s***@gmail.com

2020-02-01 13:53:50 UTC

Reply

PermalinkDiscussion:

Add Reply

s***@gmail.com

2020-02-01 13:53:50 UTC

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Permalink
s***@gmail.com

2020-02-01 15:16:14 UTC

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PermalinkAmerican English is going to be totally free of metaphor - EVERYTHING will be reduced to something a computer can understand.

"takeaway" for key ideas that emerged at a meeting.

Jack

2020-02-01 20:26:10 UTC

Reply

PermalinkAmerican English is going to be totally free of metaphor - EVERYTHING will be reduced to something a computer can understand.

else that humans might put up as their last bastion of uniqueness?

Lewis

2020-02-02 13:02:02 UTC

Reply

PermalinkAmerican English is going to be totally free of metaphor - EVERYTHING will be reduced to something a computer can understand.

else that humans might put up as their last bastion of uniqueness?

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

'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

Reply

PermalinkAmerican English is going to be totally free of metaphor - EVERYTHING will be reduced to something a computer can understand.

else that humans might put up as their last bastion of uniqueness?

incum

Lewis

2020-02-02 21:12:47 UTC

Reply

PermalinkOn Sun, 2 Feb 2020 13:02:02 -0000 (UTC), Lewis

American English is going to be totally free of metaphor - EVERYTHING will be reduced to something a computer can understand.

else that humans might put up as their last bastion of uniqueness?

incumbent on you to prove that.

--

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.

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

Reply

PermalinkAmerican English is going to be totally free of metaphor -

EVERYTHING will be reduced to something a computer can

understand.

anything else that humans might put up as their last bastion of

uniqueness?

is incumbent on you to prove that.

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

Peter Moylan http://www.pmoylan.org

Newcastle, NSW, Australia

David Kleinecke

2020-02-02 23:08:52 UTC

Reply

PermalinkAmerican English is going to be totally free of metaphor -

EVERYTHING will be reduced to something a computer can

understand.

anything else that humans might put up as their last bastion of

uniqueness?

is incumbent on you to prove that.

wouldn't be too hard to add in a recognition of metaphors.

Of course, the meaning of "understand" is open to debate.

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

Reply

PermalinkAmerican English is going to be totally free of metaphor -

EVERYTHING will be reduced to something a computer can

understand.

anything else that humans might put up as their last bastion of

uniqueness?

is incumbent on you to prove that.

wouldn't be too hard to add in a recognition of metaphors.

fairly large percentage of humans don't understand metaphor.

--

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.

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

Reply

PermalinkAmerican English is going to be totally free of metaphor -

EVERYTHING will be reduced to something a computer can

understand.

anything else that humans might put up as their last bastion of

uniqueness?

is incumbent on you to prove that.

wouldn't be too hard to add in a recognition of metaphors.

fairly large percentage of humans don't understand metaphor.

--

Peter Moylan http://www.pmoylan.org

Newcastle, NSW, Australia

Peter Moylan http://www.pmoylan.org

Newcastle, NSW, Australia

Lewis

2020-02-05 05:39:04 UTC

Reply

PermalinkAmerican English is going to be totally free of metaphor -

EVERYTHING will be reduced to something a computer can

understand.

anything else that humans might put up as their last bastion of

uniqueness?

is incumbent on you to prove that.

wouldn't be too hard to add in a recognition of metaphors.

fairly large percentage of humans don't understand metaphor.

that so far has proved exceedingly difficult and has not yet been done.

--

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.

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

Reply

PermalinkAmerican English is going to be totally free of metaphor -

EVERYTHING will be reduced to something a computer can

understand.

anything else that humans might put up as their last bastion of

uniqueness?

is incumbent on you to prove that.

wouldn't be too hard to add in a recognition of metaphors.

fairly large percentage of humans don't understand metaphor.

even if they couldn't describe what a metaphor is?

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.

