Discussion:
"meal prepping"
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s***@gmail.com
2020-02-01 13:53:50 UTC
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American English is going to be totally free of metaphor - EVERYTHING will be reduced to something a computer can understand.
s***@gmail.com
2020-02-01 15:16:14 UTC
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Post by s***@gmail.com
American English is going to be totally free of metaphor - EVERYTHING will be reduced to something a computer can understand.
"unpack" for analyze

"takeaway" for key ideas that emerged at a meeting.
Jack
2020-02-01 20:26:10 UTC
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Post by s***@gmail.com
American English is going to be totally free of metaphor - EVERYTHING will be reduced to something a computer can understand.
What makes you think computers can't understand metaphors, or anything
else that humans might put up as their last bastion of uniqueness?
Lewis
2020-02-02 13:02:02 UTC
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Post by Jack
Post by s***@gmail.com
American English is going to be totally free of metaphor - EVERYTHING will be reduced to something a computer can understand.
What makes you think computers can't understand metaphors, or anything
else that humans might put up as their last bastion of uniqueness?
If you would like to claim that computers can understand metaphor it is
incumbent on you to prove that.

Lacking any proof to the contrary it is perfectly correct to say that
computers cannot understand metaphor.
--
'They think they want good government and justice for all, Vimes, yet
what is it they really crave, deep in their hearts? Only that
things go on as normal and tomorrow is pretty much like today.'
--Feet of Clay
Jack
2020-02-02 20:45:50 UTC
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On Sun, 2 Feb 2020 13:02:02 -0000 (UTC), Lewis
Post by Lewis
Post by Jack
Post by s***@gmail.com
American English is going to be totally free of metaphor - EVERYTHING will be reduced to something a computer can understand.
What makes you think computers can't understand metaphors, or anything
else that humans might put up as their last bastion of uniqueness?
If you would like to claim that computers can understand metaphor it is
incum
Lewis
2020-02-02 21:12:47 UTC
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Post by Jack
On Sun, 2 Feb 2020 13:02:02 -0000 (UTC), Lewis
Post by Lewis
Post by Jack
Post by s***@gmail.com
American English is going to be totally free of metaphor - EVERYTHING will be reduced to something a computer can understand.
What makes you think computers can't understand metaphors, or anything
else that humans might put up as their last bastion of uniqueness?
If you would like to claim that computers can understand metaphor it is
incumbent on you to prove that.
Is not.
I see you failed logic 101.
--
The person on the other side was a young woman. Very obviously a
young woman. There was no possible way that she could have been
mistaken for a young man in any language, especially Braille.
Peter Moylan
2020-02-02 21:24:22 UTC
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Post by Lewis
Post by Jack
Post by s***@gmail.com
American English is going to be totally free of metaphor -
EVERYTHING will be reduced to something a computer can
understand.
What makes you think computers can't understand metaphors, or
anything else that humans might put up as their last bastion of
uniqueness?
If you would like to claim that computers can understand metaphor it
is incumbent on you to prove that.
A computer can "understand" anything it is programmed to handle. It
wouldn't be too hard to add in a recognition of metaphors.

Of course, the meaning of "understand" is open to debate.
--
Peter Moylan http://www.pmoylan.org
Newcastle, NSW, Australia
David Kleinecke
2020-02-02 23:08:52 UTC
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Post by Peter Moylan
Post by Lewis
Post by Jack
Post by s***@gmail.com
American English is going to be totally free of metaphor -
EVERYTHING will be reduced to something a computer can
understand.
What makes you think computers can't understand metaphors, or
anything else that humans might put up as their last bastion of
uniqueness?
If you would like to claim that computers can understand metaphor it
is incumbent on you to prove that.
A computer can "understand" anything it is programmed to handle. It
wouldn't be too hard to add in a recognition of metaphors.
Of course, the meaning of "understand" is open to debate.
I would be surprised if there doesn't already exists a typography of
metaphors which the computer could use. I think the practical problem
is likely too be stopping the computer from reading every text as a
metaphor.
Lewis
2020-02-04 19:47:17 UTC
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Post by Peter Moylan
Post by Lewis
Post by Jack
Post by s***@gmail.com
American English is going to be totally free of metaphor -
EVERYTHING will be reduced to something a computer can
understand.
What makes you think computers can't understand metaphors, or
anything else that humans might put up as their last bastion of
uniqueness?
If you would like to claim that computers can understand metaphor it
is incumbent on you to prove that.
A computer can "understand" anything it is programmed to handle. It
wouldn't be too hard to add in a recognition of metaphors.
I don't even begin to understand how you can think that considering a
fairly large percentage of humans don't understand metaphor.
Post by Peter Moylan
Of course, the meaning of "understand" is open to debate.
--
Demons have existed on the Discworld for at least as long as the
gods, who in many ways they closely resemble. The difference is
basically the same as between terrorists and freedom fighters.
Peter Moylan
2020-02-05 02:18:49 UTC
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Post by Lewis
Post by Peter Moylan
Post by Lewis
Post by Jack
Post by s***@gmail.com
American English is going to be totally free of metaphor -
EVERYTHING will be reduced to something a computer can
understand.
What makes you think computers can't understand metaphors, or
anything else that humans might put up as their last bastion of
uniqueness?
If you would like to claim that computers can understand metaphor it
is incumbent on you to prove that.
A computer can "understand" anything it is programmed to handle. It
wouldn't be too hard to add in a recognition of metaphors.
I don't even begin to understand how you can think that considering a
fairly large percentage of humans don't understand metaphor.
Computer programmers are better educated than humanity as a whole.
Post by Lewis
Post by Peter Moylan
Of course, the meaning of "understand" is open to debate.
--
Peter Moylan http://www.pmoylan.org
Newcastle, NSW, Australia
Lewis
2020-02-05 05:39:04 UTC
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Post by Peter Moylan
Post by Lewis
Post by Peter Moylan
Post by Lewis
Post by Jack
Post by s***@gmail.com
American English is going to be totally free of metaphor -
EVERYTHING will be reduced to something a computer can
understand.
What makes you think computers can't understand metaphors, or
anything else that humans might put up as their last bastion of
uniqueness?
If you would like to claim that computers can understand metaphor it
is incumbent on you to prove that.
A computer can "understand" anything it is programmed to handle. It
wouldn't be too hard to add in a recognition of metaphors.
I don't even begin to understand how you can think that considering a
fairly large percentage of humans don't understand metaphor.
Computer programmers are better educated than humanity as a whole.
Doesn't matter, you;e trying to make an intelligent computer, something
that so far has proved exceedingly difficult and has not yet been done.
Post by Peter Moylan
Post by Lewis
Post by Peter Moylan
Of course, the meaning of "understand" is open to debate.
--
It was where the city kept all those things it occasionally needed
but was uneasy about, like the Watch-house, the theatres, the
prison and the publishers. It was the place for all those things
which might go off bang in unexpected ways.
s***@gmail.com
2020-02-05 07:39:31 UTC
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Post by Lewis
Post by Peter Moylan
Post by Lewis
Post by Peter Moylan
Post by Lewis
Post by Jack
Post by s***@gmail.com
American English is going to be totally free of metaphor -
EVERYTHING will be reduced to something a computer can
understand.
What makes you think computers can't understand metaphors, or
anything else that humans might put up as their last bastion of
uniqueness?
If you would like to claim that computers can understand metaphor it
is incumbent on you to prove that.
A computer can "understand" anything it is programmed to handle. It
wouldn't be too hard to add in a recognition of metaphors.
I don't even begin to understand how you can think that considering a
fairly large percentage of humans don't understand metaphor.
But what percentage recognizes metaphors as a non-literal description,
even if they couldn't describe what a metaphor is?
Post by Lewis
Post by Peter Moylan
Computer programmers are better educated than humanity as a whole.
Doesn't matter, you;e trying to make an intelligent computer, something
that so far has proved exceedingly difficult and has not yet been done.
Recognition is a lower bar than understanding.
Post by Lewis
Post by Peter Moylan
Post by Lewis
Post by Peter Moylan
Of course, the meaning of "understand" is open to debate.
There are different levels of understanding,
but Lewis' point may carry for now ...
computers are becoming good at pattern recognition,
with large training sets,
but there are abstraction levels they don't yet manage.

(Humans are pretty good at using small training sets,
once they've reached the point of producing speech.)

/dps
Tak To
2020-02-07 20:37:57 UTC
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Post by s***@gmail.com
Post by Lewis
Post by Peter Moylan
Post by Lewis
Post by Peter Moylan
Post by Lewis
Post by Jack
Post by s***@gmail.com
American English is going to be totally free of metaphor -
EVERYTHING will be reduced to something a computer can
understand.
What makes you think computers can't understand metaphors, or
anything else that humans might put up as their last bastion of
uniqueness?
If you would like to claim that computers can understand metaphor it
is incumbent on you to prove that.
A computer can "understand" anything it is programmed to handle. It
wouldn't be too hard to add in a recognition of metaphors.
I don't even begin to understand how you can think that considering a
fairly large percentage of humans don't understand metaphor.
But what percentage recognizes metaphors as a non-literal description,
even if they couldn't describe what a metaphor is?
Post by Lewis
Post by Peter Moylan
Computer programmers are better educated than humanity as a whole.
Doesn't matter, you;e trying to make an intelligent computer, something
that so far has proved exceedingly difficult and has not yet been done.
Recognition is a lower bar than understanding.
Post by Lewis
Post by Peter Moylan
Post by Lewis
Post by Peter Moylan
Of course, the meaning of "understand" is open to debate.
There are different levels of understanding,
but Lewis' point may carry for now ...
computers are becoming good at pattern recognition,
with large training sets,
but there are abstraction levels they don't yet manage.
(Humans are pretty good at using small training sets,
once they've reached the point of producing speech.)
It has been long conjectured that an artificial intelligence
capable of passing Turing's Test cannot be programmed but has
to be "raised", very much like the mind of a human.

Drifting a bit, can we say that God's mind is a Turing machine?
--
Tak
----------------------------------------------------------------+-----
Tak To ***@alum.mit.eduxx
--------------------------------------------------------------------^^
[taode takto ~{LU5B~}] NB: trim the xx to get my real email addr
Lewis
2020-02-07 23:38:24 UTC
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Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
If unicorn's minds are made of tapioca and jellybeans then anything is
possible.
--
"Are you pondering what I'm pondering?"
Snowball: "Oh Brain, I certainly hope so."
David Kleinecke
2020-02-08 00:33:07 UTC
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Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
J. J. Lodder
2020-02-08 09:52:30 UTC
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Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
Even that wouldn't help.

Jan
Peter Moylan
2020-02-08 10:38:31 UTC
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Post by J. J. Lodder
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
Even that wouldn't help.
It's been a long time since I last looked at the theory, but I think a
single Turing machine can emulate a countable number of Turing machines.
--
Peter Moylan http://www.pmoylan.org
Newcastle, NSW, Australia
J. J. Lodder
2020-02-08 13:46:23 UTC
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Post by Peter Moylan
Post by J. J. Lodder
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
Even that wouldn't help.
It's been a long time since I last looked at the theory, but I think a
single Turing machine can emulate a countable number of Turing machines.
Exactly.
Turing machines, being of finite extent, must be countable,
so an infinity of them must be a countable infinity,
so no better than one.
So not good enough to become supreme fascist with.

