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From: William Hale on 26 Jun 2010 16:29
In article
<a6de2bf2-c455-4d95-a63a-ae71cb6b49eb(a)d8g2000yqf.googlegroups.com>,
Charlie-Boo <shymathguy(a)gmail.com> wrote:

> On Jun 26, 3:57�pm, William Hale <h...(a)tulane.edu> wrote:
> > In article
> > <7e830cd6-b698-4f19-9152-d361cbc1b...(a)a30g2000yqn.googlegroups.com>,
> >
> >
> >
> >
> >
> >
> >
> > �Charlie-Boo <shymath...(a)gmail.com> wrote:
> > > On Jun 26, 3:08 pm, William Hale <h...(a)tulane.edu> wrote:
> > > > In article
> > > > <37402b0d-57d8-479b-b6e6-11d27b6c4...(a)e5g2000yqn.googlegroups.com>,
> > > > Charlie-Boo <shymath...(a)gmail.com> wrote:
> > [cut]
> > > > > But unfortunately, people forget about Occam's Razor and in fact act
> > > > > the opposite: That a long complex proof is more impressive than the
> > > > > smarter short one.
> >
> > > > Nobody claims this.
> >
> > > People often highlight the huge proof of 1+1=2 in Principia
> > > Mathematica, as does the MetaMath web site, when Occam tells us that
> > > is a very poor design, and short proofs such as my 9 line proof are to
> > > be preferred. �It's a fairly famout bit of trivia.
> >
> > People often highlight the huge proof of 1+1=2 in Principia Mathematica
> > because it is complete in all the necessary detail, not because it is
> > huge.
>
> So the rest of the proofs aren't??

Of course the rest of the proofs are complete in all the necessary
detail.

>
> No, they say "It takes 1,000 lines!" The give the last line of the
> proof and say, "And that's on page 123!" Look at any reference to
> it. With all the proofs in Principia Mathematica, why is that one
> singled out?

I forget the exact phrasing of the well-known quote about "one plus one
equals two" as being so true that it does not require proof. Because of
that quote (I believe), that theorem of Principia Mathematica is singled
out.

>
> As far as PA vs. ZFC goes:
>
> Do you know how N is represented in ZFC? By constructing a
> representation of numbers, a set that meets Peano's Axioms, and then
> defining N to be that set.

Yes.

> So they include Peano's Axioms with a
> "definition" that N satisifies them.

ZFC doesn't include Peano's Axioms as axioms of ZFC.

> See any text - e.g. Set Theory
> for the Working Mathematician, Ciesielski pg. 26-29.

I haven't read the text, but I would suspect that they bring in Peano's
axioms for historical reasons to show how the ZFC definition of natural
numbers relate to the PA presentation of the natural numbers that was
previously given. But this is not necessary: ZFC can define the natural
numbers without mentioning anything about PA or how PA and ZFC might be
related.

> C-B
>
> > Nobody claims that a long complex proof is more impressive than the
> > smarter short one. If your 9 line proof were correct in the axiom system
> > used by the Principia Mathematica, then people would use it because they
> > prefer smart short proofs.
>
> On the MetaMath web site, the proof of 1+1=2 would take just a few
> lines, so he gives the proof of 2+2=4 that takes the 1,000 lines!
>
> > Since they don't quote your 9 line proof as
> > the preferred proof, the conclusion is not that they think a long
> > complex proof is more impressive than the smarter short one, but rather
> > that mathematicians are not good at mathematics.- Hide quoted text -
> >
> > - Show quoted text -
From: Charlie-Boo on 28 Jun 2010 21:46
On Jun 26, 4:29 pm, William Hale <h...(a)tulane.edu> wrote:
> In article
> <a6de2bf2-c455-4d95-a63a-ae71cb6b4...(a)d8g2000yqf.googlegroups.com>,
>
>
>
>
>
>  Charlie-Boo <shymath...(a)gmail.com> wrote:
> > On Jun 26, 3:57 pm, William Hale <h...(a)tulane.edu> wrote:
> > > In article
> > > <7e830cd6-b698-4f19-9152-d361cbc1b...(a)a30g2000yqn.googlegroups.com>,
>
> > > Charlie-Boo <shymath...(a)gmail.com> wrote:
> > > > On Jun 26, 3:08 pm, William Hale <h...(a)tulane.edu> wrote:
> > > > > In article
> > > > > <37402b0d-57d8-479b-b6e6-11d27b6c4...(a)e5g2000yqn.googlegroups.com>,
> > > > > Charlie-Boo <shymath...(a)gmail.com> wrote:
> > > [cut]
> > > > > > But unfortunately, people forget about Occam's Razor and in fact act
> > > > > > the opposite: That a long complex proof is more impressive than the
> > > > > > smarter short one.
>
> > > > > Nobody claims this.
>
> > > > People often highlight the huge proof of 1+1=2 in Principia
> > > > Mathematica, as does the MetaMath web site, when Occam tells us that
> > > > is a very poor design, and short proofs such as my 9 line proof are to
> > > > be preferred. It's a fairly famout bit of trivia.
>
> > > People often highlight the huge proof of 1+1=2 in Principia Mathematica
> > > because it is complete in all the necessary detail, not because it is
> > > huge.
>
> > So the rest of the proofs aren't??
>
> Of course the rest of the proofs are complete in all the necessary
> detail.
>
>
>
> > No, they say "It takes 1,000 lines!"  The give the last line of the
> > proof and say, "And that's on page 123!"  Look at any reference to
> > it.  With all the proofs in Principia Mathematica, why is that one
> > singled out?
>
> I forget the exact phrasing of the well-known quote about "one plus
one
> equals two" as being so true that it does not require proof.
Because of
> that quote (I believe), that theorem of Principia Mathematica is
singled
> out.

