On Saturday, 16 July 2022 at 02:02:41 UTC+1, Ben Bacarisse wrote:
> Malcolm McLean <malcolm.ar...@gmail.com> writes:
>
> > On Thursday, 14 July 2022 at 00:17:00 UTC+1, Ben Bacarisse wrote:
> >> Malcolm McLean <malcolm.ar...@gmail.com> writes:
> >>
> >> > On Wednesday, 13 July 2022 at 02:29:13 UTC+1, Ben Bacarisse wrote:
> >> >> Malcolm McLean <malcolm.ar...@gmail.com> writes:
> >>
> >> >> > However you can convert any parallel algorithm to a single-threaded
> >> >> > algorithm. It has been proposed that the reason that computers can't
> >> >> > match human cognitive abilities is that the algorithms are not
> >> >> > parallel enough. Whilst theoretically this is nothing to do with
> >> >> > hardware, in practice of course it is, because you can't be hanging
> >> >> > around for the results.
> >> >>
> >> >> There's no "hanging around" with a TM.
> >> >>
> >> > Of course there is. You build it, and set it running, and some time later it
> >> > comes back with the results. Or never stops. Time is inherent in the
> >> > model.
> >> No. I can't imagine how you can think this, but it seems there is no
> >> common ground for this discussion.
> >>
> > You can't distinguish, in the general case, a TM that doesn't halt from
> > one that you can't hang round for.
> What's the point of continuing the discussion when there is this
> fundamental mis-match in the use of terms? You've just talked about
> hanging around again, but there is no hanging around with TMs.
> > That's a fundamental characteristic of
> > the model and means that TMs cannot be used to prove hypotheses in
> > number theory.
> No, it's not the hanging around that's fundamental, it's the
> undecidabilty.
>
> This distinction matters because your original argument was about the
> hanging around, not the undecidabilty.
>
Turing machines can be used to disprove conjectures in number theory.
And plenty of patterns that hold for low integers never become "conjectures"
because it is so easy to find a counter-example. And how were these counter-
examples found?
And of course we can prove the conjecture for numbers up to N. But N is
determined by how well we can engineer the machine, and how long we are
prepared to hang around. N must be finite, because a calculation cannot be
instantaneous.
Time is crucial. It's present in the model,and it cannot be abstracted away by
saying that a Turing machine is a mathematical object rather than a real
object.
> > You can't get away from the fact that, to know the configuration
> > at step three, which is wholly determined by the configuration at step one,
> > you absolutely have to calculate the configuration at step two (in the
> > general case again). That's how the model works. Time is inherent to
> > it.
> Nonsense. There is a sequence. Time has nothing to do with it. If we
> apply an iterative function, say f(n) = n/2, f^1000(1) does not "take
> longer" than f^2(1). A TM is just an iterated function.
>
Of course it does. If your calculation requires an intermediate result to
be calculated before the final result, then one with more intermediate
results "takes longer" than one with fewer.
That's pretty basic. You only have to do a few months of primary school
to realise that some sums are harder than others.

On 7/16/22 12:19 AM, Malcolm McLean wrote:
> On Saturday, 16 July 2022 at 02:02:41 UTC+1, Ben Bacarisse wrote:
>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
>>
>>> On Thursday, 14 July 2022 at 00:17:00 UTC+1, Ben Bacarisse wrote:
>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
>>>>
>>>>> On Wednesday, 13 July 2022 at 02:29:13 UTC+1, Ben Bacarisse wrote:
>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
>>>>
>>>>>>> However you can convert any parallel algorithm to a single-threaded
>>>>>>> algorithm. It has been proposed that the reason that computers can't
>>>>>>> match human cognitive abilities is that the algorithms are not
>>>>>>> parallel enough. Whilst theoretically this is nothing to do with
>>>>>>> hardware, in practice of course it is, because you can't be hanging
>>>>>>> around for the results.
>>>>>>
>>>>>> There's no "hanging around" with a TM.
>>>>>>
>>>>> Of course there is. You build it, and set it running, and some time later it
>>>>> comes back with the results. Or never stops. Time is inherent in the
>>>>> model.
>>>> No. I can't imagine how you can think this, but it seems there is no
>>>> common ground for this discussion.
>>>>
>>> You can't distinguish, in the general case, a TM that doesn't halt from
>>> one that you can't hang round for.
>> What's the point of continuing the discussion when there is this
>> fundamental mis-match in the use of terms? You've just talked about
>> hanging around again, but there is no hanging around with TMs.
>>> That's a fundamental characteristic of
>>> the model and means that TMs cannot be used to prove hypotheses in
>>> number theory.
>> No, it's not the hanging around that's fundamental, it's the
>> undecidabilty.
>>
>> This distinction matters because your original argument was about the
>> hanging around, not the undecidabilty.
>>
> Turing machines can be used to disprove conjectures in number theory.
> And plenty of patterns that hold for low integers never become "conjectures"
> because it is so easy to find a counter-example. And how were these counter-
> examples found?
> And of course we can prove the conjecture for numbers up to N. But N is
> determined by how well we can engineer the machine, and how long we are
> prepared to hang around. N must be finite, because a calculation cannot be
> instantaneous.
> Time is crucial. It's present in the model,and it cannot be abstracted away by
> saying that a Turing machine is a mathematical object rather than a real
> object.
>>> You can't get away from the fact that, to know the configuration
>>> at step three, which is wholly determined by the configuration at step one,
>>> you absolutely have to calculate the configuration at step two (in the
>>> general case again). That's how the model works. Time is inherent to
>>> it.
>> Nonsense. There is a sequence. Time has nothing to do with it. If we
>> apply an iterative function, say f(n) = n/2, f^1000(1) does not "take
>> longer" than f^2(1). A TM is just an iterated function.
>>
> Of course it does. If your calculation requires an intermediate result to
> be calculated before the final result, then one with more intermediate
> results "takes longer" than one with fewer.
> That's pretty basic. You only have to do a few months of primary school
> to realise that some sums are harder than others.
>

There is order, but not necessarily physical "time" as we define it.

How "long" does it take to create the set of all natural numbers? There
are a defined sequence, and in at least one sense, 6 doesn't exist until
first 5 does, so 6 can be after it, but it doesn't take "time" to create
that set.

On Saturday, 16 July 2022 at 13:52:11 UTC+1, richar...@gmail.com wrote:
> On 7/16/22 12:19 AM, Malcolm McLean wrote:
> > On Saturday, 16 July 2022 at 02:02:41 UTC+1, Ben Bacarisse wrote:
> >> Malcolm McLean <malcolm.ar...@gmail.com> writes:
> >>
> >>> On Thursday, 14 July 2022 at 00:17:00 UTC+1, Ben Bacarisse wrote:
> >>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
> >>>>
> >>>>> On Wednesday, 13 July 2022 at 02:29:13 UTC+1, Ben Bacarisse wrote:
> >>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
> >>>>
> >>>>>>> However you can convert any parallel algorithm to a single-threaded
> >>>>>>> algorithm. It has been proposed that the reason that computers can't
> >>>>>>> match human cognitive abilities is that the algorithms are not
> >>>>>>> parallel enough. Whilst theoretically this is nothing to do with
> >>>>>>> hardware, in practice of course it is, because you can't be hanging
> >>>>>>> around for the results.
> >>>>>>
> >>>>>> There's no "hanging around" with a TM.
> >>>>>>
> >>>>> Of course there is. You build it, and set it running, and some time later it
> >>>>> comes back with the results. Or never stops. Time is inherent in the
> >>>>> model.
> >>>> No. I can't imagine how you can think this, but it seems there is no
> >>>> common ground for this discussion.
> >>>>
> >>> You can't distinguish, in the general case, a TM that doesn't halt from
> >>> one that you can't hang round for.
> >> What's the point of continuing the discussion when there is this
> >> fundamental mis-match in the use of terms? You've just talked about
> >> hanging around again, but there is no hanging around with TMs.
> >>> That's a fundamental characteristic of
> >>> the model and means that TMs cannot be used to prove hypotheses in
> >>> number theory.
> >> No, it's not the hanging around that's fundamental, it's the
> >> undecidabilty.
> >>
> >> This distinction matters because your original argument was about the
> >> hanging around, not the undecidabilty.
> >>
> > Turing machines can be used to disprove conjectures in number theory.
> > And plenty of patterns that hold for low integers never become "conjectures"
> > because it is so easy to find a counter-example. And how were these counter-
> > examples found?
> > And of course we can prove the conjecture for numbers up to N. But N is
> > determined by how well we can engineer the machine, and how long we are
> > prepared to hang around. N must be finite, because a calculation cannot be
> > instantaneous.
> > Time is crucial. It's present in the model,and it cannot be abstracted away by
> > saying that a Turing machine is a mathematical object rather than a real
> > object.
> >>> You can't get away from the fact that, to know the configuration
> >>> at step three, which is wholly determined by the configuration at step one,
> >>> you absolutely have to calculate the configuration at step two (in the
> >>> general case again). That's how the model works. Time is inherent to
> >>> it.
> >> Nonsense. There is a sequence. Time has nothing to do with it. If we
> >> apply an iterative function, say f(n) = n/2, f^1000(1) does not "take
> >> longer" than f^2(1). A TM is just an iterated function.
> >>
> > Of course it does. If your calculation requires an intermediate result to
> > be calculated before the final result, then one with more intermediate
> > results "takes longer" than one with fewer.
> > That's pretty basic. You only have to do a few months of primary school
> > to realise that some sums are harder than others.
> >
> There is order, but not necessarily physical "time" as we define it.
>
> How "long" does it take to create the set of all natural numbers? There
> are a defined sequence, and in at least one sense, 6 doesn't exist until
> first 5 does, so 6 can be after it, but it doesn't take "time" to create
> that set.
>
Write down a very large integer with digits chosen at random.
If the integer is large enough, the chance that the integer below it has ever
been represented by any human is vanishingly small. However you can
describe some properties of the number, such as its name in English words.

