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Old 2010-07-22, 15:58   #1
Damian
 
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Default When is a proof rigurous 'enough' ?

This is something that I have some time thinking, when can one be sure that a proof one has written is enough rigurous that everybody else (or mostly everyone) considers it valid?

Example:
Theorem "A"
Every polynomial P(x)=a_n x^n + a_{n-1}x^{n-1}+...+a_1x^1 + a_0 of a single variable of odd degree has at last one real root
Proof
As n is odd, if a_n > 0 clearly
\lim_{x\to+\infty}P(x) = +\infty
and
\lim_{x\to-\infty}P(x) = -\infty
because for |x|>1 the term with the n exponential grows 'faster' than those of less degree. (1)
If a_n < 0 it justs swaps +\infty with -\infty in the limits above.
And because P(x) is continuous (2) for all x (for been a polynomial), then it has to cross the x-axis at last once (3), so a real root exists.
\box

Of course, points (1), (2), and (3) could be developed further (they asume other proofs that I ommited, call them proof "B", "C", and "D"), but should one add those proofs in order for theorem "A" be complete? Could I only give bibliography where those proofs can be found?
In general, do you think I really proved Theorem "A" above? (I think most will say no)

Thanks,
Damián.

Last fiddled with by Damian on 2010-07-22 at 16:10
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Old 2010-07-22, 16:33   #2
R.D. Silverman
 
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Quote:
Originally Posted by Damian View Post
This is something that I have some time thinking, when can one be sure that a proof one has written is enough rigurous that everybody else (or mostly everyone) considers it valid?

Example:
Theorem "A"
Every polynomial P(x)=a_n x^n + a_{n-1}x^{n-1}+...+a_1x^1 + a_0 of a single variable of odd degree has at last one real root
Proof
As n is odd, if a_n > 0 clearly
\lim_{x\to+\infty}P(x) = +\infty
and
\lim_{x\to-\infty}P(x) = -\infty
because for |x|>1 the term with the n exponential grows 'faster' than those of less degree. (1)

If a_n < 0 it justs swaps +\infty with -\infty in the limits above.
And because P(x) is continuous (2) for all x (for been a polynomial),
then it has to cross the x-axis at last once (3), so a real root exists.
\box



Of course, points (1), (2), and (3) could be developed further (they asume other proofs that I ommited, call them proof "B", "C", and "D"), but should one add those proofs in order for theorem "A" be complete? Could I only give bibliography where those proofs can be found?
In general, do you think I really proved Theorem "A" above? (I think most will say no)

Thanks,
Damián.
Firstly, it does not matter what "everyone thinks". If this proof is
being done by a student, then the only thing that matters is what the
teacher will accept. If it appears in a publication, then what matters is
what professional mathematicians will accept.

As a *student*, what you have done does not prove the result.
You would indeed need to either add the proofs of (1), (2), (3), or
refer to a previously established result. And when I say "established
result", I mean a result established as part of the course. This last bit
depends a bit on the level of the course. A student in an upper-level
undergrad course can point to a known, published result that is not
part of the course itself. But in a lower-division course, I doubt whether
a teacher would accept external references.

Also, if this were done as a proof in a pre-calculus class where limits
and the intermediate value theorem were not yet established, this proof
would not be accepted by the teacher. Generally, one is not allowed to
use a more "advanced" method to prove an elementary result. The teacher
would expect a proof using results from the course or previous courses.

Of course, if given as a homework problem, then the teacher should
specify "prove all claims" as part of the problem statement.
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Old 2010-07-22, 16:37   #3
R.D. Silverman
 
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Quote:
Originally Posted by R.D. Silverman View Post
Firstly, it does not matter what "everyone thinks". If this proof is
being done by a student, then the only thing that matters is what the
teacher will accept. If it appears in a publication, then what matters is
what professional mathematicians will accept.

As a *student*, what you have done does not prove the result.
You would indeed need to either add the proofs of (1), (2), (3), or
refer to a previously established result. And when I say "established
result", I mean a result established as part of the course. This last bit
depends a bit on the level of the course. A student in an upper-level
undergrad course can point to a known, published result that is not
part of the course itself. But in a lower-division course, I doubt whether
a teacher would accept external references.