You forget it? Well known - except by you.

How many references to the 1+1=2 proof mention the quote that you say
they are in response to? I have seen about a dozen references and
none said anything about contrasting it to that.

Your choices:

1. Say it does occur frequently (you just don't have time to find me
examples - I should do that myself.)

2. It doesn't matter if they say it or not - they're certainly
thinking that.

3. "You're right - it's never mentioned."

C-B

> > As far as PA vs. ZFC goes:
>
> > Do you know how N is represented in ZFC?  By constructing a
> > representation of numbers, a set that meets Peano's Axioms, and then
> > defining N to be that set.
>
> Yes.
>
> > So they include Peano's Axioms with a
> > "definition" that N satisifies them.
>
> ZFC doesn't include Peano's Axioms as axioms of ZFC.
>
> > See any text - e.g. Set Theory
> > for the Working Mathematician, Ciesielski pg. 26-29.
>
> I haven't read the text, but I would suspect that they bring in Peano's
> axioms for historical reasons to show how the ZFC definition of natural
> numbers relate to the PA presentation of the natural numbers that was
> previously given. But this is not necessary: ZFC can define the natural
> numbers without mentioning anything about PA or how PA and ZFC might be
> related.
>
>
>
> > C-B
>
> > > Nobody claims that a long complex proof is more impressive than the
> > > smarter short one. If your 9 line proof were correct in the axiom system
> > > used by the Principia Mathematica, then people would use it because they
> > > prefer smart short proofs.
>
> > On the MetaMath web site, the proof of 1+1=2 would take just a few
> > lines, so he gives the proof of 2+2=4 that takes the 1,000 lines!
>
> > > Since they don't quote your 9 line proof as
> > > the preferred proof, the conclusion is not that they think a long
> > > complex proof is more impressive than the smarter short one, but rather
> > > that mathematicians are not good at mathematics.- Hide quoted text -
>
> > > - Show quoted text -- Hide quoted text -
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -

