On 5/25/2022 1:40 PM, Dennis Bush wrote:
> 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.

If non-equivalent computations have different behavior then this proves
that I am wrong on the basis that lying cheating bastards claim that
non-equivalent computations must have the same behavior.

--
Copyright 2022 Pete Olcott

"Talent hits a target no one else can hit;
Genius hits a target no one else can see."
Arthur Schopenhauer