Wrong Arguments - Examples Versus Proofs - Forms of Argument

Mathematical Writing - Vivaldi Franco 2014

Wrong Arguments - Examples Versus Proofs
Forms of Argument

In constructing a mathematical argument it’s easy to make mistakes. In this section we identify some common faulty arguments: confusing examples with proofs, assuming what we are supposed to prove, mishandling functions. Other mistakes will be examined in Chap. 9 and in the exercises. Awareness of these problems should help us in avoiding them.

7.7.1 Examples Versus Proofs

The verification of a statement in specific cases does not constitute a form of proof. Our study of Euler’s polynomial in Sect. 7.6 shows how misleading examples can be. This state of affairs is peculiar to mathematics; in other scientific disciplines, such as biology, a proof of a statement consists of independent experimental verifications of its validity.

In our first example the fault is easy to spot.

Theorem. For all primes $$p$$, the integer $$2^p-2$$ is divisible by $$p$$.

WRONG PROOF.

$$\begin{aligned} 2^2-2=2\cdot 1,\quad 2^3-2=3\cdot 2,\quad 2^5-2=5\cdot 6,\quad 2^7-2=7\cdot 18,\quad \text {etc}. \\ \end{aligned}$$

$$\square $$

The theorem has been proved only for $$p=2,3,5,7$$.

The next example is similar, but not so clear-cut [36, p. 138f].

Theorem. For all $$x$$, $$y$$ and $$z$$, if $$x+y < x+z$$ then $$y < z$$.

WRONG PROOF. Suppose $$x + y < x + z$$. Take $$x = 0$$. Then

$$\begin{aligned} y = 0 + y < 0 + z = z. \\ \end{aligned}$$

$$\square $$

The mistake here is that we took $$x$$ to be $$0$$, which is a special value of $$x$$. This assumption is wholly unjustified, since the quantities $$x,y,z$$ are controlled by an existential quantifier, and hence no condition may be imposed on them. By adding the assumption that $$x = 0$$, we have in fact proved the

WEAKER THEOREM. For all $$y$$ and $$z$$, if $$0+y<0+z$$, then $$y<z$$.

This is not what we claimed to prove.