Self-reference

Escher's hands

Ever since Epimenides the Cretan (7th century B.C.) declared that All Cretans are liars, the notion of self-reference became synonymous (and often erroneously) with that of paradox. However, most ubiquitous self-reference which happens each time somebody says "I", is harmless. Following are a few gems from Metamagical Themas.

  1. Thit sentence is not self-referential because "thit" is not a word.
  2. If this sentence didn't exist, somebody would have invented it.
  3. This is not a complete. Sentence. This either.
  4. This sentence will end before you can say "Jack Rob
  5. Does this sentence remind you of Agatha Cristie?

Raymond Smullyan lists what he calls self-annihilating sentences from a collection by Saul Gorn "S. Gorn's Compendium of Rarely Used Cliches." :

  1. Before I begin speaking, there is something I would like to say.
  2. I am a firm believer in optimism because without optimism, what else is there?
  3. Half the lies they tell about me are true.
  4. Every Tom, Dick, and Harry is called John.
  5. Having lost sight of our goal, we must redouble our efforts!
  6. I'll see to it that your project deserves to be funded.
  7. I've given you an unlimited budget, and you have already exceeded it!
  8. A preposition must never be used to end a sentence with.
  9. This species has always been extinct.
  10. Authorized parking forbidden!
  11. If you're not prejudiced, you just don't understand!
  12. Inflation is an economic device whereby each person earns more than the next.
  13. Superstition brings bad luck.
  14. That's a real step forward into the unknown.
  15. You've outdone yourself as usual.
  16. Every once in a while it never stops raining.
  17. Monism is the theory that anything less than everything is nothing.
  18. A formalist is one who cannot understand a theory unless it is meaningless.

David Wells asks the following questions:

How many mistakes are there in the sentence: 'This sentance contanes one misteak'? What is the answer to the same question for this sentence: 'Their are three misteaks in this sentence'?

I placed his answers at the bottom of the page. However, on my part, I can't help wondering whether one can possibly ask the same question with different objects.

From the wonderful Beyond Numeracy I gleaned the following two pearls

  1. There is the case of the voter who when asked by a pollster what were the reasons for the ignorance and apathy of the American public, responded, "I don't know and I don't care."
  2. What is the question that contains the word cantaloupe for no apparent reason?

Lest you form a self-defeating impression that thinking up self-referential sentences is an exclusive pastime of self-indulgent mathematicians, I'll cite from a small book by William Safire

  1. Don't use contractions in formal writing.
  2. Do not put statements in the negative form.
  3. Avoid run-on sentences they are hard to read.
  4. Verbs has to agree with their subjects.
  5. Avoid trendy locutions that sound flaky.

A subtlety from the Littlewood's Miscellany:

E.Harrison: "Is it true that philosophy has never proved that something exists?" Bertrand Russell: "Yes, and the evidence for it is purely empirical."

The following comes from A Whack on The Side of The Head by R. von Oech

Every rule here can be challenged except this one.

Paul R. has suggested a similar one:

There are no absolutes.

From a Russian Math Circle for grade 7:

"Hi. Do you remember me? I am the person you have noticed the other day."

And also,

Only the fool would take trouble to verify that his sentence was composed of ten a's, three b's, four d's, forty-six e's, sixteen f's, four g's, thirteen h's, .fteen i's, two k's, nine l's, four m's, twenty-five n's, twenty-four o's, five p's, sixteen r's, forty-one s's, thirty-seven t's, ten u's, eight v's, eight w's, four x's, eleven y's, twenty-seven commas, twenty-three apostrophes, seven hyphens, and, last but not least, a single !

(Compare this with Self-Descriptive Se3ntences page.)

It took me longer than I originally planned to get to the Hofstadter's Law (D. Hofstadter's Godel, Escher, Bach):

It always takes longer than you expect, even when you take into account Hofstadter's Law.

A friend of mine found the following in the Omni magazine:

Nostalgia isn't what it used to be.

There is a devilish shade of self-reference in a Piet Hein's grook, isn't there?

The Universe may
be as great as they say,
but it wouldn't be missed
if it didn't exist.

M.Minski's passage from the jacket of The Mind's I by D.Hofstadter and D.Dennett unfailingly maneuvers one's mind into reflective mood:

This great collection on reflection provides you with your own quite special ways to understand things such as why, if you don't read this book, you'll never be the same again.

Here is another one from the Littlewood's Miscellany:

In a Spectator competition the following won a prize; subject: what would you most like to read on opening the morning paper?

OUR SECOND COMPETITION

The First Prize in the second of this year's competitions goes to Mr Arthur Robinson, whose witty entry was easily the best of those we received. His choice of what he would like to read on opening his paper was headed, 'Our Second Competition', and was as follows: "The First Prize in the second of this year's competitions goes to Mr Arthur Robinson, whose witty entry was easily the best of those we received. His choice of what he would like to read on opening his paper was headed 'Our Second Competition', but owing to paper restrictions we cannot print all of it."

Krishna Kumar reminded me of the following:

Here are some self-referential statements that I came across recently.