You don't need to create integers in sequence.

Now take a Turing machine that has complex behaviour. Describe the configuration
at machine start. Now describe the configuration an any stage of your choosing,
say stage 1000. Configuration at stage 1000 is totally determined by configuration
at stage 1. But to get there, you have to calculate the intermediate stages.

The set of configurations is not like the set of the integers.

On 7/16/22 9:03 AM, Malcolm McLean wrote:
> On Saturday, 16 July 2022 at 13:52:11 UTC+1, richar...@gmail.com wrote:
>> On 7/16/22 12:19 AM, Malcolm McLean wrote:
>>> On Saturday, 16 July 2022 at 02:02:41 UTC+1, Ben Bacarisse wrote:
>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
>>>>
>>>>> On Thursday, 14 July 2022 at 00:17:00 UTC+1, Ben Bacarisse wrote:
>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
>>>>>>
>>>>>>> On Wednesday, 13 July 2022 at 02:29:13 UTC+1, Ben Bacarisse wrote:
>>>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
>>>>>>
>>>>>>>>> However you can convert any parallel algorithm to a single-threaded
>>>>>>>>> algorithm. It has been proposed that the reason that computers can't
>>>>>>>>> match human cognitive abilities is that the algorithms are not
>>>>>>>>> parallel enough. Whilst theoretically this is nothing to do with
>>>>>>>>> hardware, in practice of course it is, because you can't be hanging
>>>>>>>>> around for the results.
>>>>>>>>
>>>>>>>> There's no "hanging around" with a TM.
>>>>>>>>
>>>>>>> Of course there is. You build it, and set it running, and some time later it
>>>>>>> comes back with the results. Or never stops. Time is inherent in the
>>>>>>> model.
>>>>>> No. I can't imagine how you can think this, but it seems there is no
>>>>>> common ground for this discussion.
>>>>>>
>>>>> You can't distinguish, in the general case, a TM that doesn't halt from
>>>>> one that you can't hang round for.
>>>> What's the point of continuing the discussion when there is this
>>>> fundamental mis-match in the use of terms? You've just talked about
>>>> hanging around again, but there is no hanging around with TMs.
>>>>> That's a fundamental characteristic of
>>>>> the model and means that TMs cannot be used to prove hypotheses in
>>>>> number theory.
>>>> No, it's not the hanging around that's fundamental, it's the
>>>> undecidabilty.
>>>>
>>>> This distinction matters because your original argument was about the
>>>> hanging around, not the undecidabilty.
>>>>
>>> Turing machines can be used to disprove conjectures in number theory.
>>> And plenty of patterns that hold for low integers never become "conjectures"
>>> because it is so easy to find a counter-example. And how were these counter-
>>> examples found?
>>> And of course we can prove the conjecture for numbers up to N. But N is
>>> determined by how well we can engineer the machine, and how long we are
>>> prepared to hang around. N must be finite, because a calculation cannot be
>>> instantaneous.
>>> Time is crucial. It's present in the model,and it cannot be abstracted away by
>>> saying that a Turing machine is a mathematical object rather than a real
>>> object.
>>>>> You can't get away from the fact that, to know the configuration
>>>>> at step three, which is wholly determined by the configuration at step one,
>>>>> you absolutely have to calculate the configuration at step two (in the
>>>>> general case again). That's how the model works. Time is inherent to
>>>>> it.
>>>> Nonsense. There is a sequence. Time has nothing to do with it. If we
>>>> apply an iterative function, say f(n) = n/2, f^1000(1) does not "take
>>>> longer" than f^2(1). A TM is just an iterated function.
>>>>
>>> Of course it does. If your calculation requires an intermediate result to
>>> be calculated before the final result, then one with more intermediate
>>> results "takes longer" than one with fewer.
>>> That's pretty basic. You only have to do a few months of primary school
>>> to realise that some sums are harder than others.
>>>
>> There is order, but not necessarily physical "time" as we define it.
>>
>> How "long" does it take to create the set of all natural numbers? There
>> are a defined sequence, and in at least one sense, 6 doesn't exist until
>> first 5 does, so 6 can be after it, but it doesn't take "time" to create
>> that set.
>>
> Write down a very large integer with digits chosen at random.
> If the integer is large enough, the chance that the integer below it has ever
> been represented by any human is vanishingly small. However you can
> describe some properties of the number, such as its name in English words.
>
> You don't need to create integers in sequence.
>
> Now take a Turing machine that has complex behaviour. Describe the configuration
> at machine start. Now describe the configuration an any stage of your choosing,
> say stage 1000. Configuration at stage 1000 is totally determined by configuration
> at stage 1. But to get there, you have to calculate the intermediate stages.
>
> The set of configurations is not like the set of the integers.

Look at the foundation of numbers, 6 is DEFINED as the successor to 5.
The "Number" six only has meening as the counting number after five.

Yes, you can EXPRESS a number without expressing the previous number,
but that number itself doesn't have meaning without first having found
meaning for the numbers below it.

In the same way, you can express a Turing Machine in any configuration
that that it could possibly (or maybe not even possibly) be in without
having to go through the previous states. That configuration may not
actually have a meaning unless we know how it started.

On Saturday, 16 July 2022 at 14:22:05 UTC+1, richar...@gmail.com wrote:
> On 7/16/22 9:03 AM, Malcolm McLean wrote:
> > On Saturday, 16 July 2022 at 13:52:11 UTC+1, richar...@gmail.com wrote:
> >> On 7/16/22 12:19 AM, Malcolm McLean wrote:
> >>> On Saturday, 16 July 2022 at 02:02:41 UTC+1, Ben Bacarisse wrote:
> >>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
> >>>>
> >>>>> On Thursday, 14 July 2022 at 00:17:00 UTC+1, Ben Bacarisse wrote:
> >>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
> >>>>>>
> >>>>>>> On Wednesday, 13 July 2022 at 02:29:13 UTC+1, Ben Bacarisse wrote:
> >>>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
> >>>>>>
> >>>>>>>>> However you can convert any parallel algorithm to a single-threaded
> >>>>>>>>> algorithm. It has been proposed that the reason that computers can't
> >>>>>>>>> match human cognitive abilities is that the algorithms are not
> >>>>>>>>> parallel enough. Whilst theoretically this is nothing to do with
> >>>>>>>>> hardware, in practice of course it is, because you can't be hanging
> >>>>>>>>> around for the results.
> >>>>>>>>
> >>>>>>>> There's no "hanging around" with a TM.
> >>>>>>>>
> >>>>>>> Of course there is. You build it, and set it running, and some time later it
> >>>>>>> comes back with the results. Or never stops. Time is inherent in the
> >>>>>>> model.
> >>>>>> No. I can't imagine how you can think this, but it seems there is no
> >>>>>> common ground for this discussion.
> >>>>>>
> >>>>> You can't distinguish, in the general case, a TM that doesn't halt from
> >>>>> one that you can't hang round for.
> >>>> What's the point of continuing the discussion when there is this
> >>>> fundamental mis-match in the use of terms? You've just talked about
> >>>> hanging around again, but there is no hanging around with TMs.
> >>>>> That's a fundamental characteristic of
> >>>>> the model and means that TMs cannot be used to prove hypotheses in
> >>>>> number theory.
> >>>> No, it's not the hanging around that's fundamental, it's the
> >>>> undecidabilty.
> >>>>
> >>>> This distinction matters because your original argument was about the
> >>>> hanging around, not the undecidabilty.
> >>>>
> >>> Turing machines can be used to disprove conjectures in number theory.
> >>> And plenty of patterns that hold for low integers never become "conjectures"
> >>> because it is so easy to find a counter-example. And how were these counter-
> >>> examples found?
> >>> And of course we can prove the conjecture for numbers up to N. But N is
> >>> determined by how well we can engineer the machine, and how long we are
> >>> prepared to hang around. N must be finite, because a calculation cannot be
> >>> instantaneous.
> >>> Time is crucial. It's present in the model,and it cannot be abstracted away by
> >>> saying that a Turing machine is a mathematical object rather than a real
> >>> object.
> >>>>> You can't get away from the fact that, to know the configuration
> >>>>> at step three, which is wholly determined by the configuration at step one,
> >>>>> you absolutely have to calculate the configuration at step two (in the
> >>>>> general case again). That's how the model works. Time is inherent to
> >>>>> it.
> >>>> Nonsense. There is a sequence. Time has nothing to do with it. If we
> >>>> apply an iterative function, say f(n) = n/2, f^1000(1) does not "take
> >>>> longer" than f^2(1). A TM is just an iterated function.
> >>>>
> >>> Of course it does. If your calculation requires an intermediate result to
> >>> be calculated before the final result, then one with more intermediate
> >>> results "takes longer" than one with fewer.
> >>> That's pretty basic. You only have to do a few months of primary school
> >>> to realise that some sums are harder than others.
> >>>
> >> There is order, but not necessarily physical "time" as we define it.
> >>
> >> How "long" does it take to create the set of all natural numbers? There
> >> are a defined sequence, and in at least one sense, 6 doesn't exist until
> >> first 5 does, so 6 can be after it, but it doesn't take "time" to create
> >> that set.
> >>
> > Write down a very large integer with digits chosen at random.
> > If the integer is large enough, the chance that the integer below it has ever
> > been represented by any human is vanishingly small. However you can
> > describe some properties of the number, such as its name in English words.
> >
> > You don't need to create integers in sequence.
> >
> > Now take a Turing machine that has complex behaviour. Describe the configuration
> > at machine start. Now describe the configuration an any stage of your choosing,
> > say stage 1000. Configuration at stage 1000 is totally determined by configuration
> > at stage 1. But to get there, you have to calculate the intermediate stages.
> >
> > The set of configurations is not like the set of the integers.
> Look at the foundation of numbers, 6 is DEFINED as the successor to 5.
> The "Number" six only has meening as the counting number after five.
>
> Yes, you can EXPRESS a number without expressing the previous number,
> but that number itself doesn't have meaning without first having found
> meaning for the numbers below it.
>
> In the same way, you can express a Turing Machine in any configuration
> that that it could possibly (or maybe not even possibly) be in without
> having to go through the previous states. That configuration may not
> actually have a meaning unless we know how it started.
>
We can use the notation <Initial configuration>+Nsteps for a Turing machine
in a given state. But we can't use that notation to answer some basic questions
about the Turing machine. Like, what is the current state? How many symbols
are on the tape?
Now you might be subtle and say that there are some questions about the integers
that we can't answer without going through the previous members. E.g., what are the
prime factors of this number? That's perfectly true. And it is harder to answer the
question "what are the prime factors" for a large integer than for a small integer.

