On Wednesday, May 25, 2022 at 2:37:39 PM UTC-4, olcott wrote:
> On 5/25/2022 1:31 PM, Dennis Bush wrote:
> > On Wednesday, May 25, 2022 at 2:25:03 PM UTC-4, olcott wrote:
> >> On 5/25/2022 1:05 PM, Dennis Bush wrote:
> >>> On Wednesday, May 25, 2022 at 1:48:17 PM UTC-4, olcott wrote:
> >>>> On 5/25/2022 12:45 PM, Mr Flibble wrote:
> >>>>> On Wed, 25 May 2022 12:35:44 -0500
> >>>>> olcott <No...@NoWhere.com> wrote:
> >>>>>
> >>>>>> On 5/25/2022 6:37 AM, Ben wrote:
> >>>>>>> Andy Walker <a...@cuboid.co.uk> writes:
> >>>>>>>
> >>>>>>>> On 24/05/2022 23:59, Ben wrote:
> >>>>>>>>> Jeff Barnett <j...@notatt.com> writes:
> >>>>>>>> [...]
> >>>>>>>>>> I think, that upon reflection, you will recall that isn't true.
> >>>>>>>>>> I'm assuming that you are at least a moderate history of math
> >>>>>>>>>> (etc.) buff and have read "Proofs and Refutations" by Imre
> >>>>>>>>>> Lakatos. This book, as well as many others written by him or
> >>>>>>>>>> others, go through the history of mathematics and show examples
> >>>>>>>>>> where the whole community has bought a load of rope.
> >>>>>>>>
> >>>>>>>> Lakatos's work is interesting, but I wouldn't go overboard
> >>>>>>>> about it. It's "philosophy of maths" more than "history" of
> >>>>>>>> maths", and, AIUI, it's not so much a claim that any particular
> >>>>>>>> theorem is "wrong", rather that as maths develops, its language
> >>>>>>>> evolves and becomes refined. Of course there have been blunders,
> >>>>>>>> which may have lain undetected for a long time, but more important
> >>>>>>>> have been the "edge" cases, where what we mean by [eg] "function"
> >>>>>>>> has developed esp as we moved from the intuition of the 18thC to
> >>>>>>>> the rigour of the 19thC, so that theorems need to change to
> >>>>>>>> reflect the new language.
> >>>>>>>
> >>>>>>> I found the original essays that form the core of the book online,
> >>>>>>> and though I ave not finished reading, I think what Lakatos is
> >>>>>>> saying is somewhat tangential to my point. His main focus is on
> >>>>>>> the process of doing mathematics.
> >>>>>>>
> >>>>>>> Of course PO thinks (I should say now thinks because his claims
> >>>>>>> shift over time) that he has "refined" the idea of halting.
> >>>>>>>
> >>>>>>>> Re history -- I recall that Warwick ran a history module
> >>>>>>>> based entirely around the dead ends of geometry in the 19thC.
> >>>>>>>> Stuff that is now utterly forgotten but for a few years was all
> >>>>>>>> the rage. Had it not been so utterly forgotten, I'd be able to
> >>>>>>>> quote some examples. But it serves as a reminder that maths is
> >>>>>>>> not the "definitive" subject that is usually presented to students
> >>>>>>>> [from primary school to research] -- "Here be maths as it is and
> >>>>>>>> always has been ever since its discovery, and here be the unknown
> >>>>>>>> bits where research is ongoing" -- but contains any number of
> >>>>>>>> false starts and debates, many of which never resolve but just get
> >>>>>>>> quietly dropped.
> >>>>>>>
> >>>>>>> I think that's true of many branches of human investigation, but
> >>>>>>> since mathematics is the investigation of abstract "inventions", it
> >>>>>>> is even more prone to going off on tangents that turn out to be
> >>>>>>> fads. Then again, some fields come back to being centre stage.
> >>>>>>> Number theory, once little more than an amusement, turned into a
> >>>>>>> valuable area of research.
> >>>>>>>> Re Lakatos -- his Wiki biography is interesting, and as
> >>>>>>>> well as detailing his complicated private/political life says
> >>>>>>>> quite a lot about "Proofs ...".
> >>>>>>>>
> >>>>>>>>> I don't know that book. I'll see if I can get hold of it. It
> >>>>>>>>> sounds really interesting but, sadly, I no longer have access to
> >>>>>>>>> world-class library :-(
> >>>>>>>>
> >>>>>>>> As Jeff says, copies are available. Bookfinder lists 296
> >>>>>>>> copies for sale. But I find it hard to believe that you no longer
> >>>>>>>> have access to any library with a copy [or able to get one for
> >>>>>>>> you] -- it doesn't need to be world-class for that.
> >>>>>>>
> >>>>>>> I was just lamenting the loss as some places give retired academics
> >>>>>>> library privileges and some don't! The local council library
> >>>>>>> system is just about hanging on in most places in the UK so, yes, I
> >>>>>>> can get any book through the inter-library loan scheme. It's just
> >>>>>>> that waiting weeks is so last century.
> >>>>>>>
> >>>>>>>> Apart from that, you surely have some
> >>>>>>>> friends/acquaintances with access to a university library?
> >>>>>>>
> >>>>>>> Strictly against the rules, so I would not want to put anyone in
> >>>>>>> that position!
> >>>>>>>
> >>>>>>>> [...]
> >>>>>>>>> And of course the halting theorem and Cantor's theorem are so
> >>>>>>>>> simple they have been proved at a level of rigour that is quite
> >>>>>>>>> unprecedented, and simply not possible for the vast majority of
> >>>>>>>>> theorems.
> >>>>>>>>
> >>>>>>>> In the present context, that would be more interesting if
> >>>>>>>> PO had pointed to some flaw, or potential flaw, in the theorem.
> >>>>>>>
> >>>>>>> For a long time that was exactly his claim, though as you say he
> >>>>>>> never pointed out what he thought was the flaw.
> >>>>>>>
> >>>>>>
> >>>>>> The flaw is that no one ever bothered to think through the effects of
> >>>>>> a simulating halt decider.
> >>>>>>
> >>>>>> Instead of ever beginning this analysis simulation is simply rejected
> >>>>>> out-of-hand on the basis that some simulations would never end.
> >>>>>>
> >>>>>> A simulating halt decider recognizes the infinitely nested simulation
> >>>>>> behavior pattern of the conventional HP counter examples, thus
> >>>>>> aborted its simulation before ever reaching the "impossible" part of
> >>>>>> the "impossible" input.
> >>>>>
> >>>>> Those proofs DO NOT CONTAIN INFINITE RECURSION or an INFINITELY NESTED
> >>>>> SIMULATION. You are MAKING THIS UP AND IT IS ERRONEOUS.
> >>>>>
> >>>> When we assume that the halt decider bases its halt status decision on
> >>>> the behavior of its correctly simulated input
> >>>
> >>> By definition, the correctly simulated input is simulated by a UTM.
> >> The correctly simulated N steps of the input to a halt decider would be
> >> performed by a UTM. It must be augmented to be able to stop at any point
> >> prior to reaching the final state of its input.
> >
> > And if the UTM simulation reaches a final state then a simulating halt decider must also simulate that same input to a final state, otherwise it is wrong by definition
> You did not say that accurately. If the UTM simulation of the input
> reaches the final state of the input then a simulating halt decider
> would have this same behavior.

If it was correct it would have the same behavior. But if H(P,P)==0, then UTM(P,P) will halt, so H(P,P)==0 is wrong by definition.