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As indicates the title, this question is about "proofs" of true statements which are short and/or look elegant but are wrong.

I mean example like Cayley-Hamilton's theorem, which states that for a $n\times n$ matrix over $\Bbb C$, and $\chi$ its characteristic polynomial, then $\chi(A)=0$. The well-known fake proof consists of a substitution $\lambda=A$ in $\chi(\lambda)=\det(A-\lambda I)$, which is not allowed.

So, I think writing a big-list could be interesting, where each answer will contain:

  • the statement;
  • the fake proof;
  • an explanation of the gap in the proof;
  • if possible, a reference to a good proof.

Each one can concern any field of mathematics. It will be good to have an example in every field: real analysis, measure theory, etc...

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    $\begingroup$ The fake proof is easy to fix though: change the base field from $\mathbb{C}$ to $\overline{\mathbb{C}(x_{1,1}, \ldots, x_{n,n})}$... $\endgroup$
    – Zhen Lin
    Commented Dec 27, 2012 at 11:02
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    $\begingroup$ @ZhenLin I agree, but it's not so simple to think about such a fixation when we believed we could replace a scalar by a matrix. $\endgroup$ Commented Dec 27, 2012 at 11:06
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    $\begingroup$ Isn't this more or less the same question as Pseudo Proofs that are intuitively reasonable? It seems that many of the answers there would qualify as answers to your question. $\endgroup$
    – Martin
    Commented Dec 27, 2012 at 11:20
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    $\begingroup$ @Martin I've done a research and I didn't find it, and indeed it's related. However (it's my opinion, not necessarily true), the other thread gives "non rigorous" proofs, which are not necessarily wrong. $\endgroup$ Commented Dec 27, 2012 at 11:24
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    $\begingroup$ There is a whole book about that ranging over various fields of mathematics. Mathematical Fallacies, Flaws and Flimflam by Edward J. Barbeau. Love it! And for all of the "proofs", it takes care of the first three of your bullets. $\endgroup$ Commented Jan 18, 2013 at 20:05

5 Answers 5

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My favorite is the following:

Let $\pi$ be rational, and write $\pi = a/b$ in lowest term. Let $p \neq 2$ be a prime not dividing $a$. Then in $\Bbb{Q}_{p}$, we have

$$ 0 = \sin(pb\pi) = \sin(pa) = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)!}(pa)^{2n+1} \equiv pa \ (\mathrm{mod} \ p^2), $$

which is absurd since $a \not\equiv 0$ mod $p$. Therefore $\pi$ is irrational.

The essential gap in this too-good-to-be a proof is that a $p$-adic power series may not converge to the same value as in the real field case, even the series consists of only rational terms. Thus the value of $\sin x$ need not coincide in $\Bbb{R}$ and $\Bbb{Q}_{p}$.

This false proof appears in Neal Koblitz's p-adic Numbers, p-adic Analysis, and Zeta-Fnctions.

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Let $\displaystyle \int$ denote $\displaystyle \int_0^x (\cdot ) dx$. Consider solving the equation $$\int f = f-1.$$ Rearranging, we get that $$f - \int f = 1 \implies \left(1 -\int \right)f = 1$$ Hence, $$f = \dfrac1{1 - \displaystyle\int} = \left(1 + \int + \int \int + \int \int \int + \cdots \right)1\\ = 1 + \int_0^x 1 dx + \int_0^x \int_0^x 1 dx + \int_0^x \int_0^x \int_0^x 1 dx + \cdots = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \cdots = e^x$$which indeed satisfies the equation.

Adapted from this post. The post has lot of other interesting answers as well.

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    $\begingroup$ Is that proof really that false? As an example, on $(\mathcal C([0,1/2]),\|\cdot\|_\infty)$ the operator $\int$ you defined has norm $\|\int\|_{\mathcal L (\mathcal C([0,1/2]))}=1/2$ and all the operations you carry seem to be legitimate. Although it would be better to use two different letters for the identity on $\mathcal C([0,1/2])$ and the constant function equal to one. $\endgroup$ Commented Jan 24, 2013 at 17:16
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What I have in my mind is Wilson's theorem, which says that if $p$ is a prime number, then $$(p-1)!\equiv -1 \pmod p.$$

The fake proof I have learned is the following: Since $p=p-1+1$, by taking factorial on both sides, we have $$p!=(p-1+1)!=(p-1)!+1!=(p-1)!+1.$$ Now taking mod $p$, we obtain $$0\equiv(p-1)!+1 \pmod p.$$

Of course the "proof" is wrong. The gap occurs because factorial is not distributing in the sense that $(a+b)!\neq a!+b!$ in general. In fact, same "proof" would work without assuming $p$ is prime.

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    $\begingroup$ I think its obvious to see the mistake. $\endgroup$
    – Sawarnik
    Commented Jan 21, 2014 at 17:23
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There is a whole book about that ranging over various fields of mathematics. Mathematical Fallacies, Flaws and Flimflam by Edward J. Barbeau. Love it! And for all of the "proofs", it takes care of the first three of your bullets.

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There is simple "proof" of four color theorem:

https://superliminal.com/4color/

unfortunately I still can't see the gap in it.

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    $\begingroup$ In figure 2b, the green-red and green-blue paths can cross through each other. Therefore, after we swap yellow/blue inside the green-red path, it may be that we thereby break the green-blue path. And since there are now two blue neighbors of $V$, the argument that a green-blue path must exist cannot be repeated. $\endgroup$ Commented Feb 4, 2013 at 23:36
  • $\begingroup$ This link is now broken. Could anyone provide an updated URL? $\endgroup$
    – BSplitter
    Commented Aug 2, 2022 at 21:20
  • $\begingroup$ @BSplitter: An updated URL is here. I'm not going to edit the (community wiki) Answer above because the content is not really "simple" in my view (but your view might differ). $\endgroup$
    – hardmath
    Commented Aug 24, 2022 at 4:27

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