Comments on: When are two proofs essentially the same? http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/ Mathematics related discussions Fri, 04 Jul 2008 17:47:43 +0000 http://wordpress.org/?v=MU By: Paul http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-1084 Paul Mon, 12 May 2008 23:48:52 +0000 http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-1084 A good problem to think of in this context would be the classic theorem "Whenever a rectangle is tiled by rectangles each of which has at least one integer side, then the tiled rectangle has at least one integer side." Wagon offers 14 different proofs in "Fourteen proofs of a result about tiling of a rectangle" and he even classifies them according to how they generalize... http://www.jstor.org/stable/2322213?seq=1 A good problem to think of in this context would be the classic theorem
“Whenever a rectangle is tiled by rectangles each of which has at least one integer side, then the tiled rectangle has at least one integer side.”

Wagon offers 14 different proofs in “Fourteen proofs of a result about tiling of a rectangle” and he even classifies them according to how they generalize…

http://www.jstor.org/stable/2322213?seq=1

]]> By: Konstantin Ziegler’s Weblog http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-1083 Konstantin Ziegler’s Weblog Wed, 07 May 2008 11:17:43 +0000 http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-1083 [...] Gowers asks When are two proofs essentially the same? For example, it is often possible to convert a standard inductive proof into a proof by [...] [...] Gowers asks When are two proofs essentially the same? For example, it is often possible to convert a standard inductive proof into a proof by [...]

]]> By: elicaraq http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-1042 elicaraq Sat, 08 Mar 2008 06:18:41 +0000 http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-1042 I don't know if the following two ways to explain the coverup method for partial fraction decomposition are "interestingly different" or "based on the same underlying idea". I would like to know. Here is an elementary explanation: $latex \frac{1}{(z-1)(z-i)(z+i)} = \frac{R}{z-1} + \frac{S}{z-i} + \frac{T}{z+i}$ Multiply both sides by $latex (z-1)(z-i)(z+i)$, $latex 1 = R (z-i)(z+i) + S (z-1)(z+i) + T (z-1)(z-i)$ By setting $latex z = 1, i, -i$, we get $latex R = \left. \frac{1}{(z-i)(z+i)} \right|_{z=1} = \frac{1}{2}$ $latex S = \left. \frac{1}{(z-1)(z+i)} \right|_{z=i}= -\frac{1}{2+2i}$ $latex T = \left. \frac{1}{(z-1)(z-i)} \right|_{z=-i} =\frac{1}{-2+2i}$ Now here is a complex explanation: Consider $latex \frac{1}{2 \pi i} \int_C \frac{dz}{(z-1)(z-i)(z+i)}$ Where $latex C$ is a closed contour to be determined. We know $latex \frac{1}{2 \pi i} \int_C \frac{dz}{(z-1)(z-i)(z+i)} = \frac{1}{2 \pi i} \left(\int_C \frac{R dz}{z-1} + \int_C \frac{S dz}{z-i} + \int_C \frac{T dz}{z+i} \right)$ Set $latex C$ be a circle of radius $latex 1/2$ around $latex 1, i, -i$ respectively, and use the Cauchy's integral formula. Again, by the same process, we get: $latex R = \left. \frac{1}{(z-i)(z+i)} \right|_{z=1} = \frac{1}{2}$ $latex S = \left. \frac{1}{(z-1)(z+i)} \right|_{z=i}= -\frac{1}{2+2i}$ $latex T = \left. \frac{1}{(z-1)(z-i)} \right|_{z=-i} =\frac{1}{-2+2i}$ I don’t know if the following two ways to explain the coverup method for partial fraction decomposition are “interestingly different” or “based on the same underlying idea”. I would like to know.

Here is an elementary explanation:

\frac{1}{(z-1)(z-i)(z+i)} = \frac{R}{z-1} + \frac{S}{z-i} + \frac{T}{z+i}

Multiply both sides by (z-1)(z-i)(z+i),

1 = R (z-i)(z+i) + S (z-1)(z+i) + T (z-1)(z-i)

By setting z = 1, i, -i, we get

R = \left. \frac{1}{(z-i)(z+i)} \right|_{z=1} = \frac{1}{2}

S = \left. \frac{1}{(z-1)(z+i)} \right|_{z=i}= -\frac{1}{2+2i}

T = \left. \frac{1}{(z-1)(z-i)} \right|_{z=-i} =\frac{1}{-2+2i}

Now here is a complex explanation:

Consider

\frac{1}{2 \pi i} \int_C \frac{dz}{(z-1)(z-i)(z+i)}

Where C is a closed contour to be determined.

