Tuesday, 15 November 2011

Self-reference

I had a letter from my bank this week. On the reverse of the letter was written just this one sentence: 'This page is left blank intentionally.' 'Oh, no it isn't!' I said out loud, in my pedantic way. 'It's got that sentence written on it.'

I was reminded of this amusing example:
On a page of a book is written the heading 'Corrections' and underneath is written just the following: 1. The word at the top of this page should read 'correction' , not 'corrections'.

There is always the potential for paradox and logical fun in self-referential statements. A simple example is:
1. Statement 2 below is true.
2. Statement 1 above is false.
Think about it! You quickly find yourself in a paradoxical loop with this kind of thing.

The ultimate expression of the self-referential paradox was formulated by Bertrand Russell and is therefore known as Russell's Paradox. He proposes that there is such a thing as a set which is a member of itself. For example, the set of 'all sets that have more than ten members' is itself a set that has more than ten members, so it must be a member of itself. But there are many sets that are not members of themselves; for example, the set of mugs in my kitchen is not a mug, it's a set of mugs. OK? So, consider a set we will call set Q, which is: the set of all sets that are not members of themselves. Is Q a member of itself? If it is, then it isn't, because the members of Q are sets that are not members of themselves. But, if it isn't then it is, because all sets that are not members of themselves are in set Q.

The best book to read on self-reference, by a million miles, is Douglas Hofstadter's Gödel, Escher, Bach, which explores the idea with reference to the mathematical theory of Gödel, the art of Escher and the music of J.S. Bach. Wonderful! Apart from the Bible, it may be the best book I have ever read.

There are also some very entertaining chapters in Hofstadter's subsequent book Metamagical Themas (yes, that is an anagram of Mathematical Games), which explore the idea of self-reference further.

1 comment:

1. Respected Sir,
It is to be stated that I’ve done my Ph.D. (Edu.) on the topic, “Developing Strategy for Fostering Mathematical Creativity among Class IX Students”, from DAVV, Indore (India) in 2009, under the guidance of Prof. D. N. Sansanwal. During the study, a test, namely, Mathematical Creativity Test (MCT) was developed. Now, for, establishing Content Validly of MCT, I need your critical and valuable opinion about it. Kindly, find some time from your otherwise very busy schedule. Your expert opinion on MCT will be of immense help. Sir, Please mail me your email address so that I'm able to post MCT to you.
Thanks for time and consideration
Yogesh Sharma
mathematicalcreativity@gmail.com