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

Reply

PermalinkAmerican English is going to be totally free of metaphor -

EVERYTHING will be reduced to something a computer can

understand.

anything else that humans might put up as their last bastion of

uniqueness?

is incumbent on you to prove that.

wouldn't be too hard to add in a recognition of metaphors.

fairly large percentage of humans don't understand metaphor.

even if they couldn't describe what a metaphor is?

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.

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.)

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

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

Reply

Permalinkpossible.

--

"Are you pondering what I'm pondering?"

Snowball: "Oh Brain, I certainly hope so."

"Are you pondering what I'm pondering?"

Snowball: "Oh Brain, I certainly hope so."

David Kleinecke

2020-02-08 00:33:07 UTC

Reply

Permalinknumber?) of Turing machines operating in sync.

J. J. Lodder

2020-02-08 09:52:30 UTC

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

Jan

Peter Moylan

2020-02-08 10:38:31 UTC

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

single Turing machine can emulate a countable number of Turing machines.

--

Peter Moylan http://www.pmoylan.org

Newcastle, NSW, Australia

Peter Moylan http://www.pmoylan.org

Newcastle, NSW, Australia

J. J. Lodder

2020-02-08 13:46:23 UTC

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

single Turing machine can emulate a countable number of Turing machines.

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

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

single Turing machine can emulate a countable number of Turing machines.

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,

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

athel

Peter Moylan

2020-02-08 23:20:19 UTC

Reply

PermalinkOn Friday, February 7, 2020 at 12:38:03 PM UTC-8, Tak To

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.I doubt it. Gods's mind - like ours - might be many (an

infinite number?) of Turing machines operating in sync.

think a single Turing machine can emulate a countable number of

Turing machines.

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,

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.

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

Peter Moylan http://www.pmoylan.org

Newcastle, NSW, Australia

J. J. Lodder

2020-02-09 11:52:31 UTC

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

single Turing machine can emulate a countable number of Turing machines.

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,

Palmer, M. L., Williams, R. A., Gatherer, D., 2016. Rosen's (M, R)

system as an X-machine. J. Theor. Biol., 408, 97

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

Reply

PermalinkExactly.

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,

You must be vastly overestimating my understandings.

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.

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.

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

Reply

PermalinkExactly.

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

Tak To

2020-02-10 18:03:22 UTC

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

single Turing machine can emulate a countable number of Turing machines.

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,

Palmer, M. L., Williams, R. A., Gatherer, D., 2016. Rosen's (M, R)

system as an X-machine. J. Theor. Biol., 408, 97

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.

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

----------------------------------------------------------------+-----

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

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

single Turing machine can emulate a countable number of 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

----------------------------------------------------------------+-----

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

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

single Turing machine can emulate a countable number of 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

2020-02-12 20:20:13 UTC

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

single Turing machine can emulate a countable number of Turing machines.

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

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

"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

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

"tape".

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

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

"tape".

by meddling about with Turing machines, [1]

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

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

"tape".

by meddling about with Turing machines, [1]

Machine can determine if an arbitrary M-digit number is prime.

We can't.

--

Peter Moylan http://www.pmoylan.org

Newcastle, NSW, Australia

Peter Moylan http://www.pmoylan.org

Newcastle, NSW, Australia

J. J. Lodder

2020-02-09 12:30:20 UTC

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

"tape".

by meddling about with Turing machines, [1]

Machine can determine if an arbitrary M-digit number is prime.

We can't.

would solve it in no time.

Jan

Tak To

2020-02-10 17:59:48 UTC

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

"tape".

by meddling about with Turing machines, [1]

Machine can determine if an arbitrary M-digit number is prime.

We can't.

would solve it in no time.

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

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

"tape".

by meddling about with Turing machines, [1]

Machine can determine if an arbitrary M-digit number is prime.

We can't.

would solve it in no time.

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.

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.

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.

Jan

Lewis

2020-02-11 13:48:34 UTC

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

"tape".

by meddling about with Turing machines, [1]

Machine can determine if an arbitrary M-digit number is prime.