Brouwer still has the last laugh,

Jan
Athel Cornish-Bowden
2020-02-08 15:20:57 UTC
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Post by J. J. Lodder
Post by Peter Moylan
Post by J. J. Lodder
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
Even that wouldn't help.
It's been a long time since I last looked at the theory, but I think a
single Turing machine can emulate a countable number of Turing machines.
Exactly.
Turing machines, being of finite extent, must be countable,
so an infinity of them must be a countable infinity,
so no better than one.
So not good enough to become supreme fascist with.
Brouwer still has the last laugh,
As you understand these things, maybe you could take a look at the
following paper and let us know if you agree with its conclusion:

Palmer, M. L., Williams, R. A., Gatherer, D., 2016. Rosen’s (M, R)
system as an X-machine. J. Theor. Biol., 408, 97– 104.
doi:10.1016/j.jtbi.2016.08.007.
--
athel
Peter Moylan
2020-02-08 23:20:19 UTC
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Post by Athel Cornish-Bowden
Post by Peter Moylan
Post by J. J. Lodder
On Friday, February 7, 2020 at 12:38:03 PM UTC-8, Tak To
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing
machine?
I doubt it. Gods's mind - like ours - might be many (an
infinite number?) of Turing machines operating in sync.
Even that wouldn't help.
It's been a long time since I last looked at the theory, but I
think a single Turing machine can emulate a countable number of
Turing machines.
Exactly. Turing machines, being of finite extent, must be
countable, so an infinity of them must be a countable infinity, so
no better than one. So not good enough to become supreme fascist
with.
Brouwer still has the last laugh,
As you understand these things, maybe you could take a look at the
Palmer, M. L., Williams, R. A., Gatherer, D., 2016. Rosen’s (M, R)
system as an X-machine. J. Theor. Biol., 408, 97– 104.
doi:10.1016/j.jtbi.2016.08.007.
I'll be interested to see what Jan says. I can't get access to the
paper, but the abstract seems to contain an extraordinary claim.

At present it is generally accepted that anything that is computable can
be realised by a Turing machine. (Usually not efficiently, but that's
irrelevant to the discussion.) If Rosen's (M,R) system can't be emulated
by a Turing machine, then apparently it can compute things that are not
computable. But I don't know how deep the contradiction goes.
--
Peter Moylan http://www.pmoylan.org
Newcastle, NSW, Australia
J. J. Lodder
2020-02-09 11:52:31 UTC
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Post by Athel Cornish-Bowden
Post by J. J. Lodder
Post by Peter Moylan
Post by J. J. Lodder
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
Even that wouldn't help.
It's been a long time since I last looked at the theory, but I think a
single Turing machine can emulate a countable number of Turing machines.
Exactly.
Turing machines, being of finite extent, must be countable,
so an infinity of them must be a countable infinity,
so no better than one.
So not good enough to become supreme fascist with.
Brouwer still has the last laugh,
As you understand these things, maybe you could take a look at the
You must be vastly overestimating my understandings.
Post by Athel Cornish-Bowden
Palmer, M. L., Williams, R. A., Gatherer, D., 2016. Rosen's (M, R)
system as an X-machine. J. Theor. Biol., 408, 97
104.
Post by Athel Cornish-Bowden
doi:10.1016/j.jtbi.2016.08.007.
Hard to say from merely the abstract,
but it seems to me that the key word here is 'allegedly'.
Anyway Rosen's (M,R) systems seem to be at the base
of a large literature, which is certainly to much for me to study.

Jan
Madhu
2020-02-09 16:30:36 UTC
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Post by J. J. Lodder
Post by Athel Cornish-Bowden
Post by J. J. Lodder
Exactly.
Turing machines, being of finite extent, must be countable,
so an infinity of them must be a countable infinity,
so no better than one.
So not good enough to become supreme fascist with.
Brouwer still has the last laugh,
[please remind me - what was his claim?]
Post by J. J. Lodder
Post by Athel Cornish-Bowden
As you understand these things, maybe you could take a look at the
You must be vastly overestimating my understandings.
Post by Athel Cornish-Bowden
Palmer, M. L., Williams, R. A., Gatherer, D., 2016. Rosen's (M, R)
system as an X-machine. J. Theor. Biol., 408, 97 104.
doi:10.1016/j.jtbi.2016.08.007.
Hard to say from merely the abstract, but it seems to me that the key
word here is 'allegedly'. Anyway Rosen's (M,R) systems seem to be at
the base of a large literature, which is certainly to much for me to
study.
[My situation is worse and I can't really make a statement (but I
will.)]

The first claim in the abstract seems to be patently false. I imagine it
can be shoved under the carpet with the "allegedly" and have the paper
merely address the perception-shortcomings with ordinary mathematics.

M-R systems - whatever they are - if they are a formalism (under set
theory) are subject to the same set of limitations. Instead of talking
of Turing Machines it would be better to talk of the Turing hypothesis -
whether there are any OTHER models of computation at all. I don't think
other approaches have worked: My professor (when I started my MS in
Comp.Sci) was specializing in "Artificial Life" - I found it impossible
to look beyond the von-neumann-machine canopy that surrounds our world
(AL sounds rather blasphemous to me now but it was just only about
computation. that professor disowned me at some point and my academic
career ended there)
J. J. Lodder
2020-02-09 17:44:30 UTC
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Post by Madhu
Post by J. J. Lodder
Exactly.
Turing machines, being of finite extent, must be countable,
so an infinity of them must be a countable infinity,
so no better than one.
So not good enough to become supreme fascist with.
Brouwer still has the last laugh,
[please remind me - what was his claim?]
See unnder 'Intuitionism',

Jan
Tak To
2020-02-10 18:03:22 UTC
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Post by J. J. Lodder
Post by Athel Cornish-Bowden
Post by J. J. Lodder
Post by Peter Moylan
Post by J. J. Lodder
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
Even that wouldn't help.
It's been a long time since I last looked at the theory, but I think a
single Turing machine can emulate a countable number of Turing machines.
Exactly.
Turing machines, being of finite extent, must be countable,
so an infinity of them must be a countable infinity,
so no better than one.
So not good enough to become supreme fascist with.
Brouwer still has the last laugh,
As you understand these things, maybe you could take a look at the
You must be vastly overestimating my understandings.
Post by Athel Cornish-Bowden
Palmer, M. L., Williams, R. A., Gatherer, D., 2016. Rosen's (M, R)
system as an X-machine. J. Theor. Biol., 408, 97
104.
Post by Athel Cornish-Bowden
doi:10.1016/j.jtbi.2016.08.007.
Hard to say from merely the abstract,
but it seems to me that the key word here is 'allegedly'.
Anyway Rosen's (M,R) systems seem to be at the base
of a large literature, which is certainly to much for me to study.
It is interesting that you mentioned the complexity of the M-R
system but not that of the X-machine.
--
Tak
----------------------------------------------------------------+-----
Tak To ***@alum.mit.eduxx
--------------------------------------------------------------------^^
[taode takto ~{LU5B~}] NB: trim the xx to get my real email addr
Tak To
2020-02-08 16:03:30 UTC
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Post by Peter Moylan
Post by J. J. Lodder
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
Even that wouldn't help.
It's been a long time since I last looked at the theory, but I think a
single Turing machine can emulate a countable number of Turing machines.
No, only finitely many Turing machines.

A TM can emulate a finite number of TMs by dovetailing. One
cannot dovetails on an infinite number of TMs in finitely
many steps.
--
Tak
----------------------------------------------------------------+-----
Tak To ***@alum.mit.eduxx
--------------------------------------------------------------------^^
[taode takto ~{LU5B~}] NB: trim the xx to get my real email addr
Tak To
2020-02-08 23:42:27 UTC
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Post by Tak To
Post by Peter Moylan
Post by J. J. Lodder
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
Even that wouldn't help.
It's been a long time since I last looked at the theory, but I think a
single Turing machine can emulate a countable number of Turing machines.
No, only finitely many Turing machines.
A TM can emulate a finite number of TMs by dovetailing. One
cannot dovetails on an infinite number of TMs in finitely
many steps.
Sorry, ignore what I said.
--
Tak
----------------------------------------------------------------+-----
Tak To ***@alum.mit.eduxx
--------------------------------------------------------------------^^
[taode takto ~{LU5B~}] NB: trim the xx to get my real email addr
Tak To
2020-02-12 20:20:13 UTC
Reply
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Post by Peter Moylan
Post by J. J. Lodder
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
Even that wouldn't help.
It's been a long time since I last looked at the theory, but I think a
single Turing machine can emulate a countable number of Turing machines.
A couple of days ago I post a message refuting this but later
withdrew it because I made a faulty argument. Now that I have
had some time to think about it, I would like to add

"... only for some countably-many sets of TM's"

Note that the main idea about TM and Church's Thesis is systems
that can be *finitely* encoded. A TM by definition has only
a finite number of rules. Ditto for the number of pre-written
symbols on the tape, even though we can relax that a bit
by allowing a pre-processor TM to generate them.

To be able to emulate a (aleph-0) set of TM's one must first
transcribe the rules of all the TM's on to the tape as symbols,
and there lies the original issue. Thus a TM can emulate
(dovetail) a set of TM's only if the set (1) is finite
or (2) can be somehow generated from a finite number of rules.

Astute netters might have a deja vu feeling in the form of
Russel's Paradox...
--
Tak
----------------------------------------------------------------+-----
Tak To ***@alum.mit.eduxx
--------------------------------------------------------------------^^
[taode takto ~{LU5B~}] NB: trim the xx to get my real email addr
Tak To
2020-02-08 15:38:44 UTC
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Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
? My mind is less than a Turing machine. It has only a finite
"tape".
--
Tak
----------------------------------------------------------------+-----
Tak To ***@alum.mit.eduxx
--------------------------------------------------------------------^^
[taode takto ~{LU5B~}] NB: trim the xx to get my real email addr
J. J. Lodder
2020-02-08 19:12:25 UTC
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Post by Tak To
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
? My mind is less than a Turing machine. It has only a finite
"tape".
And so what? Mathematical theorems are not established
by meddling about with Turing machines, [1]

Jan

[1] Typical example: you (that is, Euclid) can give a generating formula
for all Pythagorean triangles with integer sides.
(3,4,5) (5,12,13) etc.
Plowing through the integers with a computer otoh
can never yield more than a finite number of examples,

Jan
Tak To
2020-02-09 02:01:32 UTC
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Post by J. J. Lodder
Post by Tak To
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
? My mind is less than a Turing machine. It has only a finite
"tape".
And so what? Mathematical theorems are not established
by meddling about with Turing machines, [1]
Let M be the number of atoms in the universe. A Turing
Machine can determine if an arbitrary M-digit number is prime.
We can't.
--
Tak
----------------------------------------------------------------+-----
Tak To ***@alum.mit.eduxx
--------------------------------------------------------------------^^
[taode takto ~{LU5B~}] NB: trim the xx to get my real email addr
Peter Moylan
2020-02-09 03:38:02 UTC
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Post by Tak To
Post by J. J. Lodder
Post by Tak To
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
? My mind is less than a Turing machine. It has only a finite
"tape".
And so what? Mathematical theorems are not established
by meddling about with Turing machines, [1]
Let M be the number of atoms in the universe. A Turing
Machine can determine if an arbitrary M-digit number is prime.
We can't.
Of course, the tape would have to be borrowed from a different universe.
--
Peter Moylan http://www.pmoylan.org
Newcastle, NSW, Australia
J. J. Lodder
2020-02-09 12:30:20 UTC
Reply
Permalink
Post by Peter Moylan
Post by Tak To
Post by J. J. Lodder
Post by Tak To
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
? My mind is less than a Turing machine. It has only a finite
"tape".
And so what? Mathematical theorems are not established
by meddling about with Turing machines, [1]
Let M be the number of atoms in the universe. A Turing
Machine can determine if an arbitrary M-digit number is prime.
We can't.
Of course, the tape would have to be borrowed from a different universe.
Just puuting all the verses from the multiverse at it
would solve it in no time.

