as a sort of 'cheat code' for mathematics. Find
> BB(27) and then the Goldbach conjecture will be trivial, etc. My
> understanding is that it's actually the other way around. To find BB(27)
> you have to actually prove or disprove the Goldbach conjecture, as well as
> countless other very similar, but less interesting problems.
>
> On Sun, 13 Dec 2020, 07:52 Bob Lyons, <boblyonsnj at gmail.com> wrote:
>
> > Dam good article on the Busy Beaver problem:
> >
> >
> >
> https://www.quantamagazine.org/the-busy-beaver-game-illuminates-the-fundamental-limits-of-math-20201210/
> >
> > Here’s the Busy Beaver sequence:
> > http://oeis.org/A060843
> >
> > Bob
> >
> >
> >
> > --
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>