All books are interesting

It’s true. All books are interesting. And I can prove it.

As you know, one standard method of proof is “Proof by contradiction” – i.e. postulate the opposite of what you’re setting out to prove, and then demonstrate that this leads to a contradiction.

Now, what I’m setting out to prove is “All books are interesting”. The opposite to that is “There exists a set of books that aren’t interesting”.

Then, either:

(i) The set contains only one book; if so, then that one book must be interesting, as it’s the only book in this set, and that itself is a point of interest. Therefore this book cannot belong to this set … Hence, we have a contradiction.

(ii) The set contains several books; if so, the book in this set that was written earliest is interesting, because its point of interest is “This is the earliest written book in this set”. Since it is interesting, we must eliminate it from this set. We may then eliminate the next earliest written book from the set by similar reasoning, and continue in this manner until there is only one member left in this set. And then, the argument given in (i) applies, and once again, we have a contradiction.

Therefore, all books are interesting.

(Yes, I know, this is a variation on a well-known mathematical paradox. But it’s quite an interesting variation all the same. Indeed, I can prove there’s no uninteresting variation of this…)

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5 responses to this post.

  1. Posted by alan on May 10, 2011 at 10:30 pm

    Isn’t your statement “All books are interesting”. The negation of “There exists a set of books that aren’t interesting”.
    Isn’t the opposite statement “No books are interesting” ?

    Reply

    • True, the statement “No book is interesting” negates the original statement. But between saying “All books are interesting” and “No book is interesting” there exists the possibility “Some books are interesting, and some aren’t”. In short, the two statements – “All books are interesting” and “No book is interesting” – while mutually exclusive, are not collectively exhaustive.

      The statements “All books are interesting” and “There exists a set of books that aren’t (or isn’t) interesting” are both mutually exlusive, and collectively exhaustive. Both are required for “proof by contradiction”.

      Reply

  2. Posted by alan on May 10, 2011 at 10:37 pm

    More substantively, it is not clear what you mean by “interesting”.
    The first statement seems to use “interesting” in the usual sense of “interesting content”, however in a later sentence you use “interesting” as an comment on existence, i.e “it is interesting that the book exists”.
    Paradoxes only serve to show the inadequacy of language and aren’t that interesting to me.

    Reply

    • Congratulations, Alan! You have just contributed he 500th comment on this blog, and you win a dram of whisky! (Remind me the next time you see me…)

      Reply

    • The first statement seems to use “interesting” in the usual sense …

      I don’t know that I can be held responsible for what something may seem to you. To quote the good Prince Hamlet, “I know not seems”.

      Reply

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