From: Charlie-Boo on 28 Jun 2010 21:49
On Jun 26, 4:29 pm, William Hale <h...(a)tulane.edu> wrote:
> In article
> <a6de2bf2-c455-4d95-a63a-ae71cb6b4...(a)d8g2000yqf.googlegroups.com>,
>
>
>
>
>
>  Charlie-Boo <shymath...(a)gmail.com> wrote:
> > On Jun 26, 3:57 pm, William Hale <h...(a)tulane.edu> wrote:
> > > In article
> > > <7e830cd6-b698-4f19-9152-d361cbc1b...(a)a30g2000yqn.googlegroups.com>,
>
> > > Charlie-Boo <shymath...(a)gmail.com> wrote:
> > > > On Jun 26, 3:08 pm, William Hale <h...(a)tulane.edu> wrote:
> > > > > In article
> > > > > <37402b0d-57d8-479b-b6e6-11d27b6c4...(a)e5g2000yqn.googlegroups.com>,
> > > > > Charlie-Boo <shymath...(a)gmail.com> wrote:
> > > [cut]
> > > > > > But unfortunately, people forget about Occam's Razor and in fact act
> > > > > > the opposite: That a long complex proof is more impressive than the
> > > > > > smarter short one.
>
> > > > > Nobody claims this.
>
> > > > People often highlight the huge proof of 1+1=2 in Principia
> > > > Mathematica, as does the MetaMath web site, when Occam tells us that
> > > > is a very poor design, and short proofs such as my 9 line proof are to
> > > > be preferred. It's a fairly famout bit of trivia.
>
> > > People often highlight the huge proof of 1+1=2 in Principia Mathematica
> > > because it is complete in all the necessary detail, not because it is
> > > huge.
>
> > So the rest of the proofs aren't??
>
> Of course the rest of the proofs are complete in all the necessary
> detail.
>
>
>
> > No, they say "It takes 1,000 lines!"  The give the last line of the
> > proof and say, "And that's on page 123!"  Look at any reference to
> > it.  With all the proofs in Principia Mathematica, why is that one
> > singled out?
>
> I forget the exact phrasing of the well-known quote about "one plus one
> equals two" as being so true that it does not require proof. Because of
> that quote (I believe), that theorem of Principia Mathematica is singled
> out.
>
>
>
> > As far as PA vs. ZFC goes:
>
> > Do you know how N is represented in ZFC?  By constructing a
> > representation of numbers, a set that meets Peano's Axioms, and then
> > defining N to be that set.
>
> Yes.
>
> > So they include Peano's Axioms with a
> > "definition" that N satisifies them.
>
> ZFC doesn't include Peano's Axioms as axioms of ZFC.

Have you heard of the Axiom of Infinity?

> > See any text - e.g. Set Theory
> > for the Working Mathematician, Ciesielski pg. 26-29.
>
> I haven't read the text, but I would suspect that they bring in
Peano's

Is there any mathematical significance to an assertion being suspected
by you?

C-B

> axioms for historical reasons to show how the ZFC definition of natural
> numbers relate to the PA presentation of the natural numbers that was
> previously given. But this is not necessary: ZFC can define the natural
> numbers without mentioning anything about PA or how PA and ZFC might be
> related.
>
>
>
> > C-B
>
> > > Nobody claims that a long complex proof is more impressive than the
> > > smarter short one. If your 9 line proof were correct in the axiom system
> > > used by the Principia Mathematica, then people would use it because they
> > > prefer smart short proofs.
>
> > On the MetaMath web site, the proof of 1+1=2 would take just a few
> > lines, so he gives the proof of 2+2=4 that takes the 1,000 lines!
>
> > > Since they don't quote your 9 line proof as
> > > the preferred proof, the conclusion is not that they think a long
> > > complex proof is more impressive than the smarter short one, but rather
> > > that mathematicians are not good at mathematics.- Hide quoted text -
>
> > > - Show quoted text -- Hide quoted text -
>
> - Show quoted text -- Hide quoted text -
>
> - Show quoted text -

From: William Hale on 28 Jun 2010 22:09
In article
<38b804b7-b1f6-48f5-bf73-f460593d8f2e(a)32g2000vbi.googlegroups.com>,
Charlie-Boo <shymathguy(a)gmail.com> wrote:

