Monday, July 02, 2007

Fun with Fallacy

A couple months back I posted on the humorous book A Random Walk in Science. One of the better contributions contained therein is a reprint of The Uses of Fallacy by Paul V. Dunmore, New Zealand Mathematics Magazine, 7, 15 (1970). In the article Dunmore explains some of the better fallacies employed by creative teachers. Here is an excerpt:

There is a whole class of methods which can be applied when a lecturer can get from his premises P to a statement A, and from another statement B to the desired conclusion C, but he cannot bridge the gap from A to B. A number of techniques are available to the aggressive lecturer in this emergency. He can write down A, and without any hesitation put "therefore B". If the theorem is dull enough, it is unlikely that anyone will question the "therefore". This is the method of Proof by Omission, and is remarkably easy to get away with (sorry, "remarkably easy to apply with success").

Alternatively, there is the Proof by Misdirection, where some statement that looks rather like "A, therefore B" is proved. A good bet is to prove the converse "B, therefore A": this will always satisfy a first-year class. The Proof by Misdirection has a countably infinite analog, if the lecturer is not pressed for time, in the method of Proof by Convergent Irrelevancies.

Proof by Definition can sometimes be used: the lecturer defines a set S of whatever entities he is considering for which B is true, and announces that in future he will be concerned only with members of S. Even an Honours class will probably take this at face value, without enquiring whether the set S might not be empty.

Proof by Assertion is unanswerable. If some vague waffle about why B is true does not satisfy the class, the lecturer simply says, "This point should be intuitively obvious. I've explained it as clearly as I can. If you still cannot see it, you will just have to think very carefully about it yourselves, and then you will see how trivial and obvious it is."

The hallmark of a Proof by Admission of Ignorance is the statement, "None of the text-books makes this point clear. The result is certainly true, but I don't know why. We shall just have to accept it as it stands." This otherwise satisfactory method has the potential disadvantage that somebody in the class may know why the result is true (or, worse, know why it is false) and be prepared to say so.

A Proof by Non-Existent Reference will silence all but the most determined troublemaker. "You will find a proof of this given in Copson on page 445", which is in the middle of the index. An important variant of this technique can be used by lecturers in pairs. Dr. Jones assumes a result which Professor Smith will be proving later in the year--but Professor Smith, finding himself short of time, omits that theorem, since the class has already done it with Dr Jones...

The entire article is available here. Great stuff.

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