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From: Bill Dubuque on 17 Nov 2009 18:11
Robert Israel <israel(a)math.MyUniversitysInitials.ca> wrote:
> Claude Girard <girardgalois(a)gmail.com> writes:
>>
>> Need help retracing the source of a quote - I've read it somewhere but
>> just can't seem to retrace it (or Google-it). It goes something like
>> "A proof is not so much about showing that something is true but WHY
>> it is true". Does someone know who has said that?
>
> I'm sure somebody has said something like that, but as it stands this
> statement is simply and blatantly false. We may like a proof better
> if it offers us some insight into "why", but that is not relevant to
> it being a proof.

While that is technically true, it is false in practice. For example,
a brute-force massively distributed computational-based proof of the
Riemann Hypothesis would probably not satisfy the requirements for
the Clay Millennium Prize nor would it be accepted by most mainstream
mathematicians (as with the Four-Color conjecture). See my old post [1]
on brute-force vs. intuition for more on this, e.g. the analogy with
humanly-incomprehensible optimal chess endgame databases.

--Bill Dubuque

[1] http://google.com/groups?selm=y8zohfumbnx.fsf%40martigny.ai.mit.edu
From: Gerry Myerson on 17 Nov 2009 18:56
In article <l2cbpj0hhll.fsf(a)shaggy.csail.mit.edu>,
Bill Dubuque <wgd(a)nestle.csail.mit.edu> wrote:

> Robert Israel <israel(a)math.MyUniversitysInitials.ca> wrote:
> > Claude Girard <girardgalois(a)gmail.com> writes:
> >>
> >> Need help retracing the source of a quote - I've read it somewhere but
> >> just can't seem to retrace it (or Google-it). It goes something like
> >> "A proof is not so much about showing that something is true but WHY
> >> it is true". Does someone know who has said that?
> >
> > I'm sure somebody has said something like that, but as it stands this
> > statement is simply and blatantly false. We may like a proof better
> > if it offers us some insight into "why", but that is not relevant to
> > it being a proof.
>
> While that is technically true, it is false in practice. For example,
> a brute-force massively distributed computational-based proof of the
> Riemann Hypothesis would probably not satisfy the requirements for
> the Clay Millennium Prize nor would it be accepted by most mainstream
> mathematicians (as with the Four-Color conjecture).

I'm not sure what you're saying here. Sure, there was some controversy
over the 4CT at the time, and there may still be a few mathematicians
who hold that it hasn't been proved, but I think the mainstream accepted
the proof long ago.

--
Gerry Myerson (gerry(a)maths.mq.edi.ai) (i -> u for email)
From: Robert Israel on 17 Nov 2009 19:18

> Robert Israel <israel(a)math.MyUniversitysInitials.ca> wrote:
> > Claude Girard <girardgalois(a)gmail.com> writes:
> >>
> >> Need help retracing the source of a quote - I've read it somewhere but
> >> just can't seem to retrace it (or Google-it). It goes something like
> >> "A proof is not so much about showing that something is true but WHY
> >> it is true". Does someone know who has said that?
> >
> > I'm sure somebody has said something like that, but as it stands this
> > statement is simply and blatantly false. We may like a proof better
> > if it offers us some insight into "why", but that is not relevant to
> > it being a proof.
>
> While that is technically true, it is false in practice. For example,
> a brute-force massively distributed computational-based proof of the
> Riemann Hypothesis would probably not satisfy the requirements for
> the Clay Millennium Prize nor would it be accepted by most mainstream
> mathematicians (as with the Four-Color conjecture). See my old post [1]
> on brute-force vs. intuition for more on this, e.g. the analogy with
> humanly-incomprehensible optimal chess endgame databases.
>
> --Bill Dubuque
>
> [1] http://google.com/groups?selm=y8zohfumbnx.fsf%40martigny.ai.mit.edu

I was not particularly thinking of computer-based proofs when I wrote that.
There are also examples of human-produced proofs that don't provide
much insight. We may not like them much, but there's no question that
they are proofs. As for computer-based proofs, I think that attitudes have
changed quite a bit since 1976 when Appel and Haken first proved the Four
Color Theorem, and I would not be at all surprised to see a
computer-based (or at least computer-assisted) proof win one of the Clay
prizes. In fifty years or so, I would guess, insisting on using human
brain-power alone in mathematical proofs will be considered as odd as
insisting on writing out manuscripts with a quill pen.