> On Jun 26, 4:29�pm, William Hale <h...(a)tulane.edu> wrote:
> > In article
> > <a6de2bf2-c455-4d95-a63a-ae71cb6b4...(a)d8g2000yqf.googlegroups.com>,
> >
> >
> >
> >
> >
> > �Charlie-Boo <shymath...(a)gmail.com> wrote:
> > > On Jun 26, 3:57 pm, William Hale <h...(a)tulane.edu> wrote:
> > > > In article
> > > > <7e830cd6-b698-4f19-9152-d361cbc1b...(a)a30g2000yqn.googlegroups.com>,
> >
> > > > Charlie-Boo <shymath...(a)gmail.com> wrote:
> > > > > On Jun 26, 3:08 pm, William Hale <h...(a)tulane.edu> wrote:
> > > > > > In article
> > > > > > <37402b0d-57d8-479b-b6e6-11d27b6c4...(a)e5g2000yqn.googlegroups.com>,
> > > > > > Charlie-Boo <shymath...(a)gmail.com> wrote:
> > > > [cut]
> > > > > > > But unfortunately, people forget about Occam's Razor and in fact
> > > > > > > act
> > > > > > > the opposite: That a long complex proof is more impressive than
> > > > > > > the
> > > > > > > smarter short one.
> >
> > > > > > Nobody claims this.
> >
> > > > > People often highlight the huge proof of 1+1=2 in Principia
> > > > > Mathematica, as does the MetaMath web site, when Occam tells us that
> > > > > is a very poor design, and short proofs such as my 9 line proof are
> > > > > to
> > > > > be preferred. It's a fairly famout bit of trivia.
> >
> > > > People often highlight the huge proof of 1+1=2 in Principia Mathematica
> > > > because it is complete in all the necessary detail, not because it is
> > > > huge.
> >
> > > So the rest of the proofs aren't??
> >
> > Of course the rest of the proofs are complete in all the necessary
> > detail.
> >
> >
> >
> > > No, they say "It takes 1,000 lines!" �The give the last line of the
> > > proof and say, "And that's on page 123!" �Look at any reference to
> > > it. �With all the proofs in Principia Mathematica, why is that one
> > > singled out?
> >
> > I forget the exact phrasing of the well-known quote about "one plus
> one
> > equals two" as being so true that it does not require proof.
> Because of
> > that quote (I believe), that theorem of Principia Mathematica is
> singled
> > out.
>
> You forget it? Well known - except by you.

I believe it is well known. It's not well known? Maybe it isn't and I am
wrong. I never claimed that I am smart. I failed first grade. I can't
answer the questions on the show "Are you smarter than a fifth grader".
Like Bush, I didn't know the wording of the famous quote about "fool me
once", but I did know it was a famous saying. Is it surprising that I
can think something is well known and not know it myself?

>
> How many references to the 1+1=2 proof mention the quote that you say
> they are in response to?

I have no idea. I never said I had a reference. I wouldn't be surprised
that it was never mentioned.

> I have seen about a dozen references and
> none said anything about contrasting it to that.
>
> Your choices:
>
> 1. Say it does occur frequently (you just don't have time to find me
> examples - I should do that myself.)

This would not be my choice.

>
> 2. It doesn't matter if they say it or not - they're certainly
> thinking that.

This would be my choice.

>
> 3. "You're right - it's never mentioned."

I would accept this as being likely.

>
> C-B
>
> > > As far as PA vs. ZFC goes:
> >
> > > Do you know how N is represented in ZFC? �By constructing a
> > > representation of numbers, a set that meets Peano's Axioms, and then
> > > defining N to be that set.
> >
> > Yes.
> >
> > > So they include Peano's Axioms with a
> > > "definition" that N satisifies them.
> >
> > ZFC doesn't include Peano's Axioms as axioms of ZFC.
> >
> > > See any text - e.g. Set Theory
> > > for the Working Mathematician, Ciesielski pg. 26-29.
> >
> > I haven't read the text, but I would suspect that they bring in Peano's
> > axioms for historical reasons to show how the ZFC definition of natural
> > numbers relate to the PA presentation of the natural numbers that was
> > previously given. But this is not necessary: ZFC can define the natural
> > numbers without mentioning anything about PA or how PA and ZFC might be
> > related.
> >
> >
> >
> > > C-B
> >
> > > > Nobody claims that a long complex proof is more impressive than the
> > > > smarter short one. If your 9 line proof were correct in the axiom
> > > > system
> > > > used by the Principia Mathematica, then people would use it because
> > > > they
> > > > prefer smart short proofs.
> >
> > > On the MetaMath web site, the proof of 1+1=2 would take just a few
> > > lines, so he gives the proof of 2+2=4 that takes the 1,000 lines!
> >
> > > > Since they don't quote your 9 line proof as
> > > > the preferred proof, the conclusion is not that they think a long
> > > > complex proof is more impressive than the smarter short one, but rather
> > > > that mathematicians are not good at mathematics.- Hide quoted text -
> >
> > > > - Show quoted text -- Hide quoted text -
> >
> > - Show quoted text -- Hide quoted text -
> >
> > - Show quoted text -
From: William Hale on 28 Jun 2010 22:19
In article
<22408c62-e769-4894-86e7-f999f3864f49(a)f17g2000vbl.googlegroups.com>,
Charlie-Boo <shymathguy(a)gmail.com> wrote:

