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 , the integer is divisible by .
WRONG PROOF.
The theorem has been proved only for .
The next example is similar, but not so clear-cut [36, p. 138f].
Theorem. For all , and , if then .
WRONG PROOF. Suppose . Take . Then
The mistake here is that we took to be , which is a special value of . This assumption is wholly unjustified, since the quantities are controlled by an existential quantifier, and hence no condition may be imposed on them. By adding the assumption that , we have in fact proved the
WEAKER THEOREM. For all and , if , then .
This is not what we claimed to prove.