On 7/16/22 12:24 PM, Malcolm McLean wrote:
> On Saturday, 16 July 2022 at 14:22:05 UTC+1, richar...@gmail.com wrote:
>> On 7/16/22 9:03 AM, Malcolm McLean wrote:
>>> On Saturday, 16 July 2022 at 13:52:11 UTC+1, richar...@gmail.com wrote:
>>>> On 7/16/22 12:19 AM, Malcolm McLean wrote:
>>>>> On Saturday, 16 July 2022 at 02:02:41 UTC+1, Ben Bacarisse wrote:
>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
>>>>>>
>>>>>>> On Thursday, 14 July 2022 at 00:17:00 UTC+1, Ben Bacarisse wrote:
>>>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
>>>>>>>>
>>>>>>>>> On Wednesday, 13 July 2022 at 02:29:13 UTC+1, Ben Bacarisse wrote:
>>>>>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
>>>>>>>>
>>>>>>>>>>> However you can convert any parallel algorithm to a single-threaded
>>>>>>>>>>> algorithm. It has been proposed that the reason that computers can't
>>>>>>>>>>> match human cognitive abilities is that the algorithms are not
>>>>>>>>>>> parallel enough. Whilst theoretically this is nothing to do with
>>>>>>>>>>> hardware, in practice of course it is, because you can't be hanging
>>>>>>>>>>> around for the results.
>>>>>>>>>>
>>>>>>>>>> There's no "hanging around" with a TM.
>>>>>>>>>>
>>>>>>>>> Of course there is. You build it, and set it running, and some time later it
>>>>>>>>> comes back with the results. Or never stops. Time is inherent in the
>>>>>>>>> model.
>>>>>>>> No. I can't imagine how you can think this, but it seems there is no
>>>>>>>> common ground for this discussion.
>>>>>>>>
>>>>>>> You can't distinguish, in the general case, a TM that doesn't halt from
>>>>>>> one that you can't hang round for.
>>>>>> What's the point of continuing the discussion when there is this
>>>>>> fundamental mis-match in the use of terms? You've just talked about
>>>>>> hanging around again, but there is no hanging around with TMs.
>>>>>>> That's a fundamental characteristic of
>>>>>>> the model and means that TMs cannot be used to prove hypotheses in
>>>>>>> number theory.
>>>>>> No, it's not the hanging around that's fundamental, it's the
>>>>>> undecidabilty.
>>>>>>
>>>>>> This distinction matters because your original argument was about the
>>>>>> hanging around, not the undecidabilty.
>>>>>>
>>>>> Turing machines can be used to disprove conjectures in number theory.
>>>>> And plenty of patterns that hold for low integers never become "conjectures"
>>>>> because it is so easy to find a counter-example. And how were these counter-
>>>>> examples found?
>>>>> And of course we can prove the conjecture for numbers up to N. But N is
>>>>> determined by how well we can engineer the machine, and how long we are
>>>>> prepared to hang around. N must be finite, because a calculation cannot be
>>>>> instantaneous.
>>>>> Time is crucial. It's present in the model,and it cannot be abstracted away by
>>>>> saying that a Turing machine is a mathematical object rather than a real
>>>>> object.
>>>>>>> You can't get away from the fact that, to know the configuration
>>>>>>> at step three, which is wholly determined by the configuration at step one,
>>>>>>> you absolutely have to calculate the configuration at step two (in the
>>>>>>> general case again). That's how the model works. Time is inherent to
>>>>>>> it.
>>>>>> Nonsense. There is a sequence. Time has nothing to do with it. If we
>>>>>> apply an iterative function, say f(n) = n/2, f^1000(1) does not "take
>>>>>> longer" than f^2(1). A TM is just an iterated function.
>>>>>>
>>>>> Of course it does. If your calculation requires an intermediate result to
>>>>> be calculated before the final result, then one with more intermediate
>>>>> results "takes longer" than one with fewer.
>>>>> That's pretty basic. You only have to do a few months of primary school
>>>>> to realise that some sums are harder than others.
>>>>>
>>>> There is order, but not necessarily physical "time" as we define it.
>>>>
>>>> How "long" does it take to create the set of all natural numbers? There
>>>> are a defined sequence, and in at least one sense, 6 doesn't exist until
>>>> first 5 does, so 6 can be after it, but it doesn't take "time" to create
>>>> that set.
>>>>
>>> Write down a very large integer with digits chosen at random.
>>> If the integer is large enough, the chance that the integer below it has ever
>>> been represented by any human is vanishingly small. However you can
>>> describe some properties of the number, such as its name in English words.
>>>
>>> You don't need to create integers in sequence.
>>>
>>> Now take a Turing machine that has complex behaviour. Describe the configuration
>>> at machine start. Now describe the configuration an any stage of your choosing,
>>> say stage 1000. Configuration at stage 1000 is totally determined by configuration
>>> at stage 1. But to get there, you have to calculate the intermediate stages.
>>>
>>> The set of configurations is not like the set of the integers.
>> Look at the foundation of numbers, 6 is DEFINED as the successor to 5.
>> The "Number" six only has meening as the counting number after five.
>>
>> Yes, you can EXPRESS a number without expressing the previous number,
>> but that number itself doesn't have meaning without first having found
>> meaning for the numbers below it.
>>
>> In the same way, you can express a Turing Machine in any configuration
>> that that it could possibly (or maybe not even possibly) be in without
>> having to go through the previous states. That configuration may not
>> actually have a meaning unless we know how it started.
>>
> We can use the notation <Initial configuration>+Nsteps for a Turing machine
> in a given state. But we can't use that notation to answer some basic questions
> about the Turing machine. Like, what is the current state? How many symbols
> are on the tape?
> Now you might be subtle and say that there are some questions about the integers
> that we can't answer without going through the previous members. E.g., what are the
> prime factors of this number? That's perfectly true. And it is harder to answer the
> question "what are the prime factors" for a large integer than for a small integer.

I think your missing my point. Before we HAVE a definition of what the
number 5 actually is, we need the definition of the number 4, because
ultimately, 5 is defined as the number after 4.

This is getting into messy detailed number theory that most people just
skip, and I only have a passing glance as a looked at it and said it
wasn't for me.

Most mathematics just takes from elementary number theory that we HAVE a
definition of things like the Natural Numbers, Rationals, etc. but these
actually DO have fundamental definitions from somewhat complicated
theory to prove that they actually exist.

This is one point where PO's idea just falls flat. If he want to try to
change the fundamental nature of what "Truth" means, he really has to go
back to that level and make sure he can still get the basic properties
of the Natural Numbers.

After all, Godel's proof is based just on having a sufficent set of
propeties of the Natural Numbers, so if he is trying to break the
ability to prove Godel's Theorem, what in the Natural Numbers is he also
breaking to do that.