> On Jun 26, 4:29�pm, William Hale <h...(a)tulane.edu> wrote:
> > In article
> > <a6de2bf2-c455-4d95-a63a-ae71cb6b4...(a)d8g2000yqf.googlegroups.com>,
> >
> >
> >
> >
> >
> > �Charlie-Boo <shymath...(a)gmail.com> wrote:
> > > On Jun 26, 3:57 pm, William Hale <h...(a)tulane.edu> wrote:
> > > > In article
> > > > <7e830cd6-b698-4f19-9152-d361cbc1b...(a)a30g2000yqn.googlegroups.com>,
> >
> > > > Charlie-Boo <shymath...(a)gmail.com> wrote:
> > > > > On Jun 26, 3:08 pm, William Hale <h...(a)tulane.edu> wrote:
> > > > > > In article
> > > > > > <37402b0d-57d8-479b-b6e6-11d27b6c4...(a)e5g2000yqn.googlegroups.com>,
> > > > > > Charlie-Boo <shymath...(a)gmail.com> wrote:
> > > > [cut]
> > > > > > > But unfortunately, people forget about Occam's Razor and in fact
> > > > > > > act
> > > > > > > the opposite: That a long complex proof is more impressive than
> > > > > > > the
> > > > > > > smarter short one.
> >
> > > > > > Nobody claims this.
> >
> > > > > People often highlight the huge proof of 1+1=2 in Principia
> > > > > Mathematica, as does the MetaMath web site, when Occam tells us that
> > > > > is a very poor design, and short proofs such as my 9 line proof are
> > > > > to
> > > > > be preferred. It's a fairly famout bit of trivia.
> >
> > > > People often highlight the huge proof of 1+1=2 in Principia Mathematica
> > > > because it is complete in all the necessary detail, not because it is
> > > > huge.
> >
> > > So the rest of the proofs aren't??
> >
> > Of course the rest of the proofs are complete in all the necessary
> > detail.
> >
> >
> >
> > > No, they say "It takes 1,000 lines!" �The give the last line of the
> > > proof and say, "And that's on page 123!" �Look at any reference to
> > > it. �With all the proofs in Principia Mathematica, why is that one
> > > singled out?
> >
> > I forget the exact phrasing of the well-known quote about "one plus one
> > equals two" as being so true that it does not require proof. Because of
> > that quote (I believe), that theorem of Principia Mathematica is singled
> > out.
> >
> >
> >
> > > As far as PA vs. ZFC goes:
> >
> > > Do you know how N is represented in ZFC? �By constructing a
> > > representation of numbers, a set that meets Peano's Axioms, and then
> > > defining N to be that set.
> >
> > Yes.
> >
> > > So they include Peano's Axioms with a
> > > "definition" that N satisifies them.
> >
> > ZFC doesn't include Peano's Axioms as axioms of ZFC.
>
> Have you heard of the Axiom of Infinity?

Yes.

>
> > > See any text - e.g. Set Theory
> > > for the Working Mathematician, Ciesielski pg. 26-29.
> >
> > I haven't read the text, but I would suspect that they bring in
> Peano's
>
> Is there any mathematical significance to an assertion being suspected
> by you?

There is no mathematical significance (in the sense that it is a
mathematical statement), but there is a significance in what I said. I
am offering a suspicion or opinion for you to consider. I hope you would
give it some consideration.

>
> C-B
>
> > axioms for historical reasons to show how the ZFC definition of natural
> > numbers relate to the PA presentation of the natural numbers that was
> > previously given. But this is not necessary: ZFC can define the natural
> > numbers without mentioning anything about PA or how PA and ZFC might be
> > related.
> >
> >
> >
> > > C-B
> >
> > > > Nobody claims that a long complex proof is more impressive than the
> > > > smarter short one. If your 9 line proof were correct in the axiom
> > > > system
> > > > used by the Principia Mathematica, then people would use it because
> > > > they
> > > > prefer smart short proofs.
> >
> > > On the MetaMath web site, the proof of 1+1=2 would take just a few
> > > lines, so he gives the proof of 2+2=4 that takes the 1,000 lines!
> >
> > > > Since they don't quote your 9 line proof as
> > > > the preferred proof, the conclusion is not that they think a long
> > > > complex proof is more impressive than the smarter short one, but rather
> > > > that mathematicians are not good at mathematics.- Hide quoted text -
> >
> > > > - Show quoted text -- Hide quoted text -
> >
> > - Show quoted text -- Hide quoted text -
> >
> > - Show quoted text -
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