Jan
Tak To
2020-02-10 17:59:48 UTC
Reply
Permalink
Post by J. J. Lodder
Post by Peter Moylan
Post by Tak To
Post by J. J. Lodder
Post by Tak To
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
? My mind is less than a Turing machine. It has only a finite
"tape".
And so what? Mathematical theorems are not established
by meddling about with Turing machines, [1]
Let M be the number of atoms in the universe. A Turing
Machine can determine if an arbitrary M-digit number is prime.
We can't.
Of course, the tape would have to be borrowed from a different universe.
Just puuting all the verses from the multiverse at it
would solve it in no time.
You seem to be missing the point of my original statement,
which was that God's mind is *not* "like ours", as DK put it.

I don't know if you are human, but human minds are not infinite
nor can they access the multiverse. There are theorems that
can be proven/verified by a TM but remain unknowable to us
humans. And I haven't even mention the time issue.

Turing-solvable is not the same as human-solvable.
--
Tak
----------------------------------------------------------------+-----
Tak To ***@alum.mit.eduxx
--------------------------------------------------------------------^^
[taode takto ~{LU5B~}] NB: trim the xx to get my real email addr
J. J. Lodder
2020-02-11 08:26:18 UTC
Reply
Permalink
Post by Tak To
Post by Peter Moylan
Post by Tak To
Post by J. J. Lodder
Post by Tak To
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
? My mind is less than a Turing machine. It has only a finite
"tape".
And so what? Mathematical theorems are not established
by meddling about with Turing machines, [1]
Let M be the number of atoms in the universe. A Turing
Machine can determine if an arbitrary M-digit number is prime.
We can't.
Of course, the tape would have to be borrowed from a different universe.
Just puting all the verses from the multiverse at it
would solve it in no time.
You seem to be missing the point of my original statement,
which was that God's mind is *not* "like ours", as DK put it.
How could this utterly silly statement have any meaning at all?
Post by Tak To
I don't know if you are human, but human minds are not infinite
nor can they access the multiverse.
-You- started the silly idea of invoking the size of the universe
for claims aboout what Turing machines can do.
BTW, PM already made fun of you
by noting that we will get the tape from another universe,
but you never notice when people make fun of you.
Post by Tak To
There are theorems that
can be proven/verified by a TM but remain unknowable to us
humans. And I haven't even mention the time issue.
Turing-solvable is not the same as human-solvable.
Why kick down this open door once again?

Jan
Lewis
2020-02-11 13:48:34 UTC
Reply
Permalink
Post by J. J. Lodder
Post by Tak To
Post by Peter Moylan
Post by Tak To
Post by J. J. Lodder
Post by Tak To
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
? My mind is less than a Turing machine. It has only a finite
"tape".
And so what? Mathematical theorems are not established
by meddling about with Turing machines, [1]
Let M be the number of atoms in the universe. A Turing
Machine can determine if an arbitrary M-digit number is prime.
We can't.
Of course, the tape would have to be borrowed from a different universe.
Just puting all the verses from the multiverse at it
would solve it in no time.
You seem to be missing the point of my original statement,
which was that God's mind is *not* "like ours", as DK put it.
How could this utterly silly statement have any meaning at all?
You could say that about *every* statement about "god".
--
I've got Mathematica 2.2 on my Quadra
Tak To
2020-02-11 18:44:35 UTC
Reply
Permalink
Post by J. J. Lodder
Post by Tak To
Post by Peter Moylan
Post by Tak To
Post by J. J. Lodder
Post by Tak To
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
? My mind is less than a Turing machine. It has only a finite
"tape".
And so what? Mathematical theorems are not established
by meddling about with Turing machines, [1]
Let M be the number of atoms in the universe. A Turing
Machine can determine if an arbitrary M-digit number is prime.
We can't.
Of course, the tape would have to be borrowed from a different universe.
Just puting all the verses from the multiverse at it
would solve it in no time.
You seem to be missing the point of my original statement,
which was that God's mind is *not* "like ours", as DK put it.
How could this utterly silly statement have any meaning at all?
What don't you ask DK?

(It seems that you thought I agreed with DK -- a
misunderstanding that could have been avoided entirely if you
would only read other people's writing carefully.)
Post by J. J. Lodder
Post by Tak To
I don't know if you are human, but human minds are not infinite
nor can they access the multiverse.
-You- started the silly idea of invoking the size of the universe
for claims aboout what Turing machines can do.
Not at all. That was for showing what *human* minds *cannot*
do.
Post by J. J. Lodder
BTW, PM already made fun of you
by noting that we will get the tape from another universe,
but you never notice when people make fun of you.
Was that PM's intention? Anyways, if he cares to explain, I
might be able to pin point where he and I disagree.
Post by J. J. Lodder
Post by Tak To
There are theorems that
can be proven/verified by a TM but remain unknowable to us
humans. And I haven't even mention the time issue.
Turing-solvable is not the same as human-solvable.
Why kick down this open door once again?
Again, why don't you ask DK?
--
Tak
----------------------------------------------------------------+-----
Tak To ***@alum.mit.eduxx
--------------------------------------------------------------------^^
[taode takto ~{LU5B~}] NB: trim the xx to get my real email addr
J. J. Lodder
2020-02-09 11:52:31 UTC
Reply
Permalink
Post by Tak To
Post by J. J. Lodder
Post by Tak To
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
? My mind is less than a Turing machine. It has only a finite
"tape".
And so what? Mathematical theorems are not established
by meddling about with Turing machines, [1]
Let M be the number of atoms in the universe. A Turing
Machine can determine if an arbitrary M-digit number is prime.
We can't.
Complete nonsense. Computability is a mathematical concept.
It has nothing to do with actual material limitations,

Jan
Anders D. Nygaard
2020-02-09 12:55:47 UTC
Reply
Permalink
Post by J. J. Lodder
Post by Tak To
Post by J. J. Lodder
Post by Tak To
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
? My mind is less than a Turing machine. It has only a finite
"tape".
And so what? Mathematical theorems are not established
by meddling about with Turing machines, [1]
Let M be the number of atoms in the universe. A Turing
Machine can determine if an arbitrary M-digit number is prime.
We can't.
Complete nonsense.
Not at all, and for exactly the reason you put forward below.
Post by J. J. Lodder
Computability is a mathematical concept.
It has nothing to do with actual material limitations,
/Anders, Denmark
J. J. Lodder
2020-02-12 10:32:50 UTC
Reply
Permalink
Post by Anders D. Nygaard
Post by J. J. Lodder
Post by Tak To
Post by J. J. Lodder
Post by Tak To
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
? My mind is less than a Turing machine. It has only a finite
"tape".
And so what? Mathematical theorems are not established
by meddling about with Turing machines, [1]
Let M be the number of atoms in the universe. A Turing
Machine can determine if an arbitrary M-digit number is prime.
We can't.
Complete nonsense.
Not at all, and for exactly the reason you put forward below.
Turing machines can determine for arbitraryly large numbers
if they are prime. This is easily proved,
for the computation involves only a finite number of steps.
The halting problem does not arise,

Jan

PS Tak is mistaken about finding large primes too.
This is routinely done in cryptography,
using standard tests for primality.
The computationally hard problem is factoring non-primes.
Post by Anders D. Nygaard
Post by J. J. Lodder
Computability is a mathematical concept.
It has nothing to do with actual material limitations,
Tak To
2020-02-12 17:12:22 UTC
Reply
Permalink
Post by J. J. Lodder
PS Tak is mistaken about finding large primes too.
This is routinely done in cryptography,
using standard tests for primality.
The computationally hard problem is factoring non-primes.
This proves conclusively that J J somehow managed to misread
my posts in more than one aspects.

To wit, I have said nothing of
(J1) what TM's cannot do
(J2) computationally hard
(J3) finding large primes

Instead, I talked about
(T1) what human minds cannot do
(T2) computationally impossible (by humans)
(T3) determining if a large (m-digit) number is prime

Note that to do (T3) one must be able to multiply two
(m/2-digit) numbers, which requires having a tape/writing pad/
register large to hold (at least) the two multiplicands as well
as the product, i,e, a total of 2*m digits.
--
Tak
----------------------------------------------------------------+-----
Tak To ***@alum.mit.eduxx
--------------------------------------------------------------------^^
[taode takto ~{LU5B~}] NB: trim the xx to get my real email addr
Arindam Banerjee
2020-02-12 00:49:58 UTC
Reply
Permalink
Computability ia methematical approach, leading to mathematical functions, to validate or clarify concepts abstract or fuzzy.
Jerry Friedman
2020-02-09 03:54:01 UTC
Reply
Permalink
Post by J. J. Lodder
Post by Tak To
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
? My mind is less than a Turing machine. It has only a finite
"tape".
And so what? Mathematical theorems are not established
by meddling about with Turing machines, [1]
Jan
[1] Typical example: you (that is, Euclid) can give a generating formula
for all Pythagorean triangles with integer sides.
(3,4,5) (5,12,13) etc.
Plowing through the integers with a computer otoh
can never yield more than a finite number of examples,
...

There have been theorem-proving programs for decades. I don't know
whether they've established anything new, though.
--
Jerry Friedman
Arindam Banerjee
2020-02-09 04:42:47 UTC
Reply
Permalink
Post by Jerry Friedman
Post by J. J. Lodder
Post by Tak To
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
? My mind is less than a Turing machine. It has only a finite
"tape".
And so what? Mathematical theorems are not established
by meddling about with Turing machines, [1]
Jan
[1] Typical example: you (that is, Euclid) can give a generating formula
for all Pythagorean triangles with integer sides.
(3,4,5) (5,12,13) etc.
Plowing through the integers with a computer otoh
can never yield more than a finite number of examples,
...
There have been theorem-proving programs for decades. I don't know
whether they've established anything new, though.
How on earth could any computer find it has found anything new unless programmed accordingly?
Post by Jerry Friedman
--
Jerry Friedman
David Kleinecke
2020-02-09 17:39:30 UTC
Reply
Permalink
Post by Arindam Banerjee
Post by Jerry Friedman
Post by J. J. Lodder
Post by Tak To
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
? My mind is less than a Turing machine. It has only a finite
"tape".
And so what? Mathematical theorems are not established
by meddling about with Turing machines, [1]
Jan
[1] Typical example: you (that is, Euclid) can give a generating formula
for all Pythagorean triangles with integer sides.
(3,4,5) (5,12,13) etc.
Plowing through the integers with a computer otoh
can never yield more than a finite number of examples,
...
There have been theorem-proving programs for decades. I don't know
whether they've established anything new, though.
How on earth could any computer find it has found anything new unless programmed accordingly?
By doing ALL of something and finding an as-yet-unnoticed case.
Peter Moylan
2020-02-10 01:00:37 UTC
Reply
Permalink
Post by David Kleinecke
Post by Arindam Banerjee
Post by Jerry Friedman
There have been theorem-proving programs for decades. I don't
know whether they've established anything new, though.
How on earth could any computer find it has found anything new
unless programmed accordingly?
By doing ALL of something and finding an as-yet-unnoticed case.
A proof was found in recent times of one of the well-known hard problems
in mathematics. It might have been Fermat's last theorem, or the
four-colour problem, or perhaps my comments below apply to both. As I
recall it, the proof was very long, involving tedious reasoning, and the
first published "proofs" were subsequently found to contain errors.

In my own work in the past, I hit problems with a similar drawback,
where it seemed that the only way forward was to do a long and tedious
series of manipulations. My experience in those cases was that there was
a very high risk of making a mistake somewhere in the middle, and
failing to see it.

