It’s noteworthy how this conversation—centered on Busy Beaver numbers, Turing machines, and the difference between finite and infinite time steps—remains tightly focused on the immediate computational (or at times proof-theoretic capabilities). In doing so, the you guys bypass the metaphysical dimension. You speak of logical formalisms (e.g. “FOL is weak”) as though they were only mundane specs—like naive users at a car dealership comparing horsepower without truly appreciating the engineering brilliance and fine grained peculiarities of every machine.
In truth, the “weakness” you ascribe to simpler formal systems—such as first-order logic without heavy axioms, or minimal forms of arithmetic—can lead to fascinating nonstandard models and subtle phenomena that arguably eclipse the raw computational yardstick you employ. The entire interplay of Gödelian incompleteness, mathematical logic, and Turing universality tells us that these systems are far more than coarse “engines” to be tested on narrow tasks mentioned here: they can reconfigure our understanding of concepts, the scope of our ideas, their granularity, and the nature of mathematical objects and properties. My concern with this purely instrumental, pragmatic, or practical approach isn’t about moralizing from on high. Rather, it’s that when you reduce mind-created constructs—especially those relating to numbers and formal proofs—to mere mechanisms, that can serve your purposes or not, we risk missing the extraordinary potential these constructs unveil about cognition (and therefore possible reconfigurations, evolution, and redefinitions of your purposes) itself. The mind that conceives these systems is neither a mechanical nor a trivial entity; it is both the architect of these tools and a citizen in the realm they inhabit. A thoughtful engineer, for example, goes beyond raw statistics of horsepower and gear ratios: they explore the design’s elegance, its broad possibilities, and limitations; understanding them as integral parts of a nuanced and creative learning process. Likewise, a reflective view of logic, arithmetic, and Turing machines recognizes that these abstractions are not just gadgets for “what we can do now,” but signposts to deeper questions of knowledge, limitation, and possibility. When interpreted with care and openness, instead of pretensions to expertise and facile reductionisms and generalizations, these constructs don’t merely compute or solve ephemeral problems; they remind us that the environment (whatever it may be), mathematics, logic, and mind are interwoven in ways that go beyond simplistic instrumental gain. And no, I don't feel a need to trumpet my approach. But I will raise and eyebrow and go: "If you're an expert in this, then show us what the f you got.'; not even fields medalists will run around claiming universal expertise in these vast fields. And this reason is why voting for populist authoritarians is worse than voting for somebody that holds themselves somewhat accountable to a a set of vague, corrupt theories: if any of them goes wrong, who is likely to change their mind and who will keep making godlike pronouncements to cover up their shit and manipulate people into believing that their shit is gold. After enough repetitions, they believe it; they'll print it and see a transperson winning a medal and going to the bathroom as a larger evil than the inequality/unfairness/injustice they actually do oppose. We don't deserve or can't even conceive of better politics and politicians if we don't afford ourselves that basic decency in our reasoning and minds. But no worries, I am starting to pick up that this is the wrong group for casual banter around ToE. On Monday, January 20, 2025 at 12:38:05 PM UTC+1 John Clark wrote: > On Sun, Jan 19, 2025 at 11:07 PM Jesse Mazer <laser...@gmail.com> wrote: > > *> Do you know if there were only 28 non-halting five-state machines >> total, or were there a lot of others but they were more easily proved to be >> non-halting, and those 28 were the last holdouts?* > > > *Most obviously would go on forever or they halted but produced fewer than > 4098 1s, those 28 were the holdouts but eventually it was proven that they > too would never halt. Many people suspect that we will never know what > BB(6) is because it will turn out to be uncomputable, we know for a fact > that BB(745) is. * > > *John K Clark See what's on my new list at Extropolis > <https://groups.google.com/g/extropolis>* > 1po > > > > >> >> On Sun, Jan 19, 2025 at 4:16 PM John Clark <johnk...@gmail.com> wrote: >> >>> On Sun, Jan 19, 2025 at 11:18 AM Jesse Mazer <laser...