On Saturday, 16 July 2022 at 17:50:53 UTC+1, richar...@gmail.com wrote:
> On 7/16/22 12:24 PM, Malcolm McLean wrote:
> > On Saturday, 16 July 2022 at 14:22:05 UTC+1, richar...@gmail.com wrote:
> >> On 7/16/22 9:03 AM, Malcolm McLean wrote:
> >>> On Saturday, 16 July 2022 at 13:52:11 UTC+1, richar...@gmail.com wrote:
> >>>> On 7/16/22 12:19 AM, Malcolm McLean wrote:
> >>>>> On Saturday, 16 July 2022 at 02:02:41 UTC+1, Ben Bacarisse wrote:
> >>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
> >>>>>>
> >>>>>>> On Thursday, 14 July 2022 at 00:17:00 UTC+1, Ben Bacarisse wrote:
> >>>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
> >>>>>>>>
> >>>>>>>>> On Wednesday, 13 July 2022 at 02:29:13 UTC+1, Ben Bacarisse wrote:
> >>>>>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
> >>>>>>>>
> >>>>>>>>>>> However you can convert any parallel algorithm to a single-threaded
> >>>>>>>>>>> algorithm. It has been proposed that the reason that computers can't
> >>>>>>>>>>> match human cognitive abilities is that the algorithms are not
> >>>>>>>>>>> parallel enough. Whilst theoretically this is nothing to do with
> >>>>>>>>>>> hardware, in practice of course it is, because you can't be hanging
> >>>>>>>>>>> around for the results.
> >>>>>>>>>>
> >>>>>>>>>> There's no "hanging around" with a TM.
> >>>>>>>>>>
> >>>>>>>>> Of course there is. You build it, and set it running, and some time later it
> >>>>>>>>> comes back with the results. Or never stops. Time is inherent in the
> >>>>>>>>> model.
> >>>>>>>> No. I can't imagine how you can think this, but it seems there is no
> >>>>>>>> common ground for this discussion.
> >>>>>>>>
> >>>>>>> You can't distinguish, in the general case, a TM that doesn't halt from
> >>>>>>> one that you can't hang round for.
> >>>>>> What's the point of continuing the discussion when there is this
> >>>>>> fundamental mis-match in the use of terms? You've just talked about
> >>>>>> hanging around again, but there is no hanging around with TMs.
> >>>>>>> That's a fundamental characteristic of
> >>>>>>> the model and means that TMs cannot be used to prove hypotheses in
> >>>>>>> number theory.
> >>>>>> No, it's not the hanging around that's fundamental, it's the
> >>>>>> undecidabilty.
> >>>>>>
> >>>>>> This distinction matters because your original argument was about the
> >>>>>> hanging around, not the undecidabilty.
> >>>>>>
> >>>>> Turing machines can be used to disprove conjectures in number theory.
> >>>>> And plenty of patterns that hold for low integers never become "conjectures"
> >>>>> because it is so easy to find a counter-example. And how were these counter-
> >>>>> examples found?
> >>>>> And of course we can prove the conjecture for numbers up to N. But N is
> >>>>> determined by how well we can engineer the machine, and how long we are
> >>>>> prepared to hang around. N must be finite, because a calculation cannot be
> >>>>> instantaneous.
> >>>>> Time is crucial. It's present in the model,and it cannot be abstracted away by
> >>>>> saying that a Turing machine is a mathematical object rather than a real
> >>>>> object.
> >>>>>>> You can't get away from the fact that, to know the configuration
> >>>>>>> at step three, which is wholly determined by the configuration at step one,
> >>>>>>> you absolutely have to calculate the configuration at step two (in the
> >>>>>>> general case again). That's how the model works. Time is inherent to
> >>>>>>> it.
> >>>>>> Nonsense. There is a sequence. Time has nothing to do with it. If we
> >>>>>> apply an iterative function, say f(n) = n/2, f^1000(1) does not "take
> >>>>>> longer" than f^2(1). A TM is just an iterated function.
> >>>>>>
> >>>>> Of course it does. If your calculation requires an intermediate result to
> >>>>> be calculated before the final result, then one with more intermediate
> >>>>> results "takes longer" than one with fewer.
> >>>>> That's pretty basic. You only have to do a few months of primary school
> >>>>> to realise that some sums are harder than others.
> >>>>>
> >>>> There is order, but not necessarily physical "time" as we define it.
> >>>>
> >>>> How "long" does it take to create the set of all natural numbers? There
> >>>> are a defined sequence, and in at least one sense, 6 doesn't exist until
> >>>> first 5 does, so 6 can be after it, but it doesn't take "time" to create
> >>>> that set.
> >>>>
> >>> Write down a very large integer with digits chosen at random.
> >>> If the integer is large enough, the chance that the integer below it has ever
> >>> been represented by any human is vanishingly small. However you can
> >>> describe some properties of the number, such as its name in English words.
> >>>
> >>> You don't need to create integers in sequence.
> >>>
> >>> Now take a Turing machine that has complex behaviour. Describe the configuration
> >>> at machine start. Now describe the configuration an any stage of your choosing,
> >>> say stage 1000. Configuration at stage 1000 is totally determined by configuration
> >>> at stage 1. But to get there, you have to calculate the intermediate stages.
> >>>
> >>> The set of configurations is not like the set of the integers.
> >> Look at the foundation of numbers, 6 is DEFINED as the successor to 5.
> >> The "Number" six only has meening as the counting number after five.
> >>
> >> Yes, you can EXPRESS a number without expressing the previous number,
> >> but that number itself doesn't have meaning without first having found
> >> meaning for the numbers below it.
> >>
> >> In the same way, you can express a Turing Machine in any configuration
> >> that that it could possibly (or maybe not even possibly) be in without
> >> having to go through the previous states. That configuration may not
> >> actually have a meaning unless we know how it started.
> >>
> > We can use the notation <Initial configuration>+Nsteps for a Turing machine
> > in a given state. But we can't use that notation to answer some basic questions
> > about the Turing machine. Like, what is the current state? How many symbols
> > are on the tape?
> > Now you might be subtle and say that there are some questions about the integers
> > that we can't answer without going through the previous members. E.g., what are the
> > prime factors of this number? That's perfectly true. And it is harder to answer the
> > question "what are the prime factors" for a large integer than for a small integer.
> I think your missing my point. Before we HAVE a definition of what the
> number 5 actually is, we need the definition of the number 4, because
> ultimately, 5 is defined as the number after 4.
>
> This is getting into messy detailed number theory that most people just
> skip, and I only have a passing glance as a looked at it and said it
> wasn't for me.
>
> Most mathematics just takes from elementary number theory that we HAVE a
> definition of things like the Natural Numbers, Rationals, etc. but these
> actually DO have fundamental definitions from somewhat complicated
> theory to prove that they actually exist.
>
If the only notation we have for a natural number is a "count the stokes" or
unary notation, then I suppose that it's true to say that before we have a
representation of 11111 we must have a representation of 1111. We can't
create the representation for five without first creating a representation for
four.
So in that sense, the set of natural numbers is like the set of configurations of
a Turing machine. You start with the machine in its initial configuration, and
you step it. To get configuration number five, we need to create configuration
number four.
I'm not sure if there is meaningful difference here. The difference I can see is
that a more sophisticated notation than unary keeps most (not all) the properties
of the integer accessible. There are very few problems where you need to convert
a natural number to unary notation and back to get the solution. However whilst
we can use the notation <initial configuration>+4 for the fifth element of our
Turing machine configurations set, we can't use it to answer very basic questions,
like "what is the current state?"
Note that we can express a configuration that isn't in the set, and it has meaning,
but it is a member of the set of "all configurations for the machine" rather than
"all reached configurations". We don't need to step the machine to do that. We
might also express a configuration that is in the set, but of we do so, it's
purely by chance. We've no way of knowing that it's in the set or its position
in the sequence, without stepping the machine (for the general case, we
can work out better rules for specific machines).
>
> This is one point where PO's idea just falls flat. If he want to try to
> change the fundamental nature of what "Truth" means, he really has to go
> back to that level and make sure he can still get the basic properties
> of the Natural Numbers.
>
> After all, Godel's proof is based just on having a sufficent set of
> propeties of the Natural Numbers, so if he is trying to break the
> ability to prove Godel's Theorem, what in the Natural Numbers is he also
> breaking to do that.
>
I must admit that I don't really understand why the claim that "true statements
must be proveable" breaks the properties of the Natural Numbers. Clearly
the claim is wrong on a general, everyday level. Clearly also there are specific,
limited, and quite advanced mathematical theories where some statement
must be either "true" or "false" and it can be shown that no proof is possible.
I'm prepared to believe that it is the case that the Natural Numbers wouldn't
exist were it other wise, but I take it on trust.