This is the sort of thing where theorem-proving software can do better
than humans, tackling calculations that require enormous patience and
extreme attention to detail.
--
Peter Moylan http://www.pmoylan.org
Newcastle, NSW, Australia
Mark Brader
2020-02-10 02:16:17 UTC
Reply
Permalink
Post by Peter Moylan
A proof was found in recent times of one of the well-known hard problems
in mathematics. It might have been Fermat's last theorem, or the
four-colour problem, or perhaps my comments below apply to both. As I
recall it, the proof was very long, involving tedious reasoning, and the
first published "proofs" were subsequently found to contain errors.
Fermat's Last Theorem. Andrew Wiles's first proof attempt, in 1993,
contained a crtical error, but it was fixed a year or two later.
Post by Peter Moylan
In my own work in the past, I hit problems with a similar drawback,
where it seemed that the only way forward was to do a long and tedious
series of manipulations. My experience in those cases was that there was
a very high risk of making a mistake somewhere in the middle, and
failing to see it.
This is the sort of thing where theorem-proving software can do better
than humans, tackling calculations that require enormous patience and
extreme attention to detail.
And that's how the 4-color theorem was proved, in 1976 by Kenneth Appel
and Wolfgang Haken. The proof involved an enormous list of cases
(called "irreducible configurations"). I think there were something
like 1,500, and it was necessary to show both that the list covered
all possible maps and that the theorem was true for all of them.
Hence the computer.
--
Mark Brader, Toronto | "To the vector go the spoils."
***@vex.net | -- Norton Juster, "The Dot and the Line"

My text in this article is in the public domain.
RH Draney
2020-02-10 07:47:05 UTC
Reply
Permalink
Post by Mark Brader
Fermat's Last Theorem. Andrew Wiles's first proof attempt, in 1993,
contained a crtical error, but it was fixed a year or two later.
And that's how the 4-color theorem was proved, in 1976 by Kenneth Appel
and Wolfgang Haken. The proof involved an enormous list of cases
(called "irreducible configurations"). I think there were something
like 1,500, and it was necessary to show both that the list covered
all possible maps and that the theorem was true for all of them.
Hence the computer.
Four-color theorem in 1976, Fermat in 1993, and next to fall was
supposed to be Goldbach, but I haven't heard anything in the news about
it yet....r
Richard Heathfield
2020-02-10 08:34:19 UTC
Reply
Permalink
Post by RH Draney
Fermat's Last Theorem.  Andrew Wiles's first proof attempt, in 1993,
contained a crtical error, but it was fixed a year or two later.
And that's how the 4-color theorem was proved, in 1976 by Kenneth Appel
and Wolfgang Haken.  The proof involved an enormous list of cases
(called "irreducible configurations").  I think there were something
like 1,500, and it was necessary to show both that the list covered
all possible maps and that the theorem was true for all of them.
Hence the computer.
Four-color theorem in 1976, Fermat in 1993, and next to fall was
supposed to be Goldbach, but I haven't heard anything in the news about
it yet....
I have discovered a truly marvelous proof of it, which this laptop has
insufficient disk space to execute.
--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within
Peter Moylan
2020-02-10 09:43:40 UTC
Reply
Permalink
Post by Richard Heathfield
Post by RH Draney
Post by Mark Brader
Fermat's Last Theorem. Andrew Wiles's first proof attempt, in
1993, contained a crtical error, but it was fixed a year or two
later.
And that's how the 4-color theorem was proved, in 1976 by Kenneth
Appel and Wolfgang Haken. The proof involved an enormous list of
cases (called "irreducible configurations"). I think there were
something like 1,500, and it was necessary to show both that the
list covered all possible maps and that the theorem was true for
all of them. Hence the computer.
Four-color theorem in 1976, Fermat in 1993, and next to fall was
supposed to be Goldbach, but I haven't heard anything in the news
about it yet....
I have discovered a truly marvelous proof of it, which this laptop
has insufficient disk space to execute.
I think I've mentioned before that one copy of my Master's thesis has a
pencilled proof in the margin. Well, actually on some facing pages,
because the thesis was typed single-sided as was the practice in those
days. The original "proof" just said "Proof. The proof is obvious and
therefore omitted." A few years later I needed that proof. It took me
two years to reconstruct it, so that time I made sure of writing it down.

(I was right, by the way. The proof was obvious.)

Now I can't find that particular copy of the thesis, but I've reached
the point where I don't worry about it any more.
--
Peter Moylan http://www.pmoylan.org
Newcastle, NSW, Australia
Lewis
2020-02-10 15:16:00 UTC
Reply
Permalink
Post by RH Draney
Post by Mark Brader
Fermat's Last Theorem. Andrew Wiles's first proof attempt, in 1993,
contained a crtical error, but it was fixed a year or two later.
And that's how the 4-color theorem was proved, in 1976 by Kenneth Appel
and Wolfgang Haken. The proof involved an enormous list of cases
(called "irreducible configurations"). I think there were something
like 1,500, and it was necessary to show both that the list covered
all possible maps and that the theorem was true for all of them.
Hence the computer.
Four-color theorem in 1976, Fermat in 1993, and next to fall was
supposed to be Goldbach, but I haven't heard anything in the news about
it yet....r
The proof for Goldbach's Conjecture seems like it should be very simple
and straightforward.

It's been shown to be true for all the numbers up to some ridiculously
large numbers, but still no proof.
--
I noticed that but was still trying to work out a way of drawing it
to everyone's attention that would be sufficiently satisfying,
combining maximum entertainment value for readers with maximum
humiliation for you. -- Laura
Jerry Friedman
2020-02-10 19:18:00 UTC
Reply
Permalink
Post by Lewis
Post by RH Draney
Post by Mark Brader
Fermat's Last Theorem. Andrew Wiles's first proof attempt, in 1993,
contained a crtical error, but it was fixed a year or two later.
And that's how the 4-color theorem was proved, in 1976 by Kenneth Appel
and Wolfgang Haken. The proof involved an enormous list of cases
(called "irreducible configurations"). I think there were something
like 1,500, and it was necessary to show both that the list covered
all possible maps and that the theorem was true for all of them.
Hence the computer.
Four-color theorem in 1976, Fermat in 1993, and next to fall was
supposed to be Goldbach, but I haven't heard anything in the news about
it yet....r
The proof for Goldbach's Conjecture seems like it should be very simple
and straightforward.
Until you think, "How would I go about proving that?"
Post by Lewis
It's been shown to be true for all the numbers up to some ridiculously
large numbers, but still no proof.
--
Jerry Friedman
Lewis
2020-02-11 01:19:00 UTC
Reply
Permalink
Post by Jerry Friedman
Post by Lewis
Post by RH Draney
Post by Mark Brader
Fermat's Last Theorem. Andrew Wiles's first proof attempt, in 1993,
contained a crtical error, but it was fixed a year or two later.
And that's how the 4-color theorem was proved, in 1976 by Kenneth Appel
and Wolfgang Haken. The proof involved an enormous list of cases
(called "irreducible configurations"). I think there were something
like 1,500, and it was necessary to show both that the list covered
all possible maps and that the theorem was true for all of them.
Hence the computer.
Four-color theorem in 1976, Fermat in 1993, and next to fall was
supposed to be Goldbach, but I haven't heard anything in the news about
it yet....r
The proof for Goldbach's Conjecture seems like it should be very simple
and straightforward.
Until you think, "How would I go about proving that?"
"It's bloody obvious!"

:)
--
'The only reason we're still alive now is that we're more fun alive
than dead,' said Granny's voice behind her. --Lords and Ladies
Peter Moylan
2020-02-11 00:31:05 UTC
Reply
Permalink
Post by Lewis
Post by RH Draney
Post by Mark Brader
Fermat's Last Theorem. Andrew Wiles's first proof attempt, in
1993, contained a crtical error, but it was fixed a year or two
later.
And that's how the 4-color theorem was proved, in 1976 by Kenneth
Appel and Wolfgang Haken. The proof involved an enormous list of
cases (called "irreducible configurations"). I think there were
something like 1,500, and it was necessary to show both that the
list covered all possible maps and that the theorem was true for
all of them. Hence the computer.
Four-color theorem in 1976, Fermat in 1993, and next to fall was
supposed to be Goldbach, but I haven't heard anything in the news
about it yet....r
The proof for Goldbach's Conjecture seems like it should be very
simple and straightforward.
Except that we know, from experience, that practically any statement
about prime numbers is hideously difficult to prove.
Post by Lewis
It's been shown to be true for all the numbers up to some
ridiculously large numbers, but still no proof.
--
Peter Moylan http://www.pmoylan.org
Newcastle, NSW, Australia
J. J. Lodder
2020-02-11 09:34:06 UTC
Reply
Permalink
Post by Peter Moylan
Post by Lewis
Post by RH Draney
Post by Mark Brader
Fermat's Last Theorem. Andrew Wiles's first proof attempt, in
1993, contained a crtical error, but it was fixed a year or two
later.
And that's how the 4-color theorem was proved, in 1976 by Kenneth
Appel and Wolfgang Haken. The proof involved an enormous list of
cases (called "irreducible configurations"). I think there were
something like 1,500, and it was necessary to show both that the
list covered all possible maps and that the theorem was true for
all of them. Hence the computer.
Four-color theorem in 1976, Fermat in 1993, and next to fall was
supposed to be Goldbach, but I haven't heard anything in the news
about it yet....r
The proof for Goldbach's Conjecture seems like it should be very
simple and straightforward.
Except that we know, from experience, that practically any statement
about prime numbers is hideously difficult to prove.
All? Only those that are hard to prove are hard to prove,

Jan
Lewis
2020-02-11 13:51:43 UTC
Reply
Permalink
Post by J. J. Lodder
Post by Peter Moylan
Post by Lewis
Post by RH Draney
Post by Mark Brader
Fermat's Last Theorem. Andrew Wiles's first proof attempt, in
1993, contained a crtical error, but it was fixed a year or two
later.
And that's how the 4-color theorem was proved, in 1976 by Kenneth
Appel and Wolfgang Haken. The proof involved an enormous list of
cases (called "irreducible configurations"). I think there were
something like 1,500, and it was necessary to show both that the
list covered all possible maps and that the theorem was true for
all of them. Hence the computer.
Four-color theorem in 1976, Fermat in 1993, and next to fall was
supposed to be Goldbach, but I haven't heard anything in the news
about it yet....r
The proof for Goldbach's Conjecture seems like it should be very
simple and straightforward.
Except that we know, from experience, that practically any statement
about prime numbers is hideously difficult to prove.
All? Only those that are hard to prove are hard to prove,
I'd agree with All, at least in so far as the common usage of
"practically all". I cannot think of a proof involving prime numbers
that is simple, and we were certainly never given examples of one (or
made to prove one) when I was still taking classes.