We can't.

would solve it in no time.

which was that God's mind is *not* "like ours", as DK put it.

--

I've got Mathematica 2.2 on my Quadra

I've got Mathematica 2.2 on my Quadra

Tak To

2020-02-11 18:44:35 UTC

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

"tape".

by meddling about with Turing machines, [1]

Machine can determine if an arbitrary M-digit number is prime.

We can't.

would solve it in no time.

which was that God's mind is *not* "like ours", as DK put it.

(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.)

I don't know if you are human, but human minds are not infinite

nor can they access the multiverse.

for claims aboout what Turing machines can do.

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.

might be able to pin point where he and I disagree.

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

2020-02-09 11:52:31 UTC

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

"tape".

by meddling about with Turing machines, [1]

Machine can determine if an arbitrary M-digit number is prime.

We can't.

It has nothing to do with actual material limitations,

Jan

Anders D. Nygaard

2020-02-09 12:55:47 UTC

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

"tape".

by meddling about with Turing machines, [1]

Machine can determine if an arbitrary M-digit number is prime.

We can't.

Computability is a mathematical concept.

It has nothing to do with actual material limitations,

J. J. Lodder

2020-02-12 10:32:50 UTC

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

"tape".

by meddling about with Turing machines, [1]

Machine can determine if an arbitrary M-digit number is prime.

We can't.

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.

Computability is a mathematical concept.

It has nothing to do with actual material limitations,

Tak To

2020-02-12 17:12:22 UTC

Reply

PermalinkPS 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.

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
Jerry Friedman

2020-02-09 03:54:01 UTC

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

"tape".

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

Jerry Friedman

Arindam Banerjee

2020-02-09 04:42:47 UTC

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

"tape".

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.

David Kleinecke

2020-02-09 17:39:30 UTC

Reply

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

"tape".

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.

Peter Moylan

2020-02-10 01:00:37 UTC

Reply

PermalinkThere have been theorem-proving programs for decades. I don't

know whether they've established anything new, though.

unless programmed accordingly?

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

Peter Moylan http://www.pmoylan.org

Newcastle, NSW, Australia

Mark Brader

2020-02-10 02:16:17 UTC

Reply

PermalinkA 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.

contained a crtical error, but it was fixed a year or two later.

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 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.

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

PermalinkFermat'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.

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

PermalinkFermat'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 wascontained 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.

supposed to be Goldbach, but I haven't heard anything in the news about

it yet....

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

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

PermalinkFermat'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.

supposed to be Goldbach, but I haven't heard anything in the news

about it yet....

has insufficient disk space to execute.

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

Peter Moylan http://www.pmoylan.org

Newcastle, NSW, Australia

Lewis

2020-02-10 15:16:00 UTC

Reply

PermalinkFermat'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.

supposed to be Goldbach, but I haven't heard anything in the news about

it yet....r

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

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

PermalinkFermat'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.

supposed to be Goldbach, but I haven't heard anything in the news about

it yet....r

and straightforward.

It's been shown to be true for all the numbers up to some ridiculously

large numbers, but still no proof.

--

Jerry Friedman

Jerry Friedman

Lewis

2020-02-11 01:19:00 UTC

Reply

PermalinkFermat'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.

supposed to be Goldbach, but I haven't heard anything in the news about

it yet....r

and straightforward.

:)

--

'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

'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

PermalinkFermat'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.

supposed to be Goldbach, but I haven't heard anything in the news

about it yet....r

simple and straightforward.

about prime numbers is hideously difficult to prove.

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

Peter Moylan http://www.pmoylan.org

Newcastle, NSW, Australia

J. J. Lodder

2020-02-11 09:34:06 UTC

Reply

PermalinkFermat'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.

supposed to be Goldbach, but I haven't heard anything in the news

about it yet....r

simple and straightforward.

about prime numbers is hideously difficult to prove.

