This is the Title of this Article

The Indefinite Article

There is a certain elegance in self-reference. This concept is most obvious in language. A well-known set of examples can be found in William Safire's Rules for Writers. This is a set of grammatical rules, each of which makes its point by breaking the rule it describes. Some of the better examples include:

Don't use no double negatives.

Verbs has to agree with their subjects.

Always avoid alliteration.

Proofread carefully to see if you any words out.

About sentence fragments.

More of these rules will be discussed as we proceed; but we must remember to avoid cliches like the plague.

There are some marvellous self-referential words in English, such as pentasyllabic. Sometimes, the more intriguing words are the "near misses", or the words that could be self-referencing, but aren't. For example, monosyllabic isn't. And why is abbreviation such a long word? (And don't start a sentence with a conjunction.) It's easy to find an antonym for "antonym", but can you think of a synonym for "synonym"? Why isn't "palindrome" a palindrome? The word "aibohphobia" has been coined for the fear of palindromes. But, of course, the passive voice is to be avoided when writing.

Sentences can be self-referencing, such as: "This sentence contains five words." To take it one step further, have you ever looked up "dictionary" in a dictionary? Is the word "encyclopedia" in your encyclopedia? Who needs rhetorical questions, anyway?

The world of computing also has some inherent self-references. You can use a Web Search Engine to find Search Engines. A strong point of some computer languages such as Pascal is an ability to use recursive procedure calls. In recursion, a procedure can call itself, with each call "stacked" on top of the last until the top one terminates.

The ability of a computer program to receive its own source code as input became a necessary step in a proof that some problems are not "computable". This was devised by Alan Turing, the father of modern computer science. The incomputable problem he found was to write a computer program that would test other computer programs to determine if they would terminate. (It is easy to write a program that will not terminate - just code in a loop back to the start.)

The proof starts with the assumption that a "termination tester" program can be written, and then proceeds to demonstrate the falsity of this assumption by showing a logical contradiction. If such a program can be written, then it would be possible to write it in such a way that the termination tester itself terminates if the input program code is proven not to terminate. It could also be made to loop forever if the input program is found to terminate. It also follows that the termination tester can then be run using itself as input. Thus, if it terminates, the test of the tester will not terminate; but if it doesn't terminate, the test will terminate.

This form of self-reference is slightly indirect; much like the sentence pair:

The following statement is false.

The preceding statement is true.

This can be made more directly self-referencing by simply stating:

This statement is false.

This is known as the Epimenides Paradox. The subject of self-reference is treated in great detail in Douglas Hofstadter's book of 1979: Gödel, Escher, Bach: An Eternal Golden Braid. In this book, Hofstadter explains the mathematical theorems of Kurt Gödel, by drawing on illustrations from the art of M. C. Escher, and the music of J. S. Bach. The briefest statement of Gödel's Incompleteness Theorem is: "All consistent axiomatic formulations of number theory include undecidable propositions." But let's eschew obfuscation. Put more simply, Gödel's work showed that any mathematical system that is powerful enough to be useful is necessarily incomplete. His proof of this is similar to Turing's non-computability proof in its use of self-reference.

Self-reference in art is shown clearly in the M. C. Escher ink drawings throughout Hofstadter's book; the famous ever-rising staircase and many others. An indirect self-reference is shown in Escher's Drawing Hands. The canons and fugues of Bach show this effect in music. In a direct self-reference, Bach encoded his own surname into the last line of Art of the Fugue. This is made possible by the German usage of 'B' for B flat, and 'H' for the English 'B'. This was the last piece of music that he wrote before his death.

An interesting form of visual self-reference is the use of a copy of a picture within itself, such as the magazine cover picture of a person reading the same magazine, with the cover picture shown. Hofstadter refers to this as "self-engulfing", and discusses the effects of a video camera focussed on a television monitor showing the picture from the camera. This produces an infinite series of smaller and smaller screens, much like parallel mirrors.

Another form of self-reference in language is in poetry. An elegant example in limerick is:

There once was a lady from Crewe
Whose limericks stopped at line two.

Or, to take it one more step: "There once was a man from Verdun." To return to grammatical examples: "One will not have needed the future perfect in one's entire life." Douglas Adams, in The Hitchhiker's Guide to the Galaxy has a prod at Star Trek with: "... to boldly split infinitives that no man has split before." In other variations on Safire:

Any noun can be verbed.

Even if a mixed metaphor sings, it should be derailed.

Analogies in writing are like feathers on a snake.

Of course, such things are not restricted to the English language. Some non-English examples do not even translate into English, such as the very German "Uemlaeute sind toell". Even in our use of the spoken language, we can pronounce "glottal stop" with a glottal stop.

In yet another consideration, if the man who invented the drawing board had got it wrong, what would he have gone back to? As a final thought, I guess that life is like a simile.

Indefinite Philosophy

There seems to be strong support for the argument that the form "two thousand and-" for naming years will shift to "twenty-" sometime within the next eleven years; that is, before "twenty-ten". Alastair Munro has correctly pointed out that even "Twenty-oh-one" is much easier to say than "Two thousand and one". However, I still believe that Arthur C. Clarke and Stanley Kubrick have firmly fixed "two thousand and one" into the English vocabulary.

Last edition, I raised the issue of the names of the decades - which decade follows the current "nineties"? Ben Deany supports the suggestion that the next decade should be called the "naughties". He also mentioned that this term is particularly in use among the younger adult age group. He sees the term as "jokingly suggesting that these years are going to be a time of fun, frivolity and naughtiness, unseen since the sixties." It is true that the current decade certainly hasn't exactly been another "naughty nineties." I have still not heard from anyone that can shed any light on the name used for the first decade of this century. Surely that knowledge must be out there somewhere?

Why is it that we want to encapsulate an entire decade in a single term, anyway? There is an innate human urge to generalise and categorise. (In terms of the subject above, all generalisations are dangerous, including this one.) Ben's suggestion that such terms may be different for different age groups is of interest. There may also be other groups in society that may originate such terminology. The computer industry has given us "Y2k"; will this perhaps give us "2k1" for 2001?

But perhaps it is time to move onto a new topic - a fundamental philosophical problem. For lunch, I have a chicken sandwich and an egg sandwich; which one do I eat first?

To join in any of these discussions, please e-mail me.

Updated: 27 Jan 2000

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