We know

\frac{1}{2 \pi i} \int_C \frac{dz}{(z-1)(z-i)(z+i)} = \frac{1}{2 \pi i} \left(\int_C \frac{R dz}{z-1} + \int_C \frac{S dz}{z-i} + \int_C \frac{T dz}{z+i} \right)

Set C be a circle of radius 1/2 around 1, i, -i respectively, and use the Cauchy’s integral formula. Again, by the same process, we get:

R = \left. \frac{1}{(z-i)(z+i)} \right|_{z=1} = \frac{1}{2}

S = \left. \frac{1}{(z-1)(z+i)} \right|_{z=i}= -\frac{1}{2+2i}

T = \left. \frac{1}{(z-1)(z-i)} \right|_{z=-i} =\frac{1}{-2+2i}

]]> By: Joe http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-900 Joe Tue, 01 Jan 2008 07:41:00 +0000 http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-900 Even more fundamentally, I have often reflected on the assertion that two mathematical statements are equivalent. Certainly if one proves that A is equivalent to B, where neither A or B is known to be true, the statement has practical value in that if one resolves the validity of one of the statements she also resolves the validity of the other. However one often finds that an author remarks that two statements known to be true are equivalent. The first example that comes to mind is: "The Prime Number Theorem is equivalent to the non-vanishing of the zeta function on the line Re(s)=1." So what does this mean? One might say that this means that if we assume the first we can prove the second and vice verse. But along these lines any two true statements are equivalent. I would protest this by saying that "The Pythagorean theorem is equivalent to the Prime Number Theorem." Now one might try to correct this by saying that we mean that A and B are equivalent if the proof of B uses A and the proof of A uses B. But, in the Prime Number Theorem example, certainly there are proofs of the Prime Number Theorem that do not make use of the fact that Re(s)=1. Now if I had tried to point to an example (say the Selberg-Erdos "Elementary" proof, or Wiener's using his Tauberian Theorem) someone might dive into a discussion about how the zeta function is really lurking in the background of these. But even if someone presented a case that every known proof of the prime number theorem relied on that fact that the zeta function doesn't vanish on the line Re(s)=1, this certainly doesn't imply that there doesn't exist a proof the proceeded without use of the fact. Moreover while I'm not sure what we do mean when we say "A is equivalent to B" I'm pretty sure it should be a mathematical assertion and not a statement about the literature in existence on the result. Even more fundamentally, I have often reflected on the assertion that two mathematical statements are equivalent. Certainly if one proves that A is equivalent to B, where neither A or B is known to be true, the statement has practical value in that if one resolves the validity of one of the statements she also resolves the validity of the other. However one often finds that an author remarks that two statements known to be true are equivalent. The first example that comes to mind is: “The Prime Number Theorem is equivalent to the non-vanishing of the zeta function on the line Re(s)=1.”

So what does this mean? One might say that this means that if we assume the first we can prove the second and vice verse. But along these lines any two true statements are equivalent. I would protest this by saying that “The Pythagorean theorem is equivalent to the Prime Number Theorem.” Now one might try to correct this by saying that we mean that A and B are equivalent if the proof of B uses A and the proof of A uses B. But, in the Prime Number Theorem example, certainly there are proofs of the Prime Number Theorem that do not make use of the fact that Re(s)=1. Now if I had tried to point to an example (say the Selberg-Erdos “Elementary” proof, or Wiener’s using his Tauberian Theorem) someone might dive into a discussion about how the zeta function is really lurking in the background of these. But even if someone presented a case that every known proof of the prime number theorem relied on that fact that the zeta function doesn’t vanish on the line Re(s)=1, this certainly doesn’t imply that there doesn’t exist a proof the proceeded without use of the fact. Moreover while I’m not sure what we do mean when we say “A is equivalent to B” I’m pretty sure it should be a mathematical assertion and not a statement about the literature in existence on the result.