  1. Break every rule.
  2. All generalizations are misleading.
  3. If somebody loves you, love them back unconditionally.
  4. Computers are like lynxes in the sense that I cannot think of a suitable analogy for either of them right now.

(The last one is by Scott Adams in one of his Dilbert books.)

Robert Arthur quoted from Desmond MacHale's "Comic Sections". Said he, one of the cleverest things I have ever seen in a book went something like this. Page 347 was devoted to ERRATA and consisted of the following

Page 347: For ERRATA read ERRATUM

John Allen Paulos opens his book Mathematics and Humor with a paragraph that contains a story of a friend of his who took a speed-reading course. The friend noted this in a letter to his mother. His mother responded with a long, chatty letter in the middle of which she wrote, "Now that you've taken that speed-reading course, you've probably already finished reading this letter."

William Dunham mentions Stephen Bock's Description of a sheltered man and his dreams: "Reading was something Jay knew about only from books, yet he was quite anxious to experience it for himself."

A superb site IMPOSSIBLE OBJECTS (which I no longer find accessible) defined Impossible objects as

An impossible object is a collection of reasonable parts, put together in an impossible way.

Now think of it. Is it possible to put anything together in an impossible way?

Lew Lefton's gem is quoted by John de Pillis (777, p. 156):

You either believe in the law of the excluded middle, or you don't.

A story is told by Steven Krantz in his Mathematical Apocrypha (p. 23):

At the end of World War II, (Stefan) Bergman found himself in France without any papers. This was always tricky, but at that time it was a matter of life and death: without papers, one could not obtain a rations card. Characteristically, Bergman found a mathematical solution to the problem. He went to the mayor of a small town outside Paris and convinced him to give Bergman a piece of paper saying, "This is to certify that Mr. Stefan Bergman has no papers." This of course was a paper, and with it Bergman was able to obtain his rations card.

Camiel Feij from the Netherlands sent me a delightful philosophical observation with a probable attribution to Isaac Bashevis Singer:

We have to believe in free-will, we have no choice.

An insert on self-reference appeared in Mathematics Magazine Vol 78, No 1, February 2005:

A Publishing Paradox

Alert reader Jack C. Abad and his brother Victor Abad send the following:

A recent Birkhäuser-Verlag book list included Unpublished Philosophical Essays/Kurt Gödel, edited by Francisco A. Rodrigez-Consuegra, 1995. If the title is accurate, it might make appropriate barbershop reading in that town where the barber shaves everyone who doesn't shave himself.

A quick search of Amazon.com turns up a wealth of similar material -- unpublished recordings of Elizabeth Schwartzkopf and Marian Anderson, unpublished letters from General Robert E. Lee to Jefferson Davis, and unpublished opinions of Warren Court. Author Michael McMullen is more scrupulously logical: The title of his book is The Blessing of God: Previously Unpublished Sermons of Jonathan Edwards.

But now come to think of it: when else but previously a book might or might not have been published? Is this specification not redundant?

Additional examples can be easily found on the Web. I came across the following at the David's Profession Jokes web site:

There are three kinds of mathematicians:
those who can count and those who can't.

There are two groups of people in the world;
those who believe that the world can be
divided into two groups of people,
and those who don't.

Another site quotes Milton Berle:

"I'm getting so absent-minded and forgetful. Sometimes in the middle of a sentence, I . . . "

In the September 2005 issue of Math Horizons the readers were asked to submit questions that contained their own answers. The results of the competition have been published in the April 2006 issue. Following are a few samples:

In 1978, Raymond Smullyan wrote a book about logical puzzles. What is the name of this book?
(Roger Nelsen)

I am the square root of -1. Who am I?

What would be the value of 190 in hexadecimal be?
(Head-Royce School Math Club)

Twenty-nine is a prime example of what kind of number?

The reciprocal of sqrt(2) is half of what number?

How many consonants are in one? How many consonants are in two? How many consonants are in three? (One can continue: how many letters are in four?)

What do you do to the length of an edge of a cube to find its volume?
(Rheta Rubenstein)

As reported by BBC, US President George W Bush was twice the man at the White House Correspondents' Association annual dinner on Saturday, 29 April, 2006. Playing up to his public persona, Mr. Bush sent himself up alongside impersonator, Steve Bridges. At some point Mr. Bridges volunteered for the president:

"Some of my critics in the international community call me arrogant. I will not even honour that with a response. Screw them!"

And here is a piece of wisdom from Afred E. Neuman [What the Numbers Say, p. 128]:

"Understatement is zillion times more effective than exaggeration."

In his The Acceptance World (part 3 of the 1st Movement in A Dance to the Music of Time), Anthony Powell treats the reader to this exchange (p. 28):

'He fell in love with himself at first sight and it is a passion to which he always remained faithful.'
Self-love seems so often unrequited.'

Dr. Mardy Grothe published several collections of quotes with syntactic and semantic twists of which Oxymoronica is especially rich with self-references:

Never take anybody's advice. [G. B. Shaw]

Be spontaneous.