On 7/16/22 4:30 PM, Malcolm McLean wrote:
> On Saturday, 16 July 2022 at 17:50:53 UTC+1, richar...@gmail.com wrote:
>> On 7/16/22 12:24 PM, Malcolm McLean wrote:
>>> On Saturday, 16 July 2022 at 14:22:05 UTC+1, richar...@gmail.com wrote:
>>>> On 7/16/22 9:03 AM, Malcolm McLean wrote:
>>>>> On Saturday, 16 July 2022 at 13:52:11 UTC+1, richar...@gmail.com wrote:
>>>>>> On 7/16/22 12:19 AM, Malcolm McLean wrote:
>>>>>>> On Saturday, 16 July 2022 at 02:02:41 UTC+1, Ben Bacarisse wrote:
>>>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
>>>>>>>>
>>>>>>>>> On Thursday, 14 July 2022 at 00:17:00 UTC+1, Ben Bacarisse wrote:
>>>>>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
>>>>>>>>>>
>>>>>>>>>>> On Wednesday, 13 July 2022 at 02:29:13 UTC+1, Ben Bacarisse wrote:
>>>>>>>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
>>>>>>>>>>
>>>>>>>>>>>>> However you can convert any parallel algorithm to a single-threaded
>>>>>>>>>>>>> algorithm. It has been proposed that the reason that computers can't
>>>>>>>>>>>>> match human cognitive abilities is that the algorithms are not
>>>>>>>>>>>>> parallel enough. Whilst theoretically this is nothing to do with
>>>>>>>>>>>>> hardware, in practice of course it is, because you can't be hanging
>>>>>>>>>>>>> around for the results.
>>>>>>>>>>>>
>>>>>>>>>>>> There's no "hanging around" with a TM.
>>>>>>>>>>>>
>>>>>>>>>>> Of course there is. You build it, and set it running, and some time later it
>>>>>>>>>>> comes back with the results. Or never stops. Time is inherent in the
>>>>>>>>>>> model.
>>>>>>>>>> No. I can't imagine how you can think this, but it seems there is no
>>>>>>>>>> common ground for this discussion.
>>>>>>>>>>
>>>>>>>>> You can't distinguish, in the general case, a TM that doesn't halt from
>>>>>>>>> one that you can't hang round for.
>>>>>>>> What's the point of continuing the discussion when there is this
>>>>>>>> fundamental mis-match in the use of terms? You've just talked about
>>>>>>>> hanging around again, but there is no hanging around with TMs.
>>>>>>>>> That's a fundamental characteristic of
>>>>>>>>> the model and means that TMs cannot be used to prove hypotheses in
>>>>>>>>> number theory.
>>>>>>>> No, it's not the hanging around that's fundamental, it's the
>>>>>>>> undecidabilty.
>>>>>>>>
>>>>>>>> This distinction matters because your original argument was about the
>>>>>>>> hanging around, not the undecidabilty.
>>>>>>>>
>>>>>>> Turing machines can be used to disprove conjectures in number theory.
>>>>>>> And plenty of patterns that hold for low integers never become "conjectures"
>>>>>>> because it is so easy to find a counter-example. And how were these counter-
>>>>>>> examples found?
>>>>>>> And of course we can prove the conjecture for numbers up to N. But N is
>>>>>>> determined by how well we can engineer the machine, and how long we are
>>>>>>> prepared to hang around. N must be finite, because a calculation cannot be
>>>>>>> instantaneous.
>>>>>>> Time is crucial. It's present in the model,and it cannot be abstracted away by
>>>>>>> saying that a Turing machine is a mathematical object rather than a real
>>>>>>> object.
>>>>>>>>> You can't get away from the fact that, to know the configuration
>>>>>>>>> at step three, which is wholly determined by the configuration at step one,
>>>>>>>>> you absolutely have to calculate the configuration at step two (in the
>>>>>>>>> general case again). That's how the model works. Time is inherent to
>>>>>>>>> it.
>>>>>>>> Nonsense. There is a sequence. Time has nothing to do with it. If we
>>>>>>>> apply an iterative function, say f(n) = n/2, f^1000(1) does not "take
>>>>>>>> longer" than f^2(1). A TM is just an iterated function.
>>>>>>>>
>>>>>>> Of course it does. If your calculation requires an intermediate result to
>>>>>>> be calculated before the final result, then one with more intermediate
>>>>>>> results "takes longer" than one with fewer.
>>>>>>> That's pretty basic. You only have to do a few months of primary school
>>>>>>> to realise that some sums are harder than others.
>>>>>>>
>>>>>> There is order, but not necessarily physical "time" as we define it.
>>>>>>
>>>>>> How "long" does it take to create the set of all natural numbers? There
>>>>>> are a defined sequence, and in at least one sense, 6 doesn't exist until
>>>>>> first 5 does, so 6 can be after it, but it doesn't take "time" to create
>>>>>> that set.
>>>>>>
>>>>> Write down a very large integer with digits chosen at random.
>>>>> If the integer is large enough, the chance that the integer below it has ever
>>>>> been represented by any human is vanishingly small. However you can
>>>>> describe some properties of the number, such as its name in English words.
>>>>>
>>>>> You don't need to create integers in sequence.
>>>>>
>>>>> Now take a Turing machine that has complex behaviour. Describe the configuration
>>>>> at machine start. Now describe the configuration an any stage of your choosing,
>>>>> say stage 1000. Configuration at stage 1000 is totally determined by configuration
>>>>> at stage 1. But to get there, you have to calculate the intermediate stages.
>>>>>
>>>>> The set of configurations is not like the set of the integers.
>>>> Look at the foundation of numbers, 6 is DEFINED as the successor to 5.
>>>> The "Number" six only has meening as the counting number after five.
>>>>
>>>> Yes, you can EXPRESS a number without expressing the previous number,
>>>> but that number itself doesn't have meaning without first having found
>>>> meaning for the numbers below it.
>>>>
>>>> In the same way, you can express a Turing Machine in any configuration
>>>> that that it could possibly (or maybe not even possibly) be in without
>>>> having to go through the previous states. That configuration may not
>>>> actually have a meaning unless we know how it started.
>>>>
>>> We can use the notation <Initial configuration>+Nsteps for a Turing machine
>>> in a given state. But we can't use that notation to answer some basic questions
>>> about the Turing machine. Like, what is the current state? How many symbols
>>> are on the tape?
>>> Now you might be subtle and say that there are some questions about the integers
>>> that we can't answer without going through the previous members. E.g., what are the
>>> prime factors of this number? That's perfectly true. And it is harder to answer the
>>> question "what are the prime factors" for a large integer than for a small integer.
>> I think your missing my point. Before we HAVE a definition of what the
>> number 5 actually is, we need the definition of the number 4, because
>> ultimately, 5 is defined as the number after 4.
>>
>> This is getting into messy detailed number theory that most people just
>> skip, and I only have a passing glance as a looked at it and said it
>> wasn't for me.
>>
>> Most mathematics just takes from elementary number theory that we HAVE a
>> definition of things like the Natural Numbers, Rationals, etc. but these
>> actually DO have fundamental definitions from somewhat complicated
>> theory to prove that they actually exist.
>>
> If the only notation we have for a natural number is a "count the stokes" or
> unary notation, then I suppose that it's true to say that before we have a
> representation of 11111 we must have a representation of 1111. We can't
> create the representation for five without first creating a representation for
> four.

Malcolm McLean <malcolm.arthur.mclean@gmail.com> writes:

> On Saturday, 16 July 2022 at 02:02:41 UTC+1, Ben Bacarisse wrote:
>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
>>
>> > On Thursday, 14 July 2022 at 00:17:00 UTC+1, Ben Bacarisse wrote:
>> >> Malcolm McLean <malcolm.ar...@gmail.com> writes:
>> >>
>> >> > On Wednesday, 13 July 2022 at 02:29:13 UTC+1, Ben Bacarisse wrote:
>> >> >> Malcolm McLean <malcolm.ar...@gmail.com> writes:
>> >>
>> >> >> > However you can convert any parallel algorithm to a single-threaded
>> >> >> > algorithm. It has been proposed that the reason that computers can't
>> >> >> > match human cognitive abilities is that the algorithms are not
>> >> >> > parallel enough. Whilst theoretically this is nothing to do with
>> >> >> > hardware, in practice of course it is, because you can't be hanging
>> >> >> > around for the results.
>> >> >>
>> >> >> There's no "hanging around" with a TM.
>> >> >>
>> >> > Of course there is. You build it, and set it running, and some time later it
>> >> > comes back with the results. Or never stops. Time is inherent in the
>> >> > model.
>> >> No. I can't imagine how you can think this, but it seems there is no
>> >> common ground for this discussion.
>> >>
>> > You can't distinguish, in the general case, a TM that doesn't halt from
>> > one that you can't hang round for.
>> What's the point of continuing the discussion when there is this
>> fundamental mis-match in the use of terms? You've just talked about
>> hanging around again, but there is no hanging around with TMs.
>> > That's a fundamental characteristic of
>> > the model and means that TMs cannot be used to prove hypotheses in
>> > number theory.
>> No, it's not the hanging around that's fundamental, it's the
>> undecidabilty.
>>
>> This distinction matters because your original argument was about the
>> hanging around, not the undecidabilty.
>>
> Turing machines can be used to disprove conjectures in number theory.

It's never a good way to do that. Finding a counterexample is always
easier that putting together a TM that finds one. But I suspect you did
not mean what you wrote. I think you are pointing out the equivalence
of two kinds of undecidability.

> And plenty of patterns that hold for low integers never become "conjectures"
> because it is so easy to find a counter-example. And how were these counter-
> examples found?

It varies. No one ever used a TM, that's for sure.

> And of course we can prove the conjecture for numbers up to N. But N is
> determined by how well we can engineer the machine,

Now machines enter the fray, but what do you mean? It can't be TMs
because they are not "engineered".