]]> By: uri http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-805 uri Sun, 16 Dec 2007 23:26:23 +0000 http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-805 Hi, Can't leave it unsaid that the second proof mentioned for the irrationality of $\sqrt{2}$ can be seen in a beautiful manner: Take an actual triangular piece of paper of sizes p,p,q (assuming $p^2=2q^2$). You can now fold one of the sides on the diagonal, and get yourself a smaller triangle of sizes $p-q,p-q,2q-p$, revealing that $2(p-q)^2=(2q-p)^2$, and contradicting the minimality of $p,q$ if you assumed that in advance. Hi,

Can’t leave it unsaid that the second proof mentioned for the irrationality of $\sqrt{2}$ can be seen in a beautiful manner:

Take an actual triangular piece of paper of sizes p,p,q (assuming $p^2=2q^2$). You can now fold one of the sides on the diagonal, and get yourself a smaller triangle of sizes $p-q,p-q,2q-p$, revealing that $2(p-q)^2=(2q-p)^2$, and contradicting the minimality of $p,q$ if you assumed that in advance.

]]> By: Maurizio http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-699 Maurizio Thu, 29 Nov 2007 04:07:47 +0000 http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-699 I'm surprised by the amount of insight about algebraic number theory that i gained reading your proof of the irrationality of $latex \sqrt{2}$, i mean it really helped me to better understand many things that i was supposed to already know. Actually the first two proofs are quite similar if you think about valuations: a valuation is a multiplicative function from a field to $latex \mathbb{R}^+$ that respects the triagular inequality, ie $latex v(ab)=v(a)v(b)$ and $latex v(a+b) \leq v(a)+v(b)$, and $latex v(a)=0$ only if $latex a=0$. It can be proved that all such valuations over the rational number are (up to equivalence) the usual absolute value ($latex v_\infty(a)=|a|$) and given by prime numbers ($latex v_p(p^k\frac{a}{b})=\frac{1}{p^k}$, where $latex a$ and $latex b$ are prime with $latex p$), so they can be thought of as a suitable generaliation of prime numers and ideals, providing additionally a "prime at infinity". All valuations share important properties, for instance they induce a topology with respect to which you can take the completion, etc. In the first proof of the irrationality of $latex \sqrt{2}$, you are taking a fraction $latex \frac{p}{q}$ where $latex p$ and $latex q$ minimize the valuation at the prime 2 (by dividing the numerator and denominator by the biggest common power of 2), and you get an absurd because $latex v_2(\frac{p^2}{q^2})$ cannot possibly be equal to $latex v_2(2)$. In the second proof, you are choosing $latex p$ and $latex q$ with smallest absolute value (and so smallest valuation at $latex \infty$), and then apply a tranformation (an element of $latex SL_2(\mathbb{Z})$) that leaves the value of the fraction, supposed to be equal to $latex \sqrt{2}$, unchanged, but decreases the valuation at $latex \infty$ of the denominator, absurd by minimality. So, by what i'm understanding, most of the difference in the proofs comes by how the "non-archimedean" valuation at 2 and the "archimedean" one at $latex \infty$ work, possibly an experienced number theorist may give us a better explaination of this fact. I’m surprised by the amount of insight about algebraic number theory that i gained reading your proof of the irrationality of \sqrt{2}, i mean it really helped me to better understand many things that i was supposed to already know.

Actually the first two proofs are quite similar if you think about valuations: a valuation is a multiplicative function from a field to \mathbb{R}^+ that respects the triagular inequality, ie v(ab)=v(a)v(b) and v(a+b) \leq v(a)+v(b), and v(a)=0 only if a=0. It can be proved that all such valuations over the rational number are (up to equivalence) the usual absolute value (v_\infty(a)=|a|) and given by prime numbers (v_p(p^k\frac{a}{b})=\frac{1}{p^k}, where a and b are prime with p), so they can be thought of as a suitable generaliation of prime numers and ideals, providing additionally a “prime at infinity”. All valuations share important properties, for instance they induce a topology with respect to which you can take the completion, etc.

In the first proof of the irrationality of \sqrt{2}, you are taking a fraction \frac{p}{q} where p and q minimize the valuation at the prime 2 (by dividing the numerator and denominator by the biggest common power of 2), and you get an absurd because v_2(\frac{p^2}{q^2}) cannot possibly be equal to v_2(2). In the second proof, you are choosing p and q with smallest absolute value (and so smallest valuation at \infty), and then apply a tranformation (an element of SL_2(\mathbb{Z})) that leaves the value of the fraction, supposed to be equal to \sqrt{2}, unchanged, but decreases the valuation at \infty of the denominator, absurd by minimality. So, by what i’m understanding, most of the difference in the proofs comes by how the “non-archimedean” valuation at 2 and the “archimedean” one at \infty work, possibly an experienced number theorist may give us a better explaination of this fact.