We had to prove the volume of a sphere, IIRC, and quite a few proofs
about conic sections in first year calculus.
--
"It's like those French have a different word for *everything*" -
Steve Martin
Tak To
2020-02-11 18:49:55 UTC
Reply
Permalink
Post by Lewis
Post by J. J. Lodder
Post by Peter Moylan
Post by Lewis
Post by RH Draney
Post by Mark Brader
Fermat's Last Theorem. Andrew Wiles's first proof attempt, in
1993, contained a crtical error, but it was fixed a year or two
later.
And that's how the 4-color theorem was proved, in 1976 by Kenneth
Appel and Wolfgang Haken. The proof involved an enormous list of
cases (called "irreducible configurations"). I think there were
something like 1,500, and it was necessary to show both that the
list covered all possible maps and that the theorem was true for
all of them. Hence the computer.
Four-color theorem in 1976, Fermat in 1993, and next to fall was
supposed to be Goldbach, but I haven't heard anything in the news
about it yet....r
The proof for Goldbach's Conjecture seems like it should be very
simple and straightforward.
Except that we know, from experience, that practically any statement
about prime numbers is hideously difficult to prove.
All? Only those that are hard to prove are hard to prove,
I'd agree with All, at least in so far as the common usage of
"practically all". I cannot think of a proof involving prime numbers
that is simple, and we were certainly never given examples of one (or
made to prove one) when I was still taking classes.
Does this count? Theorem: There are infinitely many primes.
Post by Lewis
We had to prove the volume of a sphere, IIRC, and quite a few proofs
about conic sections in first year calculus.
--
Tak
----------------------------------------------------------------+-----
Tak To ***@alum.mit.eduxx
--------------------------------------------------------------------^^
[taode takto ~{LU5B~}] NB: trim the xx to get my real email addr
Jerry Friedman
2020-02-12 15:31:42 UTC
Reply
Permalink
...
Post by Tak To
Post by Lewis
Post by J. J. Lodder
Post by Peter Moylan
Post by Lewis
The proof for Goldbach's Conjecture seems like it should be very
simple and straightforward.
Except that we know, from experience, that practically any statement
about prime numbers is hideously difficult to prove.
All? Only those that are hard to prove are hard to prove,
I'd agree with All, at least in so far as the common usage of
"practically all". I cannot think of a proof involving prime numbers
that is simple, and we were certainly never given examples of one (or
made to prove one) when I was still taking classes.
Does this count? Theorem: There are infinitely many primes.
Or this ("Fermat's Little Theorem" or a corollary of it)? For any
positive integer a and any prime p, a^p is a multiple of p. That is,
a^p = 0 (mod p). (Please feel free to use a three-bar equals sign,
meaning "is congruent to".)

I feel sure Lewis and Peter can follow that proof after some study of
modular arithmetic.

There could be different definitions of "is difficult to prove" and "is
simple" (both present-tense, by the way). Was it hard to even ask the
right question? To come up with the idea of the proof? Was the proof
difficult for the first person to prove it even after coming up with the
idea? Was the proof easily understandable to other notable
mathematicians of the time, or to amateurs of math, or to members of the
public who had learned math in school and were among the better
students? Is it easy to understand for such people now?

I think there are theorems about prime numbers that are easy to prove,
such as that a power of a prime is not divisible by any other prime.
That may be the kind Lewis was excluding with "practically all".

Of course, it depends on what you start with. If you start with the
Peano Postulates, it's all hideously difficult.
Post by Tak To
Post by Lewis
We had to prove the volume of a sphere, IIRC, and quite a few proofs
about conic sections in first year calculus.
--
Jerry Friedman
Jerry Friedman
2020-02-12 15:35:13 UTC
Reply
Permalink
...
Post by Lewis
Post by J. J. Lodder
Post by Peter Moylan
Post by Lewis
The proof for Goldbach's Conjecture seems like it should be very
simple and straightforward.
Except that we know, from experience, that practically any statement
about prime numbers is hideously difficult to prove.
All? Only those that are hard to prove are hard to prove,
I'd agree with All, at least in so far as the common usage of
"practically all". I cannot think of a proof involving prime numbers
that is simple, and we were certainly never given examples of one (or
made to prove one) when I was still taking classes.
Does this count?  Theorem: There are infinitely many primes.
Or this ("Fermat's Little Theorem" or a corollary of it)?  For any
positive integer a and any prime p, a^p is a multiple of p.  That is,
a^p = 0 (mod p).  (Please feel free to use a three-bar equals sign,
meaning "is congruent to".)
...

Or is it so complicated that I got it wrong? a^p = a (mod p). Forget
the "multiple of p" part.
--
Jerry Friedman
Lewis
2020-02-12 22:44:52 UTC
Reply
Permalink
Post by Jerry Friedman
...
Post by Tak To
Post by Lewis
Post by J. J. Lodder
Post by Peter Moylan
Post by Lewis
The proof for Goldbach's Conjecture seems like it should be very
simple and straightforward.
Except that we know, from experience, that practically any statement
about prime numbers is hideously difficult to prove.
All? Only those that are hard to prove are hard to prove,
I'd agree with All, at least in so far as the common usage of
"practically all". I cannot think of a proof involving prime numbers
that is simple, and we were certainly never given examples of one (or
made to prove one) when I was still taking classes.
Does this count? Theorem: There are infinitely many primes.
Or this ("Fermat's Little Theorem" or a corollary of it)? For any
positive integer a and any prime p, a^p is a multiple of p. That is,
a^p = 0 (mod p). (Please feel free to use a three-bar equals sign,
meaning "is congruent to".)
I feel sure Lewis and Peter can follow that proof after some study of
modular arithmetic.
The irony is thick. You go out of your way to be an ass right after
screwing up. Good job, jackass.
--
It was not, it could not be real. But in the roaring air he knew that
it was, for all who needed to believe, and in a belief so strong
that truth was not the same as fact... he knew that for now, and
yesterday, and tomorrow, both the thing, and the whole of the
thing.
s***@gmail.com
2020-02-12 23:55:27 UTC
Reply
Permalink
Post by Lewis
Post by Jerry Friedman
Or this ("Fermat's Little Theorem" or a corollary of it)? For any
positive integer a and any prime p, a^p is a multiple of p. That is,
a^p = 0 (mod p). (Please feel free to use a three-bar equals sign,
meaning "is congruent to".)
I feel sure Lewis and Peter can follow that proof after some study of
modular arithmetic.
The irony is thick. You go out of your way to be an ass right after
screwing up. Good job, jackass.
Skitt's Law is only a domain-specific application
of a General Principle.

Good day, sir!
/dps
Jerry Friedman
2020-02-13 03:35:38 UTC
Reply
Permalink
Post by Lewis
Post by Jerry Friedman
...
Post by Tak To
Post by Lewis
Post by J. J. Lodder
Post by Peter Moylan
Post by Lewis
The proof for Goldbach's Conjecture seems like it should be very
simple and straightforward.
Except that we know, from experience, that practically any statement
about prime numbers is hideously difficult to prove.
All? Only those that are hard to prove are hard to prove,
I'd agree with All, at least in so far as the common usage of
"practically all". I cannot think of a proof involving prime numbers
that is simple, and we were certainly never given examples of one (or
made to prove one) when I was still taking classes.
Does this count? Theorem: There are infinitely many primes.
Or this ("Fermat's Little Theorem" or a corollary of it)? For any
positive integer a and any prime p, a^p is a multiple of p. That is,
a^p = 0 (mod p). (Please feel free to use a three-bar equals sign,
meaning "is congruent to".)
I feel sure Lewis and Peter can follow that proof after some study of
modular arithmetic.
The irony is thick. You go out of your way to be an ass right after
screwing up. Good job, jackass.
Happy to oblige, but would you rather I'd said you and he might consider
that proof hideously difficult?
--
Jerry Friedman
Peter Moylan
2020-02-13 10:17:40 UTC
Reply
Permalink
Post by Jerry Friedman
Post by Tak To
Post by Lewis
Post by J. J. Lodder
Post by Peter Moylan
Post by Lewis
The proof for Goldbach's Conjecture seems like it should be
very simple and straightforward.
Except that we know, from experience, that practically any
statement about prime numbers is hideously difficult to
prove.
All? Only those that are hard to prove are hard to prove,
I'd agree with All, at least in so far as the common usage of
"practically all". I cannot think of a proof involving prime
numbers that is simple, and we were certainly never given
examples of one (or made to prove one) when I was still taking
classes.
Does this count? Theorem: There are infinitely many primes.
That's easy to prove, and I think I considered it easy even when I first
met that statement. But I still assert that the easy theorems about
primes are by far outnumbered by the extremely difficult ones.
Post by Jerry Friedman
Or this ("Fermat's Little Theorem" or a corollary of it)? For any
positive integer a and any prime p, a^p is a multiple of p. That is,
a^p = 0 (mod p). (Please feel free to use a three-bar equals sign,
meaning "is congruent to".)
Since you followed up with a correction almost immediately, let's assume
that we're talking about the correct version.
Post by Jerry Friedman
I feel sure Lewis and Peter can follow that proof after some study of
modular arithmetic.
Follow the proof, yes. But I'm less confident about being able to come
up with the proof (without looking it up) in less than several months.
Perhaps much longer.

And that's after having met a proof in the past.
--
Peter Moylan http://www.pmoylan.org
Newcastle, NSW, Australia
J. J. Lodder
2020-02-13 11:39:09 UTC
Reply
Permalink
Post by Peter Moylan
Post by Tak To
Post by Lewis
Post by J. J. Lodder
Post by Peter Moylan
Post by Lewis
The proof for Goldbach's Conjecture seems like it should be
very simple and straightforward.
Except that we know, from experience, that practically any
statement about prime numbers is hideously difficult to
prove.
All? Only those that are hard to prove are hard to prove,
I'd agree with All, at least in so far as the common usage of
"practically all". I cannot think of a proof involving prime
numbers that is simple, and we were certainly never given
examples of one (or made to prove one) when I was still taking
classes.
Does this count? Theorem: There are infinitely many primes.
That's easy to prove, and I think I considered it easy even when I first
met that statement. But I still assert that the easy theorems about
primes are by far outnumbered by the extremely difficult ones.
How would you go about defining a measure
on the set of all theorems about primes?

Jan
Adam Funk
2020-02-13 13:10:31 UTC
Reply
Permalink
Post by J. J. Lodder
Post by Peter Moylan
Post by Tak To
Post by Lewis
Post by J. J. Lodder
Post by Peter Moylan
Post by Lewis
The proof for Goldbach's Conjecture seems like it should be
very simple and straightforward.
Except that we know, from experience, that practically any
statement about prime numbers is hideously difficult to
prove.
All? Only those that are hard to prove are hard to prove,
I'd agree with All, at least in so far as the common usage of
"practically all". I cannot think of a proof involving prime
numbers that is simple, and we were certainly never given
examples of one (or made to prove one) when I was still taking
classes.
Does this count? Theorem: There are infinitely many primes.
That's easy to prove, and I think I considered it easy even when I first
met that statement. But I still assert that the easy theorems about
primes are by far outnumbered by the extremely difficult ones.
How would you go about defining a measure
on the set of all theorems about primes?
If the easy theorems are finite, job done. Otherwise you'd have to
come up with a diagonalization (or worse) to show that the difficult
ones are more infinite than the easy ones.
--
No right of private conversation was enumerated in the Constitution.
I don't suppose it occurred to anyone at the time that it could be
prevented. ---Whitfield Diffie
J. J. Lodder
2020-02-13 15:00:24 UTC
Reply
Permalink
Post by Adam Funk
Post by J. J. Lodder
Post by Peter Moylan
Post by Tak To
Post by Lewis
Post by J. J. Lodder
Post by Peter Moylan
Post by Lewis
The proof for Goldbach's Conjecture seems like it should be
very simple and straightforward.
Except that we know, from experience, that practically any
statement about prime numbers is hideously difficult to
prove.
All? Only those that are hard to prove are hard to prove,
I'd agree with All, at least in so far as the common usage of
"practically all". I cannot think of a proof involving prime
numbers that is simple, and we were certainly never given
examples of one (or made to prove one) when I was still taking
classes.
Does this count? Theorem: There are infinitely many primes.
That's easy to prove, and I think I considered it easy even when I first
met that statement. But I still assert that the easy theorems about
primes are by far outnumbered by the extremely difficult ones.
How would you go about defining a measure
on the set of all theorems about primes?
If the easy theorems are finite, job done. Otherwise you'd have to
come up with a diagonalization (or worse) to show that the difficult
ones are more infinite than the easy ones.
I guess PM's 'easy theorems' means the easy ones we know about,
which makes the statement trivially true,
and utterly uninteresting,