I may have found the source of Claude's quote, though. This is
from pp. 856-7 of "The collected papers of Stephen Smale, Vol 2"
<http://books.google.ca/books?id=BigUKkT1S4UC>
in a panel discussion on "What is Chaos". The speaker is Heinz-Otto Peitgen:

PEITGEN: I think that James Gleick has touched a very important point
about mathematics. I think that proof, in the first place, does not desire to
find out whether something is right or wrong. That's not the role of
mathematics. I think the role of proof is to find out why something is right
or wrong, and in that sense, of course, proof means to deliver insight into
something.
--
Robert Israel israel(a)math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
From: David Bernier on 18 Nov 2009 09:24
Bill Dubuque wrote:
> Robert Israel <israel(a)math.MyUniversitysInitials.ca> wrote:
>> Claude Girard <girardgalois(a)gmail.com> writes:
>>> Need help retracing the source of a quote - I've read it somewhere but
>>> just can't seem to retrace it (or Google-it). It goes something like
>>> "A proof is not so much about showing that something is true but WHY
>>> it is true". Does someone know who has said that?
>> I'm sure somebody has said something like that, but as it stands this
>> statement is simply and blatantly false. We may like a proof better
>> if it offers us some insight into "why", but that is not relevant to
>> it being a proof.
>
> While that is technically true, it is false in practice. For example,
> a brute-force massively distributed computational-based proof of the
> Riemann Hypothesis would probably not satisfy the requirements for
> the Clay Millennium Prize nor would it be accepted by most mainstream
> mathematicians (as with the Four-Color conjecture). See my old post [1]
> on brute-force vs. intuition for more on this, e.g. the analogy with
> humanly-incomprehensible optimal chess endgame databases.
>
> --Bill Dubuque
>
> [1] http://google.com/groups?selm=y8zohfumbnx.fsf%40martigny.ai.mit.edu

I had a look at what's known about Brun's constant B_2,

B_2 = sum_{p such that p and p+2 are prime} (1/p + 1/(p+2)).

B_2 < oo (Brun).

B_2 > 1.83 unconditionally [ Pascal Sebah and Xavier Gourdon, 2002 ].

B_2 ~= 1.90216 heuristically justified [ Sebah, Gourdon based on
a conjecture of Hardy and
Littlewood].

B_2 < ? [ I didn't find anything on unconditional upper bounds].

Sebah and Gourdon estimate that the partial sums conjecturally
first surpass 1.9 for p ~ 10^530 .

Cf.:
< http://numbers.computation.free.fr/Constants/Primes/twin.html > .

So unconditionally, 1.83 < B_2 < ? .

I don't know how Brun's proof goes, or if some unconditional
upper bound on B_2 follows from it.

David Bernier
From: Claude Girard on 18 Nov 2009 14:00
On Nov 16, 1:16 pm, Claude Girard <girardgal...(a)gmail.com> wrote:
> Hello
>
> Need help retracing the source of a quote - I've read it somewhere but
> just can't seem to retrace it (or Google-it). It goes something like
> "A proof is not so much about showing that something is true but WHY
> it is true". Does someone know who has said that?
>
> Thx
> Claude

By making more Google searches I finally came across the quote I'm
looking for:

To quote from George Simmons' introductory calculus text,
presentations like those of Rudin "produce belief without insight, and
are therefore fundamentally unsatisfying. It is important to know that
a mathematical theorem is true, but it is often more important to
understand why it is true" (p.852 with original italics; he is
discussing a weakness of using mathematical induction in explaining a
theorem).

What I find odd is that I have never read G Simmons' calculus text!
Someone, somewhere else, must have quoted him (and that's where I've
first read it). The exact words Simmons used will help settle the
meaning which has been the source of some debate here... :-)

Thanks everyone.
Claude
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