@gmail.com> wrote: >>> >>> *>> I think Alan Turing's Proof that there is no general solution to the >>>>> halting problem and it's corollary that some things are true but >>>>> uncomputable, is more relevant than Godel's theorem. In general there is >>>>> no >>>>> way to determine if a cellular automation, such as Conway's Game Of Life, >>>>> will keep on growing forever, or start repeating itself, or die out >>>>> completely; all you can do is watch it and see what it does, and you >>>>> might >>>>> need to watch it forever. * >>>>> *And the first five Busy Beaver Numbers are 1, 6, 21 , 107 and >>>>> 47,176,870, and we know for a fact that even if we had a mechanical >>>>> computer that strictly followed Newton's laws and even if we had an >>>>> infinite amount of time at our disposal, we will NEVER find the 745'th >>>>> Busy >>>>> Beaver Number even though it's well defined and it definitely exists. The >>>>> same thing could probably be said about the sixth busy Beaver Number, >>>>> although we are not certain about that, all we know for sure is that it's >>>>> greater than 10^15. * >>>>> >>>> >>>> *> I gather finding the Nth Busy Beaver number requires knowing whether >>>> all possible N-state Turing Machines halt or not (I don't understand the >>>> exact definition of N-state Turing machine,* >>>> >>> >>> *A**n easy way to understand is to separate the program and the data in >>> a Turing Machine. The data is on the tape and the program is on a series of >>> cards that determine what state the Turing machine is in. You always start >>> at card #1 and, depending on if you are reading a 1 or a 0 , that card >>> tells you to print a one or a zero, move left or right, and the number of >>> the card to read next. A N card Turing Machine has N states plus a halt >>> state. (The information on the cards could have originally come from the >>> tape but we’ll assume that step has already happened.) For a start >>> let's look at a one card machine, there are 8 different actions a card >>> could tell you to perform if you first read a zero on the tape :* >>> >>> *1) Write a 1 or a 0.* >>> >>> *2) Move left or right.* >>> >>> *3) Stay on card#1 (the only card) or halt.* >>> >>> *So there are 2^3=2*2*2 =8 things you can do if you read a zero. But >>> there are also 8 things you can do if you first read a 1 on the tape, so >>> there are 8*8= 64 possible one card (aka one state) Turing Machines. * >>> >>> *Now let’s look at a 2 card Turing Machine, if you’re currently reading >>> 0 there are 12 things the first card could tell you to do; write a zero or >>> a one, shift left or right, go to card 1 or card 2 or halt, 2*2*3= 12. And >>> if you’re currently reading 1 there are also 12 things the first card could >>> tell you to do*. >>> *So there are 12*12=144 ways that the first card could be. However each >>> of those 144 cards must be paired up with a second card, and there are also >>> 144 things that the second card could tell you to do. So there are 144*144= >>> 20,736 different 2 card (a.k.a. 2 state) Turing Machines. * >>> >>> *The general formula is there are (2 N s +1) ^ s N Turing machines with >>> N states and with s symbols. A two symbol machine is the simplest and can >>> do everything a machine with more symbols can do so if s=2 the formula for >>> the number of 2 symbol N state (card Turing Machines) is [4(N+1)]^2N. * >>> >> >> Thanks, nice explanation--I was familiar with the general idea that >> Turing machines have internal states which tell them what to print, how to >> move, and what internal state to go into next, but seeing the possibilities >> listed this way makes it clear where that [4(N+1)]^2N formula comes from. >> >> >> >>> *So the number of Turing Machines increases exponentially with N, but >>> that's not the Busy Beaver function, BB increases far far faster than >>> exponentially, in fact BB increases faster than ANY computable function. >>> Starting with an all zero tape, BB(N) is the number of moves a N state >>> Turing machine makes while printing out the largest FINITE number of 1's >>> before it halts.* >>> >> >> Ah, so the newly confirmed BB number was the maximum number of 1's >> printed by a halting 5-state machine, not the maximum number of steps >> before halting as I had imagined--I see from the wolfram mathworld article >> at https://mathworld.wolfram.com/BusyBeaver.html that some authors do >> define them in terms of number of steps but it sounds like the definition >> in terms of number of 1's is more commonly used. >> >> >> >>> * Some Turing Machines print out an infinite number of 1's because they >>> never halt but they do NOT count, it must halt.* >>> >>> >>>> *> the fact that they found the first five must mean that for all the >>>> non-halting machines with 5 or less states, they were able to find proofs >>>> that they never halt?* >>>> >>> >>> *Yes. For several decades people suspected that the fifth busy beaver >>> number was 47176870 because they found a five State Turing machine that >>> after making 47176870 steps and printing 4098 1s it halted, but they >>> weren't certain because there were 28 candidate five state (card) Turing >>> machines that were still running and printed many more than 4098 1s and >>> MIGHT eventually HALT. But one by one people were able to prove that they >>> would never halt, the last one fell just a few months ago , so there's no >>> longer any doubt, Busy Beaver six is 47176870. Turing never said you can >>> never predict what a Turing machine will do, he said you can't always >>> predict it .* >>> >> >> Do you know if there were only 28 non-halting five-state machines total, >> or were there a lot of others but they were more easily proved to be >> non-halting, and those 28 were the last holdouts? The formula indicates >> there would be 24^10 different five-state machines, about 63.4 trillion, I >> guess I have no particular reason to expect any particular ratio of halting >> to non-halting machines but it'd seem sort of intuitively surprising to me >> if the ratio were so lopsided. >> >> >> -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To view this discussion visit https://groups.google.com/d/msgid/everything-list/eb52b07b-03b8-434f-b5cb-21eba4b28431n%40googlegroups.com.