> and how long we are
> prepared to hang around.

And since there is no hanging around with TMs you must be talking about
some physical machines that really do execute in time.

> N must be finite, because a calculation cannot be
> instantaneous.
> Time is crucial.

Not for TMs.

> It's present in the model, and it cannot be abstracted away by
> saying that a Turing machine is a mathematical object rather than a real
> object.

It simply is /not/ present in the model. You are talking about
something other that a mathematical function. Ordering does not imply
time.

>> > You can't get away from the fact that, to know the configuration
>> > at step three, which is wholly determined by the configuration at step one,
>> > you absolutely have to calculate the configuration at step two (in the
>> > general case again). That's how the model works. Time is inherent to
>> > it.
>> Nonsense. There is a sequence. Time has nothing to do with it. If we
>> apply an iterative function, say f(n) = n/2, f^1000(1) does not "take
>> longer" than f^2(1). A TM is just an iterated function.
>>
> Of course it does. If your calculation requires an intermediate result to
> be calculated before the final result, then one with more intermediate
> results "takes longer" than one with fewer.

The scare quotes give it away. It's a metaphor we use because we find
is hard to think about a sequence without time. But it if helps, think
of the sequence going /back/ in time. Then every TM halts earlier than
is starts. Both uses of "before" and "after" are metaphorical.

> That's pretty basic. You only have to do a few months of primary school
> to realise that some sums are harder than others.

Yes, but that's because we really do "take time".

Anyway, you seem to have lost your original claim in engineering. It
was about what a TM can do and the does not depend on time, unless you
want to claim some actual speed. Do your TMs "run slower" than mine?
Are they "faster" than Turing's? The whole notion is absurd.

--
Ben.

On Saturday, 16 July 2022 at 23:35:12 UTC+1, richar...@gmail.com wrote:
> On 7/16/22 4:30 PM, Malcolm McLean wrote:
> > On Saturday, 16 July 2022 at 17:50:53 UTC+1, richar...@gmail.com wrote:
> >> On 7/16/22 12:24 PM, Malcolm McLean wrote:
> >>> On Saturday, 16 July 2022 at 14:22:05 UTC+1, richar...@gmail.com wrote:
> >>>> On 7/16/22 9:03 AM, Malcolm McLean wrote:
> >>>>> On Saturday, 16 July 2022 at 13:52:11 UTC+1, richar...@gmail.com wrote:
> >>>>>> On 7/16/22 12:19 AM, Malcolm McLean wrote:
> >>>>>>> On Saturday, 16 July 2022 at 02:02:41 UTC+1, Ben Bacarisse wrote:
> >>>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
> >>>>>>>>
> >>>>>>>>> On Thursday, 14 July 2022 at 00:17:00 UTC+1, Ben Bacarisse wrote:
> >>>>>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
> >>>>>>>>>>
> >>>>>>>>>>> On Wednesday, 13 July 2022 at 02:29:13 UTC+1, Ben Bacarisse wrote:
> >>>>>>>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
> >>>>>>>>>>
> >>>>>>>>>>>>> However you can convert any parallel algorithm to a single-threaded
> >>>>>>>>>>>>> algorithm. It has been proposed that the reason that computers can't
> >>>>>>>>>>>>> match human cognitive abilities is that the algorithms are not
> >>>>>>>>>>>>> parallel enough. Whilst theoretically this is nothing to do with
> >>>>>>>>>>>>> hardware, in practice of course it is, because you can't be hanging
> >>>>>>>>>>>>> around for the results.
> >>>>>>>>>>>>
> >>>>>>>>>>>> There's no "hanging around" with a TM.
> >>>>>>>>>>>>
> >>>>>>>>>>> Of course there is. You build it, and set it running, and some time later it
> >>>>>>>>>>> comes back with the results. Or never stops. Time is inherent in the
> >>>>>>>>>>> model.
> >>>>>>>>>> No. I can't imagine how you can think this, but it seems there is no
> >>>>>>>>>> common ground for this discussion.
> >>>>>>>>>>
> >>>>>>>>> You can't distinguish, in the general case, a TM that doesn't halt from
> >>>>>>>>> one that you can't hang round for.
> >>>>>>>> What's the point of continuing the discussion when there is this
> >>>>>>>> fundamental mis-match in the use of terms? You've just talked about
> >>>>>>>> hanging around again, but there is no hanging around with TMs.
> >>>>>>>>> That's a fundamental characteristic of
> >>>>>>>>> the model and means that TMs cannot be used to prove hypotheses in
> >>>>>>>>> number theory.
> >>>>>>>> No, it's not the hanging around that's fundamental, it's the
> >>>>>>>> undecidabilty.
> >>>>>>>>
> >>>>>>>> This distinction matters because your original argument was about the
> >>>>>>>> hanging around, not the undecidabilty.
> >>>>>>>>
> >>>>>>> Turing machines can be used to disprove conjectures in number theory.
> >>>>>>> And plenty of patterns that hold for low integers never become "conjectures"
> >>>>>>> because it is so easy to find a counter-example. And how were these counter-
> >>>>>>> examples found?
> >>>>>>> And of course we can prove the conjecture for numbers up to N. But N is
> >>>>>>> determined by how well we can engineer the machine, and how long we are
> >>>>>>> prepared to hang around. N must be finite, because a calculation cannot be
> >>>>>>> instantaneous.
> >>>>>>> Time is crucial. It's present in the model,and it cannot be abstracted away by
> >>>>>>> saying that a Turing machine is a mathematical object rather than a real
> >>>>>>> object.
> >>>>>>>>> You can't get away from the fact that, to know the configuration
> >>>>>>>>> at step three, which is wholly determined by the configuration at step one,
> >>>>>>>>> you absolutely have to calculate the configuration at step two (in the
> >>>>>>>>> general case again). That's how the model works. Time is inherent to
> >>>>>>>>> it.
> >>>>>>>> Nonsense. There is a sequence. Time has nothing to do with it. If we
> >>>>>>>> apply an iterative function, say f(n) = n/2, f^1000(1) does not "take
> >>>>>>>> longer" than f^2(1). A TM is just an iterated function.
> >>>>>>>>
> >>>>>>> Of course it does. If your calculation requires an intermediate result to
> >>>>>>> be calculated before the final result, then one with more intermediate
> >>>>>>> results "takes longer" than one with fewer.
> >>>>>>> That's pretty basic. You only have to do a few months of primary school
> >>>>>>> to realise that some sums are harder than others.
> >>>>>>>
> >>>>>> There is order, but not necessarily physical "time" as we define it.
> >>>>>>
> >>>>>> How "long" does it take to create the set of all natural numbers? There
> >>>>>> are a defined sequence, and in at least one sense, 6 doesn't exist until
> >>>>>> first 5 does, so 6 can be after it, but it doesn't take "time" to create
> >>>>>> that set.
> >>>>>>
> >>>>> Write down a very large integer with digits chosen at random.
> >>>>> If the integer is large enough, the chance that the integer below it has ever
> >>>>> been represented by any human is vanishingly small. However you can
> >>>>> describe some properties of the number, such as its name in English words.
> >>>>>
> >>>>> You don't need to create integers in sequence.
> >>>>>
> >>>>> Now take a Turing machine that has complex behaviour. Describe the configuration
> >>>>> at machine start. Now describe the configuration an any stage of your choosing,
> >>>>> say stage 1000. Configuration at stage 1000 is totally determined by configuration
> >>>>> at stage 1. But to get there, you have to calculate the intermediate stages.
> >>>>>
> >>>>> The set of configurations is not like the set of the integers.
> >>>> Look at the foundation of numbers, 6 is DEFINED as the successor to 5.
> >>>> The "Number" six only has meening as the counting number after five.
> >>>>
> >>>> Yes, you can EXPRESS a number without expressing the previous number,
> >>>> but that number itself doesn't have meaning without first having found
> >>>> meaning for the numbers below it.
> >>>>
> >>>> In the same way, you can express a Turing Machine in any configuration
> >>>> that that it could possibly (or maybe not even possibly) be in without
> >>>> having to go through the previous states. That configuration may not
> >>>> actually have a meaning unless we know how it started.
> >>>>
> >>> We can use the notation <Initial configuration>+Nsteps for a Turing machine
> >>> in a given state. But we can't use that notation to answer some basic questions
> >>> about the Turing machine. Like, what is the current state? How many symbols
> >>> are on the tape?
> >>> Now you might be subtle and say that there are some questions about the integers
> >>> that we can't answer without going through the previous members. E.g., what are the
> >>> prime factors of this number? That's perfectly true. And it is harder to answer the
> >>> question "what are the prime factors" for a large integer than for a small integer.
> >> I think your missing my point. Before we HAVE a definition of what the
> >> number 5 actually is, we need the definition of the number 4, because
> >> ultimately, 5 is defined as the number after 4.
> >>
> >> This is getting into messy detailed number theory that most people just
> >> skip, and I only have a passing glance as a looked at it and said it
> >> wasn't for me.
> >>
> >> Most mathematics just takes from elementary number theory that we HAVE a
> >> definition of things like the Natural Numbers, Rationals, etc. but these
> >> actually DO have fundamental definitions from somewhat complicated
> >> theory to prove that they actually exist.
> >>
> > If the only notation we have for a natural number is a "count the stokes" or
> > unary notation, then I suppose that it's true to say that before we have a
> > representation of 11111 we must have a representation of 1111. We can't
> > create the representation for five without first creating a representation for
> > four.
> No, I am NOT talking about representation, but the definitions of the
> numbers themselves.
>
> Try to define the number two without using the number one or words which
> depend on already knowing what two is (like pair).
>
The smallest prime number.
The smallest integer that can be used as a base such that the size of the
representation grows as the logarithm of the value.
What hands, eyes, feet, horns, parents, scissors and straits have in
common. (This one doesn't work so well in English. However some languages
have a special grammatical form for these words).
The smallest number of people who can share a cake, such that each
slice is smaller than the whole cake.
The number of signs created by allowing the inverse operation to addition.
The number of dimensions needed to construct a triangle.