Jan
Jerry Friedman
2020-02-13 14:42:58 UTC
Reply
Permalink
Post by J. J. Lodder
Post by Peter Moylan
Post by Tak To
Post by Lewis
Post by J. J. Lodder
Post by Peter Moylan
Post by Lewis
The proof for Goldbach's Conjecture seems like it should be
very simple and straightforward.
Except that we know, from experience, that practically any
statement about prime numbers is hideously difficult to
prove.
All? Only those that are hard to prove are hard to prove,
I'd agree with All, at least in so far as the common usage of
"practically all". I cannot think of a proof involving prime
numbers that is simple, and we were certainly never given
examples of one (or made to prove one) when I was still taking
classes.
Does this count? Theorem: There are infinitely many primes.
That's easy to prove, and I think I considered it easy even when I first
met that statement. But I still assert that the easy theorems about
primes are by far outnumbered by the extremely difficult ones.
How would you go about defining a measure
on the set of all theorems about primes?
Use the samples in textbooks?
--
Jerry Friedman
Jerry Friedman
2020-02-13 15:00:15 UTC
Reply
Permalink
Post by Peter Moylan
Post by Lewis
Post by J. J. Lodder
Post by Peter Moylan
Post by Lewis
The proof for Goldbach's Conjecture seems like it should be
very simple and straightforward.
Except that we know, from experience, that practically any
statement about prime numbers is hideously difficult to
prove.
All? Only those that are hard to prove are hard to prove,
I'd agree with All, at least in so far as the common usage of
"practically all". I cannot think of a proof involving prime
numbers that is simple, and we were certainly never given
examples of one (or made to prove one) when I was still taking
classes.
Does this count?  Theorem: There are infinitely many primes.
That's easy to prove, and I think I considered it easy even when I first
met that statement. But I still assert that the easy theorems about
primes are by far outnumbered by the extremely difficult ones.
Or this ("Fermat's Little Theorem" or a corollary of it)?  For any
positive integer a and any prime p, a^p is a multiple of p.  That is,
 a^p = 0 (mod p).  (Please feel free to use a three-bar equals sign,
 meaning "is congruent to".)
Since you followed up with a correction almost immediately, let's assume
that we're talking about the correct version.
I feel sure Lewis and Peter can follow that proof after some study of
 modular arithmetic.
Follow the proof, yes.
I should have said "Some or all of those proofs," since there seem to be
quite a few.
Post by Peter Moylan
But I'm less confident about being able to come
up with the proof (without looking it up) in less than several months.
Perhaps much longer.
And that's after having met a proof in the past.
I was in the same situation before I looked them up last night. I guess
it depends on what you mean by "hideously difficult to prove". I was
thinking of something like the theorem on the asymptotic distribution of
prime numbers that Jan mentioned. In the context of Goldbach's
conjecture, I'd think that if there were a proof that you or I could
follow, it wouldn't be difficult for a specialist to come up with.
--
Jerry Friedman
Peter Moylan
2020-02-13 10:05:35 UTC
Reply
Permalink
Post by Lewis
Post by J. J. Lodder
Post by Peter Moylan
Post by Lewis
The proof for Goldbach's Conjecture seems like it should be
very simple and straightforward.
Except that we know, from experience, that practically any
statement about prime numbers is hideously difficult to prove.
All? Only those that are hard to prove are hard to prove,
I wrote "practically any", which is not quite the same as "all". I'm
asserting that the easy-to-prove group is only a small subset of the whole.
Post by Lewis
I'd agree with All, at least in so far as the common usage of
"practically all". I cannot think of a proof involving prime numbers
that is simple, and we were certainly never given examples of one
(or made to prove one) when I was still taking classes.
We had to prove the volume of a sphere, IIRC, and quite a few proofs
about conic sections in first year calculus.
Calculus is easier than the higher arithmetic. I was first introduced to
calculus in the final year of high school, and went on to things like
the volume of a sphere in first year university. I didn't hit arithmetic
(in the sense used here) until third year, and it was possibly the
toughest subject I'd ever faced.

The first assignment I was given in that topic, as I recall it, was to
prove that every natural number has a unique decomposition into primes.
All that I managed to prove was that the textbook proof was invalid,
because it was circular in a non-obvious way. (A rather sophisticated
example of begging the question.) That was just for one textbook, but I
think I subsequently came across other "proofs" that fell into the same
subtle trap.

I've managed to forget much of what I learnt back then. In contrast, I'm
confident that I could still breeze through any first year calculus
problem with hardly any effort.
--
Peter Moylan http://www.pmoylan.org
Newcastle, NSW, Australia
Jerry Friedman
2020-02-13 15:04:14 UTC
Reply
Permalink
Post by Peter Moylan
Post by Lewis
Post by J. J. Lodder
Post by Peter Moylan
Post by Lewis
The proof for Goldbach's Conjecture seems like it should be
very simple and straightforward.
Except that we know, from experience, that practically any
statement about prime numbers is hideously difficult to prove.
All? Only those that are hard to prove are hard to prove,
I wrote "practically any", which is not quite the same as "all". I'm
asserting that the easy-to-prove group is only a small subset of the whole.
Post by Lewis
I'd agree with All, at least in so far as the common usage of
"practically all". I cannot think of a proof involving prime numbers
that is simple, and we were certainly never given examples of one
(or made to prove one) when I was still taking classes.
We had to prove the volume of a sphere, IIRC, and quite a few proofs
about conic sections in first year calculus.
Calculus is easier than the higher arithmetic.
AmE "number theory", as we've mentioned before.
Post by Peter Moylan
I was first introduced to
calculus in the final year of high school, and went on to things like
the volume of a sphere in first year university. I didn't hit arithmetic
(in the sense used here) until third year, and it was possibly the
toughest subject I'd ever faced.
The first assignment I was given in that topic, as I recall it, was to
prove that every natural number has a unique decomposition into primes.
But we do call that "the fundamental theorem of arithmetic".
Post by Peter Moylan
All that I managed to prove was that the textbook proof was invalid,
because it was circular in a non-obvious way. (A rather sophisticated
example of begging the question.) That was just for one textbook, but I
think I subsequently came across other "proofs" that fell into the same
subtle trap.
After looking at Wikipedia last night, I guess you can avoid the trap by
using infinite descent.
Post by Peter Moylan
I've managed to forget much of what I learnt back then. In contrast, I'm
confident that I could still breeze through any first year calculus
problem with hardly any effort.
Same, although problems in first-year calculus (which I'm currently
teaching) can be quite tedious, with plenty of room for mistakes. In my
case I used calculus a lot in physics, but proofs in number theory for
nothing.
--
Jerry Friedman
Richard Heathfield
2020-02-11 11:56:29 UTC
Reply
Permalink
<snip>
Post by Peter Moylan
Post by Lewis
Post by RH Draney
Four-color theorem in 1976, Fermat in 1993, and next to fall was
supposed to be Goldbach, but I haven't heard anything in the news
about it yet....r
The proof for Goldbach's Conjecture seems like it should be very
simple and straightforward.
Except that we know, from experience, that practically any statement
about prime numbers is hideously difficult to prove.
There are infinitely many of them.

Euclid proved this in Elements IX Proposition 20, and the proof is
remarkably simple to follow. (Even I can understand it.)
--
Richard Heathfield
Email: rjh at cpax dot org dot uk
"Usenet is a strange place" - dmr 29 July 1999
Sig line 4 vacant - apply within
J. J. Lodder
2020-02-11 13:34:46 UTC
Reply
Permalink
Post by Richard Heathfield
<snip>
Post by Peter Moylan
Post by Lewis
Post by RH Draney
Four-color theorem in 1976, Fermat in 1993, and next to fall was
supposed to be Goldbach, but I haven't heard anything in the news
about it yet....r
The proof for Goldbach's Conjecture seems like it should be very
simple and straightforward.
Except that we know, from experience, that practically any statement
about prime numbers is hideously difficult to prove.
There are infinitely many of them.
Euclid proved this in Elements IX Proposition 20, and the proof is
remarkably simple to follow. (Even I can understand it.)
Yes, and the asymptotic distribution is also known.
But for more detail it will be necessary
to settle the Riemann hypothesis...

Jan
Lewis
2020-02-11 14:00:38 UTC
Reply
Permalink
Post by J. J. Lodder
Post by Richard Heathfield
<snip>
Post by Peter Moylan
Post by Lewis
Post by RH Draney
Four-color theorem in 1976, Fermat in 1993, and next to fall was
supposed to be Goldbach, but I haven't heard anything in the news
about it yet....r
The proof for Goldbach's Conjecture seems like it should be very
simple and straightforward.
Except that we know, from experience, that practically any statement
about prime numbers is hideously difficult to prove.
There are infinitely many of them.
Euclid proved this in Elements IX Proposition 20, and the proof is
remarkably simple to follow. (Even I can understand it.)
Yes, and the asymptotic distribution is also known.
But for more detail it will be necessary
to settle the Riemann hypothesis...
I expect that will not fall until the Twin Prime Conjecture falls, and we
are oh so very close on that one.

Not there, but I suspect a lot of mathematicians in the field can
practically taste it.
--
Vampires are [...] by nature as co-operative as sharks. Vampyres are
just the same, the only real difference being that they can't
spell properly. --Carpe Jugulum
Athel Cornish-Bowden
2020-02-11 16:08:35 UTC
Reply
Permalink
Post by Richard Heathfield
<snip>
Post by Peter Moylan
Post by Lewis
Post by RH Draney
Four-color theorem in 1976, Fermat in 1993, and next to fall was
supposed to be Goldbach, but I haven't heard anything in the news
about it yet....r
The proof for Goldbach's Conjecture seems like it should be very
simple and straightforward.
Except that we know, from experience, that practically any statement
about prime numbers is hideously difficult to prove.
There are infinitely many of them.
Euclid proved this in Elements IX Proposition 20, and the proof is
remarkably simple to follow. (Even I can understand it.)
Yes, but can Lewis?
--
athel
Lewis
2020-02-12 05:36:29 UTC
Reply
Permalink
Post by Athel Cornish-Bowden
Post by Richard Heathfield
<snip>
Post by Peter Moylan
Post by Lewis
Post by RH Draney
Four-color theorem in 1976, Fermat in 1993, and next to fall was
supposed to be Goldbach, but I haven't heard anything in the news
about it yet....r
The proof for Goldbach's Conjecture seems like it should be very
simple and straightforward.
Except that we know, from experience, that practically any statement
about prime numbers is hideously difficult to prove.
There are infinitely many of them.
Euclid proved this in Elements IX Proposition 20, and the proof is
remarkably simple to follow. (Even I can understand it.)
Yes, but can Lewis?
Right now? Not a chance.
--
You are twisted and sick; I like that in a person.
Mark Brader
2020-02-11 22:30:24 UTC
Reply
Permalink
Post by Richard Heathfield
Post by Peter Moylan
Except that we know, from experience, that practically any statement
about prime numbers is hideously difficult to prove.
There are infinitely many of them.
Euclid proved this in Elements IX Proposition 20, and the proof is
remarkably simple to follow. (Even I can understand it.)
Simple to follow, but how simple to *think of*? This proof is
acclaimed to this day, and I say it's precisely because it's so
simple to follow *but must not have been simple to think of*.
--
Mark Brader, Toronto | "Wait, was that me? That was pretty good!"
***@vex.net | --Steve Summit

My text in this article is in the public domain.
J. J. Lodder
2020-02-12 10:32:50 UTC
Reply
Permalink
Post by Mark Brader
Post by Richard Heathfield
Post by Peter Moylan
Except that we know, from experience, that practically any statement
about prime numbers is hideously difficult to prove.
There are infinitely many of them.
Euclid proved this in Elements IX Proposition 20, and the proof is
remarkably simple to follow. (Even I can understand it.)
Simple to follow, but how simple to *think of*? This proof is
acclaimed to this day, and I say it's precisely because it's so
simple to follow *but must not have been simple to think of*.
Not at the time, because ancient Greeks were not accustomed
to manipulating large numbers.
It is simple for us,

Jan
Rich Ulrich
2020-02-12 17:58:43 UTC
Reply
Permalink
Post by J. J. Lodder
Post by Mark Brader
Post by Richard Heathfield
Post by Peter Moylan
Except that we know, from experience, that practically any statement
about prime numbers is hideously difficult to prove.
There are infinitely many of them.
Euclid proved this in Elements IX Proposition 20, and the proof is
remarkably simple to follow. (Even I can understand it.)
Simple to follow, but how simple to *think of*? This proof is
acclaimed to this day, and I say it's precisely because it's so
simple to follow *but must not have been simple to think of*.
Not at the time, because ancient Greeks were not accustomed
to manipulating large numbers.
It is simple for us,
Jan
Oh! Kuhn used the word "paradigm" - the idea or tool
that enables advances.