Eschew obfuscation. [W. Safire]

Last, but not least, avoid clichés like the plague [W. Safire].

Writing of the Icelanders Eric Weiner [The Geography of Bliss, p. 144] observes:

Everything in moderation, they believe, including moderation.

And apparently the opposite

Moderation in all things!

My little son, Eli, found on the web a collection of writing samples by high school students. Here is just one:

Her vocabulary was as bad as, like, whatever.

Here is a 21st century upgrade (twitted by https://twitter.com/#!/caalie):

"The trouble with quotes on the internet is that you never know if they are genuine" - Abraham Lincoln

Alfred E. Neumann is credited with the following piece of wisdom:

In retrospect it becomes clear that hindsight is definitely overrated!

A very clever post on Linkedin:

If you choose an answer to this question at random, what is the chance that you will be correct?

a) 25%   b) 50%   c) 33%   d) 25%

Robin Wilson (Lewis Carroll in Numberland, p. 194) quotes a self-referential paradox by Lewis Carroll:

Small girl: I'm so glad I don't like asparagus!

Friend: Why, my dear?

Small girl: Because, if I did, I should have to eat it - and I can't bear it!

A quote from "Conversations on Game Theory" by Haim Shapira (I translate from Hebrew):

Most things are more complex than it appears to you even if it appears to you that you understand this sentence.

Now returning to the origin of all the troubles, Epimenides the Cretan, you have probably heard of the island where each person is either a Knight (and speaks only truth) or a Knave (and hence always lies). Does it follow that Epimenides could not possibly live in such a place?

References

  1. B. Bollobas, Littlewood's Miscellany, Cambridge University Press, 1990
  2. W. Dunham, The Mathematical Universe, John Wiley & Sons, NY, 1994.
  3. D. Fomin, S. Genkin, I. Itenberg, Mathematical Circles (Russian Experience), AMS, 1996
  4. M. Gardner, aha! Gotcha. Paradoxes to puzzle and delight, Freeman & Co, NY, 1982
  5. D. R. Hofstadter, Metamagical Themas, Basic Books, Inc., 1985, Chapter 16.
  6. D. Hofstadter and D. Dennett, The Mind's I, Basic Books, 1981
  7. D. Niederman, D. Boyum, What the Numbers Say?, Broadway Books, 2003
  8. S. Krantz, Mathematical Apocrypha, MAA, 2002
  9. J. A. Paulos, Beyond Numeracy, Vintage Books, 1992
  10. J. de Pillis, 777 Mathematical Conversation Starters, MAA, 2002
  11. W. Safire, Fumblerules, Doubleday, 1990
  12. R.Smullyan, 5000 B.C. and Other Philosophical Fantasies, St. Martin's Press, NY, 1983
  13. R. von Oech, A Whack on The Side of The Head, Warner Books, 1990
  14. D.Wells, The Penguin Book of Curious and Interesting Puzzles, Penguin Books, 1992
  • Self-reference and apparent self-reference
    1. Does It Blink?
    2. Apparent paradox
    3. Set of all subsets
    4. An Impossible Page
    5. Russell's paradox
    6. An Impossible Machine
    7. A theorem with an obvious proof
    8. The Diagonal Argument
    9. A link to a very similar puzzle

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    Copyright © 1996-2017 Alexander Bogomolny

    Four. The sentence contains three spelling mistakes, plus the false claim that it only contains one mistake, making a total of four mistakes.

    The second question, paradoxically, cannot be answered. It contains only two spelling mistakes but claims to contain three mistakes; therefore that claim is wrong and it actually contains three mistakes - except that if it contains three mistakes then the claim that it contains three mistakes is correct, and so it only contains the two spelling mistakes, in which case...!

    Here is a remark by Richard Bissell

    Regarding your answer in the Self-reference section I believe that you are in error when you give the answer that the statement, "Their are three misteaks to this sentence", is unanswerable.

    The three mistakes are:

    1. Spelling error; Their for there.
    2. Spelling error; Misteak for mistake.
    3. Grammatical error; Use of Their for there.

    I appreciate the work that has been done on all the sections of this web page. Keep up the work.

    Richard Bissell

    A response came from Italy:

    hallo!

    I just wanted to point out something about Richard Bissel remark to your answer concerning the sentence "Their are three misteaks in this sentence".

    He says that error three is: "3. Grammar error, use of their for there."

    I think that if we accept this as an error then the word 'their' is no longer mispelled and therefore error one isn't really an error.

    Thanks again for your great site.

    Rita

    Here's a reaction to Rita's letter:

    Nice site.

    I was looking at your page on self-reference and I came upon the David Wells questions and the additional readers' comments.

    First, Richard Bissell did not say that all three mistakes coexisted, so Rita's comment is not a valid criticism.

    Mine, however, is. There are no mistakes in either sentence. While they contain incorrect spellings, falsehoods, and a violation of a grammatical rule, none of these were accidental, and therefore they cannot legitimately be considered mistakes.

    -Frank Schwartz

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