On 7/17/22 6:34 AM, Malcolm McLean wrote:
> On Saturday, 16 July 2022 at 23:35:12 UTC+1, richar...@gmail.com wrote:
>> On 7/16/22 4:30 PM, Malcolm McLean wrote:
>>> On Saturday, 16 July 2022 at 17:50:53 UTC+1, richar...@gmail.com wrote:
>>>> On 7/16/22 12:24 PM, Malcolm McLean wrote:
>>>>> On Saturday, 16 July 2022 at 14:22:05 UTC+1, richar...@gmail.com wrote:
>>>>>> On 7/16/22 9:03 AM, Malcolm McLean wrote:
>>>>>>> On Saturday, 16 July 2022 at 13:52:11 UTC+1, richar...@gmail.com wrote:
>>>>>>>> On 7/16/22 12:19 AM, Malcolm McLean wrote:
>>>>>>>>> On Saturday, 16 July 2022 at 02:02:41 UTC+1, Ben Bacarisse wrote:
>>>>>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
>>>>>>>>>>
>>>>>>>>>>> On Thursday, 14 July 2022 at 00:17:00 UTC+1, Ben Bacarisse wrote:
>>>>>>>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
>>>>>>>>>>>>
>>>>>>>>>>>>> On Wednesday, 13 July 2022 at 02:29:13 UTC+1, Ben Bacarisse wrote:
>>>>>>>>>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
>>>>>>>>>>>>
>>>>>>>>>>>>>>> However you can convert any parallel algorithm to a single-threaded
>>>>>>>>>>>>>>> algorithm. It has been proposed that the reason that computers can't
>>>>>>>>>>>>>>> match human cognitive abilities is that the algorithms are not
>>>>>>>>>>>>>>> parallel enough. Whilst theoretically this is nothing to do with
>>>>>>>>>>>>>>> hardware, in practice of course it is, because you can't be hanging
>>>>>>>>>>>>>>> around for the results.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> There's no "hanging around" with a TM.
>>>>>>>>>>>>>>
>>>>>>>>>>>>> Of course there is. You build it, and set it running, and some time later it
>>>>>>>>>>>>> comes back with the results. Or never stops. Time is inherent in the
>>>>>>>>>>>>> model.
>>>>>>>>>>>> No. I can't imagine how you can think this, but it seems there is no
>>>>>>>>>>>> common ground for this discussion.
>>>>>>>>>>>>
>>>>>>>>>>> You can't distinguish, in the general case, a TM that doesn't halt from
>>>>>>>>>>> one that you can't hang round for.
>>>>>>>>>> What's the point of continuing the discussion when there is this
>>>>>>>>>> fundamental mis-match in the use of terms? You've just talked about
>>>>>>>>>> hanging around again, but there is no hanging around with TMs.
>>>>>>>>>>> That's a fundamental characteristic of
>>>>>>>>>>> the model and means that TMs cannot be used to prove hypotheses in
>>>>>>>>>>> number theory.
>>>>>>>>>> No, it's not the hanging around that's fundamental, it's the
>>>>>>>>>> undecidabilty.
>>>>>>>>>>
>>>>>>>>>> This distinction matters because your original argument was about the
>>>>>>>>>> hanging around, not the undecidabilty.
>>>>>>>>>>
>>>>>>>>> Turing machines can be used to disprove conjectures in number theory.
>>>>>>>>> And plenty of patterns that hold for low integers never become "conjectures"
>>>>>>>>> because it is so easy to find a counter-example. And how were these counter-
>>>>>>>>> examples found?
>>>>>>>>> And of course we can prove the conjecture for numbers up to N. But N is
>>>>>>>>> determined by how well we can engineer the machine, and how long we are
>>>>>>>>> prepared to hang around. N must be finite, because a calculation cannot be
>>>>>>>>> instantaneous.
>>>>>>>>> Time is crucial. It's present in the model,and it cannot be abstracted away by
>>>>>>>>> saying that a Turing machine is a mathematical object rather than a real
>>>>>>>>> object.
>>>>>>>>>>> You can't get away from the fact that, to know the configuration
>>>>>>>>>>> at step three, which is wholly determined by the configuration at step one,
>>>>>>>>>>> you absolutely have to calculate the configuration at step two (in the
>>>>>>>>>>> general case again). That's how the model works. Time is inherent to
>>>>>>>>>>> it.
>>>>>>>>>> Nonsense. There is a sequence. Time has nothing to do with it. If we
>>>>>>>>>> apply an iterative function, say f(n) = n/2, f^1000(1) does not "take
>>>>>>>>>> longer" than f^2(1). A TM is just an iterated function.
>>>>>>>>>>
>>>>>>>>> Of course it does. If your calculation requires an intermediate result to
>>>>>>>>> be calculated before the final result, then one with more intermediate
>>>>>>>>> results "takes longer" than one with fewer.
>>>>>>>>> That's pretty basic. You only have to do a few months of primary school
>>>>>>>>> to realise that some sums are harder than others.
>>>>>>>>>
>>>>>>>> There is order, but not necessarily physical "time" as we define it.
>>>>>>>>
>>>>>>>> How "long" does it take to create the set of all natural numbers? There
>>>>>>>> are a defined sequence, and in at least one sense, 6 doesn't exist until
>>>>>>>> first 5 does, so 6 can be after it, but it doesn't take "time" to create
>>>>>>>> that set.
>>>>>>>>
>>>>>>> Write down a very large integer with digits chosen at random.
>>>>>>> If the integer is large enough, the chance that the integer below it has ever
>>>>>>> been represented by any human is vanishingly small. However you can
>>>>>>> describe some properties of the number, such as its name in English words.
>>>>>>>
>>>>>>> You don't need to create integers in sequence.
>>>>>>>
>>>>>>> Now take a Turing machine that has complex behaviour. Describe the configuration
>>>>>>> at machine start. Now describe the configuration an any stage of your choosing,
>>>>>>> say stage 1000. Configuration at stage 1000 is totally determined by configuration
>>>>>>> at stage 1. But to get there, you have to calculate the intermediate stages.
>>>>>>>
>>>>>>> The set of configurations is not like the set of the integers.
>>>>>> Look at the foundation of numbers, 6 is DEFINED as the successor to 5.
>>>>>> The "Number" six only has meening as the counting number after five.
>>>>>>
>>>>>> Yes, you can EXPRESS a number without expressing the previous number,
>>>>>> but that number itself doesn't have meaning without first having found
>>>>>> meaning for the numbers below it.
>>>>>>
>>>>>> In the same way, you can express a Turing Machine in any configuration
>>>>>> that that it could possibly (or maybe not even possibly) be in without
>>>>>> having to go through the previous states. That configuration may not
>>>>>> actually have a meaning unless we know how it started.
>>>>>>
>>>>> We can use the notation <Initial configuration>+Nsteps for a Turing machine
>>>>> in a given state. But we can't use that notation to answer some basic questions
>>>>> about the Turing machine. Like, what is the current state? How many symbols
>>>>> are on the tape?
>>>>> Now you might be subtle and say that there are some questions about the integers
>>>>> that we can't answer without going through the previous members. E.g., what are the
>>>>> prime factors of this number? That's perfectly true. And it is harder to answer the
>>>>> question "what are the prime factors" for a large integer than for a small integer.
>>>> I think your missing my point. Before we HAVE a definition of what the
>>>> number 5 actually is, we need the definition of the number 4, because
>>>> ultimately, 5 is defined as the number after 4.
>>>>
>>>> This is getting into messy detailed number theory that most people just
>>>> skip, and I only have a passing glance as a looked at it and said it
>>>> wasn't for me.
>>>>
>>>> Most mathematics just takes from elementary number theory that we HAVE a
>>>> definition of things like the Natural Numbers, Rationals, etc. but these
>>>> actually DO have fundamental definitions from somewhat complicated
>>>> theory to prove that they actually exist.
>>>>
>>> If the only notation we have for a natural number is a "count the stokes" or
>>> unary notation, then I suppose that it's true to say that before we have a
>>> representation of 11111 we must have a representation of 1111. We can't
>>> create the representation for five without first creating a representation for
>>> four.
>> No, I am NOT talking about representation, but the definitions of the
>> numbers themselves.
>>
>> Try to define the number two without using the number one or words which
>> depend on already knowing what two is (like pair).
>>
> The smallest prime number.