]]> By: Simon Morris http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-374 Simon Morris Thu, 25 Oct 2007 18:43:37 +0000 http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-374 Sorry - the "generalization" I had imprecisely in mind was just that $latex G $ consists of $latex n $ th roots of unity, not that it is all of them, which isn't a generalization of the result you gave. I think the first proof shows in passing, as part of its argument, that the group is cyclic, whereas with the second proof, you have to go back at the end and look at the group you've proved you have, and check that it's cyclic. I'm not sure whether that's a real distinction between the proofs. My rather poor point about the quaternions was just that $latex G $ needn't be cyclic there, so there can't be any generalization of the first proof there that uses some other definition of minimal element. Now that I've looked at some of your web pages, I see that I was thinking vaguely along the lines of "quaternions are another model for the complex numbers", in a way slightly reminiscent of your page about why it isn't obvious that the integers are bounded in the reals. Sorry - the “generalization” I had imprecisely in mind was just that G consists of n th roots of unity, not that it is all of them, which isn’t a generalization of the result you gave.

I think the first proof shows in passing, as part of its argument, that the group is cyclic, whereas with the second proof, you have to go back at the end and look at the group you’ve proved you have, and check that it’s cyclic. I’m not sure whether that’s a real distinction between the proofs. My rather poor point about the quaternions was just that G needn’t be cyclic there, so there can’t be any generalization of the first proof there that uses some other definition of minimal element. Now that I’ve looked at some of your web pages, I see that I was thinking vaguely along the lines of “quaternions are another model for the complex numbers”, in a way slightly reminiscent of your page about why it isn’t obvious that the integers are bounded in the reals.

]]> By: gowers http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-373 gowers Thu, 25 Oct 2007 17:29:50 +0000 http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-373 Simon, I don't think I understand what you are saying about quaternions. For instance, $latex G$ could be the group $latex \null \{\pm 1, \pm i, \pm j, \pm k\}$, which is not cyclic. The second proof would break down because there are more than eight 8th roots of unity, and the first hardly gets started. But perhaps you had a different generalization in mind. I also don't quite understand what it means to say that the second proof doesn't show that the group is cyclic: if it shows that it's the group of $latex n$th roots of unity then surely it does. Simon, I don’t think I understand what you are saying about quaternions. For instance, G could be the group \null \{\pm 1, \pm i, \pm j, \pm k\}, which is not cyclic. The second proof would break down because there are more than eight 8th roots of unity, and the first hardly gets started. But perhaps you had a different generalization in mind.

I also don’t quite understand what it means to say that the second proof doesn’t show that the group is cyclic: if it shows that it’s the group of nth roots of unity then surely it does.

]]> By: Simon Morris http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-352 Simon Morris Wed, 24 Oct 2007 09:16:35 +0000 http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-352 The second proof generalises to (for example) the quaternions, but I can't see how the first proof would. More convincingly (to me), the first proof also shows that the group is cyclic, whereas the second proof doesn't. I don't think I'm missing some small variation of the second proof that would show this, because the transliteration of the second proof to the quaternions certainly couldn't. But I'm not sure which subset, if any, of those observations really shows that the proofs are different. The second proof generalises to (for example) the quaternions, but I can’t see how the first proof would. More convincingly (to me), the first proof also shows that the group is cyclic, whereas the second proof doesn’t. I don’t think I’m missing some small variation of the second proof that would show this, because the transliteration of the second proof to the quaternions certainly couldn’t.

But I’m not sure which subset, if any, of those observations really shows that the proofs are different.

]]> By: E.Bz. http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-347 E.Bz. Tue, 23 Oct 2007 22:42:39 +0000 http://gowers.wordpress.com/2007/10/04/when-are-two-proofs-essentially-the-same/#comment-347 I think you can get enough of Lagrange from the first argument in this situation. Your Euclidean algorithm/ minimal argument is basically exactly how one proves that subgroups of cyclic groups are cyclic - it is then pretty easy to prove that you have Lagrange for the cyclic groups so I suppose there is some similarity. Beyond that I guess they are intrinsically different. I think you can get enough of Lagrange from the first argument in this situation. Your Euclidean algorithm/ minimal argument is basically exactly how one proves that subgroups of cyclic groups are cyclic - it is then pretty easy to prove that you have Lagrange for the cyclic groups so I suppose there is some similarity. Beyond that I guess they are intrinsically different.

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