Big numbers were also difficult for the ancient Greeks
because they did not have "zero" and the positional
notation system that requires a zero.

I've wondered how much of the increase in "measured IQ"
in the last century owes to the spread of literacy, or to
the spread of particular habits of thinking. Literacy, by
itself, affects performance on IQ tests. (Lurie's old work
with Russian peasants.)
--
Rich Ulrich
J. J. Lodder
2020-02-13 11:27:50 UTC
Reply
Permalink
Post by Rich Ulrich
Post by J. J. Lodder
Post by Mark Brader
Post by Richard Heathfield
Post by Peter Moylan
Except that we know, from experience, that practically any statement
about prime numbers is hideously difficult to prove.
There are infinitely many of them.
Euclid proved this in Elements IX Proposition 20, and the proof is
remarkably simple to follow. (Even I can understand it.)
Simple to follow, but how simple to *think of*? This proof is
acclaimed to this day, and I say it's precisely because it's so
simple to follow *but must not have been simple to think of*.
Not at the time, because ancient Greeks were not accustomed
to manipulating large numbers.
It is simple for us.
Oh! Kuhn used the word "paradigm" - the idea or tool
that enables advances.
Kuhn used 'paradigm' with many different meanings.
Post by Rich Ulrich
Big numbers were also difficult for the ancient Greeks
because they did not have "zero" and the positional
notation system that requires a zero.
Big numbers for the Greeks start with Archimedes.
He just did it.
Calling that a new paradigm is highly misleading.
Post by Rich Ulrich
I've wondered how much of the increase in "measured IQ"
in the last century owes to the spread of literacy, or to
the spread of particular habits of thinking. Literacy, by
itself, affects performance on IQ tests. (Lurie's old work
with Russian peasants.)
IQ has no objective existence. It can't be measured.
Your " " are there withgood reason,

JAN
J. J. Lodder
2020-02-10 10:54:16 UTC
Reply
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Post by Peter Moylan
Post by David Kleinecke
Post by Arindam Banerjee
Post by Jerry Friedman
There have been theorem-proving programs for decades. I don't
know whether they've established anything new, though.
How on earth could any computer find it has found anything new
unless programmed accordingly?
By doing ALL of something and finding an as-yet-unnoticed case.
A proof was found in recent times of one of the well-known hard problems
in mathematics. It might have been Fermat's last theorem, or the
four-colour problem, or perhaps my comments below apply to both.
It was the four colour theorem for the sphere.
(or infinite plane, or the cylinder)
The proof is not difficult at all, conceptually.
It is just that it takes an enormous number of steps.
Perhaps surprisingly, it is that hard only on the sphere.
Higher topological varieties, such as the torus, are much easier.
(seven colours for the torus)
Post by Peter Moylan
As I recall it, the proof was very long, involving tedious reasoning, and
the first published "proofs" were subsequently found to contain errors.
In my own work in the past, I hit problems with a similar drawback,
where it seemed that the only way forward was to do a long and tedious
series of manipulations. My experience in those cases was that there was
a very high risk of making a mistake somewhere in the middle, and
failing to see it.
This is the sort of thing where theorem-proving software can do better
than humans, tackling calculations that require enormous patience and
extreme attention to detail.
There is theorem checking software,
beginning with 'Automath' by De Bruijn,
that checks proofs by writing out all steps explicitly,
down to the axioms,

Jan
Lewis
2020-02-10 15:06:35 UTC
Reply
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Post by Peter Moylan
Post by David Kleinecke
Post by Arindam Banerjee
Post by Jerry Friedman
There have been theorem-proving programs for decades. I don't
know whether they've established anything new, though.
How on earth could any computer find it has found anything new
unless programmed accordingly?
By doing ALL of something and finding an as-yet-unnoticed case.
A proof was found in recent times of one of the well-known hard problems
in mathematics. It might have been Fermat's last theorem, or the
four-colour problem, or perhaps my comments below apply to both. As I
recall it, the proof was very long, involving tedious reasoning, and the
first published "proofs" were subsequently found to contain errors.
That sounds like Fermat's last theorem.

Numberphile has a good and short video that covers the theorem and the
solution.


Post by Peter Moylan
In my own work in the past, I hit problems with a similar drawback,
where it seemed that the only way forward was to do a long and tedious
series of manipulations. My experience in those cases was that there was
a very high risk of making a mistake somewhere in the middle, and
failing to see it.
Absolutely, which is why proof's are reviewed and reviewed and
reviewed.
Post by Peter Moylan
This is the sort of thing where theorem-proving software can do better
than humans, tackling calculations that require enormous patience and
extreme attention to detail.
Maybe. So much depends on the underlying code and the methodologies
used. Relying on computer code you didn't write or don't understand
*perfectly* is going to lead to failures.

Andrew Wiles proved Fermat's last theorem, but there was an error
discovered in it. Wiles worked for another year and fixed the error.

This related video might be more relevant to the discussion, but it is
considerably longer.



(Anyone with even a passing appreciation for mathematics should watch
all of Numberphile)
--
It was intended that when Newspeak had been adopted once and for all
and Oldspeak forgotten, a heretical thought...should be literally
unthinkable, at least so far as thought is dependent on words.
Tak To
2020-02-09 20:48:47 UTC
Reply
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Post by Arindam Banerjee
Post by Jerry Friedman
Post by J. J. Lodder
Post by Tak To
Post by David Kleinecke
Post by Tak To
Drifting a bit, can we say that God's mind is a Turing machine?
I doubt it. Gods's mind - like ours - might be many (an infinite
number?) of Turing machines operating in sync.
? My mind is less than a Turing machine. It has only a finite
"tape".
And so what? Mathematical theorems are not established
by meddling about with Turing machines, [1]
Jan
[1] Typical example: you (that is, Euclid) can give a generating formula
for all Pythagorean triangles with integer sides.
(3,4,5) (5,12,13) etc.
Plowing through the integers with a computer otoh
can never yield more than a finite number of examples,
...
There have been theorem-proving programs for decades. I don't know
whether they've established anything new, though.
How on earth could any computer find it has found anything new unless programmed accordingly?
According to the common theory of evolution, the first
(biological) TM.FT[1] came into being spontaneously and
"natural selection" makes TM.FT's smarter and smarter. At
some point they became aware that they are in fact TM.FT's.

TM.FT's eventually create non-biological TM.FT's with
cognitive capabilities that rival themselves. In one segment
of the film /Doomsday Book/, a non-biological TM.FT achieves
Nirvana.

[1] Turing Machine with finite tape
--
Tak
----------------------------------------------------------------+-----
Tak To ***@alum.mit.eduxx
--------------------------------------------------------------------^^
[taode takto ~{LU5B~}] NB: trim the xx to get my real email addr
RH Draney
2020-02-09 07:50:17 UTC
Reply
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There have been theorem-proving programs for decades.  I don't know
whether they've established anything new, though.
I remember something about one of those programs that proved something
about isosceles triangles in a way no human had ever thought of...the
proof had something to do with superimposing the triangle on its mirror
image....r
Arindam Banerjee
2020-02-09 07:55:10 UTC
Reply
Permalink
Post by RH Draney
There have been theorem-proving programs for decades.  I don't know
whether they've established anything new, though.
I remember something about one of those programs that proved something
about isosceles triangles in a way no human had ever thought of...the
proof had something to do with superimposing the triangle on its mirror
image....r
Nope, Euclid's Theorem 4 (as i remember) gives that method as a proof for SAS equivalence ot triangles.
Not that they teach Euclid these days in school. But I loved geometry!
Peter Moylan
2020-02-09 09:29:43 UTC
Reply
Permalink
Post by RH Draney
Post by Jerry Friedman
There have been theorem-proving programs for decades. I don't know
whether they've established anything new, though.
I remember something about one of those programs that proved
something about isosceles triangles in a way no human had ever
thought of...the proof had something to do with superimposing the
triangle on its mirror image....r
That's the one I was thinking of. If the two sides are equal, then ABC
is congruent to ACB, so the two relevant angles are equal. A proof
that's now known to almost everyone, but it was new to me when I first
read that a theorem-proving program had found it.

It is possibly an urban legend that software was the first entity to
find this proof. One reference I came across

<URL:https://www.math.uga.edu/sites/default/files/inline-files/10.pdf>

first gives Euclid's unnecessarily complicated proof, then the shorter
proof we're talking about. About this, it says

"This much shorter argument was found later, perhaps by Proclus."

I don't know whether it's certain that Proclus knew about the shorter proof.
--
Peter Moylan http://www.pmoylan.org
Newcastle, NSW, Australia
Anders D. Nygaard
2020-02-09 12:53:12 UTC
Reply
Permalink
Post by Peter Moylan
Post by RH Draney
There have been theorem-proving programs for decades.  I don't know
 whether they've established anything new, though.
I remember something about one of those programs that proved
something about isosceles triangles in a way no human had ever
thought of...the proof had something to do with superimposing the
triangle on its mirror image....r
That's the one I was thinking of. If the two sides are equal, then ABC
is congruent to ACB, so the two relevant angles are equal. A proof
that's now known to almost everyone, but it was new to me when I first
read that a theorem-proving program had found it.
It is possibly an urban legend that software was the first entity to
find this proof.
Almost certainly.

The way I remember this story it that it came as a great surprise to
the implementers of the theorem-proving program that it came up with
what appeared to them to be an original thought - the proof was known,
but at the time not generally known and taught

My take on it is that the "originality" of the proof lies in it being
a fairly simple symbol manipulation which is not usually found by
humans, not because it is difficult, but because it is non-intuitive.
In a sense, the machine is not limited by human intuition.
Post by Peter Moylan
One reference I came across
<URL:https://www.math.uga.edu/sites/default/files/inline-files/10.pdf>
first gives Euclid's unnecessarily complicated proof, then the shorter
proof we're talking about. About this, it says
"This much shorter argument was found later, perhaps by Proclus."
I don't know whether it's certain that Proclus knew about the shorter proof.
/Anders, Denmark.
J. J. Lodder
2020-02-09 11:52:31 UTC
Reply
Permalink
Post by RH Draney
Post by Jerry Friedman
There have been theorem-proving programs for decades. I don't know
whether they've established anything new, though.
I remember something about one of those programs that proved something
about isosceles triangles in a way no human had ever thought of...the
proof had something to do with superimposing the triangle on its mirror
image....r
Yes, the computer said what amounts to:
Triangle (ABC) congruent to Triangle (ACB)
which violates some fundamentalist rules
about what is allowed in the way of 'proof'
in Euclidean geometry.
So go drop a perpendicular instead,

Jan
J. J. Lodder
2020-02-08 09:52:33 UTC
Reply
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Post by Tak To
Post by s***@gmail.com
Post by Lewis
Post by Peter Moylan
Post by Lewis
Post by Peter Moylan
Post by Lewis
Post by Jack
Post by s***@gmail.com
American English is going to be totally free of metaphor -
EVERYTHING will be reduced to something a computer can
understand.
What makes you think computers can't understand metaphors, or
anything else that humans might put up as their last bastion of
uniqueness?
If you would like to claim that computers can understand metaphor it
is incumbent on you to prove that.
A computer can "understand" anything it is programmed to handle. It
wouldn't be too hard to add in a recognition of metaphors.
I don't even begin to understand how you can think that considering a
fairly large percentage of humans don't understand metaphor.
But what percentage recognizes metaphors as a non-literal description,
even if they couldn't describe what a metaphor is?
Post by Lewis
Post by Peter Moylan
Computer programmers are better educated than humanity as a whole.
Doesn't matter, you;e trying to make an intelligent computer, something
that so far has proved exceedingly difficult and has not yet been done.
Recognition is a lower bar than understanding.
Post by Lewis
Post by Peter Moylan
Post by Lewis
Post by Peter Moylan
Of course, the meaning of "understand" is open to debate.
There are different levels of understanding,
but Lewis' point may carry for now ...
computers are becoming good at pattern recognition,
with large training sets,
but there are abstraction levels they don't yet manage.
(Humans are pretty good at using small training sets,
once they've reached the point of producing speech.)
It has been long conjectured that an artificial intelligence
capable of passing Turing's Test cannot be programmed but has
to be "raised", very much like the mind of a human.
Drifting a bit, can we say that God's mind is a Turing machine?
Of course not.
By a postulate due to Hilbert
god knows the answers to all of Brouwers' questions.