Jan

Lewis

2020-02-11 13:51:43 UTC

Reply

PermalinkFermat'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.

supposed to be Goldbach, but I haven't heard anything in the news

about it yet....r

simple and straightforward.

about prime numbers is hideously difficult to prove.

"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

"It's like those French have a different word for *everything*" -

Steve Martin

Tak To

2020-02-11 18:49:55 UTC

Reply

PermalinkFermat'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.

supposed to be Goldbach, but I haven't heard anything in the news

about it yet....r

simple and straightforward.

about prime numbers is hideously difficult to prove.

"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.

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

PermalinkThe proof for Goldbach's Conjecture seems like it should be very

simple and straightforward.

about prime numbers is hideously difficult to prove.

"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.

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.

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

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".)

...The proof for Goldbach's Conjecture seems like it should be very

simple and straightforward.

about prime numbers is hideously difficult to prove.

"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.

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

Jerry Friedman

Lewis

2020-02-12 22:44:52 UTC

Reply

Permalink...

The proof for Goldbach's Conjecture seems like it should be very

simple and straightforward.

about prime numbers is hideously difficult to prove.

"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.

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.

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.

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

PermalinkOr 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.

screwing up. Good job, jackass.

of a General Principle.

Good day, sir!

/dps

Jerry Friedman

2020-02-13 03:35:38 UTC

Reply

Permalink...

The proof for Goldbach's Conjecture seems like it should be very

simple and straightforward.

about prime numbers is hideously difficult to prove.

"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.

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.

screwing up. Good job, jackass.

that proof hideously difficult?

--

Jerry Friedman

Jerry Friedman

Peter Moylan

2020-02-13 10:17:40 UTC

Reply

PermalinkThe proof for Goldbach's Conjecture seems like it should be

very simple and straightforward.

statement about prime numbers is hideously difficult to

prove.

"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.

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".)

that we're talking about the correct version.

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

modular arithmetic.

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

Peter Moylan http://www.pmoylan.org

Newcastle, NSW, Australia

J. J. Lodder

2020-02-13 11:39:09 UTC

Reply

PermalinkThe proof for Goldbach's Conjecture seems like it should be

very simple and straightforward.

statement about prime numbers is hideously difficult to

prove.

"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.

met that statement. But I still assert that the easy theorems about

primes are by far outnumbered by the extremely difficult ones.

on the set of all theorems about primes?

Jan

Adam Funk

2020-02-13 13:10:31 UTC

Reply

PermalinkThe proof for Goldbach's Conjecture seems like it should be

very simple and straightforward.

statement about prime numbers is hideously difficult to

prove.

"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.

met that statement. But I still assert that the easy theorems about

primes are by far outnumbered by the extremely difficult ones.

on the set of all theorems about primes?

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

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

PermalinkThe proof for Goldbach's Conjecture seems like it should be

very simple and straightforward.

statement about prime numbers is hideously difficult to

prove.

"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.

met that statement. But I still assert that the easy theorems about

primes are by far outnumbered by the extremely difficult ones.

on the set of all theorems about primes?

come up with a diagonalization (or worse) to show that the difficult

ones are more infinite than the easy ones.

which makes the statement trivially true,

and utterly uninteresting,

Jan

Jerry Friedman

2020-02-13 14:42:58 UTC

Reply

PermalinkThe proof for Goldbach's Conjecture seems like it should be

very simple and straightforward.

statement about prime numbers is hideously difficult to

prove.

"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.

met that statement. But I still assert that the easy theorems about

primes are by far outnumbered by the extremely difficult ones.

on the set of all theorems about primes?

--

Jerry Friedman

Jerry Friedman

Jerry Friedman

2020-02-13 15:00:15 UTC

Reply

PermalinkThe proof for Goldbach's Conjecture seems like it should be

very simple and straightforward.

statement about prime numbers is hideously difficult to

prove.

"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.

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 assumepositive 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".)

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.modular arithmetic.

quite a few.

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.