On Monday, 18 July 2022 at 13:25:45 UTC+1, richar...@gmail.com wrote:
> On 7/17/22 6:34 AM, Malcolm McLean wrote:
> > On Saturday, 16 July 2022 at 23:35:12 UTC+1, richar...@gmail.com wrote:
> >> On 7/16/22 4:30 PM, Malcolm McLean wrote:
> >>> On Saturday, 16 July 2022 at 17:50:53 UTC+1, richar...@gmail.com wrote:
> >>>> On 7/16/22 12:24 PM, Malcolm McLean wrote:
> >>>>> On Saturday, 16 July 2022 at 14:22:05 UTC+1, richar...@gmail.com wrote:
> >>>>>> On 7/16/22 9:03 AM, Malcolm McLean wrote:
> >>>>>>> On Saturday, 16 July 2022 at 13:52:11 UTC+1, richar...@gmail.com wrote:
> >>>>>>>> On 7/16/22 12:19 AM, Malcolm McLean wrote:
> >>>>>>>>> On Saturday, 16 July 2022 at 02:02:41 UTC+1, Ben Bacarisse wrote:
> >>>>>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
> >>>>>>>>>>
> >>>>>>>>>>> On Thursday, 14 July 2022 at 00:17:00 UTC+1, Ben Bacarisse wrote:
> >>>>>>>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
> >>>>>>>>>>>>
> >>>>>>>>>>>>> On Wednesday, 13 July 2022 at 02:29:13 UTC+1, Ben Bacarisse wrote:
> >>>>>>>>>>>>>> Malcolm McLean <malcolm.ar...@gmail.com> writes:
> >>>>>>>>>>>>
> >>>>>>>>>>>>>>> However you can convert any parallel algorithm to a single-threaded
> >>>>>>>>>>>>>>> algorithm. It has been proposed that the reason that computers can't
> >>>>>>>>>>>>>>> match human cognitive abilities is that the algorithms are not
> >>>>>>>>>>>>>>> parallel enough. Whilst theoretically this is nothing to do with
> >>>>>>>>>>>>>>> hardware, in practice of course it is, because you can't be hanging
> >>>>>>>>>>>>>>> around for the results.
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>>> There's no "hanging around" with a TM.
> >>>>>>>>>>>>>>
> >>>>>>>>>>>>> Of course there is. You build it, and set it running, and some time later it
> >>>>>>>>>>>>> comes back with the results. Or never stops. Time is inherent in the
> >>>>>>>>>>>>> model.
> >>>>>>>>>>>> No. I can't imagine how you can think this, but it seems there is no
> >>>>>>>>>>>> common ground for this discussion.
> >>>>>>>>>>>>
> >>>>>>>>>>> You can't distinguish, in the general case, a TM that doesn't halt from
> >>>>>>>>>>> one that you can't hang round for.
> >>>>>>>>>> What's the point of continuing the discussion when there is this
> >>>>>>>>>> fundamental mis-match in the use of terms? You've just talked about
> >>>>>>>>>> hanging around again, but there is no hanging around with TMs.
> >>>>>>>>>>> That's a fundamental characteristic of
> >>>>>>>>>>> the model and means that TMs cannot be used to prove hypotheses in
> >>>>>>>>>>> number theory.
> >>>>>>>>>> No, it's not the hanging around that's fundamental, it's the
> >>>>>>>>>> undecidabilty.
> >>>>>>>>>>
> >>>>>>>>>> This distinction matters because your original argument was about the
> >>>>>>>>>> hanging around, not the undecidabilty.
> >>>>>>>>>>
> >>>>>>>>> Turing machines can be used to disprove conjectures in number theory.
> >>>>>>>>> And plenty of patterns that hold for low integers never become "conjectures"
> >>>>>>>>> because it is so easy to find a counter-example. And how were these counter-
> >>>>>>>>> examples found?
> >>>>>>>>> And of course we can prove the conjecture for numbers up to N. But N is
> >>>>>>>>> determined by how well we can engineer the machine, and how long we are
> >>>>>>>>> prepared to hang around. N must be finite, because a calculation cannot be
> >>>>>>>>> instantaneous.
> >>>>>>>>> Time is crucial. It's present in the model,and it cannot be abstracted away by
> >>>>>>>>> saying that a Turing machine is a mathematical object rather than a real
> >>>>>>>>> object.
> >>>>>>>>>>> You can't get away from the fact that, to know the configuration
> >>>>>>>>>>> at step three, which is wholly determined by the configuration at step one,
> >>>>>>>>>>> you absolutely have to calculate the configuration at step two (in the
> >>>>>>>>>>> general case again). That's how the model works. Time is inherent to
> >>>>>>>>>>> it.
> >>>>>>>>>> Nonsense. There is a sequence. Time has nothing to do with it. If we
> >>>>>>>>>> apply an iterative function, say f(n) = n/2, f^1000(1) does not "take
> >>>>>>>>>> longer" than f^2(1). A TM is just an iterated function.
> >>>>>>>>>>
> >>>>>>>>> Of course it does. If your calculation requires an intermediate result to
> >>>>>>>>> be calculated before the final result, then one with more intermediate
> >>>>>>>>> results "takes longer" than one with fewer.
> >>>>>>>>> That's pretty basic. You only have to do a few months of primary school
> >>>>>>>>> to realise that some sums are harder than others.
> >>>>>>>>>
> >>>>>>>> There is order, but not necessarily physical "time" as we define it.
> >>>>>>>>
> >>>>>>>> How "long" does it take to create the set of all natural numbers? There
> >>>>>>>> are a defined sequence, and in at least one sense, 6 doesn't exist until
> >>>>>>>> first 5 does, so 6 can be after it, but it doesn't take "time" to create
> >>>>>>>> that set.
> >>>>>>>>
> >>>>>>> Write down a very large integer with digits chosen at random.
> >>>>>>> If the integer is large enough, the chance that the integer below it has ever
> >>>>>>> been represented by any human is vanishingly small. However you can
> >>>>>>> describe some properties of the number, such as its name in English words.
> >>>>>>>
> >>>>>>> You don't need to create integers in sequence.
> >>>>>>>
> >>>>>>> Now take a Turing machine that has complex behaviour. Describe the configuration
> >>>>>>> at machine start. Now describe the configuration an any stage of your choosing,
> >>>>>>> say stage 1000. Configuration at stage 1000 is totally determined by configuration
> >>>>>>> at stage 1. But to get there, you have to calculate the intermediate stages.
> >>>>>>>
> >>>>>>> The set of configurations is not like the set of the integers.
> >>>>>> Look at the foundation of numbers, 6 is DEFINED as the successor to 5.
> >>>>>> The "Number" six only has meening as the counting number after five.
> >>>>>>
> >>>>>> Yes, you can EXPRESS a number without expressing the previous number,
> >>>>>> but that number itself doesn't have meaning without first having found
> >>>>>> meaning for the numbers below it.
> >>>>>>
> >>>>>> In the same way, you can express a Turing Machine in any configuration
> >>>>>> that that it could possibly (or maybe not even possibly) be in without
> >>>>>> having to go through the previous states. That configuration may not
> >>>>>> actually have a meaning unless we know how it started.
> >>>>>>
> >>>>> We can use the notation <Initial configuration>+Nsteps for a Turing machine
> >>>>> in a given state. But we can't use that notation to answer some basic questions
> >>>>> about the Turing machine. Like, what is the current state? How many symbols
> >>>>> are on the tape?
> >>>>> Now you might be subtle and say that there are some questions about the integers
> >>>>> that we can't answer without going through the previous members. E.g., what are the
> >>>>> prime factors of this number? That's perfectly true. And it is harder to answer the
> >>>>> question "what are the prime factors" for a large integer than for a small integer.
> >>>> I think your missing my point. Before we HAVE a definition of what the
> >>>> number 5 actually is, we need the definition of the number 4, because
> >>>> ultimately, 5 is defined as the number after 4.
> >>>>
> >>>> This is getting into messy detailed number theory that most people just
> >>>> skip, and I only have a passing glance as a looked at it and said it
> >>>> wasn't for me.
> >>>>
> >>>> Most mathematics just takes from elementary number theory that we HAVE a
> >>>> definition of things like the Natural Numbers, Rationals, etc. but these
> >>>> actually DO have fundamental definitions from somewhat complicated
> >>>> theory to prove that they actually exist.
> >>>>
> >>> If the only notation we have for a natural number is a "count the stokes" or
> >>> unary notation, then I suppose that it's true to say that before we have a
> >>> representation of 11111 we must have a representation of 1111. We can't
> >>> create the representation for five without first creating a representation for
> >>> four.
> >> No, I am NOT talking about representation, but the definitions of the
> >> numbers themselves.
> >>
> >> Try to define the number two without using the number one or words which
> >> depend on already knowing what two is (like pair).
> >>
> > The smallest prime number.
> Define PRIME without using the definition of the number one.
>
Given a heap of stones, we distribute the stones amongst a group of people.
If we can find a group and distribution such that all members hold the same
number of stones, and all members hold fewer stones than in the pile, then
the number of stones, the number of stones in the pile is not prime. If additionally
we can find a group and distribution in which all members hold fewer stones
than in the pile, we can say that the number of stones in the pile is prime.