Brouwer's questions are what they are
precisely because Turing machines can't answer them,

Jan
Anders D. Nygaard
2020-02-09 12:58:37 UTC
Reply
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Post by J. J. Lodder
Brouwer's questions are what they are
precisely because Turing machines can't answer them,
What are Brouwer's questions? I can easily find Hilbert's
(they are usually known as "problems", though), but not his.

/Anders, Denmark.
J. J. Lodder
2020-02-09 14:06:23 UTC
Reply
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Post by Anders D. Nygaard
Post by J. J. Lodder
Brouwer's questions are what they are
precisely because Turing machines can't answer them,
What are Brouwer's questions? I can easily find Hilbert's
(they are usually known as "problems", though), but not his.
A Brouwerian question is a question
that can in principle be settled by computation,
but not in practice.
It is easy to invent infinitely many of them,
so if one happens to get settled, just take another one.
Obsolete and no longer needed,
for the halting problem for Turing machines
does just the same for you in a more general way.
In this context Brouwerian questions can be seen
as illustrative examples for the undecidability of the halting problem.

Brouwer used them to construct other counter-examples,
such as a number that is neither smaller than zero, nor equal to zero,
nor greater than zero.

Jan

PS Brouwer's original question was:
Are there seven consecutive zeroes in \pi?
This prompted Hilbert's reply: "God knows!"
This question seemed preposterously difficult to settle in 1925,
but nowadays we know that the answer is yes.
(which is not surprising, assuming normality of \pi)

A later question, also due to Brouwer is:
We define a number B constructively as follows:
The first decimal will be zero,
unless the first decimal of \pi equals 7, otherwise it will be 1
The second decimal will be zero, unless the second and third decimals of
\pi both equal 7, else 1
And so on, with the nth decimal equal to zero,
unless decimals n to 2n-1 of \pi are all equal to 7.
We know nowadays that B begins with 10 trillion zeroes,
but is B equal to zero?
In words: is there an n,
such that there are n consecutive zeros in \pi,
starting at decimal position n.

Best guess is that we will never know.
Not even Marvin can help us out.
Anders D. Nygaard
2020-02-11 23:46:50 UTC
Reply
Permalink
Post by J. J. Lodder
Post by Anders D. Nygaard
Post by J. J. Lodder
Brouwer's questions are what they are
precisely because Turing machines can't answer them,
What are Brouwer's questions? I can easily find Hilbert's
(they are usually known as "problems", though), but not his.
A Brouwerian question is a question
that can in principle be settled by computation,
but not in practice.
But that is different from what you described previously:
"not in practice" is not the same as "can't answer";
it is more like "haven't yet answered" - and in fact
your (Brouwer's) example below (about seven consecutive zeros
in the decimal expansion of pi) has, in fact, been settled;
presumably even to Brouwer's satisfaction.
Post by J. J. Lodder
[...]
Brouwer used them to construct other counter-examples,
such as a number that is neither smaller than zero, nor equal to zero,
nor greater than zero.
AFAIU, his constructions provided numbers which are not, at present,
*known* to obey the standard trichotomy, and in fact provide a large,
perhaps even inexhaustible source of such numbers, every single one
of which might at any time (given enough computation) be decided.

If he had known about the halting problem or Chaitin's omega, he
could AIUI have defined numbers where we will provably never know
whether they are smaller than, greater than or equal to zero.

Which according to his constructivist position would mean that
it is neither, but according to more standard mathematics means
just that we don't know which of the options apply.

/Anders, Denmark.
J. J. Lodder
2020-02-12 10:32:50 UTC
Reply
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Post by Anders D. Nygaard
Post by J. J. Lodder
Post by Anders D. Nygaard
Post by J. J. Lodder
Brouwer's questions are what they are
precisely because Turing machines can't answer them,
What are Brouwer's questions? I can easily find Hilbert's
(they are usually known as "problems", though), but not his.
A Brouwerian question is a question
that can in principle be settled by computation,
but not in practice.
"not in practice" is not the same as "can't answer";
it is more like "haven't yet answered" - and in fact
your (Brouwer's) example below (about seven consecutive zeros
in the decimal expansion of pi) has, in fact, been settled;
presumably even to Brouwer's satisfaction.
The point is not the settling of individual questions.
If it is settled, just take another one,
there is an infinity to choose from.
Post by Anders D. Nygaard
Post by J. J. Lodder
[...]
Brouwer used them to construct other counter-examples,
such as a number that is neither smaller than zero, nor equal to zero,
nor greater than zero.
AFAIU, his constructions provided numbers which are not, at present,
*known* to obey the standard trichotomy, and in fact provide a large,
perhaps even inexhaustible source of such numbers, every single one
of which might at any time (given enough computation) be decided.
If he had known about the halting problem or Chaitin's omega, he
could AIUI have defined numbers where we will provably never know
whether they are smaller than, greater than or equal to zero.
Of course. Brouwer was a forerunner,
who made it plausible.
He reformulated the same points later
in terms of the halting problem for Turing machines.
Post by Anders D. Nygaard
Which according to his constructivist position would mean that
it is neither, but according to more standard mathematics means
just that we don't know which of the options apply.
Yes, and there is an infinite number of those.
So standard mathematics really needs an axiom
(such as the axiom of choice, or something equivalent)
that claims that we can do by axiom what we can't really do,

Jan
J. J. Lodder
2020-02-08 09:52:33 UTC
Reply
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Post by Peter Moylan
Post by Lewis
Post by Peter Moylan
Post by Lewis
Post by Jack
Post by s***@gmail.com
American English is going to be totally free of metaphor -
EVERYTHING will be reduced to something a computer can
understand.
What makes you think computers can't understand metaphors, or
anything else that humans might put up as their last bastion of
uniqueness?
If you would like to claim that computers can understand metaphor it
is incumbent on you to prove that.
A computer can "understand" anything it is programmed to handle. It
wouldn't be too hard to add in a recognition of metaphors.
I don't even begin to understand how you can think that considering a
fairly large percentage of humans don't understand metaphor.
Computer programmers are better educated than humanity as a whole.
But they are also better mis-educated in many ways,

Jan
Paul Carmichael
2020-02-10 11:53:57 UTC
Reply
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Post by Peter Moylan
Computer programmers are better educated than humanity as a whole.
I was a (successful) computer programmer and my education was shite.
--
Paul.

https://paulc.es/elpatio
Mark Brader
2020-02-10 12:54:29 UTC
Reply
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Post by Paul Carmichael
Post by Peter Moylan
Computer programmers are better educated than humanity as a whole.
I was a (successful) computer programmer and my education was shite.
Are you suggesting that this contradicts where Peter said?
--
Mark Brader | "To a security officer the ideal world was one where
Toronto | nobody talked to anyone else... [But] of course...
***@vex.net | such a world rarely did anything worth securing
| in the first place." -- Tom Clancy
J. J. Lodder
2020-02-10 14:19:54 UTC
Reply
Permalink
Mark Brader <***@vex.net> wrote:
[false quote level restored to what it should have been]
Post by Mark Brader
Post by Paul Carmichael
Post by Peter Moylan
Computer programmers are better educated than humanity as a whole.
I was a (successful) computer programmer and my education was shite.
Are you suggesting that this contradicts where Peter said?
Some education may be worse than useless,

Jan
Peter Moylan
2020-02-11 00:50:35 UTC
Reply
Permalink
Post by J. J. Lodder
it should have been]
Post by Mark Brader
Post by Paul Carmichael
Post by Peter Moylan
Computer programmers are better educated than humanity as a
whole.
I was a (successful) computer programmer and my education was shite.
Are you suggesting that this contradicts where Peter said?
Some education may be worse than useless,
I realised in hindsight that the two schools I went to were not as good
as one could have hoped. The nuns teaching in the convent school
probably hadn't finished primary school themselves. In my final year of
high school, I had teachers for only three of my six subjects. I had to
do both mathematics subjects by correspondence, and physics by
self-study (with no decent texts in the library). In earlier high
school years, I had a number of teachers who had no tertiary quaifications.

Yet I managed to get a first class honours degree at university, with
first place in a number of subjects. Meanwhile, friends of mine who had
been to expensive high-status schools failed their subjects in first or
(more rarely) second year. Those schools of high reputation had an
excellent record for getting their pupils into university, but they
weren't so good at producing students who could finish university.

Even then I could see what was going wrong. Those kids had been coached
and coached over again to pass the high school exams. And then they went
to university and were thrown on their own resources, with no crutch.
They hadn't been trained to work independently.

Meanwhile, my crap schools taught me to rely on my own resources, to
work everything out by myself. If there was something I didn't
understand, I just had to try harder until I did understand it. In some
cases I even needed to understand things that my teachers didn't
understand. Ultimately, that was a good foundation for life.
--
Peter Moylan http://www.pmoylan.org
Newcastle, NSW, Australia
s***@gmail.com
2020-02-02 00:59:58 UTC
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Post by s***@gmail.com
American English is going to be totally free of metaphor - EVERYTHING will be reduced to something a computer can understand.
"comfort zone".

American "culture" is killing everything - but it is happening slowly enough so that world is fast asleep - even with the advent of Trump.

If a man woos a woman and she rebuffs him with "lets be friends" - in American culture he has been put in "the friend zone".
RH Draney
2020-02-02 19:11:15 UTC
Reply
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Post by s***@gmail.com
American English is going to be totally free of metaphor - EVERYTHING will be reduced to something a computer can understand.
Darmok and Jalad at Tenagra!...r
RH Draney
2020-02-04 18:30:06 UTC
Reply
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Post by RH Draney
Post by s***@gmail.com
American English is going to be totally free of metaphor - EVERYTHING will be reduced to something a computer can understand.
Darmok and Jalad at Tenagra!...r
It seems like it should take historical Data to translate that.
I don't remember the resolution of that plot.
I do...Picard told his opposite number the story of Gilgamesh and how he
became friends with Enki-du, showing that he had figured out how to
think like a Tamarian....r
b***@shaw.ca
2020-02-04 23:12:07 UTC
Reply
Permalink
Words like "newt" and "sere" are not accepted in an US word game. On the other > hand "login" and "multi" are accepted. Abbreviations are often taken as fill > words.
"Newt and "sere" are both in the current edition of the North American official
Scrabble dictionary, as well as in any Oxford. Perhaps you need
a better quality word game.

bill
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