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

Jerry Friedman

Peter Moylan

2020-02-13 10:05:35 UTC

Reply

PermalinkThe proof for Goldbach's Conjecture seems like it should be

very simple and straightforward.

statement about prime numbers is hideously difficult to prove.

asserting that the easy-to-prove group is only a small subset of the whole.

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 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

Peter Moylan http://www.pmoylan.org

Newcastle, NSW, Australia

Jerry Friedman

2020-02-13 15:04:14 UTC

Reply

PermalinkThe proof for Goldbach's Conjecture seems like it should be

very simple and straightforward.

statement about prime numbers is hideously difficult to prove.

asserting that the easy-to-prove group is only a small subset of the whole.

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.

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.

using infinite descent.

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.

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

Jerry Friedman

Richard Heathfield

2020-02-11 11:56:29 UTC

Reply

PermalinkFour-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

simple and straightforward.

about prime numbers is hideously difficult to prove.

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

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<snip>

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

simple and straightforward.

about prime numbers is hideously difficult to prove.

Euclid proved this in Elements IX Proposition 20, and the proof is

remarkably simple to follow. (Even I can understand it.)

But for more detail it will be necessary

to settle the Riemann hypothesis...

Jan

Lewis

2020-02-11 14:00:38 UTC

Reply

Permalink<snip>

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

simple and straightforward.

about prime numbers is hideously difficult to prove.

Euclid proved this in Elements IX Proposition 20, and the proof is

remarkably simple to follow. (Even I can understand it.)

But for more detail it will be necessary

to settle the Riemann hypothesis...

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

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<snip>

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

simple and straightforward.

about prime numbers is hideously difficult to prove.

Euclid proved this in Elements IX Proposition 20, and the proof is

remarkably simple to follow. (Even I can understand it.)

--

athel

athel

Lewis

2020-02-12 05:36:29 UTC

Reply

Permalink<snip>

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

simple and straightforward.

about prime numbers is hideously difficult to prove.

Euclid proved this in Elements IX Proposition 20, and the proof is

remarkably simple to follow. (Even I can understand it.)

--

You are twisted and sick; I like that in a person.

You are twisted and sick; I like that in a person.

Mark Brader

2020-02-11 22:30:24 UTC

Reply

PermalinkExcept that we know, from experience, that practically any statement

about prime numbers is hideously difficult to prove.

Euclid proved this in Elements IX Proposition 20, and the proof is

remarkably simple to follow. (Even I can understand it.)

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.

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

PermalinkExcept that we know, from experience, that practically any statement

about prime numbers is hideously difficult to prove.

Euclid proved this in Elements IX Proposition 20, and the proof is

remarkably simple to follow. (Even I can understand it.)

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*.

to manipulating large numbers.

It is simple for us,

Jan

Rich Ulrich

2020-02-12 17:58:43 UTC

Reply

PermalinkExcept that we know, from experience, that practically any statement

about prime numbers is hideously difficult to prove.

Euclid proved this in Elements IX Proposition 20, and the proof is

remarkably simple to follow. (Even I can understand it.)

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*.

to manipulating large numbers.

It is simple for us,

Jan

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

Rich Ulrich

J. J. Lodder

2020-02-13 11:27:50 UTC

Reply

PermalinkExcept that we know, from experience, that practically any statement

about prime numbers is hideously difficult to prove.

Euclid proved this in Elements IX Proposition 20, and the proof is

remarkably simple to follow. (Even I can understand it.)

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*.

to manipulating large numbers.

It is simple for us.

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.

He just did it.

Calling that a new paradigm is highly misleading.

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.)

Your " " are there withgood reason,

JAN

J. J. Lodder

2020-02-10 10:54:16 UTC

Reply

PermalinkThere have been theorem-proving programs for decades. I don't

know whether they've established anything new, though.

unless programmed accordingly?

in mathematics. It might have been Fermat's last theorem, or the

four-colour problem, or perhaps my comments below apply to both.

(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)

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.

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

PermalinkThere have been theorem-proving programs for decades. I don't

know whether they've established anything new, though.

unless programmed accordingly?