Also, if this were done as a proof in a pre-calculus class where limits
and the intermediate value theorem were not yet established, this proof
would not be accepted by the teacher. Generally, one is not allowed to
use a more "advanced" method to prove an elementary result. The teacher
would expect a proof using results from the course or previous courses.

Of course, if given as a homework problem, then the teacher should
specify "prove all claims" as part of the problem statement.
An omission on my part:

Even if this proof were done in a calculus course, you would still need to
refer to the intermediate value theorem to establish the claim that the
polynomial does indeed cross the axis. Merely stating that it crosses
the axis would be insufficient.
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Old 2010-07-23, 20:18   #4
Damian
 
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Thank you Dr. Silverman for your reply.
I'm not interested in that particular proof, it was just an example.

Do you think that a proof of a theorem can always be made more rigurous? Or there is a limit on how rigurous a proof can be?
I mean, what is today considered a rigurous proof of an established theorem, may not be a rigurous proof of tomorrow with more advanced techniques?

Or do you (anyone who reads this) think that a proof can be so rigurous that it can not be enhanced in any way?
What about alternative proofs? Do you think it is useless to find other proof of an established theorem? Do you think one proof will always be the better/more rigurous? Or that two proofs of a given theorem can have relative advantages in some sense over the other? (rigurous vs simple, etc)

What would be the point of finding a proof of an already established theorem? Can it be better in some way? In wich sense? (elemental, simple, rigurous, short, elegant?) (It may be a matter of taste if a proof is more elegant than another, isn't it?)

One think that surprised me was to read that Gauss gave 6 different proofs of the fundamental theorem of algebra. Wasn't one good enough?

Thanks.

Last fiddled with by Damian on 2010-07-23 at 20:28
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Old 2010-07-23, 20:25   #5
R.D. Silverman
 
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Quote:
Originally Posted by Damian View Post
Thank you Dr. Silverman for your reply.
I'm not interested in that particular proof, it was just an example.

Do you think that a proof of a theorem can always be made more rigurous? Or there is a limit on how rigurous a proof can be?
A proof is either rigorous or it is not. AFAIK, "degrees of rigor" do not
exist. Certainly a formal proof in first/second order logic is as rigorous as
can be.
Quote:

I mean, what is today considered a rigurous proof of an established theorem, may not be a rigurous proof of tomorrow with more advanced techniques?
This is poorly defined nonsense.
Quote:
Or do you (anyone who reads this) think that a proof can be so rigurous that it can not be enhanced in any way?
Define 'enhanced'.

You are bandying about informal English words in a way that is not applicable
to mathematics.
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Old 2010-07-23, 20:49   #6
CRGreathouse
 
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It's a huge, poorly-understood topic. Here's 200+ pages of thesis on the topic that I'm coincidentally reading:
Understanding informal mathematical discourse
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Old 2010-07-23, 20:51   #7
CRGreathouse
 
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Quote:
Originally Posted by Damian View Post
One think that surprised me was to read that Gauss gave 6 different proofs of the fundamental theorem of algebra. Wasn't one good enough?
One proof was good enough to show that the result was true, but it's common to produce alternate proofs. They may be more elegant, or they may show connections that were obscure in the original.

There are hundreds of proofs of quadratic reciprocity, half a dozen or more due to Gauss.
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Old 2010-07-23, 22:40   #8
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Quote:
Originally Posted by CRGreathouse View Post
One proof was good enough to show that the result was true, but it's common to produce alternate proofs. They may be more elegant, or they may show connections that were obscure in the original.
More importantly, different proofs can use different sets of axioms, in which case they will be applicable in different situations. The simplest example I can think of is multiplicative cancellation (i.e. ab=ac implies b=c) which can be proven in (at least) two very different ways.
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Old 2010-07-25, 10:05   #9
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As an outsider (I did chemistry at university) my understanding is:

A new proof is most useful if significantly shorter and/or easier to follow. This makes it easier to check there are no errors in it and easier to teach.

A second way to prove something is also useful in that is is less likely there is an error in both proofs. This is most useful if the proof is too long to easily check it for errors.

Chris K
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