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.

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

solution.

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.

reviewed.

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.

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.

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

PermalinkI doubt it. Gods's mind - like ours - might be many (an infinite

number?) of Turing machines operating in sync.

"tape".

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.

(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

PermalinkThere 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 somethingwhether they've established anything new, though.

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

PermalinkThere 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 somethingwhether they've established anything new, though.

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

Not that they teach Euclid these days in school. But I loved geometry!

Peter Moylan

2020-02-09 09:29:43 UTC

Reply

PermalinkThere have been theorem-proving programs for decades. I don't know

whether they've established anything new, though.

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

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

Peter Moylan http://www.pmoylan.org

Newcastle, NSW, Australia

Anders D. Nygaard

2020-02-09 12:53:12 UTC

Reply

PermalinkThere 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 provedwhether they've established anything new, though.

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

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.

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.

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.

J. J. Lodder

2020-02-09 11:52:31 UTC

Reply

PermalinkThere have been theorem-proving programs for decades. I don't know

whether they've established anything new, though.

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

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

PermalinkAmerican English is going to be totally free of metaphor -

EVERYTHING will be reduced to something a computer can

understand.

anything else that humans might put up as their last bastion of

uniqueness?

is incumbent on you to prove that.

wouldn't be too hard to add in a recognition of metaphors.

fairly large percentage of humans don't understand metaphor.

even if they couldn't describe what a metaphor is?

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.

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.)

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?

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

PermalinkBrouwer's questions are what they are

precisely because Turing machines can't answer them,

(they are usually known as "problems", though), but not his.

/Anders, Denmark.

J. J. Lodder

2020-02-09 14:06:23 UTC

Reply

PermalinkBrouwer's questions are what they are

precisely because Turing machines can't answer them,

(they are usually known as "problems", though), but not his.

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

PermalinkBrouwer's questions are what they are

precisely because Turing machines can't answer them,

(they are usually known as "problems", though), but not his.

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.

[...]

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.

*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

PermalinkBrouwer's questions are what they are

precisely because Turing machines can't answer them,

(they are usually known as "problems", though), but not his.

that can in principle be settled by computation,

but not in practice.

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.

If it is settled, just take another one,

there is an infinity to choose from.

[...]

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.

*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.

who made it plausible.

He reformulated the same points later

in terms of the halting problem for Turing machines.

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.

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

PermalinkAmerican English is going to be totally free of metaphor -

EVERYTHING will be reduced to something a computer can

understand.

anything else that humans might put up as their last bastion of

uniqueness?

is incumbent on you to prove that.

wouldn't be too hard to add in a recognition of metaphors.

fairly large percentage of humans don't understand metaphor.

Jan

Paul Carmichael

2020-02-10 11:53:57 UTC

Reply

Permalink--

Paul.

https://paulc.es/elpatio

Paul.

https://paulc.es/elpatio

Mark Brader

2020-02-10 12:54:29 UTC

Reply

Permalink--

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

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[false quote level restored to what it should have been]

Are you suggesting that this contradicts where Peter said?

Jan

Peter Moylan

2020-02-11 00:50:35 UTC

Reply

Permalinkit should have been]

Are you suggesting that this contradicts where Peter said?

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

Peter Moylan http://www.pmoylan.org

Newcastle, NSW, Australia

s***@gmail.com

2020-02-02 00:59:58 UTC

Reply

PermalinkAmerican English is going to be totally free of metaphor - EVERYTHING will be reduced to something a computer can understand.

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

PermalinkAmerican English is going to be totally free of metaphor - EVERYTHING will be reduced to something a computer can understand.

RH Draney

2020-02-04 18:30:06 UTC

Reply

PermalinkAmerican English is going to be totally free of metaphor - EVERYTHING will be reduced to something a computer can understand.

I don't remember the resolution of that plot.

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

PermalinkWords 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 officialScrabble dictionary, as well as in any Oxford. Perhaps you need

a better quality word game.

bill

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